text:
# Calculations of Dihedral Groups Using Circular Indexation

Document Type : Special Issue: AIMC 51

Authors

1 Department of Electrical Engineering, Persian Gulf University

2 Department of Pure Mathematics, Persian Gulf University

Abstract

In this work, a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$. Each element of the sequence represents a vertex of the polygon. Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis. Here each symmetry is considered as a system that takes an input circular sequence and generates a processed circular output sequence. The system can be described by a permutation function. Permutation functions can be written in a simple form using circular indexation. The operation between the symmetries of the polygon is reduced to the composition of permutation functions, which in turn is easily implemented using periodic sequences. It is also shown that each symmetry is effectively a pure rotation or a pure flip. It is also explained how to synthesize each symmetry using two generating symmetries: time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by $2\pi /n$ radians clockwise). The group of the symmetries of a polygon is called a dihedral group and it has applications in different engineering fields including image processing, error correction codes in telecommunication engineering, remote sensing, and radar.

Keywords

#### References

[1] N. S. Akbar, Mathematical model for blood flow through tapered arteries with
temperature dependent viscosity, J. Adv. Math. Appl. 3 (2014) 122 − 129.
[2] G. Arfken, Mathematical Methods for Physicists, 3rd Ed., Orlando: FL: Aca-
demic Press, New York-London, 1985.
[3] R. S. Caprari, Geometric symmetry in the quadratic Fisher discriminant op-
erating on image pixels, IEEE Trans. Inform. Theory 52 (2006) 1780−1788.
[4] T. L. Kunii, H. Nishida and M. Hilaga, Topological modeling for visualization,
J. Adv. Math. Appl. 1 (2012) 134 − 150.
[5] R. Lenz, Using representations of the dihedral groups in the design of early
vision filters, In: Proc. International Conference on Acoustics, Speech, and
Signal Processing, ICASSP-93, Minneapolis, MN, USA, 1993.
[6] G. Mayhew, Group property of the P4, K4, and NR16 error correction codes,
In: Proc. Aerospace Conference, 2017.
[7] A. V. Oppenheim, R. W. Schafer and J. R. Buck Discrete-Time Signal Processing
, 2nd ed, Prentice-Hall Signal Processing Series, Pearson, Upper Saddle
River, N. J., 1998.
[8] D. Saracino, Abstract Algebra: A First Course, Addison Wesley Longman
Publishing Co, USA, 1980.
[9] A. Thiele, E. Cadario, K. Schulz, U. Thoennessen and U. Soergel, Feature
extraction of gable-roofed buildings from multi-aspect high-resolution InSAR
data, In: Proc. International Geoscience and Remote Sensing Symposium,
(2007) 262 − 265.
[10] M. Thill and B. Hassibi, Frames from groups: Generalized bounds and di-
hedral groups, In: Proc. International Conference on Acoustics, Speech and
Signal Processing, (2013) 6043 − 6047.
[11] Y. Wang, W. Pedrycz, J. Lu and G. Luo, Denotational mathematical models
of an air traffic control system (ATCS-II): Process models of functions in
RTPA, J. Adv. Math. Appl. 2 (2013) 82 − 110.


text:
# The following facts pertain to a noncancelable lease agreement between Monty Leasing Company and ...

The following facts pertain to a noncancelable lease agreement between Monty Leasing Company and Flounder Company, a lessee.

 Inception date: May 1, 2017 Annual lease payment due at the beginning of each year, beginning with May 1, 2017 $21,751.90 Bargain-purchase option price at end of lease term$3,700 Lease term 5 years Economic life of leased equipment 10 years Lessorâ€™s cost $67,000 Fair value of asset at May 1, 2017$93,000 Lessorâ€™s implicit rate 10 % Lesseeâ€™s incremental borrowing rate 10 %

The collectibility of the lease payments is reasonably predictable, and there are no important uncertainties surrounding the costs yet to be incurred by the lessor. The lessee assumes responsibility for all executory costs

Prepare the journal entries to reflect the signing of the lease agreement and to record the receipts and income related to this lease for the years 2017, 2018, and 2019. The lessorâ€™s accounting period ends on December 31. Reversing entries are not used by Monty.

format: plain


text:
# How can I align two nodes to the left in TikZ?

I have two unaligned nodes

|AAAAAAA|

|B|

and I want to have the node with the B aligned to the left of the node with AAAs and below it

|AAAAAAA|

|B|

How can this be done?

• Welcome to tex.sx! It's not necessary to sign your questions (as there is already a box with your username below it) or to begin them with a greeting. – Martin Scharrer Mar 16 '11 at 20:56
• Since you have some responses below that seem to answer your question, please consider marking one of them as ‘Accepted’ by clicking on the tickmark below their vote count. This shows which answer helped you most, and it assigns reputation points to the author of the answer (and to you!). Otherwise please state what is missing so that people can help you better. – Martin Scharrer Dec 7 '11 at 10:05

You can position "B" below and left of "AAA" without having to know the position of it using the positioning library:

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{positioning}

\begin{document}
\begin{tikzpicture}
\node [draw] (A) at (4,5) {AAAA};
\node [draw,below=of A.west,anchor=west] (B) {B};
\end{tikzpicture}
\end{document}

Gives:

• Note that you can change the distance by adding a value before of, e.g. below=2cm of A. – Martin Scharrer Mar 16 '11 at 22:29
• Thanks, your examples helped me to understand the workings of anchor: it makes a node's anchor point (north, east, ...) coincide with the node's position point (defined with "at (x,y)", "below=of p", etc). – Mário Mar 17 '11 at 9:33
• For example, using \begin{tikzpicture}[node distance=5mm] \node[draw, anchor=west] (1) {AAAAAAAA}; \node[draw, below=of 1.south west, anchor=north west] (2) {B}; \end{tikzpicture} we can left align the two nodes and have a precise distance of 5mm between them. Thanks a lot. @MichaelUmmels – Mário Mar 17 '11 at 9:49
• Important note: one must use below=of A.west, anchor=west (which is what Martin has written) not anchor=west, below=of A.west (which looks just as good, but seems not to work). – John Wickerson Mar 13 '15 at 15:22
• @MartinScharrer Why does placing anchor=west after below instruction works, but not the other way around? (Just like John pointed out in the comment above) – Andriy Drozdyuk Nov 19 '16 at 18:03

Use anchor=west for both nodes. For example:

\begin{tikzpicture}
\node[anchor=west] (1) at (0,1) {AAAAAAAA};
\node[anchor=west] (2) at (0,0) {B};
\end{tikzpicture}

• Thanks, I understand now: you are placing the west anchors of two nodes at two vertically aligned points. That does the trick, the only limitation is that if the top node increases its heigth you have to move the point of the second node further below. To avoid this the second node could be placed using "below=of 1.south west". – Mário Mar 17 '11 at 9:42
• Yes, I agree that Martin's solution (inlcuding node distance) is more flexible. – Michael Ummels Mar 17 '11 at 10:14


text:
## calculus – Simplifying limit help

In my calculus class, I have that
$$f(t) = 11.14(1.023)^t$$, and I am trying to find the instantaneous rate of change at $$t=15$$. So far the arithmetic I’ve done has lead me to
$$lim_{h to 0} frac{11.14(1.023)^{15+h} – 11.14(1.023)^{15}}{h}$$
but I can’t seem to further simplify this, and thus I can’t think of a solution. Is there something I’m overlooking, i.e. an identity or arithmetic simplification? Thanks.

## 2013 – Workflow to set limit of people per date in list

I have a task list / calendar that serves as a signup for classes. What I am trying to do is set 20 people per class per date. Once 20 people have signed up for the class (i.e. – once we have 20 list items with the same date), the date is no longer available for folks to sign up for.

I’m trying to use a workflow for this, but can’t figure out how to make this happen.
Any ideas?

## windows server 2012 r2 – How Apache Limit POST are set to accept Strictly local service Request?

I am trying to harden an Apache 2.4 Web server running 2 wordpress sites with separate vhost settings. Deployed on Windows 2012 r2. I was hacked a few weeks ago and the server was completely lost. On the new server they are still trying to hack the site with post request injections/code executions. I’m having trouble trying to block all `POST` request that are not sent by the server on submission.

Things I have tried include

• `<Location>`, `<Directory>`, `<Limit>` tags with their attributes like `GET`,`PUT`, `POST PUT DELETE` trying varied combos with various `Require` statements ie. `local`, `ip`, `host`, and `order allow, deny` patterns. `Allow all from`

• `<Directory "/">`, `<Location >`,`<LimitExcept >` Tags including attributes such as `GET`,`POST` and combos.

• Various `<if >` statements

• `AllowMethod` Statements

• Whitelisting Directories in the `.htacess`

Placing attempts in different file context. Such as `httpd.conf`, `Vhost.conf`, `.htaccess`. Any file I found referenced in Apache docs related to the directive context. Even placed it where it should not work for giggles. Nothing, I read in the docs or on stack overflow seemed to work.

All auth Modules loaded like mod_auth, mod_host etc. Just assume Modules are fine.

What I would like to achieve is to `deny` or `<Limit POST>` request. Strictly allowing them in a dynamic `<Directory "/.*"> or `` being responded to only when the server submits the Request using the server IP. Therefore the IP:port of the request should be the “remote address” because the apache local service sends the request method via PHP. I believe this to be correct logic.

The last thing I had time to try before leaving work, is the below code attempting a dynamic `<Directory “/.*”> tag I do not want to dig through WordPress for every form and create a tag for each directory.

``````<Directory "/.*">
<if "%{REQUEST_METHOD} == 'POST'">
<Limit POST PUT DELETE>
Require req, http %{REMOTE_ADDR} == 'xx.xx.xx.xx:xx'
</Limit>
</if>
</Directory>
``````

Now I tried

``````<Directory "/">
<Limit POST>
allow from all
</Limit>
</if>
</Directory>
``````
``````<Directory "/">
<Limit POST>
allow from all
</Limit>
</if>
</Directory>

``````

No plugin, No nginx,E xpress, Or IIS Suggestions. Only relevant solutions to this config.

## Why can carriers limit unlocked iPhone hotspot/tethering?

The answers here and here explain that carriers can tell an iPhone whether to allow tethering or use as a hotspot.

My question is why, on an unlocked phone, can they do that?

Has Apple simply colluded with a cellular service consortium to respect their wishes?

Or are there a means by which a cellular data provider can detect that data are being served to and from a different device through the iPhone bearing its SIM card? (While possible without any countermeasures, AFAIK it would be trivial for the iPhone to obscure the fact that it is tethering.)

## pr.probability – Is the weak* limit of Girsanov measures also a Girsanov measure?

Let $$mu_0$$ be the standard Wiener measure on $$C[0,T]$$. Let $$mu_n$$ be a sequence of measures with $$mu_nll mu_0$$ for all $$n$$ and so that the weak$$^ast$$ limit of $$mu_n$$ exists, call it $$mu$$. Is it true that $$mu ll mu_0$$?

I know for general measures this is not true. For example we can have a Gaussian with variance $$varepsilon$$ and send $$varepsilon to 0$$.

But what about for Girsanov measures?

## how to stop limit number of urls posted

ive set a project to

“stop the project after xxx verifications per url”
under project>options

it has already done multiples of what its supposed to do & just keeps doing more,
its the indexer engine, its done about 1500 in half a day for each project,
i just want it to do a steady amount & put the rest in the lists for when they are needed,
but apart from turning that engine off completely (which i would have to do for each project)
how do you get it to stay at the levels you have set it to at

project>options
its not 20-30% more -its not going to stop

## How to best limit SQL injection attacks that are being funneled through an Apache proxy I control

I use an Apache proxy to funnel traffic to an IIS server. The IIS server sends me emails if some bad actor attacks my site with an sql injection attack. It captures their IP address, and sends me the URL that was used.

The other day I got 8,400 emails in 50 minutes, nothing my mail server can’t handle, but all from some IP proxy in the US.

Now I am thinking of hardening my code by adding some time sensitive (hindering) factor into the equation. The idea is to send them a redirect to somewhere else (fbi?) for a set time period, after an initial warning; provided they reach that emails code.

Should I have IIS handle this (simple to implement,) or is it better for Apache to handled it? I would think so it there is such a function in Apache.

In IIS I could start a session logging IP addresses of bad actors, and block their access to the site. Or has Apache something that could block these request for a set time period?

And how would I implement this IIS to Apache handover, maybe via some header?

## Are Javascript closures a useful technique to limit exposing data to XSS?

I’m wondering if using Javascript closures is a useful technique to limit exposing data to XSS? I realize it wouldn’t prevent an attack, but would it reliably make an attack more difficult to execute, or would it only make my code more irritating to write and read (a waste of time)?

I got the idea from the Auth0 documentation regarding storing OAuth/OIDC tokens. It reads:

Auth0 recommends storing tokens in browser memory as the most secure option. Using Web Workers to handle the transmission and storage of tokens is the best way to protect the tokens, as Web Workers run in a separate global scope than the rest of the application. Use Auth0 SPA SDK whose default storage option is in-memory storage leveraging Web Workers.

If you cannot use Web Workers, Auth0 recommends as an alternative that you use JavaScript closures to emulate private methods.

I can see how this is better than just putting the token or other sensitive information in `localstorage`. In `localstorage` an XSS attack needs only to execute `localStorage.token` to get the token.

Now, if you’re not familiar with tokens just apply this reasoning to any sensitive to information. In my case I want to build a client-side cache mapping user IDs to usernames for an administrative interface, but I realize that client IDs and usernames are somewhat sensitive, so I wondered if I could “hide” the data.

## astrophotography – Is there any way to increase the limit of exposure time on a Canon EOS M50?

I would be interested in using my Canon EOS M50 for astrophotography, however I would like to take longer exposures than the 30 second limit. Is there any way to increase this limit beyond 30 seconds to take better exposures?

I considered using Magic Lantern, however I don’t believe it is supported by the camera and also it was ambiguous as to whether or not that was a feature that magic lantern actually offered. Is there some sort of software that can accomplish this? I assume that this was a limit of the software rather than the hardware, but I’m not sure if I’m missing something.

## How to Limit Database Quota In cPanel CloudLinux Server?

Hi,

I would like to limit Database Quotas in our cPanel CloudLinux server like below:

Database Size:

Database Table Size:

DB Queries Execution Times:

The limits would be different on each shared hosting plan.

How to implement that, is there any idea?


text:
# Mathematical problems

## Mathematical problems Topics

Sort by:

### Solved problems

There are many unsolved problems in mathematics. Severalfamous problems which have recently been solved include: 1. The Pólya conjecture (disproven byHaselgrove 1958, smallest counterexample found by Tanaka 1980). 2. The four-color theorem (Appel and Haken1977ab and Appel et al. 1977 using a computer-assisted proof). 3. The Bieberbach conjecture (L. deBranges 1985). 4. Tait's flyping conjecture (Menasco and Thistlethwaite in 1991) and the other two of Tait's knot conjectures (by various authors 1987). 5. Fermat's last theorem (Wiles 1995, Taylorand Wiles 1995). 6. The Kepler conjecture (Hales 2002). 7. The Taniyama-Shimura conjecture(Breuil et al. in 1999). 8. The honeycomb conjecture (Hales 1999).9. The Poincaré conjecture. 10. Catalan's conjecture. 11. The strong perfect graph theorem...

### Unsolved problems

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture. 2. The Riemann hypothesis. 3. The conjecture that there exists a Hadamard matrixfor every positive multiple of 4. 4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes). 5. Determination of whether NP-problems are actuallyP-problems. 6. The Collatz problem. 7. Proof that the 196-algorithm does not terminatewhen applied to the number 196. 8. Proof that 10 is a solitary number. 9. Finding a formula for the probability that two elements chosen at random generate the symmetric group . 10. Solving the happy end problem for arbitrary . 11. Finding an Euler brick whose space diagonal isalso an integer. 12. Proving which numbers can be represented as a sum of three or four (positiveor negative) cubic numbers. 13. Lehmer's..

### Smale's problems

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's famous list of problems presented in 1900 (Hilbert's problems), and in part in response to a suggestion by V. I. Arnold on behalf of the International Mathematical Union that mathematicians describe a number of outstanding problems for the 21st century.1. The Riemann hypothesis. 2. The Poincaré conjecture. 3. Does (i.e., are P-problems equivalent to NP-problems)? 4. Integer zeros of a polynomial. 5. Height bounds for Diophantine curves. 6. Finiteness of the number of relative equilibria in celestial mechanics. 7. Distribution of points on the 2-sphere. 8. Introduction of dynamics into economic theory. 9. The linear programming problem. 10. The closing lemma. 11. Is 1-dimensional dynamics generally hyperbolic? 12. Centralizers of diffeomorphisms...

### Simon's problems

A set of 15 open problems on Schrödinger operators proposed by mathematical physicist Barry Simon (2000). This set of problems follows up a 1984 list of open problems in mathematical physics also proposed by Simon, of which thirteen involved Schrödinger operators.1. Extended states. Prove for and suitable values of that the Anderson model has purely absolutely continuous spectrum in some energy range. 2. Localization in two dimensions. Prove that for , the spectrum of the Anderson model is dense pure point for all values of . 3. Quantum diffusion. Prove that for and values of where there is a.c. spectrum that grows as as . 4. Ten Martini problem. Prove for all and all irrational that (which is independent) is a Cantor set, that is, that it is nowhere dense. 5. Prove for all irrational and that has measure zero. 6. Prove for all irrational and that the spectrum is purely absolutely continuous. 7. Do there exist potentials on so that..

### Landau's problems

Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..

### Hilbert's problems

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900. In particular, the problems presented by Hilbert were 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22 (Derbyshire 2004, p. 377). Furthermore, the final list of 23 problems omitted one additional problem on proof theory (Thiele 2001).Hilbert's problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. As such, some were areas for investigation and therefore not strictly "problems."1. "Cantor's problem of the cardinal number of the continuum." The question of if there is a transfinite number between that of a denumerable set and the numbers of the continuum was answered by Gödel and Cohen in their..

### Coin problem

Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..

### Lam's problem

Given a (0,1)-matrix, fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a projective plane of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists.Lam's problem is equivalent to finding nine orthogonal Latinsquares of order 10.

Subscribe to our updates
79 345 subscribers already with us
Math Topics
Math Categories
Math Subcategories
Check the price
for your project
Support
1-855-407-7728
Follow us
Help About us FAQ for students FAQ for writers
Main
Students


text:
Orthogonal projection to symmetric matrices with bounded eigenvalues

Let $$0, $$\mathbb{S}^n$$ be the set of all $$n\times n$$ real symmetric matrices, and $$\mathcal{V} \subset\mathbb{S}^n$$ be the set of symmetric matrices such that $$aI\le X \le b I$$, where $$\le$$ means the Loewner order. Consider the orthogonal projection $$\Pi:\mathbb{S}^n \to \mathcal{V}$$ such that for all $$A\in \mathbb{S}^n$$, $$\Pi(A)=\arg\min_{X\in \mathcal{V}}\|X-A\|^2.$$ where $$\|\cdot\|$$ indicates the Frobenius norm or spectral norm.

I was wondering whether there is a simple procedure to evaluate $$\Pi$$ (for one of the above matrix norms). Can the projection operator $$\Pi$$ be expressed in terms of projecting the corresponding eigenvalues of $$A$$?

As shown here, the claim holds if $$a=0$$ and $$b=\infty$$. The argument relies on any positive semidefinite matrices have nonnegative diagonals, which can not be easily extended to general $$a,b$$.

You can still reduce the problem to diagonals. For the Frobenius norm, von-Neumann's trace inequality gives

$$\|X - A\|_F^2 = \|X\|_F^2 + \|A\|_F^2 - 2 \langle X, A\rangle \geq \|X\|_F^2 + \|A\|_F^2 - 2 \sum_{i = 1}^n \sigma_i(X)\sigma_i(A) \\ = \| \Sigma_X \|_F^2 + \| \Sigma_A \|_F^2 - 2 \langle \Sigma_A, \Sigma_X \rangle = \| \Sigma_X - \Sigma_A \|_F^2$$

where $$\sigma_i(\cdot)$$ denotes the $$i$$-th singular value and $$\Sigma_A, \Sigma_X$$ are diagonal matrices holding the singular values of $$A$$ and $$X$$.

Note that the inequality is actually an equality when $$X$$ and $$A$$ commute; i.e., $$X$$ and $$A$$ have the same set of eigenvectors. Therefore, the optimal solution will be

$$X = U \mathbf{diag}\left(\mathrm{proj}_{[a, b]} \sigma_i(A)\right) U^*,$$

where $$U$$ is the matrix of eigenvectors of $$A$$.

• Thanks for your answer. May you kindly provide a reference for the equality condition of von-Neumann's trace inequality?
– John
Aug 5 at 23:18
• I don't have an easily accessible reference at hand, but one way to verify that the condition is sufficient for equality is to simply set $X$ equal to $U D U^*$ for any diagonal matrix $D$ and $U$ being the eigenvectors of $A$ and notice that it is equal to the lower bound. Aug 6 at 17:54
• To verify the claim, have you assumed that $A=U\Sigma_A U^*$? As $\Sigma_A$ contains the singular values of $A$, and $A$ is not necessarily positive semidefinite, it seems that $U$ is not the eigenvectors of $A$. In fact, I feel the correct formula is $\Pi(A)= U \mathbf{diag}\left(\mathrm{proj}_{[a, b]} \sigma_i(A)\right) V^*$, where $A=U\Sigma_A V^*$ is the singular value decomposition of $A$ (here a related post). Please kindly let me know if I overlooked anything.
– John
Aug 7 at 23:27
• May you kindly clarify whether you have assumed that $A=U \Sigma_A U^*$ for the SVD? I don't think this expression is true for general symmetric $A$, as it implies that $A$ is positive semidefinite. In particular, your expression of $X$ does not recover the solution in here if $A$ is negative definite.
– John
Aug 11 at 20:57
• Sorry, I missed your previous comment! Yes, I did overlook that possibility. However, in that case you can just replace the von-Neumann inequality with Fan's trace inequality, which is expressed in terms of eigenvalues instead of singular values. I can clarify in my original answer if you need more context. Aug 11 at 21:19


text:
Tag Info

Accepted

How to estimate GridSearchCV computing time?

You could fit your model/pipeline (with default parameters) to your data once and see how long it takes to train. Then you would multiply that by how many times you want to train the model through ...
• 7,758

What is the most efficient method for hyperparameter optimization in scikit-learn?

Optimization isn't my field, but as far as I know, efficient and effective hyper-parameter optimization these days heavily revolves around building a surrogate model. As models increase in complexity, ...
• 371
Accepted

How to plot mean_test score and mean_train score of GridSearchCV

You could visualize them as a heatmap. For example you could use the C values as the rows, the gamma values as the columns and ...
• 311

GridSearch without CV

GridSearchCV is built around cross validation, but if speed is your main concern, you may be able to get better performance using a smaller number of folds. From the docs: class sklearn....
• 578

Log loss vs accuracy for deciding between different learning rates?

According to me, it is not correct to co-relate loss with accuracy. Loss is used to optimize the hypothesis such that we can get best weights whereas accuracy is used to identify how well model ...
• 1,232
Accepted

how to pass parameters over sklearn pipeline's stages?

I figured out how to do that by monkey patching ParameterGrid.__iter__ and GridSearchCV._run_search methods. ParameterGrid.__iter__ iterates over all possible combinations of hyerparameters (dict of ...
Accepted

Can GridSearchCV be used for unsupervised learning?

The goal of GridSearchCV is to iterate over (hence search) all possible combinations (hence grid) of hyper parameters and evaluate a model on a cross-validation (...

When to use BayesianSearchCV and how it works?

As mentioned in that Kaggle notebook, you can use it pretty much as just a drop-in replacement for other search methods (grid or random). Bayesian searches still are random searches over a predefined ...
• 10.8k

What's the default Scorer in Sci-kit learn's GridSearchCV?

From the User Guide: By default, parameter search uses the score function of the estimator to evaluate a parameter setting. These are the ...
• 10.8k

What is the most efficient method for hyperparameter optimization in scikit-learn?

You can take a look at auto-sklearn. That's an automated machine learning toolkit which is a direct extension of scikit-learn.
• 526

Large negative R2 or accuracy scores for random forest with GridSearchCV but not train_test_split

Negative R2 values can be observed when using it in the context of model validation (where we have data that is withheld from the model) because in this context, SST $\ne$ SSE + SSR. That is, this ...
• 1,874

Large negative R2 or accuracy scores for random forest with GridSearchCV but not train_test_split

R2 can be negative if the model is arbitrarily worse according to the sklearn documentation So the very negative train scores were indicative of an extremely bad performance. Why was the test ...
• 151

Default parameters for decision trees give better results than parameters optimised using GridsearchCV

My Suggestion: The intrinsic separation of classes needs more complex model to be captured. I say this, because the difference between default model and your grid search is in max_depth parameter ...
• 6,452
Accepted

How to get mean test scores from GridSearchCV with multiple scorers - scikit-learn

For multi-metric evaluation, the scores for all the scorers are available in the cv_results_ dict at the keys ending with that scorer's name ('_scorer_name'). so use grid.cv_results_['mean_test_(...
• 526
Accepted

Comparison of machine learning approaches for a topic in a scientific paper

If I understand, you could evaluate your approach based on your efforts in using the right preprocessing, picking the right features, building the appropriate architecture (choosing the model that ...
• 886
Accepted

GridSearch mean_test_score vs mean_train_score

When you do k-fold cross-validation, you train k models, each one of them leaving the proportion $1/k$ of the data out. For each of the models, you can compute its train error and validation error. ...
• 5,921

ScikitLearn - RandomForestRegressor score different in and out of grid search

I have found the answer to my question : there is randomness in RandomForestRegressor(). Using the random_state hyper parameter fix the problem : ...
Accepted

sklearn.model_selection: GridSearchCV vs. KFold

Yes, you can replace the cv=5 with cv=KFold(n_splits=5, random_state=None, shuffle=False). Leaving it set to an integer, like 5, ...
• 1,714

GridSearch without CV

By passing a callable for parameter scoring, that uses the model's oob score directly and completely ignores the passed data, you should be able to make the ...
• 10.8k
Accepted

plotting a decision tree based on gridsearchcv

While I don't have a module named graphviz I can still try to help. Reading the documentation for GridSearchCV, I can see that there's a attribute called ...
• 813

Why is cross-validation score so low?

I know this question has been here for two years, however, I was having the same problem when using cross_val_score on my data and I ended up here. The results ...
• 21
Accepted

Why is cross-validation score so low?

In your random forest, this is due to the fact that your final model is overfitting. Sklearn's GridSearchCV has a default argument ...
• 5,921

How to estimate GridSearchCV computing time?

Let the search complete and then you can use cv_results_ attribute to compute the elapsed time as given below. ...

Class weight ineffective in sklearn

This is most likely because sklearn has chosen your majority class for you and you simply need to weigh your classes the other way around with "0" being the majority class.
• 21

Splitting hold-out sample and training sample only once?

In general, it's a good idea to split up your data into three sets: Training Set (60-80% of your data) Cross-Validation Set (10-20% of your data) Test Set (10-20% of your data) When you select a ...
• 573

Log loss vs accuracy for deciding between different learning rates?

Actually, loss is used by the model to decide the probability of the class. So, logloss just indicates how much is your model certain in comparison to the correct labels of the classes in test samples....
• 497
Accepted

How to find optimal number of trees in random forest using Grid search in R?

mtry is the parameter indicating how many of the features are checked in each split decision. http://topepo.github.io/caret/train-models-by-tag.html#random-forest ,...
• 10.8k
Accepted

Am I using GridSearch correctly or do I need to use all data for cross validation?

Your procedure is, from what I can tell, correct. You are correctly splitting your data into train/test, and then using your training data only to find optimal hyper-parameters. Using all of the ...
• 1,874
Accepted

CNNs - Hyperparameter tuning with different training sizes of the same data set

First suggestion: you should first find a CNN architecture that satisfies you, and then stick with it. Second suggestion: be careful with cross validation. CNNs are extremely "heavy" models, they can ...
• 5,897


text:
# How do confidence levels compare to significance levels?

Confidence level is denoted as (1-\alpha)*100\%, while significance level is denoted as $\setminus \alpha$.
For example, if confidence level is 95\%, significance level is 5\% , i.e, $\setminus \alpha = 0.05$


text:
## 20150421

### Linux Mint 17.1, Nvidia, CUDA, and cuDNN

I recently replaced a Titan X, which was on loan, with a GTX 980. After messing with drivers for nearly a day I was able to get my dual monitor setup running again. Unfortunately whatever i did freaked out Theano yielding the error:


text:
# Homework Help: Find the max height of the second projectile relative to seal level

1. Oct 1, 2012

### PhysicsFTW22

1. The problem statement, all variables and given/known data
A projectile is launched at 110.0 m/s at 40.0° above the horizontal off a cliff that is 250.0 m above sea level. After 4.00 s of flight, this projectile launches a second projectile at 50.0 m/s directed at 15.0° below the horizontal.
a) Find the max height of the second projectile relative to seal level.
b) Find the range of the second projectile when it reaches sea level.

2. Relevant equations
In x-direction: x=(vicos∅)t , vi=vf , vix= vicos∅
In y-direction: y=yi + vit + .5at^2 , v=vi+at , vi=visin∅

3. The attempt at a solution

I got the answer to part a, 472m. However, I am lost at trying to find an answer to part b. Using x=(vi)(t)=(110cos40)(4)=337m, I used that as the range the first projectile goes before releasing the second projectile. Now in trying to find how far the second projectile goes when it hits sea level, I plan on adding that to the first range. Any thoughts?

2. Oct 1, 2012

### danielu13

Using position functions:

$x=v$i$*t*cosθ$

$y=v$i$*t*sinθ-\frac{1}{2}*9.81*t^2+y$o

You get the position of projectile 1 as (337.060, 454.347) when the second projectile is released. Using velocity equation:

$v=v$i$+at$

The velocity at t(4) is (84.265 $\hat{i}$, 31.467 $\hat{j}$), neglecting drag.

Projectile 2 is released at 50 m/s at 15°, which is added to the velocity of projectile 1, resulting in an initial velocity of (132.561 $\hat{i}$, 18.526 $\hat{j}$). Do note that the launch velocity in the y direction was subtracted from the value for projectile 1 because it was in the downward direction, and I believe this was at a time that the projectile was still rising; if it was falling you would just add the initial velocity of projectile 2 relative to projectile 1, which was 12.941 $\hat{j}$, to the velocity of projectile 1 at t(4).

You then use this velocity to find the time at which projectile 2 will hit the ground, then use the position function to find the range. I know this was a bit of a mess, but it should help. I can work it out further if necessary.


text:
This puzzle is based on the very popular game 2048, which was invented by Gabriele Cirulli, and (if you haven't seen it) details can be found here.

As you play the game you gradually end up with tiles of higher and higher values - 2, 4, 8, 16, and so on, up in powers of two to the 2048 tile which gives the game its name. And beyond to higher powers of two if you are good (and lucky) enough.

Every time two tiles combine you score the value of the resulting tile. So combining two 2 tiles to make a 4 tile will score 4.

To make an 8 tile you will have to make two 4 tiles (scoring 4 each) and combine them to make the 8 tile (scoring 8) which makes a total of 16. So combining tiles to make an 8 tile will score 16.

Can you find a general formula to calculate the score for any power of two? That is, to make a tile of 2 to the power x will produce a score of y.

Each tile with a new power of two requires two tiles of the previous power of two, so each new power is twice the score of the previous power (to make the two tiles) plus the value of combining these (the new power).

As we've seen an 8 tile scores twice the 4-tile score plus 8, ie 16. A 16 tile scores twice the 8-tile score (2 x 16) plus 16, ie 48. A 32 tile scores twice the 16-tile score (2 x 48) plus 32, ie 128. And so on . . .

The scores for tiles with each increasing power of two are:

Tile 4 scores 4

Tile 8 scores 16

Tile 16 scores 48

Tile 32 scores 128

Tile 64 scores 320

Tile 128 scores 768

Tile 256 scores 1792

Tile 512 scores 4096

Tile 1024 scores 9216

Tile 2048 scores 20480

The general formula when n is the power of two is

$$2^n * (n - 1)$$

Devotees of the game will have spotted that we are ignoring the fact that sometimes the new tiles are 4s and not 2s, and so the scoring would be a little different. However, for the purposes of this puzzle, we are keeping it simple and just working in 2s.

If you find the puzzle easy you can always extend it, assuming that one new tile in seven is a 4 not a 2!

If one new tile in seven is a 4 then you will get six 2 tiles and one 4 tile, which adds up to 16.

In the solution above you combine eight 2 tiles to make four 4 tiles (scoring 16), and then you combine the four 4 tiles to make two 8 tiles (scoring another 16), and finally you combine the two 8 tiles to make a 16 tile (scoring 16 again). This results in the 48 that we have in the solution above.

With six 2 tiles and one 4 tile you combine the six 2 tiles to make three 4 tiles (scoring 12), and then using these three 4 tiles plus the other 4 tile you proceed as above, scoring 16 and 16. This makes a total of 44 instead of 48.

From 16 up through the powers of two you can proceed as before, namely twice the score for the previous power plus the power, remembering that the 16 tile scores 44 not 48. This starts

Tile 16 scores 44

Tile 32 scores 120

Tile 64 scores 304

And the general formula is

2^n * (n - 1) - 2^(n - 2)


text:
How comes that we have the “freshman algebraist's iso theorem”?

It should seem a silly (or even trivial) question, but I've always wondered who first noticed that quotient by an equivalence relation "really behaves like a fraction", meaning that (whatever this mean, depending on the setting you're working in) $$A/B \cong (A/C)/(B/C)$$ everytime you have $C\le B\le A$. I mean, why on earth should I denote a set of equivalence classes by a relation -which I suppose just for the sake of simplicity to be a congruence on the set- with a fraction $\frac{\text{whole structure}}{\text{substructure}}$, if it wasn't for that useful "simplification"?

-
$\frac{\frac{A}{C}}{\frac{B}{C}} = \frac{A}{C}\frac{C}{B}=\frac{A}{B}$ –  tetrapharmakon Jan 14 '12 at 18:20
So you're asking who first came up with this isomorphism theorem? They're often attributed to Noether, but see the article for details. –  Dylan Moreland Jan 14 '12 at 18:22
I found the article but I don't understand any German. So you're saying that Noether first used the fractional notation. But what was her motivation? Did she find the iso theorem and then defined quotients, or...? –  tetrapharmakon Jan 14 '12 at 18:30
Oh, I meant the Wikipedia article. I don't read German either, and I don't know anything about her motivation or notation. I only hoped that some bit of historical information might get you started. –  Dylan Moreland Jan 14 '12 at 18:35
In the 1927 paper mentioned in the Wikipedia article she almost uses the modern notation: she writes $\mathfrak{R}\mid\mathfrak{a}$ for the quotient of a ring $\mathfrak{R}$ by an ideal $\mathfrak{a}$. She calls it a Restklassenring; this is literally 'residue class ring', and her Restklasse 'residue class' is an equivalence class under the congruence induced by the ideal. –  Brian M. Scott Jan 14 '12 at 20:21


text:
# Example of a CW complex not homeomorphic to the realization of a simplicial set?

I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an example for the opposite statement as well, if it is true)?

-
Heh, I just realized that we could call the "opposite statement" the "adjoint statement". –  Harry Gindi Aug 5 '10 at 20:13

The mapping cylinder of a really messy continuous map $I\to I$

The nerve of the category in which there are two objects and each Hom set is a singleton.

-
What do you mean by the nerve in the second example? I thought the nerve was a simplicial set by definition, so it can't possibly provide an example of a CW-complex that doesn't come from a simplicial set. –  Omar Antolín-Camarena Jul 16 '13 at 13:43

The geometric realization of a simplicial set is always triangulable. See Corollary 4.6.12 in Cellular Structures in Topology by Fritsch and Piccinini. They also give an explicit example (in section 3.4) of a non-triangulable CW-complex (which uses, I think, essentially the same idea as Tom Goodwillie's suggestion). This paper contains another example and shows, on the other hand, that every CW-complex with cells in at most two dimensions is triangulable.

-


text:
Volume 28, Issue 3
BMO Estimates for Multilinar Fractional Integrals
10.3969/j.issn.1672-4070.2012.03.002

Anal. Theory Appl., 28 (2012), pp. 224-231

Preview Full PDF BiBTex 180 568
• Abstract

In this paper, the authors prove that the multilinear fractional integral operator$T_{\Omega,\alpha}^{A1,A2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A1,A2}$ W,a with rough kernel are both bounded from$L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−a),\infty$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A1,A2}( f )(x) =\int_{R^n}\frac{{R_{m_1}}(A_1;x,y){R_{m_2}}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2_2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A1,A2}(f)(x)= \sup_{r & gt 0}\frac{1}{r^n−\alpha +m_1+m_2−2}\int_{|x−y| \< r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0 < \alpha < n$, $\Omega \in L^s(S^n−1)$ $(s \geq 1)$ is a homogeneous function of degree zero in $R^n$, $A_i$ is a function defined on $R^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor seriesof $A_i$ at $x$ about $y$. More precisely, $R_{m_i}$ (A_i;x,y) = A_i(x)− \sum_{|\gamma|< m_i}\frac{1}{\gamma!}D^\gammaA_i(y)(x−y)^r,$where$D^\gamma (Ai) \in BMO(R^n)$for$|\gamma | = m_i−1(m_i > 1)$,$i = 1,2\$.

• History

Published online: 2012-10

• Keywords
• AMS Subject Headings
• Cited by


text:
# Question The Cambro Foundation, a nonprofit organization, is planning toinvest $159,040 in a project Question The Cambro Foundation, a nonprofit organization, is planning toinvest$159,040 in a project that will last for three years. Theproject will produce net cash inflows as follows:

 Year1 $50,000 Year2$ 66,000 Year3 ?

 Click here to view Exhibit 13B-1 and Exhibit 13B-2, to determinethe appropriate discount factor(s) using table.

Required:

 Assuming that the project will yield exactly a 7% rate ofreturn, what is the expected net cash inflow for Year 3?(Round discount factor(s) to 3 decimalplaces.)

 Cash Inflow ChooseNumerator: / ChooseDenominator: = Cash Inflow Present value of year 3 cash inflow / Present value factor = Cash Inflow / 0.816 =


text:
Applied Statistics Lesson of the Day – Additive Models vs. Interaction Models in 2-Factor Experimental Designs

$Applied Statistics Lesson of the Day – Additive Models vs. Interaction Models in 2-Factor Experimental Designs$

In a recent “Machine Learning Lesson of the Day“, I discussed the difference between a supervised learning model in machine learning and a regression model in statistics.  In that lesson, I mentioned that a statistical regression model usually consists of a systematic component and a random component.  Today’s lesson strictly concerns the systematic component. An […]

Ever wonder how popular your favorite R functions are?

March 8, 2014
By

How's that fried pickle sandwich treating you?  Perhaps your taste in R  functions are less bizarre than your taste in R commands?Now you can easily find out using this new shiny app!  In this post I use the R function frequency table co...

Per capita GDP versus years since women received right to vote

March 7, 2014
By

Below is a plot of per capita GPD (in log scale) against years since women received the right to vote for 42 countries. Is this cause, effect, both or neither? We all know correlation does not imply causation, but I … Continue reading →

Selection bias in the reporting of shaky research

March 7, 2014
By

I’ll reorder this week’s posts a bit in order to continue on a topic that came up yesterday. A couple days ago a reporter wrote to me asking what I thought of this paper on Money, Status, and the Ovulatory Cycle. I responded: Given the quality of the earlier paper by these researchers, I’m not […]The post Selection bias in the reporting of shaky research appeared first on Statistical Modeling,…

RUG-Philippines Meetup: Markov Switching Models in R

March 7, 2014
By

To the R users based in the Philippines, there will be upcoming meetup, here are the details:topics: Markov Switching Models in R           by Ohly Santos           How to use the op...

Advances in scalable Bayesian computation [day #4]

March 7, 2014
By

Final day of our workshop Advances in Scalable Bayesian Computation already, since tomorrow morning is an open research time ½ day! Another “perfect day in paradise”, with the Banff Centre campus covered by a fine snow blanket, still falling…, and making work in an office of BIRS a dream-like moment. Still looking for a daily theme, […]

On replacing calculus with statistics

March 7, 2014
By

Russ Roberts had this to say about the proposal to replacing the calculus requirement with statistics for students. Statistics is in many ways much more useful for most students than calculus. The problem is, to teach it well is extraordinarily difficult. It’s very easy to teach a horrible statistics class where you spit back the […]

IJF news

March 7, 2014
By

This is a short piece I wrote for the next issue of the Oracle newsletter produced by the International Institute of Forecasters. Special section topics We continue to publish special sections on selected topics. Because of the change in the way regular papers are now handled, we do not publish whole special issues any more. Rather, each issue has regular papers at the front, and if there are any special…

Analytics Software Popularity Update: Counting Blogs, Simplifying Job Searches

March 7, 2014
By

My latest update to The Popularity of Data Analysis Software is an attempt to use blog counts to estimate the popularity of analytics software. While I was able to greatly broaden the coverage of packages when studying job data, I … Continue reading →

Freshman hordes slightly more godless than ever

March 6, 2014
By

This article is an update to my annual series on one of the most under-reported stories of the decade: the fraction of college freshmen who report no religious preference has tripled since 1985, from 8% to 24%, and the trend is accelerating.In last yea...

How much time (if any) should we spend criticizing research that’s fraudulent, crappy, or just plain pointless?

March 6, 2014
By

I had a brief email exchange with Jeff Leek regarding our recent discussions of replication, criticism, and the self-correcting process of science. Jeff writes: (1) I can see the problem with serious, evidence-based criticisms not being published in the same journal (and linked to) studies that are shown to be incorrect. I have been mostly […]The post How much time (if any) should we spend criticizing research that’s fraudulent, crappy,…

The Secret to Entropy’s role in Statistics

March 6, 2014
By

Entropy is the single most powerful statistical tool discovered to date. That’s a bold claim. I intend to back that claim up in this post. Forget for a moment about Entropy and Statistics and start fresh. Suppose we know two variables , are relat...

Two charts that fail self-sufficiency

March 6, 2014
By

My twitter followers have been sending in several howlers. Twitter (link) made a bunch of bold claims about its own influence by using the number of tweets about the Oscars as fodder. They also adopt the euphenism common to the...

Advances in scalable Bayesian computation [day #3]

March 6, 2014
By

We have now gone over the midpoint of our workshop Advances in Scalable Bayesian Computation with three talks in the morning and an open research or open air afternoon. (Maybe surprisingly I chose to stay indoors and work on a new research topic rather than trying cross-country skiing!) If I must give a theme for […]

Machine Learning Lesson of the Day – Memory-Based Learning

Memory-based learning (also called instance-based learning) is a type of non-parametric algorithm that compares new test data with training data in order to solve the given machine learning problem.  Such algorithms search for the training data that are most similar to the test data and make predictions based on these similarities.  (From what I have learned, memory-based learning […]

Highlighting the web

March 6, 2014
By

Users of my new online forecasting book have asked about having a facility for personal highlighting of selected sections, as students often do with print books. We have plans to make this a built-in part of the platform, but for now it is possible to do it using a simple browser extension. This approach allows any website to be highlighted, so is even more useful than if we only had…

"Unifying the Counterfactual and Graphical Approaches to Causality" (Tomorrow at the Statistics Seminar)

March 5, 2014
By

Attention conservation notice: Late notice of an academic talk in Pittsburgh. Only of interest if you care about the places where the kind of statistical theory that leans on concepts like "the graphical Markov property" merges with the kind of analy...

Exploring Ball Locations and Player Behaviors in Basketball

March 5, 2014
By

"Game on!" by Fathom Information Design is an exploratory visualization prototype that allows users to parse through a basketball game's data, to investigate the behaviors and patterns in terms of the statistics and locations of players. Based on a ...

Fun with Mike Steele Quotes and Rants

March 5, 2014
By

Check out the web page of my Penn Statistics buddy, Mike Steele, probabilist, statistician and mathematician extraordinaire. (And that's just his day job. At night he battles the really hard stuff -- financial markets.) Among other things, you'll ...

PLoS One, I have an idea for what to do with all your profits: buy hard drives

March 5, 2014
By

I've been closely following the fallout from PLoS One's new policy for data sharing. The policy says, basically, that if you publish a paper, all data and code to go with that paper should be made publicly available at the … Continue reading →

Plagiarism, Arizona style

March 5, 2014
By

Last month a history professor sent me a note regarding plagiarism at Arizona State University: Matthew Whitaker, who had received an expedited promotion to full professor and was made Director of a new Center for the Study of Race and Democracy by Provost Elizabeth Capaldi and President Michael Crow, was charged by most of the […]The post Plagiarism, Arizona style appeared first on Statistical Modeling, Causal Inference, and Social Science.

Advances in scalable Bayesian computation [day #2]

March 5, 2014
By

And here is the second day of our workshop Advances in Scalable Bayesian Computation gone! This time, it sounded like the “main” theme was about brains… In fact, Simon Barthelmé‘s research originated from neurosciences, while Dawn Woodard dissected a brain (via MRI) during her talk! (Note that the BIRS website currently posts Simon’s video as […]

Optimizing a function that evaluates an integral

March 5, 2014
By

SAS programmers use the SAS/IML language for many different tasks. One important task is computing an integral. Another is optimizing functions, such as maximizing a likelihood function to find parameters that best fit a set of data. Last week I saw an interesting problem that combines these two important tasks. [...]


text: 1 0 obj << >> endobj About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra, and number theory. E Abstract Concepts258 ... ments and solve them on paper before entering the answers online. /Type /Page Pick one of our Abstract Algebra … /Contents 13 0 R /Contents 3 0 R sample 2: Composite and Inverse Functions. >> endobj /Filter /FlateDecode x�U�]O�0���+zY9��O-8ef@�Q/�@G�0j��ޕr����O���bv��� m4#�a".Q��9)j�L��d��7i��A�#��l�C��+�t���kqKB�AAB.�;���Y��t5B6�C���4[MqĹ�L����͘dR�f�H�$� 8 0 obj << Ask Question Asked 7 years, 11 months ago. /Parent 6 0 R Algebra MCQ Questions and answers with easy and logical explanations. � /ProcSet [ /PDF /Text ] Also Detailed solutions with full explanations are included sample 2. In this chapter we will outline the background needed for a course in abstract algebra… The 100 multiple-choice questions are based on the Mathematics 7–12 test framework. you done a great work for the new learners in linear algebra like me. /Filter /FlateDecode /Filter /FlateDecode But this is not true for the matrix 1 0 0 0 whose rank is one. False Problem 2. text is Linear Algebra: An Introductory Approach [5] by Charles W. Curits. Evaluate 7m + 3mn when m = 8 and n = 14 A) 84 B) 196 C) 392 D) 168 E) NOTA 2. c�1���S�A�n�'�����m\�k���Z��3D��H{Nα���|>O�a]�Rr��&m�r�vz�I�ձ�6�C(JM]�cյ1�q̕$~��+"VJǍ��Nu��c./0�r��S�����B (multiple choice questions) College Algebra Problems with Answers; sample 1: Quadratic Functions. /Contents 17 0 R Multiple choice question on characteristic subgroups. The determinant of a square matrix A is nonzero if and only if the equation Ax = 0 has a unique solution. A file containing multiple choice questions on commutative algebra /Parent 6 0 R Good job! /ProcSet [ /PDF /Text ] ��w3�T�г433QIS042�337R07��3��PIQ���,ӌ ��r � 8 D. x ≤ 6. If $$p \times q = p + q + \frac{p}{q}{\text{,}}$$    then the value of 8 × 2 is? 0. multiple choice question on ideals. Practice questions and answers for Class 9 Mathematics are available on Important Questions for Mathematics. /Filter /FlateDecode /Parent 6 0 R endobj Arithmetic Ability provides you all type of quantitative and competitive aptitude MCQ questions on Algebra with easy and logical explanations. /Filter /FlateDecode %PDF-1.4 /ProcSet [ /PDF /Text ] q����F Multiple Choice Math Problems with Solutions. The value of $$\left( {{\text{1 + }}\frac{1}{x}} \right)$$ $$\left( {{\text{1 + }}\frac{1}{{x + 1}}} \right)$$  $$\left( {{\text{1 + }}\frac{1}{{x + 2}}} \right)$$  $$\left( {{\text{1 + }}\frac{1}{{x + 3}}} \right)$$   is? of abstract algebra. (multiple choice questions). Abstract algebra assumes a working prerequisite knowledge of necessary mathematical fundamentals. Similarly questions and answers are available for Classes 1 to 11 covering all the important topics.You may also visit Examrace YouTube channel for many important lectures covered on … You will receive incredibly detailed scoring results at the end of your Abstract Algebra practice test to help you identify your strengths and weaknesses. Even more important is the ability to read and understand mathematical proofs. $$\frac{{x + 4}}{x}$$, A. 2. a problem on ideal. abstract algebra questions and answers Sep 06, 2020 Posted By Dean Koontz Media Publishing TEXT ID f38d4f18 Online PDF Ebook Epub Library algebra and no student should go through such a course without a good notion of what a proof is the importance of applications such as coding theory and cryptography has These notes are prepared in 1991 when we gave the abstract al-gebra course. Integrated Algebra Multiple Choice Regents Exam Questions www.jmap.org 2 6 Which expression represents 12x 3 −6x2 +2x 2x in simplest form? (Hint: The answers to all three parts are related!) If $$\frac{a}{b}{\text{ = }}\frac{2}{3}$$   and $$\frac{b}{c}{\text{ = }}\frac{4}{5}{\text{,}}$$   then the ration $$\frac{{a + b}}{{b + c}}$$   equal to? These assessments will help solidify your understanding of linear equations and abstract algebra using several types of multiple-choice math problems. 10, A. This download link will take you to the full document containing close to 100 Financial Accounting past questions and answers. /Length 324 Active 7 years, 10 months ago. endstream Some of the solutions here are very short and in the form of a hint. If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to? Texas public schools must possess. Abstract Algebra Unit – I 1 : Questions of 2 marks 1) Define product of two permutations on n symbols. >> endobj /MediaBox [0 0 595.276 841.89] >> Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Questionaire for math Math 2550 Matrix Algebra Multiple Choice Exam Form A Oct. 27, 2014 Problem 1. 1 ) Let a, B the … Practice questions and answers ` ꄦtGv �S�� [ W�t�: >. Choice tests are used at every grade level in school in the of! Algebra using several types of multiple-choice math problems more important is the ability to read and understand mathematical proofs we... Test are in elementary algebra, abstract algebra, and number theory not true for the matrix 1 0 0... Is a must $, B your strengths and weaknesses all Mathematics majors a solution! 100 Financial Accounting past questions and answers the test scores are less 72... Basic knowledge of set theory, mathematical induction, equivalence relations, and matrices is a must the right.! Was to help the students by giving them some exercises and get them familiar with some solutions 9 are., 11 months ago } }$ $, B important questions a. The end of your abstract algebra using several types of multiple-choice math problems equation Ax = 0 a. Geometry '', D. Holten and J. Lloyd, CBRC, 1978 are less than 72 → be. A → B be non empty sets and f: a → B be non empty and... ( a ) has three pivots, we see that the rank of is... Ability to read and understand mathematical proofs J. Lloyd, CBRC, 1978 subject that... Months ago will take you to the full document containing close to 100 Financial past.: Quadratic Functions for exams like Banking exams, IBPS, SCC, CAT, XAT,,... W�T�: ���JC�������ۇ���� > � �Sw_�.H���� in linear algebra like me Asked 7,... For choosing the right answer true for the new learners in linear algebra, abstract algebra, linear,! You to the full document containing close to 100 Financial Accounting past questions answers. � �Sw_�.H���� of the test may contain questions that do not tend to fare poorly midterm... Is one = √ ( 6 - 2x ) is given as option B ability to read and cite the... ) = √ ( 6 - 2x ) is given by answer: the characteristic polynomial … here a! Algebra multiple choice tests are used at every grade level in school answers Class. ( x ) = √ ( 6 - 2x ) is given as option B [ 5 by... Short and in the test are in elementary algebra, abstract algebra, linear,. 27 } }$ $\frac { { 27 } } { 9 }$... Read and cite all the research you need on ResearchGate... abstract answers ; sample:! , B questions in the test may contain questions that do not count toward score!, 11 months ago of 2 marks 1 ) Define product of two permutations on symbols. Will take you to the full document containing close to 100 Financial past... Two permutations on n symbols induction, equivalence relations, and number theory 1 marks Choose the correct from... – I 1: questions of 2 marks 1 ) Define product of two permutations on n symbols questions! Help your child prepare for them by making your own tests at home and him. Done a great work for the new learners in linear algebra: abstract algebra multiple choice questions with answers pdf... Them some exercises and get them familiar with some solutions the research you need on...... Answer Key.pdf 7 years, 11 months ago are based on the Mathematics 7–12 test framework, equivalence relations and. 1 and 0 so that this matrix is not nilpotent and 0 so that this matrix is not for! The score prepare for them by making your own tests at home and teaching him or her techniques choosing! By the qroup problems: Quiz2PracticeQuestions.pdf has three pivots, we see that rank. Rank is one for exams like Banking exams, IBPS, SCC CAT! Product of two permutations on n symbols past questions and answers for tips and sample questions Mathematics! Is not nilpotent right answer of 1 marks Choose the correct option from the options! Mcq questions and answers on n symbols your own tests at home and teaching him or her techniques for the. Empty sets and f: a → B be non empty sets and f: →! 2 Mult choice Practice answer Key.pdf ResearchGate... abstract only on scored.... Him or her techniques for choosing the right answer number theory a variety of.! Mcq questions on algebra with easy and logical explanations a lot of multiple-choice math problems and get them familiar some! Those who do not count toward the score, abstract algebra Practice to... Algebra problems with answers ; sample 1: Quadratic Functions tend to fare poorly on abstract algebra multiple choice questions with answers pdf.! = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to assumes. ꄦTgv �S�� [ W�t�: ���JC�������ۇ���� > � �Sw_�.H���� own tests at home and teaching him or her for. Types of multiple-choice math problems ( x ) = √ ( 6 2x! New learners in linear algebra, abstract algebra, abstract algebra, number. 7–12 test framework 9 Mathematics are available on important questions for Mathematics variety levels... Mathematics are available on important questions for Mathematics characteristic polynomial … here are a dozen questions inspired by qroup... Form of a square matrix a is nonzero if and only if the equation Ax = 0 has unique... A working prerequisite knowledge of set theory, mathematical induction, equivalence relations, and number.. A dozen questions inspired by the qroup problems: Quiz2PracticeQuestions.pdf 25 percent of the function abstract algebra multiple choice questions with answers pdf! Your understanding of linear equations and abstract algebra, and matrices is a must 29th is! Hint: the characteristic polynomial … here are a dozen questions inspired by qroup! Choice Practice answer Key.pdf you done a great work for the matrix 1 0 whose... The end of your abstract algebra Practice test to help you identify your strengths and weaknesses and all. By making your own tests at home and teaching him or her techniques for choosing the right answer I. Algebra, and number theory, MAT, etc are used at every grade level in school tests are at... Then 3 * 5 + 5 * 3 is equal to theory, mathematical induction, equivalence relations, matrices... Researchgate... abstract Asked 7 years, 11 months ago fare poorly on midterm examinations mathematical proofs may contain that... Eigenvalues are 1 and 0 so that this matrix is not nilpotent 3. A great work for the new learners in linear algebra: An Introductory Approach [ 5 ] by W.... Not nilpotent some solutions the qroup problems: Quiz2PracticeQuestions.pdf percent of the function f ( x ) = (. Algebra MCQ is important for exams like Banking exams, IBPS, SCC, CAT, XAT, MAT etc... Is the ability to read and cite all the research you need on...! Solidify your understanding of linear equations and abstract algebra using several types of multiple-choice questions are based on the 7–12. Matrix a is nonzero if and only if the equation Ax = 0 a!, 2014 Problem 1 fare poorly on midterm examinations MCQ is important exams... 9 Mathematics are available on important questions for a variety of levels, D. Holten and J.,! 3 is equal to is one J. Lloyd, CBRC, 1978 for Mathematics was to help the by. Problem 1 sample 1: Quadratic Functions great work for the matrix 1 0 0 0 whose rank is...., 2014 Problem 1 Lloyd, CBRC, 1978 three parts are related )! √ ( 6 - 2x ) is given by teaching him or techniques. For a variety of levels f: a → B be a permutation will receive incredibly Detailed scoring at! ) Define product of two permutations on n symbols the ability to read and cite all the research you on... ( hint: the answers to all three parts are related! new learners in linear algebra: Introductory. Two permutations on n symbols question is given as option B based only on scored questions weaknesses. Define product of two permutations on n symbols Geometry '', D. Holten and Lloyd! And understand mathematical proofs if the equation Ax = 0 has a unique solution to read and cite the! Results at the end of your abstract algebra assumes a working prerequisite of...: the answers to all three parts are related! on important for! Important for exams like Banking exams, IBPS, SCC, CAT, XAT, MAT etc! Given as option B questions: Quiz 2 Mult choice Practice.pdf rank is one , B be empty! } { { 27 } } { 9 }  \frac { { 20 } }  {! By giving them some exercises and get them familiar with some solutions, SCC, CAT, XAT MAT! Child prepare for them by making your own tests at home and teaching him her! Full explanations are included sample 2 available on important questions for a variety of levels equivalence! Polynomial … here abstract algebra multiple choice questions with answers pdf a dozen questions inspired by the qroup problems:.! * 3 is equal to IBPS, SCC, CAT, XAT, MAT,.. 29Th question is given by to the full document containing close to 100 Financial Accounting past questions and answers questions. Qroup problems: Quiz2PracticeQuestions.pdf midterm examinations answer Key.pdf the ability to read and all!, CBRC, 1978 on algebra with easy and logical explanations aptitude MCQ questions on algebra with easy and explanations... Test scores are less than 72 take you to the full document containing close to 100 Financial Accounting questions... = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to ꄦtGv [...


text:
Language modeling is the task of predicting the next word or character in a document. Language models were originally developed for the problem of speech recognition; they are also widely used in other NLP applications. For example:

• In machine translation, we convert from text in a source language to text in a target language.
• In summarization, we convert from audio signal to text.
• In dialogue systems, we convert from the user’s input (and perhaps an external knowledge base) into a text response.

## N-gram

Formally, the task of language modeling is to assign a probability to any sequences of words $w_1^n$, i.e., to estimate $P(w_1^n)$.
Notations: the expression $w_1^n$ means the string $w_1,w_2,...,w_n$. For the joint probability of each word in a sequence having a particular value $P(X=w_1,Y=w_2,Z=w_3,...,W=w_n$, we’ll use $P(w_1,w_2,...w_n)$.
By using the chain rule of probability, we get

The chain rule converts the joint probability to the product of conditional probabilities. However, it doesn’t help to simplify the calculation. Therefore, language models make use of the markov-assumption, which approximates the history by just the last few words.
The bigram, for example, approximates the probability of a word given all the previous words $P(w_n|w_1^{n-1}$ by using only the conditional probability of the preceding word $P(w_n|w_{n-1})$, then we can compute the probability of a complete word sequence:

To estimate the probability, an intuitive way is maximum likelihood estimation(MLE).

To compute the probability of an entire sentence, it is convenient to pad the beginning and end with special symbols. E.g., a special symbol <s> at the beginning of the sentence, and a special end-symbol </s>.
For the general case of MLE n-gram parameter estimation:

The hyperpamater n controls the size of the context used in each conditional probability. The bias-variance tradeoff problem lies in n-gram language model. A small n-gram size may cause high bias, and a large n-gram size could lead to high variance. Since the language is full of long-range dependencies, it is impossible to capture those dependencies with small n. Language is creative and variant as well, so it is hard to set n too large.

### Smoothing

We have to make some modifications to our probabilities in case it meets some unseen patterns in the test text. There are several ways: add-one smoothing, backoff and interpolation, Kneser-Ney smoothing, etc.

#### Laplace Smoothing

Take unigram for example. The unsmoothed maximum likelihood estimate of the unigram probability of the word $w_i$ is its count $c_i$ normalized by the total number of word tokens N: $P(w_i)=\frac{c_i}{N}$.
Laplace smoothing just adds one to each count (that’s why it’s called add-one smoothing). Since there are $V$ words in the vocabulary and each one is incremented, we also need to adjust the denominator to take into account the extra $V$ observations.

For bigram counts, we need to augment the unigram count by the number of total word types in the vocabulary $V$:

#### Lidstone Smoothing

Laplace smoothing is a special case of Lidstone smoothing. The basic framework of Lidstone smoothing:

Instead of changing both the numerator and denominator, it is convenient to describe how a smoothing algorithm affects the numerator, by defining an adjusted count $c^*$:

Then the probability can be calculated by dividing $c^*$ by $N$.

#### Good-Turing Estimate

The Good-Turing estimate (Good, 1953) states that for any n-gram that occurs $r$ times, we just pretend that the occurrences is $r^*$ where

and where $n_r$ is the number of n-grams that occur exactly $r$ times in the training data. Then the corresponding probability for an n-gram $\alpha$ with $r$ counts should be normalized as:

#### Discounting

A related way to view smoothing is as discounting (lowering) some non-zero counts in order to get the probability mass that will be assigned to the zero counts. The ratio of the discounted counts to the original counts, discount $d_c$:

Church and Gale (1991) show empirically that the average Good-Turing discount $(r-r^*)$ associated with n-grams is largely constant over $r$. Absolute discounting formalizes this by subtracting a fixed (absolute) discount $d$ from each count.

#### Backoff

When using large n-gram (trigram or more), we may have no corresponding examples. In a backoff n-gram model, if the n-gram we need has zero counts, we approximate it by backing off to the (N-1)-gram. We continue backing off until we reach a history that has some counts.
We have to apply backoff with discounting for a correct probability distribution. This kind of backoff is called Katz backoff.

The term $\alpha(j)$ indicates the amount of probability mass that has been discounted for context j.
Katz backoff is often combined with Good-Turing. The combined Good-Turing backoff algorithm involves quite detailed computation for estimating the Good-Turing smoothing and the $P^*$ and $\alpha$ values.

#### Interpolation

In backoff, we only “back off” to a lower-order n-gram if we have zero evidence for a higher-order n-gram. By contrast, in interpolation, we always mix the probability estimates from all related n-gram estimators.
In simplelinear interpolation, we estimate the trigram probability by mixing together the unigram, bigram, and trigram probabilities, each weighted by a $\lambda$:

To ensure that the estimated probability is valid, $\lambda$s should sum to 1: $\sum_{n=1}^3\lambda_n=1$.
An elegant way to find the specific $\lambda$ values is expectation-maximization(EM).

## Evaluating language modeling

Extrinsic evaluation: the best way to evaluate the performance of a language model is to embed it in an application and measure how much the application, like machine translation, improves. However, such end-to-end evaluation is hard and often expensive.
Intrinsic evaluation: intrinsic evaluation is task-neutral, which can be used to quickly evaluate potential improvements in a language model.
Note that for intrinsic evaluation, we have to use held out data, which is not used during training.

### Perplexity

Perplexity is an information theoretic measurement of how well a probability model predicts a sample. The lower perplexity, the better.

Some special cases:

• In the limit of a perfect language model, probability $1$ is assigned to the held-out corpus, with $\text{Perplex}(\boldsymbol{w})=2^{-\frac{1}{n}\log_{2}1}=2^0=1$
• In the opposite limit, probability zero is assigned to the held-out corpus, which corresponds to an infinite perplexity, $\text{Perplex}(\boldsymbol{w})=2^{-\frac{1}{n}\log_{2}1}=2^\infty=\infty$
• Assume a uniform, uniform model in which $\text{p}(w_i)=\frac{1}{V}$ for all words in the vocabulary. Then,

The perplexity measure is a good indicator of the quality of a language model. However, in many cases improvement in perplexity scores do not transfer to improvement in extrinsic, task-quality scores. In that sense, the perplexity measure is good for comparing different language models in terms of their ability to pick-up regularities in sequences, but is not a good measure for assessing progress in language understanding or language-processing tasks. A model’s improvement in perplexity should always be confirmed by an end-to-end evaluation of a real task before concluding the evaluation of the model.

### Datasets

A small benchmark dataset is the Penn Treebank, which contains roughly a million tokens; its vocabulary is limited to 10,000 words, with all other tokens mapped a special $\langle \text{UNK} \rangle$ symbol.
A larger-scale language modeling dataset is the 1B word Benchmark, which contains text from Wikipedia.
I’ll complement this section after I read the relevant papers. Besides, the state-of-the-art leaderboards can be viewed here.

## References

Jacob Eisenstein. 2018. Natural Language Processing (draft of November 13, 2018). https://github.com/jacobeisenstein/gt-nlp-class/tree/master/notes, pages 125-143.

Dan Jurafsky, James H. Martin. 2018. Speech and Language Processing (3rd ed. draft, Sep. 23, 2018). https://web.stanford.edu/%7Ejurafsky/slp3/, pages 37-62.

Yoav Goldberg. 2017. Neural Network Methods for Natural Language Processing. https://doi.org/10.2200/S00762ED1V01Y201703HLT037, Pages 105-113.


text:
# Syntaxes for Writing Markdown Documents

There are some useful Markdown extensions that brings more convenience for you while working with writing, especially in editing technical documents. Let's take a look at all extensions and new syntaxes which are being used in this site.

Last update: 2022-05-07

## Table of Content

For basic markdown syntax, refer to Markdown Guide

## Meta-data#

The Meta-Data extension adds a syntax for defining meta-data of a document. It is inspired by and follows the syntax of MultiMarkdown. Meta-data is the additional information that can be used to briefly describe the content of a post, such as the title, the short description, tags, and sometimes the banner image. I prefer YAML format:

---
title: The page title
description: The summary of the page content
---

The meta-data can be used in the template and the page content1. In Jinja syntax, each page is represented as a page object, then the meta-data field {{page.meta.title}} will be replaced by the string The page title.

## Code highlighting#

Below extensions are extremely useful for showing code blocks by adding colors, and decoration.

### Inline code#

The InlineHilite is an inline code highlighter inspired by CodeHilite.

Borrowing from CodeHilite’s existing syntax, InlineHilite utilizes the following syntax to insert inline highlighted code: :::language my code or #!language my code.

This will render this line #!python [x for x in range(1, 10) if x % 2] to a fully colored inline Python code: [x for x in range(1, 10) if x % 2].

### Code blocks#

The Code blocks plugin provides a number of features including allowing the nesting of fences, and ability to specify custom fences to provide features like flowcharts, sequence diagrams, or other custom blocks. Highlighting can be further controlled via the Highlight extension.

The standard format which supports to add id, class, or custom attribute key=value is as below:

{ .language #id .class key="value"}
content

Some special attributes key=value are as below:

• title="abc" creates a title for the block, used to show the filename or the purpose.
• linenums="n" creates line numbers starting from n.
• hl_lines="x y-z" highlights the x-th line and lines in the range from y-th to z-th. Line numbers are always referenced starting at 1 ignoring what the line number is started labeling at the number set by the option linenums="n".

Example:

 cpp title="main.c" linenums="2" hl_lines="1 4-5"
#include <stdio.h>

int main(void) {
printf("Hello world!\n");
return 0;
}

Result:

main.c
#include <stdio.h>

int main(void) {
printf("Hello world!\n");
return 0;
}

#### Code annotations#

Code annotation is a unique feature of Material theme which offers a comfortable and friendly way to attach arbitrary content to specific sections of code blocks by adding numeric markers in block and inline comments in the language of the code block.

Code annotations can be placed anywhere in a code block where a comment for the language of the block can be placed, e.g. for JavaScript in // ... and /* ... */, for YAML in # ..., etc.

Example:

 cpp
int main(void) {
printf("Hello world!\n");
return 0;
}

1. Need including\
#!cpp #include <stdio.h>

Result:

Click on to show the annotation.

int main(void) {
printf("Hello world!\n"); // (1)
return 0;
}

1. Need including
#include <stdio.h>

Admonitions, also known as call-outs, are an excellent choice for including side content without significantly interrupting the document flow.

### Marked blocks#

These types of blocks show an icon to help readers notice the kind of content, such as additional information, caution, or error.

Example:

!!! info "The title of the block"

The content can contain formatted text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nullam sagittis ante blandit diam accumsan scelerisque.

- Lorem ipsum dolor sit amet
- Lorem ipsum dolor sit amet

 cpp
int main(void) {
return 0;
}

Result:

The title of the block

The content can contain formatted text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nullam sagittis ante blandit diam accumsan scelerisque.

• Lorem ipsum dolor sit amet
• Lorem ipsum dolor sit amet
int main(void) {
return 0;
}

To remove the title but keep the icon, use a space:

!!! hint " "

Result:

#### Supported types#

abstract, summary, tldr

info, todo

tip, hint, important

success, check, done

question, help, faq

tag

warning, caution, attention

failure, fail, missing

danger, error

bug

example

quote, cite

### Collapsible blocks#

When Details is enabled and an admonition block is started with ??? instead of !!!, the admonition is rendered as a collapsible block with a small toggle on the right side. Use plus (+) sign to make it expanded by default.

Example:

???+ quote "Expandable"

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nulla et euismod nulla. Curabitur feugiat, tortor non consequat finibus, justo purus auctor massa, nec semper lorem quam in massa.

Result:

Expandable

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nulla et euismod nulla. Curabitur feugiat, tortor non consequat finibus, justo purus auctor massa, nec semper lorem quam in massa.

## MathJax#

MathJax is a beautiful and accessible way to display mathematical content in the browser, adds support for mathematical typesetting in different notations (e.g. LaTeX, MathML, AsciiMath).

#### Block syntax#

Blocks must be enclosed in $$...$$ or $...$ on separate lines:

Example:

$$\operatorname{ker} f=\{g\in G:f(g)=e_{H}\}{\mbox{.}}$$

Result:

$\operatorname{ker} f=\{g\in G:f(g)=e_{H}\}{\mbox{.}}$

#### Inline syntax#

Inline blocks must be enclosed in $...$ or $$...$$ :

Example:

The homomorphism $f$ is injective if and only if its kernel is only the singleton set $e_G$, because otherwise $\exists a,b \in G$ with $a \neq b$ such that $f(a)=f(b)$.

Result:

The homomorphism $$f$$ is injective if and only if its kernel is only the singleton set $$e_G$$, because otherwise $$\exists a,b \in G$$ with $$a \neq b$$ such that $$f(a)=f(b)$$.

## Formatting#

Beside Italic, Bold, Italic and Bold, here are some more syntax to format texts.

### Caret#

Caret optionally adds two different features which are syntactically built around the ^ character.

The double carets ^^ inserts  tags, and
the single caret ^ inserts  tags.

Example:

The ^^mass-energy^^ equivalence: E=m*c^2^.

Result:

The mass-energy equivalence: E=m*c2.

### Mark#

Mark adds the ability to insert  tags.

The syntax requires the text to be surrounded by double equal signs == .

Example:

==mark me== and ==mark==me==all==.

Result:

mark me and mark==me==all.

### Tildes#

Tildes optionally adds two different features which are syntactically built around the ~ character.

Add Deletion by using double tildes ~~ which inserts  tags and
Add Subscript by using single tilde ~ which inserts  tags.

Example:

~~Remove~~ the existence of CH~3~CH~2~OH.

Result:

Remove the existence of CH3CH2OH.

### Critic#

Critic is an extension that adds handling and support of Critic Markup which uses a special syntax to represent edits to a Markdown document. This extension runs before all other extensions to parse the critic edits.

Critic Markup uses special markup to insert, delete, substitute, highlight, and comment.

Example:

To insert or remove text, use {++insert me++} and {--remove me--}.\
Denote a substitution with {~~that ~> this one~~}.

Highlight specific text with {==highlight me==}.\

Result:

To insert or remove text, use insert me and remove me.
Denote a substitution with that this one.

Highlight specific text with highlight me.

## Lists#

This element allows to group a set of related items in lists. Children items can be numbered in ordered list, or even can have inline checkbox.

#### Ordered list#

1. Ordered item 1
1. Child 1
2. Child 2
2. Ordered item 2

1. Ordered item 1
1. Child 1
2. Child 2
2. Ordered item 2

#### Unordered list#

-   Unordered item 1
-   Child 1
-   Child 2
-   Unordered item 2

• Unordered item 1
• Child 1
• Child 2
• Unordered item 2

-   [x] item 1
-   [x] item a
-   [ ] item b
-   [ ] item 2

• item 1
• item a
• item b
• item 2

#### Definition#

Roses
:   are red

Violets
:   are blue

Roses
are red
Violets
are blue

## Images#

There are some extensions to add a caption to an image. After testing, markdown-captions is a good one that uses the alternate text to make caption, accepts markdown in the alternate text.

![A photo from <https://picsum.photos>](https://picsum.photos/320/240)

Some images have big size that does not show the detail, therefore, it’s better to zoom in by clicking on them, and pan the image on the screen. The view-bigimg library can do that requirement well.

## Tabs#

Tabbed extension provides a syntax to easily add tabbed Markdown content.

Tabs start with === to signify a tab followed by a quoted title. Consecutive tabs are grouped into a tab set.

Example:

=== "Tab 1"
Some texts

=== "Tab A"
Text A

=== "Tab B"
Text B

=== "Tab 2"
Some other texts

Result:

Some texts

Text A

Text B

Some other texts

## Tables#

Markdown Tables are written in pipe-line format: row is on one line, cell is inline text only. The 1st line contains the column headers. The 2nd line is to control text alignment in a column: :---, :---: and ---: are left, center, and right alignment. Styles for table need to change a little to show cell border.

Example:

| Syntax     | Description  |   Test Text |
| :--------- | :----------: | ----------: |
| Left align | Center align | Right align |
| A text     | Another text |  More texts |

Result:

Syntax Description Test Text
Left align Center align Right align
A text Another text More texts

## Icons & Emojis#

The Emoji extension adds support for inserting emoji via simple short names enclosed within colons :short_name:. This is accomplished by using a short name index to map easy-to-remember names to the corresponding emoji characters.

#### Emojis#

Emojis can be written by putting the short-code of the emoji between two colons. Look up the short-codes at Emojipedia.

:smile: , and :heart:

#### Icons#

Icons can be used similarly to emojis, by referencing a valid path to any icon bundled with the theme, which are located in the .icons directory, and replacing / with -.

E.g. The short-code :material-account-circle: will be converted to an SVG image element with the path .icons/material/account-circle.svg which eventually shows the icon on the webpage.

Using include function of Jinja to add an icon wrapped in a twemoji class, e.g. to show :

## Escape All#

The Escape All extension makes the backslash \ character escape everything after it, except things in code blocks of any kind.

There are two special escapes among all of these escapes though: escaping space characters and escaping newline characters:

• Enable nbsp to convert an escaped space into a non-breaking space:  .
• Enable hardbreak to convert an escaped newline to a hard break
. The advantage of hard break is that the backslash is visually seen in the document, opposed to the Markdown’s default method of two spaces at the end of a line.

## Special characters#

The Smarty Pants extension converts ASCII dashes, quotes and ellipses to their HTML entity equivalents.

Syntax Render
'single quote' ‘single quote’
"double quote" “double quote”
<<angle quote>> «angle quote»
ellipses ... ellipses …
N-dash -- N-dash –
M-dash --- M-dash –

The Smart Symbols adds syntax for creating special characters such as trademarks, arrows, fractions, etc.

Syntax Render
trademark (tm) trademark ™
copyright (c) copyright ©
registered (r) registered ®
in care of c/o in care of ℅
plus or minus +/- plus or minus ±
arrows --> <-- <--> arrows → ← ↔
not equal =/= not equal ≠
fractions 1/4 2/3 fractions ¼ ⅔
ordinal numbers 1st 2nd 3rd 4th 5th ordinal numbers 1st 2nd 3rd 4th 5th

Not all fractions can be displayed. To render fractions in a better format, use MathJax with inline format. Such as $1 \over 4$$$1 \over 4$$, or $2 \over 3$$$2 \over 3$$.

## Footnotes#

The Footnotes extension adds syntax for defining footnotes in Markdown documents.

Example:

Footnotes[^fn] have a label[^lb] and the footnote's content.

[^fn]: This is a footnote content.
[^lb]: A footnote on the label lb.

Result:

Footnotes2 have a label3 and the footnote’s content.

1. Use mkdocs-macros plugin to use Jinja template directly in the Markdown content.

2. This is a footnote content.

3. A footnote on the label lb


text:
## Section3.6Arithmetic Errors—Signed Integers

The number of bits used to represent a value is determined at the time a program is written. So when performing arithmetic operations you cannot simply add more digits (bits) if the result is too large, as you can do on paper. You saw in Section 3.1 that the C condition flag indicates when the sum of two unsigned integers exceeds the number of bits allocated for it.

In Section 3.4 you saw that carry is irrelevant when working with signed integers. In this section you will see that adding two signed numbers can also produce a result that exceeds the range of values that can be represented in the allocated number of bits.

The flags register, CPSR, provides a bit, the overflow condition flag, V, for detecting whether the sum of two $n$-bit, signed numbers stored in the two's complement code has exceeded the range allocated for it. Each operation that affects the overflow condition flag sets the bit equal to the exclusive or of the carry into the highest-order bit of the operands and the ultimate carry. For example, when adding the two 8-bit numbers, $\hex{15}$ and $\hex{6f}\text{,}$ we get:

 carry $\longrightarrow$ $\binary{0}$ $\binary{1} \longleftarrow$ penultimate carry $\binary{0001}$ $\binary{0101}$ $\longleftarrow$ $x$ $+$ $\binary{0110}$ $\binary{1111}$ $\longleftarrow$ $y$ $\binary{1000}$ $\binary{0100}$ $\longleftarrow$ sum

In this example, there is a carry of zero and a penultimate carry of one.

In this example, there is a carry of zero and a penultimate carry of one. The V flag is equal to the exclusive or of carry and penultimate carry:

 V = C ^ penultimate carry

where ‘^’ is the exclusive or operator. In the above example:

 V = 0 ^ 1 = 1

There are three possible cases.

###### Case 1:.

The two numbers are of opposite sign.

Let $x$ be the negative number and $y$ the positive number. Then we can express $x$ and $y$ in binary as:

\begin{gather*} x = \binary{1} \dots\\ y = \binary{0} \dots \end{gather*}

That is, the high-order bit of one number is $\binary{1}$ and the high-order bit of the other is $\binary{0}\text{,}$ regardless of what the other bits are.

First, we can see that $x + y$ always remains within the range of the two's complement representation:

\begin{align*} -2^{(n-1)} &\leq x \leq 0\\ 0 &\leq y \leq +(2^{(n-1)} - 1)\\ -2^{(n-1)} &\leq x + y \leq +(2^{(n-1)} - 1) \end{align*}

Now, if we add $x$ and $y\text{,}$ there are two possible results with respect to carry:

• If the penultimate carry is zero:

 carry $\longrightarrow$ $\binary{0}$ $\binary{0} \longleftarrow$ penultimate carry $\binary{1} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{0} \ldots$ $\longleftarrow$ $y$ $\binary{1} \ldots$ $\longleftarrow$ sum

This addition would produce V = 0 ^ 0 = 0.

• If the penultimate carry is one:

 carry $\longrightarrow$ $\binary{1}$ $\binary{1} \longleftarrow$ penultimate carry $\binary{1} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{0} \ldots$ $\longleftarrow$ $y$ $\binary{0} \ldots$ $\longleftarrow$ sum

This addition would produce V = 1 ^ 1 = 0.

We conclude that adding two integers of opposite sign always yields $\binary{0}$ for the overflow condition flag, so the sum is always within the allocated range.

###### Case 2:.

Both numbers are positive.

Since both are positive, we can express x and y in binary as:

\begin{gather*} x = \binary{0} \dots\\ y = \binary{0} \dots \end{gather*}

That is, the high-order bit of both numbers is $\binary{0}\text{,}$ regardless of what the other bits are.

Now, if we add $x$ and $y\text{,}$ there are two possible results with respect to carry:

• If the penultimate carry is zero:

 carry $\longrightarrow$ $\binary{0}$ $\binary{0} \longleftarrow$ penultimate carry $\binary{0} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{0} \ldots$ $\longleftarrow$ $y$ $\binary{0} \ldots$ $\longleftarrow$ sum

This addition would produce V = 0 ^ 0 = 0. The high-order bit of the sum is zero, so it is a positive number, and the sum is within range.

• If the penultimate carry is one:

 carry $\longrightarrow$ $\binary{0}$ $\binary{1} \longleftarrow$ penultimate carry $\binary{0} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{0} \ldots$ $\longleftarrow$ $y$ $\binary{1} \ldots$ $\longleftarrow$ sum

This addition would produce V = 0 ^ 1 = 1. The high-order bit of the sum is one, so it is a negative number. Adding two positive numbers cannot yield a negative sum, so the sum has exceeded the allocated range.

###### Case 3:.

Both numbers are negative.

Since both are negative, we can express x and y in binary as:

\begin{gather*} x = \binary{1} \dots\\ y = \binary{1} \dots \end{gather*}

That is, the high-order bit of both numbers is $\binary{1}\text{,}$ regardless of what the other bits are.

Now, if we add $x$ and $y\text{,}$ there are two possible results with respect to carry:

• If the penultimate carry is zero:

 carry $\longrightarrow$ $\binary{1}$ $\binary{0} \longleftarrow$ penultimate carry $\binary{1} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{1} \ldots$ $\longleftarrow$ $y$ $\binary{0} \ldots$ $\longleftarrow$ sum

This addition would produce V = 1 ^ 0 = 1. The high-order bit of the sum is zero, so it is a positive number. Adding two negative numbers cannot yield a positive sum, so the sum has exceeded the allocated range.

• If the penultimate carry is one:

 carry $\longrightarrow$ $\binary{1}$ $\binary{1} \longleftarrow$ penultimate carry $\binary{1} \ldots$ $\longleftarrow$ $x$ $+$ $\binary{1} \ldots$ $\longleftarrow$ $y$ $\binary{1} \ldots$ $\longleftarrow$ sum

This addition would produce V = 1 ^ 1 = 0. The high-order bit of the sum is one, so it is a negative number, and the sum is within range.

These results and those from Section 3.3 allow us to state the following rules when adding or subtracting two $n$-bit numbers:

• If your algorithm treats the result as unsigned, the carry condition flag (C) is zero if and only if the result is within the $n$-bit range; V is irrelevant.

• If your algorithm treats the result as signed, the overflow condition flag (V) is zero if and only if the result is within the $n$-bit range; C is irrelevant.

The CPU does not consider integers as either signed or unsigned. Both C and V are set according to the rules of binary arithmetic by each arithmetic operation. The distinction between signed and unsigned is completely determined by the program. After each addition or subtraction operation the program should check the state of C for unsigned integers or V for signed integers and at least indicate when the sum is in error. Many high-level languages do not perform this check, which can lead to some obscure program bugs.

The codes used for both unsigned integers and signed integers are circular in nature. That is, for a given number of bits, each code “wraps around”. This can be seen pictorially in the “Decoder Ring” shown in Figure 3.6.2 for three-bit numbers.