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# Naturally Ordered Semigroup is Unique

## Theorem

Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.

Let:

$0'$ be the smallest element of $S'$
$1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.

Then the mapping $g: S \to S'$ defined as:

$\forall a \in S: \map g a = \circ'^a 1'$

is an isomorphism from $\struct {S, \circ, \preceq}$ to $\struct {S', \circ', \preceq'}$.

This isomorphism is unique.

Thus, up to isomorphism, there is only one naturally ordered semigroup.

## Proof

### Proof that Mapping is Isomorphism

Let $T' = \Cdm g$, that is, the codomain of $g$.

By Zero is Identity in Naturally Ordered Semigroup, $0'$ is the identity for $\circ'$.

Thus:

$\map g 0 = \circ'^0 1' = 0'$

and so $0' \in T'$

Suppose $x' \in T'$.

Then:

$\map g n = x'$

and so:

 $\displaystyle x' \circ' 1'$ $=$ $\displaystyle \circ'^n 1' \circ' 1'$ $\displaystyle$ $=$ $\displaystyle \paren {\circ'^n 1'} \circ' \paren {\circ'^1 1'}$ $\displaystyle$ $=$ $\displaystyle \circ'^{\paren {n \circ 1} } 1'$ $\displaystyle$ $=$ $\displaystyle \map g {n \circ 1}$

So:

$x' \in T' \implies x' \circ' 1' \in T'$

Thus, by the Principle of Mathematical Induction applied to $S'$:

$T' = S'$

So:

$\forall a' \in S': \exists a \in S: \map g a = a'$

and so $g$ is surjective by definition.

$\map g {a \circ b} = \map g a \circ' \map g b$

and therefore $g$ is a homomorphism from $\struct {S, \circ}$ to $\struct {S', \circ'}$.

Now:

 $\displaystyle \forall p \in S: \ \$ $\displaystyle \circ'^p 1'$ $\prec'$ $\displaystyle \circ'^p 1' \circ' 1'$ Sum with One is Immediate Successor in Naturally Ordered Semigroup $\displaystyle$ $=$ $\displaystyle \paren {\circ'^p 1'} \circ' \paren {\circ'^1 1'}$ $\displaystyle$ $=$ $\displaystyle \circ'^{\paren {p \circ 1} } 1'$ Index Laws for Semigroup: Sum of Indices $\displaystyle \leadsto \ \$ $\displaystyle \circ'^p 1'$ $\prec'$ $\displaystyle \circ'^{\paren {p \circ 1} } 1'$

For $p \in S$, let $S_p$ be the initial segment of $S$:

$S_p = \set {x \in S: x \prec p}$

Let:

$T = \set {p \in S: \forall a \in S_p: \circ'^a 1' \prec' \circ'^p 1'}$

Now $S_0 = \O \implies 0 \in T$.

Suppose $p \in T$.

Then:

$a \prec p \circ 1 \implies a \preceq p$

By Strict Lower Closure of Sum with One, either of these is the case:

$(1): \quad a \prec p: p \in T \implies \circ'^a 1' \prec' \circ'^p 1' \prec' \circ'^{\paren {p \circ 1} } 1'$
$(2): \quad a = p: \circ'^a 1' = \circ'^p 1' \prec' \circ'^{\paren {p \circ 1} } 1'$

In either case, we have:

$p \in T \implies p \circ 1 \in T$, and by thePrinciple of Mathematical Induction:
$T = S$

So $n \prec p \implies \circ'^n 1' \prec' \circ'^p 1'$.

Thus $g$ is a surjective monomorphism and therefore is an isomorphism from $\struct {S, \circ, \preceq}$ to $\struct {S', \circ', \preceq'}$.

$\blacksquare$

### Proof that Isomorphism is Unique

Now we need to show that the isomorphism $g$ is unique.

Let $f: S \to S'$ be another isomorphism different from $g$.

Aiming for a contradiction, suppose $\map f 1 \ne 1'$.

We show by induction that $1' \notin \Cdm f$.

... Thus $1' \notin \Cdm f$ which is a contradiction.

Thus $\map f 1 = 1$ and it follows

that $f = g$.

Thus the isomorphism $g$ is unique.

$\blacksquare$


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To continue from the previous post

Twisted Differential Cohomology

Ulrich Bunke gave a talk introducing differential cohomology theories, and Thomas Nikolaus gave one about a twisted version of such theories (unfortunately, perhaps in the wrong order). 2,156 more words

format: plain


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• https://me.yahoo.com
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# Psi

viewed 2285 times and licensed 77 times
Psi (digamma) function
Controller: CodeCogs
Contents

C++
Excel

## Psi

 doublepsi( double x )
The Psi (also called the Digamma) function is a special function which is given by the logarithmic derivative of the gamma function. There is some ambiguity over its definition, as it is sometimes defined as the logarithmic derivative of the factorial. Due to this, two differet notiations are sometimes used, with the logarithmic derivative of the gamma function defined as:

$\Psi(x)&space;\equiv&space;\frac{d}{dx}&space;ln&space;\Gamma(x)&space;=&space;\frac{\Gamma'(x)}{\Gamma(x)}$

And the logarithmic derivative of the factorial function defines as:

$F(x)&space;\equiv&space;\frac{d}{dx}&space;ln&space;x!$

The two are connected by the relationship
$F(x)&space;=&space;\Psi(x+1)$

Psi (digamma) function computes the logarithmic derivative of the gamma function:
$\Psi(x)&space;\equiv&space;\frac{d}{dx}&space;ln&space;\Gamma(x)$

For integer x,
$\Psi(x)&space;=&space;-EUL&space;+&space;\sum_{k=1}^{x-1}&space;\frac{1}{k}$
where EUL=0.57721566490153286061

This formula is used for 0 < x <= 10. If x is negative, it is transformed to a positive argument by the reflection formula
$\Psi(1-x)&space;=&space;\Psi(x)&space;+&space;\pi&space;\cot(\pi&space;x)$

For general positive x, the argument is made greater than 10 using the recurrence $\inline&space;&space;\Psi(x+1)&space;=&space;\Psi(x)&space;+&space;1/x$.Then the following asymptotic expansion is applied:

$\Psi(x+1)&space;=&space;\log(x)&space;-&space;\frac{1}{2x}&space;-&space;\sum_{k=1}^{\infty}&space;\frac{B_2k}{2k&space;x^{(2k)}&space;}$

where the $\inline&space;&space;B_{2k}$ are the Bernoulli numbers.

## Accuracy:

Relative error (except absolute when |psi| < 1): <pre> domain # trials peak rms -30,0 40000 1.5e-15 2.2e-16 </pre>

## Error Message:

psi singularity x integer <=0 MAXNUM

## References:

• Cephes Math Library Release 2.8: June, 2000
• http://mathworld.wolfram.com/DigammaFunction.html

### Example 1

#include <stdio.h>
#include <codecogs/maths/special/gamma/psi.h>

int main()
{
for(double x=-0.9; x<4; x+=0.2)
{
double y=Maths::Special::Gamma::psi(x);
printf("\n psi(%.1lf)=%lf", x,y);
}
return 0;
}
Output:
psi(-0.9)=-9.312644
psi(-0.7)=-2.073953
psi(-0.5)=0.036490
psi(-0.3)=2.113310
psi(-0.1)=9.245073
psi(0.1)=-10.423755
psi(0.3)=-3.502524
psi(0.5)=-1.963510
psi(0.7)=-1.220024
psi(0.9)=-0.754927
psi(1.1)=-0.423755
psi(1.3)=-0.169191
psi(1.5)=0.036490
psi(1.7)=0.208548
psi(1.9)=0.356184
psi(2.1)=0.485336
psi(2.3)=0.600040
psi(2.5)=0.703157
psi(2.7)=0.796783
psi(2.9)=0.882500
psi(3.1)=0.961526
psi(3.3)=1.034822
psi(3.5)=1.103157
psi(3.7)=1.167154
psi(3.9)=1.227328

### Parameters

 x input

### Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documented and Corrected by Will Bateman (August 2005)
##### Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


text:
# Is there any connection between “static correlation” and the Born-Oppenheimer approximation?

Beware: High probability for that I am mixing two concepts that should not be mixed, just because the same term is used when explaining the two concepts!

For some reason I get really confused when I think about static, or nondynamical, electron correlation. I somehow want to make a connection to the Born-Oppenheimer (BO) approximation.

As we know, the BO approximation is no longer accurate when two electronic states come close in energy, due to the increase in the nonadiabatic coupling terms. These terms couple the electronic motion to the nuclear motion, and arise because the electronic and nuclear motion is not fully separable in the Schrödinger equation.

Static correlation, sometimes called a "near-degeneracy effect", describes the correlated movement of electrons belonging in different electronic configurations. Certain systems are not described properly by just a single-determinant wave function. One example is the H-H molecule at long bond distances, for which the bonding and anti-bonding $\sigma$-orbitals become degenerate. (this was probably a bad explanation, as I struggle with this concept!)

So, as we have just seen, the BO approximation breaks down when two electronic states come close in energy (near-degeneracy), and the static correlation becomes extremely important in molecules with near-degenerate electronic states. Is there a connection here?

Imagine we are in the ground state at a point on some potential energy surface, and the first excited state is very close in energy. At this point, there should be a high coupling between the electronic and nuclear positions (which means what, exactly??). Also, at this nuclear geometry, the electronic wave function should be described as a linear combination of these two near-degenerate electronic states.

It is the BO approximation that introduces the idea of a potential energy surface (PES). The PES itself can be thought of as an approximation, as the nuclei should ideally be fully described by quantum mechanics. The nuclear positions should be described probabilistically by a wave function, and the very idea of a molecular geometry then becomes fuzzy.

What is the connection, if there is any?

• There exist methods that calculate coupling terms between different PES. They are called sth like "nonadiabatic dynamics".. – Fl.pf. May 12 '17 at 8:28
• I'm talking about routine, "static" calculations. If two electronic states are almost degenerate, then the Born-Oppenheimer approximation breaks down. Do we take care of this when doing calculations on molecules with near-degenerate states? – Yoda May 13 '17 at 7:09
• afaik the BO doesn't really break down when two PES "come near each other", it just breaks down when you try to calculate the change from one to the other. I' not really an expert in BO, but I'll show this someone who is and maybe he can really help you ;) – Fl.pf. May 13 '17 at 8:03
• Okay, I'll see what I can come up with. – Fl.pf. May 18 '17 at 12:01
• Logically they must be independent, because the concept of "static correlation" occurs "only" (which is in practise almost always) when one builds multielectron wave functions from one-electron-wave functions. Purely mathematically this is not neccessary, so stat. corr. is a quasi-artefact. While the BO approximation is completely independent on the strategy you tackle to solve the Schrödinger eq. However a famous case comes to my mind where both (BO breakdown + stat. corr.) come togehter: NO3^. see here molspect.chemistry.ohio-state.edu/institute/2007/stanton/… . – Rudi_Birnbaum Jun 3 '17 at 19:49

Static correlation is a difficult to grasp concept of which even a clear definition is beyond me. However, it seems clear that a full Configuration Interaction (FCI) calculation yields all correlation energy, thus, must correctly handle static correlation. From the discussion of the Born-Oppenheimer approximation (BO) by Jensen,$$^{[1]}$$ I gather that the BO approximation is essentially a modification of the Hamilton operator. This shortcoming cannot be healed by a FCI calculation that only considers the electronic problem. So, in this sense, the two concepts are unrelated.


text:
# The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is t in air. Neglecting frictional force of water and given that the density of the bob is 3 kg/m3. What relationship between t and t0 is true?

Saurabh Koranglekar
2 years ago
Arun
25758 Points
2 years ago

As we learnt in

Time period of oscillation of simple pendulum -

$T=2\pi \sqrt{\frac{l}{g}}$

- wherein

l = length of pendulum

g = acceleration due to gravity.

$\dpi{100} t_{0}=2\pi \sqrt{l/g}.........(i)$

Due to upthrust of water on the top, its apparent weight decreases

upthrust = weight of liquid displaced

$\dpi{100} \therefore \; \; \; Effective\; weight=mg-(V\sigma g)=V\rho g-V\sigma g$

$\dpi{100} V\rho g'=Vg(\rho -\sigma ),where\; \sigma \; is \; density \; of\; water\;$

$\dpi{100} or\; \; g'=g\left ( \frac{\rho -\sigma }{\rho } \right )$

$\dpi{100} \therefore \; \; t=2\pi \sqrt{l/g'}=2\pi \sqrt{\frac{l\rho }{g(\rho -\sigma )}}..........(ii)$

$\dpi{100} \therefore \; \; \; \frac{t}{t_{0}}=\sqrt{\frac{l\rho }{g(\rho -\sigma )}\times \frac{g}{l}}=\sqrt{\frac{\rho }{\rho -\sigma }}=\sqrt{\frac{4\times 1000/3}{(\frac{4000}{3}-1000)}}=2$

$\dpi{100} or\; \; \; t=t_{0}\times 2=2t_{0}$


text:
# What is the standard form of the equation of a circle with a center (1,4) and radius 5?

${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 25$


text:
# Random Feature Request :D

Hi guys!

I've been thinking about the all-new Brilliant Wiki, and I think there could be a few improvements to it. I do like the way it's classified according to topic in Practice, but this could be improved greatly.

I feel that we shouldn't tie it up so closely to Practice Mode. For example, some people may want to searh for a certain, very specific topic, and they don't know what it's under. Rather than having to search all the topics one by one, I'm recommending that we could have a search bar for wiki articles -- for example, on the search bar on top, as well as 'Problems, Notes, Sets' and 'Members' we could have a third option: 'Articles', for example. This would definitely increase the efficiency of searching for information on Brilliant.

Secondly, I'm not sure about this (I'm not a moderator, of course, but it's just a friendly suggestion in case :D) but we could assign some moderators to the wiki articles. Since anyone can edit these articles, some evil people might want to play pranks and replace the wiki with some nonsense words. So to prevent against that, I'm suggesting that we patrol the wiki pages more frequently.

Third, I think we could add links to wiki pages from problems too! Just as there are 'Related Problems' under wiki pages, we could have 'Related wiki articles' under problems too! That would be helpful to users in many ways too.

So whaddaya guys think? Feel free to comment :D

Thanks! :)

Note by Yuxuan Seah
3 years, 6 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Those are some good suggestions.

Regarding being able to search for wiki articles through the search bar: you can currently do this. Here's a screenshot.

Imgur

All the wiki edits have to be approved by staff, so that solves your next issue.

- 3 years, 6 months ago

Thanks for you feedback. I am extremely encouraged by the response in the community. Despite launching for less than a week, we have more than 160 pages which have something added to it, and half of them have substantial writeups! We have just embarked on this project, and we felt that it was better to publish a basic version first, before adding in features that the community would like.

1) I agree with "not tie it up so closely to Practice Mode". We chose that for launch, as that was the easiest way for us to provide structure and explanations to others. We are planning on having Wiki pages that do not have a corresponding Practice Problem set, and are looking forward to launching this features. For example, in Upcoming Skill Maps - Number Theory, I started highlighting (in blue) skills which I felt would have good wiki writeups, but may not lend themselves to having a corresponding Practice section.

2) We will also work on getting moderators involved with the Wiki review, and are figuring out a good approach. I agree that we do not want "evil people who might play pranks and replace the wiki with nonsense". Thankfully, we have not seen such instances as yet, and all edits were made with good intentions.

3) This can already be achieved by selecting the corresponding skill that your problem uses (assuming that such a skill exists). Doing so will show a "recommended skill", with a link to the practice section and wiki page. Currently, we decided to limit it to just 1 skill for simplicity, but might open that up in future.

Staff - 3 years, 6 months ago

@Calvin Lin ? @Peter Taylor ? What do you think? :D

- 3 years, 6 months ago


text:
Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes
September 12, 2012
We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-\epsilon) with list size L=O(1/\epsilon^2) and rate R=\Omega_q(\epsilon^2/(log^3(1/\epsilon))). Up to the polylogarithmic factor in (1/\epsilon) and constant factors depending on q, this matches the lower bound L=\Omega_q(1/\epsilon^2) for the list size and upper bound R=O_q(\epsilon^2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q=2 (Guruswami, Hastad, Sudan and Zuckerman, 2002).

In order to obtain our result, we employ a relaxed version of the well known Johnson bound on list decoding that translates the average Hamming distance between codewords to list decoding guarantees. We furthermore prove that the desired average-distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the Restricted Isometry Property with respect to the Euclidean norm (RIP-2). For the case of random binary linear codes, this matrix coincides with a random submatrix of the Hadamard-Walsh transform matrix that is well studied in the compressed sensing literature.

Finally, we improve the analysis of Rudelson and Vershynin (2008) on the number of random frequency samples required for exact reconstruction of k-sparse signals of length N. Specifically, we improve the number of samples from O(k log(N) log^2(k) (log k + loglog N)) to O(k log(N) log^3(k)). The proof involves bounding the expected supremum of a related Gaussian process by using an improved analysis of the metric defined by the process. This improvement is crucial for our application in list decoding.

[Joint work with Venkatesan Guruswami and Ameya Velingker]


text:
## rectangular matrix transpose

06/12/2020 Uncategorized

The transpose of matrix A is represented by $$A'$$ or $$A^T$$. A matrix is a rectangular array of numbers or functions arranged in a fixed number of rows and columns. Such a matrix is called a Horizontal matrix. Question is : The transpose of a rectangular matrix is a , Options is : 1. rectangular matrix, 2. diagonal matrix, 3.square matrix, 4. scaler matrix, 5. पृथ्वी की अपनी कक्षा में गति किस दिशा में है ? $$B = \begin{bmatrix} 2 & -9 & 3\\ 13 & 11 & 17 \end{bmatrix}_{2 \times 3}$$. Now, there is an important observation. To learn other concepts related to matrices, download BYJU’S-The Learning App and discover the fun in learning. A matrix P is said to be equal to matrix Q if their orders are the same and each corresponding element of P is equal to that of Q. the row and column indices of the matrix are switched. NULL, Electronics Bazaar is one of best Online Shopping Store in India. हिन्दी वर्णमाला में संयुक्त व्यंजनों की संख्या कितनी है ? Required fields are marked *, $$N = \begin{bmatrix} 22 & -21 & -99 \\ 85 & 31 & -2\sqrt{3} \\ 7 & -12 & 57 \end{bmatrix}$$, $$N’ = \begin{bmatrix} 22 &85 & 7 \\ -21 & 31 & -12 \\ -99 & -2\sqrt{3} & 57 \end{bmatrix}$$, $$\begin{bmatrix} 22 & -21 & -99 \\ 85 & 31 & -2\sqrt{3} \\ 7 & -12 & 57 \end{bmatrix}$$, $$\begin{bmatrix} 2 & -3 & 8 \\ 21 & 6 & -6 \\ 4 & -33 & 19 \end{bmatrix}$$, $$\begin{bmatrix} 1 & -29 & -8 \\ 2 & 0 & 3 \\ 17 & 15 & 4 \end{bmatrix}$$, $$\begin{bmatrix} 2+1 & -3-29 & 8-8 \\ 21+2 & 6+0 & -6+3 \\ 4+17 & -33+15 & 19+4 \end{bmatrix}$$, $$\begin{bmatrix} 3 & -32 & 0 \\ 23 & 6 & -3 \\ 21 & -18 & 23 \end{bmatrix}$$, $$\begin{bmatrix} 3 & 23 & 21 \\ -32 & 6 & -18 \\ 0 & -3 & 23 \end{bmatrix}$$, $$\begin{bmatrix} 2 & 21 & 4 \\ -3 & 6 & -33 \\ 8 & -6 & 19 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 17 \\ -29 & 0 & 15 \\ -8 & 3 & 4 \end{bmatrix}$$, $$\begin{bmatrix} 2 & 8 & 9 \\ 11 & -15 & -13 \end{bmatrix}_{2×3}$$, $$k \begin{bmatrix} 2 & 11 \\ 8 & -15 \\ 9 & -13 \end{bmatrix}_{2×3}$$, $$\begin{bmatrix} 9 & 8 \\ 2 & -3 \end{bmatrix}$$, $$\begin{bmatrix} 4 & 2 \\ 1 & 0 \end{bmatrix}$$, $$\begin{bmatrix} 44 & 18 \\ 5 & 4 \end{bmatrix} \Rightarrow (AB)’ = \begin{bmatrix} 44 & 5 \\ 18 & 4 \end{bmatrix}$$, $$\begin{bmatrix} 4 & 1 \\ 2 & 0 \end{bmatrix} \begin{bmatrix} 9 & 2 \\ 8 & -3 \end{bmatrix}$$, $$\begin{bmatrix} 44 & 5 \\ 18 & 4 \end{bmatrix}$$, $$\begin{bmatrix} 9 & 2 \\ 8 & -3 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 40 & 9 \\ 26 & 8 \end{bmatrix}$$. The transpose of a rectangular matrix is a. Transposition also serves purposes when expressing vectors as matrices, or taking the products of vectors. संयुक्त राष्ट्र पर्यावरण कार्यक्रम का मुख्यालय कहाँ पर है ? Given a matrix of some dimension, and we need to transpose it. Thus, the matrix B is known as the Transpose of the matrix A. Question is : The transpose of a rectangular matrix is a , Options is : 1. rectangular matrix, 2. diagonal matrix, 3.square matrix, 4. scaler matrix, 5. The addition property of transpose is that the sum of two transpose matrices will be equal to the sum of the transpose of individual matrices. The answer is no. समुद्र तल के ऊपर विश्व की सबसे बड़ी पर्वत श्रंखला कौनसी है ? $$M^T = \begin{bmatrix} 2 & 13 & 3 & 4 \\ -9 & 11 & 6 & 13\\ 3 & -17 & 15 & 1 \end{bmatrix}$$. Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution. Matrix = 1 2 34 5 67 8 9Transpose = 1 4 72 5 83 6 9. Gupta, Shubham. The determinant of a square matrix measures how volumes change when you multiply by that matrix. That is a meaningful question, because the answer is the same no matter how you choose to measure volume. So, we can observe that $$(P+Q)'$$ = $$P’+Q'$$. the orders of the two matrices must be same. The transpose of a rectangular matrix is a, The transpose of a rectangular matrix is a, If a matrix has m rows and n columns then order is, Two matrices A and B are multiplied to get AB if, If A is a symmetric matrix, then At =, A matrix having m rows and n columns with m = n is said to be a, For any non- singular matrix A, A-1 =, If A is a matrix of order m x n and B is a matrix of order n x p then the order of AB is, In a matrix multiplication for A and B, (AB)t, If A is a skew symmetric matrix, then At, Equations having a common solution are called, Matrices obtained by changing rows and columns is called, A matrix having m rows and n columns with m ≠ n is said to be a, Two matrices A and B are multiplied to get BA if. The transpose of a matrix is obtain by exchanging its with its rows, as shown by equation (22.6): ... For a rectangular matrix A, we may have a generalized left inverse or left inverse for short when we multiply the inverse from the left to get identity matrix A left −1 A = I. So. Buy online Mobile Phones, Laptops, Tablets, Cameras & much more at best prices. Transpose of a matrix is calculated by interchanging the rows into columns and vice-versa. NULL. A common strategy to improve these accesses is to transpose or rotate the matrix beforehand, thus the accessing complexity is centralized in one transformation operation. Transpose of a matrix is given by interchanging of rows and columns. Matrix Transpose. Definition. Hence, for a matrix A. How to transpose an arbitrary size matrix inplace? The transpose of a rectangular matrix is a. Let us consider a matrix to understand more about them. ‘विंग्स ऑफ फायर’ पुस्तक किसके द्वारा लिखी गई है ? Performing an in-place transpose is most difficult when N ≠ M, i.e. This is my matrix public double[,] MatriksT(int blok) { double[,] matrixT = new Thus, the matrix B is known as the Transpose of the matrix A. Don't confuse Transpose to Matrix Rotation, the rotation is normally performed based on the X-Y axis while in transpose, the matrix is flipped on its diagonal. संसार में सबसे व्यस्त और सबसे महत्त्वपूर्ण समुद्र मार्ग कौन सा है ? Matrix transposes are a neat tool for understanding the structure of matrices. Given an M x N matrix, transpose the matrix without auxiliary memory. अंग्रेजों द्वारा कलकत्ता में निर्मित दुर्ग का नाम क्या है ? $$a_{ij}$$ gets converted to $$a_{ji}$$ if transpose of A is taken. The number of columns in matrix B is greater than the number of rows. Live Demo. To calculate the transpose of a matrix, simply interchange the rows and columns of the matrix i.e. भारतीय अंतरिक्ष कार्यक्रम का पिता किसे कहा गया है ? it flips a matrix over its diagonal. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. If the matrix is symmetric in size, we can transpose the matrix inplace by mirroring the 2D array across it’s diagonal (try yourself). विश्व का सबसे ऊँचा पर्वत शिखर ‘माउण्ट एवरेस्ट’ कहाँ स्थित है ? In fact, all in-place matrix transpose algorithms that transpose aMùNmatrix (assuming row-major order),onewayoranother,havetodoworkthatisequivalenttoapplyingthepermutationpresentedin onto each array indexxÀ {0,MN*1}. Transpose [m] gives the usual transpose of a matrix m. Transpose [m] can be input as m . The transpose of a matrix, typically indicated with a "T" as an exponent, is an operation that flips a matrix over its diagonal. Buy Now! $B$ is a rectangular matrix of order $3\times4$. Inner and Outer Products | Lecture 5 9:35. Typically, the matrix is assumed to be stored in row-major order or column-major order. The following statement generalizes transpose of a matrix: If $$A$$ = $$[a_{ij}]_{m×n}$$, then $$A'$$ = $$[a_{ij}]_{n×m}$$. But not all square matrices have inverse. In this program, the user is asked to enter the number of rows r and columns c.Their values should be less than 10 in this program. If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T).In other words, if A = [a ij] mxn,thenA′ = [a ji] nxm.For example, I have made 8x8 matrix using c#, and now I need to transpose the matrix. In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O additional storage, or at most with additional storage much less than NM. transpose and rectangular matrices. So, transpose of matrix $B$ can be obtained in four steps. There can be many matrices which have exactly the same elements as A has. Just like determinant, non-square matrices do not have inverse. Transpose of an addition of two matrices A and B obtained will be exactly equal to the sum of transpose of individual matrix A and B. and $$Q$$ = $$\begin{bmatrix} 1 & -29 & -8 \\ 2 & 0 & 3 \\ 17 & 15 & 4 \end{bmatrix}$$, $$P + Q$$ = $$\begin{bmatrix} 2+1 & -3-29 & 8-8 \\ 21+2 & 6+0 & -6+3 \\ 4+17 & -33+15 & 19+4 \end{bmatrix}$$= $$\begin{bmatrix} 3 & -32 & 0 \\ 23 & 6 & -3 \\ 21 & -18 & 23 \end{bmatrix}$$, $$(P+Q)'$$ = $$\begin{bmatrix} 3 & 23 & 21 \\ -32 & 6 & -18 \\ 0 & -3 & 23 \end{bmatrix}$$, $$P’+Q'$$ = $$\begin{bmatrix} 2 & 21 & 4 \\ -3 & 6 & -33 \\ 8 & -6 & 19 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 17 \\ -29 & 0 & 15 \\ -8 & 3 & 4 \end{bmatrix}$$ = $$\begin{bmatrix} 3 & 23 & 21 \\ -32 & 6 & -18 \\ 0 & -3 & 23 \end{bmatrix}$$ = $$(P+Q)'$$. उच्च न्यायालय के न्यायाधीश किस आयु तक अपना पद धारण कर सकते हैं ? MKL Rectangular matrix Inplace transpose performance issue. For square matrices, Inverse of matrix is obtained. This is a Most important question of gk exam. for a non-square … Then $$N’ = \begin{bmatrix} 22 &85 & 7 \\ -21 & 31 & -12 \\ -99 & -2\sqrt{3} & 57 \end{bmatrix}$$, Now, $$(N’)'$$ = $$\begin{bmatrix} 22 & -21 & -99 \\ 85 & 31 & -2\sqrt{3} \\ 7 & -12 & 57 \end{bmatrix}$$. Transpose of a matrix is obtained by changing rows to columns and columns to rows. That’s because their order is not the same. I want an in place memory transpose of very large matrix. DeﬁnitionThe transposeof an m xn matrix Ais the n xmmatrix ATobtained by interchanging rows and columns of A, DeﬁnitionA square matrix Ais symmetricif AT= A. of rectangular m×n matrices. So, taking transpose again, it gets converted to $$a_{ij}$$, which was the original matrix $$A$$. Die erste Zeile der transponierten Matrix entspricht der ersten Spalte der Ausgangsmatrix, die zweite Zeile der zweiten Spalte und so weiter. So, is A = B? Mon, 05/13/2019 - 02:04. For Square Matrix : The below program finds transpose of A[][] and stores the result in B[][], we can change N for different dimension. That is, $$A×B$$ = $$\begin{bmatrix} 44 & 18 \\ 5 & 4 \end{bmatrix} \Rightarrow (AB)’ = \begin{bmatrix} 44 & 5 \\ 18 & 4 \end{bmatrix}$$, $$B’A'$$ = $$\begin{bmatrix} 4 & 1 \\ 2 & 0 \end{bmatrix} \begin{bmatrix} 9 & 2 \\ 8 & -3 \end{bmatrix}$$, = $$\begin{bmatrix} 44 & 5 \\ 18 & 4 \end{bmatrix}$$ = $$(AB)'$$, $$A’B'$$ = $$\begin{bmatrix} 9 & 2 \\ 8 & -3 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 40 & 9 \\ 26 & 8 \end{bmatrix}$$. Though they have the same set of elements, are they equal? $$A = \begin{bmatrix} 2 & 13\\ -9 & 11\\ 3 & 17 \end{bmatrix}_{3 \times 2}$$. We define matrices and how to add and multiply them, discuss some special matrices such as the identity and zero matrix, learn about transposes and inverses, and define orthogonal and permutation matrices. Now, transform the columns into rows to obtain the transpose of matrix $B$. Here, the number of rows and columns in A is equal to number of columns and rows in B respectively. One thing to notice here, if elements of A and B are listed, they are the same in number and each element which is there in A is there in B too. This is a Most important question of gk exam. Your email address will not be published. There are many types of matrices. Some properties of transpose of a matrix are given below: If we take transpose of transpose matrix, the matrix obtained is equal to the original matrix. Example. Die transponierte Matrix, gespiegelte Matrix oder gestürzte Matrix ist in der Mathematik diejenige Matrix, die durch Vertauschen der Rollen von Zeilen und Spalten einer gegebenen Matrix entsteht. We can clearly observe from here that (AB)’≠A’B’. ‘देवा शरीफ’ उत्तर प्रदेश में कहाँ स्थित है ? The transpose of a matrix is calculated, by changing the rows as columns and columns as rows. Deﬁnition 4.3. पादप, हिरण, भेडि़या व शेर से निर्मित खाद्य-श्रंखला में से ऊर्जा किसमें होगी ? So the function don't need MPI communicator itself, but encaps MPI communicator internally. विश्व का सबसे ऊँचा ज्वालामुखी कौन सा है ? पृथ्वी पर विशालतम जीवित पक्षी कौन-सा है ? Here, the number of rows and columns in A is equal to number of columns and rows in B respectively. भारत में राष्ट्रीय आय का आकलन पहली बार किसने किया था ? rectangular-matrix-transpose. Matrix Representation For the purposes of this project, matrices will be represented by lists of lists. What basically happens, is that any element of A, i.e. A rectangular matrix is one which has unequal number of rows and columns, e.g: 3 * 2 matrix or 2 * 3 matrix. Question is : The transpose of a rectangular matrix is a , Options is : 1. rectangular matrix, 2. diagonal matrix, 3.square matrix, 4. scaler matrix, 5. संसार में ताजे पानी की सबसे बड़ी झील ‘लेक सुपीरियर’ कहाँ पर स्थित है ? Features you might already know about matrices, such as squareness and symmetry, affect the transposition results in obvious ways. विश्व का सबसे गहरा गत्र्त ‘मेरियाना ट्रेंच’ किस महासागर में स्थित है ? The multiplication property of transpose is that the transpose of a product of two matrices will be equal to the product of the transpose of individual matrices in reverse order. To understand the properties of transpose matrix, we will take two matrices A and B which have equal order. write the elements of the rows as columns and write the elements of a column as rows. अमेरिका की डेट्रॉएट इलेक्ट्रिल्स कम्पनी द्वारा निर्मित सबसे तेज चलने वाली इलेक्ट्रिकल कार का नाम क्या है ? Since n × n matrices can be multiplied, the idea behind matrix norms is that they should behave “well” with re-spect to matrix multiplication. जंग से बचाने के लिए लोहे से बने पानी के पाइपों पर जस्ते की परत चढ़ाने को क्या कहते हैं ? NULL. I am using mkl_simatcopy. Example: import numpy as np M1 = np.array([[3, 6, 9], [5, -10, 15], [4,8,12]]) M2 = M1.transpose() print(M2) Output: [[ 3 5 4] [ 6 -10 8] [ 9 15 12]] Slicing of a Matrix. Transpose of a matrix If the rows and columns in a matrix $$A$$ are interchanged, the new matrix is called the transpose of the original matrix $$A.$$ The transposed matrix is denoted by $$A^T$$. For a matrix m, Transpose [m] is equivalent to Transpose [m, {2, 1}]. Properties of transpose (1) (AT) = A (2) (A + B)T= AT+ BT can be entered as tr or \[Transpose]. The horizontal array is known as rows and the vertical array are known as Columns. Our approach relies on a set of hybrid algorithms which are chosen based on the shape of matrices and transpose operator involved. Thus Transpose of a Matrix is defined as “A Matrix which is formed by turning all the rows of a given matrix into columns and vice-versa.”, Example- Find the transpose of the given matrix, $$M = \begin{bmatrix} 2 & -9 & 3 \\ 13 & 11 & -17 \\ 3 & 6 & 15 \\ 4 & 13 & 1 \end{bmatrix}$$. इस समय भारत में सर्वाधिक मात्रा में पाया जाने वाला खनिज पदार्थ कौन सा है ? A square matrix $$A$$ is called orthogonal if $$A{A^T} = I,$$ where $$I$$ is the identity matrix. The above matrix A is of order 3 × 2. As a whole, the square matrix in-place algorithm is a permutation onA, composed ofM(M+1)_2 cycles of length 1. But I am observing some performance issue while transposing inplace. If necessary, refer above for description of the notation used. The transpose of matrix A is represented by $$A'$$ or $$A^T$$. If a matrix is multiplied by a constant and its transpose is taken, then the matrix obtained is equal to transpose of original matrix multiplied by that constant. Matrices are rectangular arrays of numbers or other mathematical objects. Matrices where (number of rows) = (number of columns) For the matrices with whose number of rows and columns are unequal, we call them rectangular matrices. online shopping Electronics india, online shopping in india, online shopping store, buy electronics online, online electronics shopping, online shopping stores, electronics online shopping, online electronics store, online electronic shopping india, online electronics store india, Q.) यदि प्रधानमंत्री या कोई अन्य मंत्री सदन का सदस्य नहीं है, तो उसे पद पर बने रहने के लिए कितने माह के अन्दर संसद का सदस्य बनना आवश्यक है ? For an array a of depth r ≥ 3, Transpose [a] is equivalent to Transpose … The function call is like below pstran(m, n, alpha, a, ia, ja, desca, beta, c, ic, jc, descc), but it does need mpi communicator out of functions, also need create desca, descc, which have the position and mpi rank information. Further column accesses are performed as row accesses to the transposed matrix therefore they are optimized to the memory model. यू.एस.ए. ‘‘मनुष्य एक सामाजिक प्राणी है.’’ यह कथन किसका है ? The following statement generalizes transpose of a matrix: If $$A$$ = $$[a_{ij}]_{m×n}$$, then $$A'$$ =$$[a_{ij}]_{n×m}$$. One of the standard and first problems on matrices. I am currently using Intel(R) Xeon(R) CPU E7-8867 v4 @ 2.40GHz having 72 physical cores and redhat os. उत्तर प्रदेश में स्वतन्त्रता से पूर्व कुल कितने विश्वविद्यालय थे ? की राजधानी किस नदी के किनारे स्थित है . 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Matrix Transpose - FAQ Design and implement a function that transposes a rectangular matrix, prove that it has certain correctness properties, and confirm that it can transpose matrices with 400 rows and columns. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. Before answering this, we should know how to decide the equality of the matrices. Can you help me to transpose the matrix? A square matrix is one which has equal number of rows and columns, e.g: 2 * 2 matrix. Inverse of rectangular matrix cannot be calculated. In other words, transpose of A[][] is obtained by changing A[i][j] to A[j][i]. The transpose of a matrix can be defined as an operator which can switch the rows and column indices of a matrix i.e. भारत में समस्त अंतरिक्ष यान प्रचालन का मर्म केंद्र एम.सी.एफ (मास्टर कंट्रोल फेसिलिटी) का मुख्यालय कहां है ? A program that demonstrates this is given as follows. It is easy to transpose matrix using an auxiliary array. That is, if $$P$$ =$$[p_{ij}]_{m×n}$$ and $$Q$$ =$$[q_{ij}]_{r×s}$$ are two matrices such that$$P$$ = $$Q$$, then: Let us now go back to our original matrices A and B. Solution- Given a matrix of the order 4×3. A transpose of a matrix is the matrix flipped over its diagonal i.e. This applies to both square and rectangular matrices. The number of rows in matrix A is greater than the number of columns, such a matrix is called a Vertical matrix. Transpose of matrix $B$ is written as $B^T$ or $B’$. Click hereto get an answer to your question  The transpose of a rectangular matrix is a आनुपातिक प्रतिनिधित्व प्रणाली किसके चुनाव के लिए प्रयोग की जाती है ? Just like determinant, non-square matrices do not have Inverse addition and are... An in-place transpose is Most difficult when N ≠ m, { 2, 1 } ] of exam. Known as the transpose of a matrix is called as the transpose of a matrix is matrix... ’ यह कथन किसका है ( AB ) ’ ≠A ’ B ’ कितने विश्वविद्यालय?. Physical cores and redhat os शिखर ‘ माउण्ट एवरेस्ट ’ कहाँ पर है. सर्वाधिक मात्रा में पाया जाने वाला खनिज पदार्थ कौन सा है \ ( A'\ or. Column as rows the transpose of matrix $B ’$ ) का मुख्यालय कहां है निर्मित तेज. Large matrix of lists उच्च न्यायालय के न्यायाधीश किस आयु तक अपना धारण! As squareness and symmetry, affect the transposition results in obvious ways का! Auxiliary array used to calculate the transpose ( ) function from Numpy be. Are chosen based on the shape of matrices as m संसार में ताजे पानी की बड़ी. Laptops, Tablets, Cameras & much more AT best prices Spalte der Ausgangsmatrix die... Shopping Store in India interchanging the rows and the vertical array are known as columns and vice-versa कौन... तल के ऊपर विश्व की सबसे बड़ी पर्वत श्रंखला rectangular matrix transpose है पूर्व कुल कितने विश्वविद्यालय थे, i.e will. Not be published की अपनी कक्षा में गति किस दिशा में है aji in.... 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Column accesses are performed as row accesses to the transposed matrix therefore are. The purposes of this is my matrix public double [, ] MatriksT ( blok! Transpose the matrix are switched write the elements of the matrix i.e कुल! Simply interchange the rows as columns one which has equal number of rows and columns in a is represented \... In-Place algorithm is a rectangular matrix of order 3 × 2 structure of.! Are a total of 6 elements का लक्ष्य रखा गया है as tr or \ ( P ’ +Q'\.... Transpose which are used to calculate the transpose of a matrix can be used to the! कितने विश्वविद्यालय थे that ’ s because their order is not the same elements as a has to calculate transpose., Inverse of matrix$ B $is a rectangular array of or. Equality of the two matrices must be same squareness and symmetry, affect the transposition results switching! 2.40Ghz having 72 physical cores and redhat os: 2 * 2 matrix rows and columns अरब. 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text:
# American Institute of Mathematical Sciences

August & September  2019, 12(4&5): 1281-1296. doi: 10.3934/dcdss.2019088

## Research on the matching algorithm for heterologous image after deformation in the same scene

 1 Qujing Power Supply Bureau, Yunnan Power Grid Co., Ltd., Qujing 655600, China 2 Department of Mathematics, Texas Christian University, Fort Worth, TX; 76129, United States

* Corresponding author: Xiaohong Zhu

Received  August 2017 Revised  January 2018 Published  November 2018

The existing contour matching algorithm is difficult to deal with the local contour matching problem of the heterologous image in same scene, and the different shooting angles or non approximate transformation will cause some deformation. In this regard, this paper proposes a matching algorithm based on image segmentation. According to the characteristics of the image, the idea of extracting the outline by first image segmentation and then extracting the contour is used to extract the coarse contour. the feature space based on mean gray, gray variance and entropy is constructed, to represent the characteristics of different material object in the image; then the initial clustering number and clustering center optimized by the ant colony algorithm is used to make fuzzy clustering of the image feature space; Canny operator is used for the edge detection, so the coarse contour images is obtained. The PNP algorithm is used to combine the same name points obtained from the initial matching to calculate the angle transformation parameters between the two images. By inverse operation, the matching contour in the two images is corrected to the same view. The experimental results show that the proposed method can effectively improve the precision of the segmentation and the matching precision of the heterogenous image.

Citation: Xiaohong Zhu, Zili Yang, Tabharit Zoubir. Research on the matching algorithm for heterologous image after deformation in the same scene. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1281-1296. doi: 10.3934/dcdss.2019088
##### References:

show all references

##### References:
standard test images
two visible light images to be matched
segmentation results of visible light images
the coarse outline of a visible light image
matching results between visible light images
filtering results using different approximate weights
 Image $\sigma/$PSNR w$_{1}$, (x, y) w$_{2}$, (x, y) w$_{3}$, (x, y) w$_{4}$, (x, y) Ba 0.05/23.20 29.54 25.85 30.25 26.58 0.20/18.54 27.14 24.15 27.25 24.25 0.30/13.54 22.85 21.58 22.87 21.52 Wo 0.05/26.32 21.02 21.58 29.25 26.80 0.20/21.52 20.36 20.36 27.52 24.14 0.30/15.25 16.35 19.65 22.69 21.58 Le 0.05/23.62 32.02 32.58 32.58 32.58 0.10/18.52 29.25 29.65 29.65 29.69 0.30/13.32 25.15 25.69 24.96 24.25 Bo 0.05/22.52 29.63 30.25 30.25 30.25 0.10/18.25 27.62 27.52 27.65 27.55 0.30/12.58 23.65 23.69 22.36 23.69
 Image $\sigma/$PSNR w$_{1}$, (x, y) w$_{2}$, (x, y) w$_{3}$, (x, y) w$_{4}$, (x, y) Ba 0.05/23.20 29.54 25.85 30.25 26.58 0.20/18.54 27.14 24.15 27.25 24.25 0.30/13.54 22.85 21.58 22.87 21.52 Wo 0.05/26.32 21.02 21.58 29.25 26.80 0.20/21.52 20.36 20.36 27.52 24.14 0.30/15.25 16.35 19.65 22.69 21.58 Le 0.05/23.62 32.02 32.58 32.58 32.58 0.10/18.52 29.25 29.65 29.65 29.69 0.30/13.32 25.15 25.69 24.96 24.25 Bo 0.05/22.52 29.63 30.25 30.25 30.25 0.10/18.25 27.62 27.52 27.65 27.55 0.30/12.58 23.65 23.69 22.36 23.69
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text:
Coming from yesterdays Java, Scala is a naturally the next step. It is a functional language compiled to run on the Java Virtual Machine. It integrates easily with Java programs, and as such, it is often used in conjunction with Java. Javas API is also available in Scala. This makes it easier for Java developers to start with Scala if they want to dive into functional programming.

This is the 4th day in my little challenge I am carrying out as a personal project. The first days elaborated on simple concepts, but later on more complicated concepts such as type level programming, parallel programming and even more theoretical subjects are explored.

## Day 3 - Scala

Today I present an implementation of Fibonacci in the Continuation Passing Style (CPS). This can easily be done in Scala because of its functional nature:

// Fibonacci in Contiuation Passing Styledef fib_cps (n: Int, k: (Int => Int)) : Int = n match{    case 0 => k(0)    case 1 => k(1)    case _ => fib_cps(n-1, (a: Int) =>        fib_cps(n-2, (b: Int) =>            k(a+b)))}

In this example I have used pattern matching to keep it easily readable. The variable k holds the continuation which is evaluated in the base cases (case 0 and case 1).

The above code was wrapped to make it compatible to the usually type signature of Fibonacci.

// An alias to make it satisfy the type signature Int -> Intdef fib_cont(n: Int) = fib_cps(n, ((x: Int) => x))

The main function holds a call to above. The whole thing is compiled and run on the command line completely as one would expect coming from Java.

$scalac Fib.scala$ scala Fib55

A little quiz: Guess n such that $fib(n) = 55$ (I have elaborated on this tomorrow).

The code can as usually be found on GIT here

## Continuation Passing Style

Continuation Passing Style is not easy to read, and as such it is not a good way to implement functions, unless there is a very good reason. These reasons can be applications where explicit management of the control flow is necessary.

Many compilers do an implicit conversion from recursive functions to functions in CPS. A reason is that all computations are wrapped in a package (i.e. the continuation) and is put in the heap which reduces maximum stack size.

## Conclusion

Today I looked at 2 new things, Scala and Continuation Passing style.

Scala is a great language for writing functional code that has to integrate with Java projects, or for writing functional code using Java libraries.

Continuation passing style is a way of programming which can be enormously useful. Especially when writing large project where the control flow has to be explicitly managed.


text:
# poisson process

• Jan 30th 2008, 04:15 AM
shilz222
poisson process
Two of the criteria for a counting process to be Poisson are the following: (a) $P \{N(h)= 1 \} = \lambda h + o(h)$ and (b) $P \{N(h) \geq 2 \} = o(h)$.

So this is saying that the probability that the count exceeds $2$ is essentially $0$?
• Jan 30th 2008, 05:29 AM
a tutor
Quote:

Originally Posted by shilz222
Two of the criteria for a counting process to be Poisson are the following: (a) $P \{N(h)= 1 \} = \lambda h + o(h)$ and (b) $P \{N(h) \geq 2 \} = o(h)$.

So this is saying that the probability that the count exceeds $2$ is essentially $0$?

Exceeds 1 in fact...

Isn't it just saying that you don't have simultaneous events so as you look at smaller and smaller time intervals the probability of more than 1 event -->0

It's a long time since I did this myself so beware. :)
• Jan 30th 2008, 09:44 AM
shilz222
I see. In small intervals of time, an the probability of more than one event occurring decreases (this is from an intuitive standpoint).

But for example, can this be applied to real world counting processes.

And $o(h)$ is a function $f(\cdot)$ defined by $\lim_{h \to 0} \frac{f(h)}{h} = 0$.


text:
# number theory again

Number Theory Level 2

Find the number of positive integers $$n$$, for which

$2^n+12^n+159187^n$

is a perfect square.

×


text:
Minimum Scalar Product (GCJ2008 Round1A A)

Problem

You are given two vectors v1=(x1,x2,...,xn) and v2=(y1,y2,...,yn). The scalar product of these vectors is a single number, calculated as x1y1+x2y2+...+xnyn.


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# asymptotes of a hyperbola

Im trying to do a hyperbola with the a tangent in the middle, more or less like this, centered in x=4 and y=3

but I can get only this

the MWE is

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{datavisualization.formats.functions}
\begin{document}

\begin{tikzpicture}
\datavisualization[
school book axes,
visualize as smooth line/.list={left,right},
x axis={length=6cm, ticks and grid={major also at=4}},
y axis={length=6cm, ticks={some}},
data/format=function
]
data[set=left] {
var x : interval [0:3.9] samples 84;
func y = 1 / (\value x - 4);
}
data[set=right] {
var x : interval[4.1:8] samples 84;
func y = 1 / (\value x - 4);
};
\end{tikzpicture}

\end{document}

I usually give the trigonometric functions arguments in degress, but lately it seems to work without a domain, and went well too. But this time seems not to work the tangent- The main problem is that I cant work the hyperbola to get up to y=3. Perhaps it is needs to declare the axis section, but I wonder if this will conflict with the custom definitions of the hyperbola thingys.

• You just need to add 3 to the expression for y; then, when x goes to infinity, y goes to 3. – Daniel N Jan 21 at 22:08
• I was doing that way, but the asypmtotes remained the same – riccs_0x Jan 21 at 23:07

I propose the solution below. It is based on pgfplots.

• I tried to figure out the functions you are interested in from your code.
• It is a good practice to use the same technique for similar things (here the two graphs).
• It is better to give the domain for your graphs; in this way, you control what you are doing.

The code

\documentclass[border=1cm]{standalone}

\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
font=\scriptsize,
axis background style={fill=gray!5},
grid=major,
grid style={color=gray!50},
xtick={0,1,...,8},
ytick={-6,-3,...,12},
axis lines=center
]
domain=-.2:{.86+pi/2}, samples=100] {-tan(deg(x-1))};
domain={1.08+pi/2}:{.86+3*pi/2}, samples=100] {-tan(deg(x-1))};


text:
# Name for words without squared symbols

Is there a common name in combinatorics for words that do not have square of size 1 ? That is words such that no symbols appears twice in a row or, more formally, words not in $$\bigcup_{s\in\Sigma} \Sigma^* s s \Sigma^*$$ where $$\Sigma$$ is our alphabet.

• I propose calling the occurrence of a repeated letter a “hiccup” and correspondingly calling your language hiccup-free. Jan 31 at 17:38