text:
# Electromagnetism

## In Text Solution 2

A positively-charged particle (alpha-particle) projected towards west is deflected towards north by a magnetic field. The direction of magnetic field is

1. Towards south
2. Towards east
3. Downward
4. Upward

Answer: This question can be solved by using Fleming’s Left Hand Rule. We know that the direction of current is opposite to the direction of electron’s movement and hence it would be same as the direction of proton’s movement. So, the direction of current is towards west. As per Fleming’s Left Hand Rule; the middle finger shows the direction of current, the forefinger shows the direction of magnetic field and the thumb shows the direction of motion. Here, the deflection is towards north, i.e. in north westerly direction and hence, the direction of magnetic field would be towards north, i.e. upward.

State Fleming’s Left Hand Rule.

Answer: Fleming’s Left Hand Rule states that if the left hand is stretched in a way that the index finger, the middle finger and the thumb are in mutually perpendicular directions; then the index finger and middle finger of a stretched left hand show the direction of magnetic field and direction of electric current respectively and the thumb shows the direction of motion or force acting on the conductor. The directions of electric current, magnetic field and force are similar to three mutually perpendicular axes, i.e. x, y and z axes.

What is the principle of an electric motor?

Answer: The electric motor works on the principle of Fleming’s Left Hand Rule. When a rectangular coil is placed within a magnetic field and current is passed through the coil, there is deflection in the coil. The deflection changes into rotation of coil because of split ring commutator in the motor.

What is the role of a split ring in an electric motor?

Answer: In an electric motor, after every half rotation the direction of coil gets reversed due to change in orientation of the magnetic field. To ensure a continuous rotation; a split ring is attached to the coil so that the polarity of the coil changes after every half rotation. This changes the direction of current and thus the armature keeps on rotating continuously.

Explain different ways to induce current in a coil.

Answer: For electromagnetic induction; the coil and the magnet should be in relative motion. This can be ensured by any of the following two ways:

1. The coil should be moved within a magnetic field.
2. The magnet should be moved and coil can be kept static.

State the principle of an electric generator.

Answer: Electric Generator works on the principle of electromagnetic induction which obeys Fleming’s Right Hand Rule. When coil is moved inside a magnetic field a current is induced in the coil. The electric current is thus ‘generated’ by electric generator.

Name some sources of direct current.

Answer: Electrochemical cell, DC generator, photovoltaic cell, etc.

Which sources produce alternating current?

A rectangular coil of copper wires is rotated in a magnetic field. The direction of the induced current changes once in each

1. two revolutions
2. one revolution
3. half revolution
4. one-fourth revolution

Name two safety measures commonly used in electric circuits and appliances.

Answer: Earth wire and electric fuse

An electric oven of 2 kW is operated in a domestic electric circuit (220 V) that has a current rating of 5 A. What results do you expect? Explain.

Answer: The current drawn by the electric oven can be calculated using following formula: P = VI

Given; P = 2 kW = 2000 W and V = 220 V
Since, P = VI hence, I = P/V
Or, I = 2000 W ÷ 220 V = 9.09 A

Here, the oven is drawing 9.09 ampere of current from a 5 ampere source.

This means there would be an overload on the circuit. This can result in accidental fire in the circuit.

What precaution should be taken to avoid the overloading of domestic electric circuits?


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# Integral of Arcsin[x]

by Jameson
Tags: arcsinx, integral
 P: 789 I just can't see it. I would think that it would involve some form of trig substitution, but I'm just drawing a blank. I'll do the work if someone can please give me a nice little hint. I know that $$\int \arcsin{(x)}dx = \sqrt{1-x^2}+ x\arcsin{(x}) + C$$ from my calculator and mathematica. Hint please.
 Emeritus Sci Advisor PF Gold P: 16,101 If only you were working with its derivative! I bet you know how to integrate something that looks like its derivative!
 P: 789 Sure... I'd draw a nice triangle and do trig substitution if it was $$\int \frac{dx}{\sqrt{1-x^2}}$$ Hmmm...I'll think out loud here. If the integral has a square-root and is in the form of $c^2-x^2$ then x is one leg, c is the hypotenuse, and the other leg is the previously mentioned radical. Is trig substitution with right triangles on the right track? Since the hypotenuse is $\sqrt{a^2+b^2}$, it seems that one leg might need to be $\sqrt{\arcsin{(x)}}$ Somehow I don't think I'm on track.
P: 669

## Integral of Arcsin[x]

Use integration by parts. Remember to let u=arcsin(x) and v=x.

u sub: InverseLogAlgebraicTrigExp
 Emeritus Sci Advisor PF Gold P: 16,101 Sheesh, just give him the answer, why don't ya?
P: 789
 Quote by apmcavoy Use integration by parts. Remember to let u=arcsin(x) and v=x. u sub: InverseLogAlgebraicTrigExp
Ah, thank you! It's so simple.

And thank you Hurkyl as well. I still have lots to learn.
 P: 669 I apologize Hurkyl
 P: 84 well, you know the integral of sinx with limits. Now arcsin x will be the limits, and you can make a rectangle.
 P: n/a Or you could just take the derivative of the right hand side and go "ta da!" and that's proof enough for me.
 P: 3 equate the arc sine to another variable e.g y.making it a sine fxn.e.g the arc sine of 0.5=30,while sine30=0.5.this will simplify the integral and further substitution will conclude it
 Sci Advisor HW Helper P: 3,149 Just for the fun of it ... The sum of the integrals $$\int \sin^{-1} x dx + \int \sin y dy$$ is just the area of the bounding rectangle: $x \times \sin^{-1} x$ Since $$\int \sin y dy = -\cos y + C$$ and $$\cos y = \cos \sin^{-1} x = \sqrt {1-x^2}$$ it follows that $$\int \sin^{-1} x dx = \sqrt {1-x^2} + x \sin^{-1} x + C$$
 P: 17 Hey, Im really sorry to arise dead threads from the past (which i have seen though google) but somehting really weird happened me when I tried to use integration by parts on arcsinx. let me show you: S(arcsinx)= {v'(x)=1} {u(x)=arcsinx} xarcsinx-S(x*d(arcsinx))= xarcsinx-S(x/(1-x^2)^0.5= {u(x)=x v'(x)=arcsinx} xarcsinx-xarcsinx+S(arcsin)dx ==> S(arcsinx)=S(arcsinx) :\ I know I have done something really stupid here, but please be easy on me since I started studying Integrals only three days ago. In the second time I used integration by parts, do I miss something , is there another efficient choice of v and u? Thanks in advance, Aviv p.s: I will edit it better to use normal math signs once I figure out how.
 HW Helper P: 3,353 For the second time you integrate by parts, swap your choices.
 P: 17 Im sorry, Tried it also and all i got is: xarcsine x -(x^2/2)(1/(Sqrt(1-x^2)))-(x/4)(sqrt(1/(1-x^2))+1/4(arcsinx) this isn't going anywhere :(
 Quote by Jameson I just can't see it. I would think that it would involve some form of trig substitution, but I'm just drawing a blank. I'll do the work if someone can please give me a nice little hint. I know that $$\int \arcsin{(x)}dx = \sqrt{1-x^2}+ x\arcsin{(x}) + C$$ from my calculator and mathematica. Hint please.
Make $x=\sin t$. Then apply part integration on the resulting integral. It's just a way to avoid the simple solution of part integrating directly.

format: plain


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# Recursive marginal quantization of the Euler scheme of a diffusion process

Abstract : We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t_0=0$ to time $t_k$. For numerics, we restrict our analysis to the one dimensional setting and show how to compute the optimal grids using a Newton-Raphson algorithm. We then propose a closed formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.
keyword :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00809313
Contributor : Abass Sagna <>
Submitted on : Monday, April 20, 2015 - 2:02:35 PM
Last modification on : Friday, March 27, 2020 - 3:06:43 AM
Document(s) archivé(s) le : Wednesday, April 19, 2017 - 12:23:20 AM

### Files

MargQuantPagesSagna_revD.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00809313, version 3
• ARXIV : 1304.2531

### Citation

Gilles Pagès, Abass Sagna. Recursive marginal quantization of the Euler scheme of a diffusion process. 2014. ⟨hal-00809313v3⟩

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text:
Lemma 63.11.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$, $G$ be algebraic spaces over $S$. Let $G \to F$ be a representable morphism. Let $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and $q : U \to F$ surjective and étale. Set $V = G \times _ F U$. Finally, let $\mathcal{P}$ be a property of morphisms of schemes as in Definition 63.5.1. Then $G \to F$ has property $\mathcal{P}$ if and only if $V \to U$ has property $\mathcal{P}$.

Proof. (This lemma follows from Lemmas 63.5.5 and 63.5.6, but we give a direct proof here also.) It is clear from the definitions that if $G \to F$ has property $\mathcal{P}$, then $V \to U$ has property $\mathcal{P}$. Conversely, assume $V \to U$ has property $\mathcal{P}$. Let $T \to F$ be a morphism from a scheme to $F$. Let $T' = T \times _ F G$ which is a scheme since $G \to F$ is representable. We have to show that $T' \to T$ has property $\mathcal{P}$. Consider the commutative diagram of schemes

$\xymatrix{ V \ar[d] & T \times _ F V \ar[d] \ar[l] \ar[r] & T \times _ F G \ar[d] \ar@{=}[r] & T' \\ U & T \times _ F U \ar[l] \ar[r] & T }$

where both squares are fibre product squares. Hence we conclude the middle arrow has property $\mathcal{P}$ as a base change of $V \to U$. Finally, $\{ T \times _ F U \to T\}$ is a fppf covering as it is surjective étale, and hence we conclude that $T' \to T$ has property $\mathcal{P}$ as it is local on the base in the fppf topology. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Given that $\binom{16}r =\binom{16}{2r + 1}$, what is the value of $r$?

Solve $$\displaystyle\binom{16}r =\binom{16}{2r + 1}.$$

From what I understand:

$$\displaystyle\frac{16!}{r!(16-r)!}=\frac{16!}{(2r+1)!(16-(2r+1))!}$$

$$\implies r!(16-r)! = (2r+1)!(16-(2r+1))!$$

But I am stuck at this point. If there is any straightforward way to solving this problem, please describe it.

• algebra.com/algebra/homework/Permutations/… – lab bhattacharjee Dec 15 '18 at 9:24
• Use ${n\choose m}={n\choose n-m}$ and $r\ne 2r+1$ to find $r=16-(2r+1)\Rightarrow r=5$. – farruhota Dec 15 '18 at 9:32
• Don't forget that any negative integer $r$ is a trivial solution. – user10354138 Dec 16 '18 at 8:46
• @user10354138, and any integer greater than 16. – Peter Taylor Dec 17 '18 at 14:50

This is a great question! If all you have learnt is the definition of $$\binom nm$$ as $$\binom nm = \frac{n!}{m!(n-m)!}\tag{\dagger}$$ then this problem seems quite imposing. How is one to check for which values of $$r$$ the equation $$r!(16-r)! = (2r+1)! (16-(2r+1))!$$ holds?!

However, if you have the correct tool in hand then this problems opens up to your efforts very easily. The basic idea is given by @farruhota in the comments (and also in @AmbretteOrrisey's answer), but I believe there is one key ingredient still missing, so I will elaborate on that in this answer.

Now, it is easy to see from the definition of $$\binom nm$$ that $$\binom nm = \binom{n}{n-m}.$$ Indeed, just plug into the formula $$(\dagger)$$ and check that LHS = RHS.

However, you cannot immediately use this to solve $$r = 2r+1$$ and thereby get the desired value of $$r$$. Because, perhaps there also exist some other integers $$0 \leq r < l \leq n$$ such that $$\binom nr = \binom nl$$ is true? One still needs to rule out that possibility.

As it happens, it is not too difficult to prove that $$\binom nr = \binom nl$$ holds if and only if $$r = l$$ or $$r = n - l$$. So, here goes: say $$0 \leq r < l \leq n$$ and $$\binom nr = \binom nl.$$ Then, using the definition we get \begin{align} &&\frac{n!}{r! (n-r)!} &= \frac{n!}{l!(n-l)!}\\ \iff&& l!(n-l)! &= r!(n-r)!\\ \iff&& \frac{l!}{r!} &= \frac{(n-r)!}{(n-l)!}\\ \iff&& l(l-1)(l-2) \cdots (r+1) &= (n-r)(n-r-1)(n-r-2) \cdots (n-l+1).\tag{*} \end{align}

Now, in the last equation, both the LHS and the RHS are a product of $$l-r$$ consecutive integers. When can they be equal to each other? Well, for any positive integers $$a,b,k$$, we have $$a(a+1)(a+2) \cdots (a+k) = b(b+1)(b+2) \cdots (b+k) \iff a = b. \qquad (\text{Why? Check this!})$$

And, et voila! We can now conclude that $$(*)$$ holds if and only if $$r+1 = n-l+1 \iff r = n-l$$ as we wanted to show.

With this result in hand, it is now easy to solve this problem. I believe you can take it from here. :)

Binomial coefficients are duplicated at the 'mirror-image' argument - by which I mean $$\binom{n}{k}=\binom{n}{n-k} .$$ The binomial 'function' of order n, with k considered to be a variable, is exactly symmetrical about the centreline. So in this case we have $$16-r=2r+1$$$$\therefore$$$$3r=15$$$$\therefore$$$$r=5 .$$


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# 5 Minds For The Future

2013 April 30
/*
author: john
title: 5 Minds For The Future
amazon_asins=1422145352
date: 2013-04-30
wikipedia: http://en.wikipedia.org/wiki/
*/

I recently read "5 Minds for the Future" ([Gardner, Howard. 5 Minds for the Future. Harvard Business Review Press 2008](http://www.amazon.com/Five-Minds-Future-Howard-Gardner/dp/1422145352) ) by Howard Gardner. This page is just my notes and outline of ideas I read.

I see it as 5 types of Knowledge, Skills, Abilities, and Attitudes that he feels are essential in order for someone to be productive, in demand, and enjoy life. He calls these things "minds". The 5 minds he identifies are
[[#ENDSUMMARY]]

* [1. Disciplined Mind](#disciplined)
* [2. Synthesizing Mind](#synthesizing)
* [3. Creating Mind](#creating)
* [4. Respectful Mind](#respectful)
* [5. Ethical Mind](#ethical)

## Education at Large
pg8: everyone needs to be a teacher and role model.  Education needs to be lifelong

pg10: Due to vast changes in the world, educational practices should change. i.e. why memorize everything when you have google at your fingertips.  Also need to decide where to focus on the 5 minds in education.  Respect is up and coming in education. Can;t just focus on discipline knowledge.  quote Winston Churchill: "The EMPIRES OF THE FUTURE WILL BE THE EMPIRES OF THE MIND".

pg12: assertions that most sciences are the same regardless of culture.  There is only one mathematics, on biology,etc... technology maybe different in different cultures.  technology does not have to have science behind it.  but when it does, it advances.  society today though must have science to advance its technology.  So he asserts science/math are critical to students, but goes on about the limits...

pg 13:-14 emphasis on various other disciplines in addition to science/math: literature, history, economics, etc...  Don't make the focus only science, even though science seems like the most imp, don;t make it the only!

pg 17: he believes current formal education prepares students for the world of the past and not the world of the future where everything is global (people, knowledge, capital, and information).

## Disciplined
pg16:Individuals without one or more disciplines will not be able to succeed at any demanding workplace and will be restricted to menial tasks.

pg21: experienced students of a discipline and inexperienced students answer questions the same way!  So what have those experienced students really studied? Many examples provided.

pg 22: "Thinking in a disciplinary manner".  Humans didn't evolve with the purpose to understand the physical, biological worlds.  Did not need that to survive.

pg25: The [trivium](http://en.wikipedia.org/wiki/Trivium) and [quadrivium](http://en.wikipedia.org/wiki/Quadrivium) which comprised the classical seven [Liberal Arts](http://en.wikipedia.org/wiki/Liberal_arts|)

pg27: Starts getting into what is discipline, the difference between subject matter and discipline; a discipline has a distinctive way of looking at the world. "signature pedagogies of a profession".  learning a discipline requires feedback/mentors.

So how do we develop disciplined mind?  Begin at page 31.
He believes that everyone at the PRE-COLLEGE LEVEL should be able to execute and think in the major disciplines of mathematics, science, history, and one art form.  i.e. we should be able to think in the accepted ways that mathematicians, scientists, historians, an an artist thinks.  Gardner really believes that a single person should master these??  He calls the "gateways" to all the other disciplines that someone may choose to specialize in for the life work.

pg 32.  4 steps to acquiring a disciplined mind:

* Identify truly important topics within the discipline'
* Spend a significant amount of time on it
* Approach the topic in many different ways/viewpoints/angles,methods
* use "Performance Understanding" under a variety of conditions.  i.e. it is not about memorizing, it is about performing/thinking about a new subject using what you learned in the previous topic.

Pg 35: studies show folks that studied deeply often give the same answers as novices when they have a topic they did not specifically study before!  There is an absence of disciplinary thinking.

memorizing is not discipline.

reflective practitioners

pg 41:

Plato: " Through Education we need to help students find plkeasure in what they have to learn"

beware discipline can take on a pathological form...

two meanings of discipline: to master a craft and the capacity to renew that craft through regular application over the years.

## Synthesizing
pg6: managers need synthesis

pg16:Individuals without synthesizing capabilities will be over- whelmed by information and unable to make judicious decisions about personal or professional matters.

The Main Chapter here begins Chapter 3 page 45
The ability to knit together information from various sources into a coherent whole.  How do we get the skill? How do we know who does it well?

Types of Synthesis with great descriptions:

1. Narratives
2. Taxonomies
3. Complex Concepts
4. Rules and Aphorisms
5. Powerful metaphors, images, and themes
5. Embodiments without words
5. Theories
5. Metatheory

All types of synthesis have these components:
1. A goal
2. A starting Point
3. Selection of strategy, method, and approach
4. Drafts and Feedback

Interdisciplinary synthesis is difficult, but when skilled folks can do it, good stuff.  Three reasons for interdisciplinary synthesis.

Check out Bill Bryson, A Short History of Nearly Everything

Lumpers and Splitters

When we are young, we really want to find connections between new things and things we already know.  This works well, but as you get older you loose that mind-set.  It doesn't always work for complex ideas anyway; however, the mind set is still useful, how can we keep it?

Laser Intelligence / Search Light Intelligence

Aristotle deemed the capacity to create apt metaphors as a sign of genius.

We need role models of people who are good synthesizers,

## Creating

pg16:Individuals without creating capabilities will be replaces by computers and will drive away those who do have the creative spark.

Starts in depth on page 77
How do we mature creativity?  Creativity in history was often not rewarded, often persecuted.
pg 80: Creativity is the occasional emergent from the interaction of 3 autonomous elements:

* The Individual
* The cultural domain
* The social field.

Know what is meant by domains and fields.  Representatives of the field are the judges of what is good creativity.  Creativity exists if it exerts a genuine detectable influence on subsequent work in the domain.
So that is the test for creativity:  has the domain been altered by your creation?

How are creators different from experts?

Creators aren't bothered by failure, they keep going keep trying.

creativity is peaking at a 5 year old!  How do you maintain that childlike creativity into adulthood?
8 SHOW DIFFERENT EQUALLY VIABLE SOLUTIONS TO A SINGLE POSED PROBLEM.
Lear to take and give criticism.

Creativity of groups is questionable, bit possible.

Neuro-,  Geno-, Silicon  how are we going to be creative with these.

## Respectful
pg8: all make efforts to seek common cause and trust regardless of differences

pg16:Individuals without respect will not be worthy of respect by others and will poison the workplace and the commons.

Page 106:
Humans do have affinity for groups.  But how to respect between groups?
More critical in the non-science math disciplines.  Science/Math should be a universal language, but for example history may be subject to other interpretations.

parents,parents, parents, role models

## Ethical
pg8: really just a do unto others as you want them to do to you ?

pg16:Individuals without ethics will yield a world devoid of decent workers And responsible citizens none of us will want to live on the desolate planet.

ethical has to do with the discipline versus respect which is more individual.

## Conclusion

He wraps it up well.  Read the Chapter first and then the rest of the book. Or reread this when i need a refresher.

## Resources
[http://www.slideshare.net/LisaBerg/5-minds-pp-presentation]()

[https://wiki.bath.ac.uk/display/charlescornelius/Howard+Gardner's+Five+Minds]()


text:
Tag Info

29

One good piece of evidence that all particles of a given type are identical is the exchange interaction. The exchange symmetry (that one can exchange any two electrons and leave the Hamiltonian unchanged) results in the Pauli exclusion principle for fermions. It also is responsible for all sorts of particle statistics effects (particles following the ...

5

BebopButUnsteady answer is obviously correct. Actually most wave functions cannot be written as single Slater determinants and certainly no energy eigenstates of N-fermion systems (atom, molecules or solids) interacting by two-body potentials (like the Coulomb potential). The Slater determinant is just a nice simple approximation to ground state ...

3

Exchange interaction is an addition to other interactions between identical particles caused by permutation symmetry. This addition is a result of specific form of multi-particle wave function. It gives no contribution to Hamiltonian unlike "usual" interactions but appears as an additional term in equations for single-particle wave functions (e.g. ...

2

Interactions have to conserve angular momentum and spin is angular momentum, so trivially "Yes". In your hypothetical with two uncharged (I assuming you mean the electric charge here) particles there will be no electromagnetic repulsion because there will be no electromagnetic interaction. But Hartree-Fock is used for interacting systems...

2

Assuming that: by "a perfect DFT solution" you mean the set of Kohn-Sham orbitals for the exact (presently unknown) denistiy functional that existance of which Hohenberg and Kohn have proven; you are asking whether the individual energies of these "perfect" Kohn-Sham orbitals can be used as estimates of exact ionization and/or excitation energies, the ...

1

"The nonrelativistic theory has no necessary relation between spin and statistics, as you can understand from the example of a nonrelativistic 2d spinless fermion (imagine 2d confined electrons with a huge magnetic field in the z-direction which forces the spin to always be along the z-axis). The electrons behave as spinless particles with respect to the 2d ...

1

Hole is not a particle. Microscopically, hole is a quasiparticle perturbation of many-electron system. And what is called electron in semiconductors is also a perturbation in the same many-electron system. Exchange interaction within this many-electron system results in a complicated interaction between quasiparticles which live there. Which is called ...

1

It's a good question. The answer is that the bound on the density is given by the requirement that the interactions between the bosons have to remain weak for the Bose-Einstein condensate to exist. In practice, the helium-4 atoms have to be further away from each other than their radius. Why it is so? Well, if you're talking about the bosons occupying the ...

1

There's a repulsive van der Waals force between the atoms. This shows up in the second quantized formalism as H = \int d^3x \left[ \frac{\hbar^2}{2m} \nabla \Psi^\dagger \cdot \nabla \Psi - \mu\Psi^\dagger \Psi \right] + \int d^3x\, d^3y\, \frac{1}{2}V\left( \vec{y} - \vec{x} \right) \Psi^\dagger\left( \vec{x} \right) \Psi^\dagger\left( \vec{y} \right) ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
MeatAxe  2.4 Programs for working with modular representations

# Command Line

zad [Options] [-]Mat [-]Mat ... Result
Options
Standard options, see Standard Command Line Options
Mat
Input matrix
Result
Result matrix

Mat
Input matrix

Result
Result matrix

# Description

This program reads two or more input matrices, calculates their sum or difference and writes the result to a file. The input matrices must be compatible, i.e., they must be over the same field and have the same dimensions. zad is designed to work with very large matrices without running out of memory. Only two rows are allocated as working memory.

By default, all input matrices are added. For example,

zad A B C

calculates the sum of A and B, and writes the result to C. If a file name is preceeded by a minus sign, this matrix is subtracted. For example,

zad A -B -C D

calculates $D=A-B-C$. To subtract the first matrix, you must insert an extra "--" before the file names. Otherwise the first argument would be interpreted as a program option. For example, to calculate $C=-A-B$, use

zad -- -A -B C

If a file name starts with "-", preceed the file name by "+" to add, or by "-" to subtract the matrix. If, for example, the second input file is "-B", use the following syntax:

zad A +-B C
zad A –B C

MeatAxe 2.4 documentation, generated on Wed Jan 7 2015 08:38:36


text:
# How to refer to section numbers in a citation call-out?

In the example bellow, the citation call-out to entry 58 includes also the section 12.3.2. Is there a package or a built-in way to create call-outs in this format? I'm using bibtex to manage the bibliography.

For example, I expected that I could use a command like:

default graph \cite{foo}[12.3.2]. This...

If you load the natbib citation management package (with the option numbers, since it looks like you use a numeric citation call-out style), you could issue the following citation command to achieve your objective:

\cite[\S12.3.2]{<foo>}

where <foo> is the key of the bib entry you wish to cite.

Here's the output of a minimal working example that illustrates these points:

\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{abcdefgh.bib}
@article{xyz,
author = "Anne Author",
title  = "Thoughts",
year   = "3001",
journal= "Circularity Today",
volume = "111",
number = 1,
pages  = "1-100",
}
\end{filecontents}
\usepackage[numbers]{natbib}
\bibliographystyle{plainnat}
\begin{document}
\cite[\S12.3.2]{xyz}
\bibliography{abcdefgh}
\end{document}

• I found that natlib have some incompatibility with algorithm2e. I got an error when using both libraries in the same document. – Daniel Hernández Feb 4 '15 at 20:30
• I found the solution here: tex.stackexchange.com/questions/15098/… – Daniel Hernández Feb 4 '15 at 20:34
• @DanielHernández - The solution is simple: be sure to load natbib before algorithm2e. – Mico Feb 4 '15 at 20:39
• Thanks! My solution was including the package titlesec before both packages. Your solution is better. – Daniel Hernández Feb 4 '15 at 22:14


text:
# User talk:Jshflynn

## Welcome

Welcome to ProofWiki! Since you're new, you may want to check out the general help page. It's the best first stop to see how things are done (next to reading proofs, of course!). Please feel free to contribute to whichever area of mathematics interests you, either by adding new proofs, or fixing up existing ones. If you have any questions please feel free to contact one of the administrators, or post your question on the questions page.

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• Recent Changes - To keep up with what's new, and what's being added.
• Check out our house style if you are keen on contributing.
• Main Page talk - This is where most of the main discussions regarding the direction of the site take place. If you have any ideas, please share them!

Cheers!

--Your friendly ProofWiki WelcomeBot 10:10, 2 May 2012 (EDT)

## Title style

Note the house style when it comes to titles: important words are capitalised. What will happen is that someone will go through and change the titles of some of your recent pages as appropriate. Getting a good title for a page is an art so you may find the pages you've written may get namechanged.

Please keep up the good work - you are filling gaps we did not know we had. --prime mover 23:57, 19 July 2012 (UTC)

Thanks prime.mover. It is indeed difficult especially for something as unnameable as a B-Algebra identity. The true credit goes to others though. Someday I hope to publish my own paper so that I can see my own ideas on relation theory on this wiki. For now though I will continue research as too often I have found that any original idea of mine has been analysed to death in an obscure research paper. --Jshflynn 00:23, 20 July 2012 (UTC)

## Editing section-wise

Although convenient, the section-wise editing feature destroys the carefully set up large whitespace separating sections. This then requires a subsequent edit to recover the desired amount of white. It is therefore generally preferable (except on talk/user pages, where house style is not enforced) to edit the page as a whole using the link at the top. --Lord_Farin 21:48, 21 July 2012 (UTC)

I see. Will do so for my next edit. --Jshflynn 21:53, 21 July 2012 (UTC)

## Nudge

To avoid reinventing the wheel, consider Category:Definitions/Formal Systems. --Lord_Farin 23:48, 9 August 2012 (UTC)

That's alright. If I make something worthwhile then I will. --Jshflynn 23:49, 9 August 2012 (UTC)

## Motivation?

"I honestly wish I had never discovered mathematics."

I'm glad I did, because I'm even more incompetent at everything else. --prime mover 06:40, 10 August 2012 (UTC)

## Rework?

Not meaning to pour cold wet water-based liquid on your flames, but a lot of the stuff you're adding to your "Definitions" section has already been documented in the Category:Definitions/Formal Systems category (admittedly not that well structured at the moment). I'm not sure whether you're trying deliberately to reinvent this area of mathematics from a different viewpoint (full respect due) or whether you're not familiar with this area yet. Just sayin' ... --prime mover 11:53, 10 August 2012 (UTC)

Sorry - I see L_F has already been here. No worries. --prime mover 11:54, 10 August 2012 (UTC)
I am reading a book on Formal languages and computation at the moment. It says that a lot of concepts are clear/obvious. I also noticed you, L_F and a few others seem to process your books by putting them on a completely rigorous basis on this site. As you are so successful at doing so and I am so unsuccessful at finishing books I decided to adopt the same approach.
The definitions on the section of this site clashed with a lot of the things I was reading (most likely because the field in question isn't that standardised). So, I decided to create my own user-based section in order to not mess up the work someone else did. Right now, it's really just the theory of finite sequences. But in due time it might look better. --Jshflynn 12:24, 10 August 2012 (UTC)
It is often the case that different books do things in different ways, partirulcarly in evolving fields like mathematical logic and formal systems. What the approach is that we want on this site is to gather the various different approaches into one by using the "also known as" and "also defined as" technique: the first if the same concept is called by something different, and the second if either (a) the same name is used for two different things, and (b) (more tricky to do rigorously) if it's the same concept but defined with a subtle difference in the conditions (something in topology comes to mind where certain objects are defined on "open sets" by one source and "not necessarily open sets" in another).
The problem of course lies in the fact that proofs based on the definition in format A may well not work using the definition in the format B. Then we get angry emails from people who assume (without checking) that the proof assumes format B. The only way round this is to ensure rigorous proofwriting so that when an ambiguous property of an object is used, it is stated as well as being linked to. --prime mover 07:38, 16 August 2012 (UTC)
Oh yes, and it's also got to be pointed out that some books are not very good (in terms of rigor), if they are designed as a basic introduction for popular consumption. IMO this over-simplification approach might not be appropriate for ProofWiki. But as the book you are quoting from hasn't been specified, it is not possible to say whether this applies in this particular circumstance. --prime mover 07:42, 16 August 2012 (UTC)

The link you put on the community portal does not work. --Lord_Farin 06:33, 16 August 2012 (UTC)

Fixed it, moved it into the appropriate place. --prime mover 07:38, 16 August 2012 (UTC)

## Other Languages

I'm so happy you took the initiative with German here on ProofWiki. I know I'd love to do something with the Hindi or Spanish portions, but I am currently tied up at work. My whole goal was to diversify ProofWiki and you've helped it get started. Maybe after things settle I could ask ask you again. I thank you once again. --Smettems (talk) 01:09, 8 October 2012 (UTC)

## Appointment

You seem to have missed our virtual appointment; it happens. It's just that I had been looking forward to it. --Lord_Farin (talk) 22:23, 27 October 2012 (UTC)

## Dictionary of Mathematics

You are (to considerable benefit) adding references to the "Dictionary of Mathematics". As you may or may not have noticed, it is standard that some indication is provided as to where in a source the concept under consideration is (first) encountered. IMO, it would benefit the significance of your exercise if you included this. --Lord_Farin (talk) 09:27, 6 December 2012 (UTC)

No problem. In the dictionary I found "exterior multiplication" in the entry "module". So would:
Become:
What would the house prefer? --Jshflynn (talk) 10:18, 6 December 2012 (UTC)
That seems like an adequate solution. --Lord_Farin (talk) 10:21, 6 December 2012 (UTC)
with perhaps highlighting of the entry itself by making it italic (? maybe). Also note that in the case (like this) where the reference is to a concept held on ProofWiki on a different page, make it a link:
Suggestions only. --prime mover (talk) 11:37, 6 December 2012 (UTC)
The "Entry:" addition is clarifying IMO. The amalgamation:
has my vote (possibly without italics). --Lord_Farin (talk) 12:25, 6 December 2012 (UTC)
Might be worth adding an option to the BookReference template, e.g. "entry=..." which, if present as e.g. "entry=Module", will automatically add ": Entry: Module". Mind, last time I tried editing this template it started adding spurious linefeeds in the rendition, which I couldn't work out how to eliminate, so that may not be immediate. Then, if we decide on a different format, we need to change it in one place only. --prime mover (talk) 12:54, 6 December 2012 (UTC)

Due to some spare time being consumed atm anyway, I have implemented the suggested feature:

* {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2=Jonathan M. Borwein|entry=Module|entryref=Definition:Module}}

ought to produce:

...and after a subsequent edit, it does. --Lord_Farin (talk) 13:12, 6 December 2012 (UTC)

It appears the spurious line breaks are due to MediaWiki interpreting a colon at the start of some piece of text (e.g. in an if clause) as were it positioned at the start of a line, causing a newline and indentation. Nowiki tags resolve the issue. --Lord_Farin (talk) 13:14, 6 December 2012 (UTC)

Excellent. --prime mover (talk) 14:23, 6 December 2012 (UTC)

10/10 for enthusiasm, but preferable to post up the details of the book (and its authors)you are adding citations for before you add the citations - otherwise it's a great big pile of redlinks. I know, it can be a tedious task (specially if you don't have an ocr) to fill in the contents list, but you'll probably find that you may be the only active ProofWiki worker who has the book in question. --prime mover (talk) 06:12, 10 December 2012 (UTC)

In addition to that, please note that the sources are ought to be in chronological order. --Lord_Farin (talk) 09:04, 10 December 2012 (UTC)
I found the time to correct the order on all pages you have added the source for now. --Lord_Farin (talk) 09:11, 10 December 2012 (UTC)
Noted and thanks. On a similar note, lately I have found some books introduce theorems about a concept and name it afterwards. They have their reasons but it's quite inconvenient for the sources of PW. --Jshflynn (talk) 20:22, 10 December 2012 (UTC)
Why is it inconvenient? You enter the definition, you enter the theorem, and you thread the links in the opposite order. Where's the problem? --prime mover (talk) 21:22, 10 December 2012 (UTC)
My mistake. I was under the impression that the (Previous)...(Next) feature had to respect the presentation of the book exactly. Perhaps it strays from the aims of ProofWiki but is there someway to see a list of all definition pages with no sources subsection? --Jshflynn (talk) 22:26, 10 December 2012 (UTC)
The way I implement it, the (Previous)...(Next) feature does respect the presentation of the book exactly. So the (previous) link of (definition of concept A) is (theorem proving existence of concept A) and the (next) link of (theorem proving existence of concept A) is (definition of concept A). As I say, where's the problem? --prime mover (talk) 22:32, 10 December 2012 (UTC)
Why do you need a list of all the definition pages with no sources subsection? --prime mover (talk) 22:33, 10 December 2012 (UTC)
Would you rather learn theorems pertaining to A or the definition of A first? If it is immaterial to you then there is no problem and I see now it's just personal preference that I would rather learn a definition first.
Perfectly rational to prove the existence of an object, then after you've done that, give it a name. That's how the mathematician who discovered such an object probably worked it. Oh look, if you perform that operation to this object you get such-and-such an object. I'll define it thus-and-so. --prime mover (talk) 23:22, 10 December 2012 (UTC)
Ah. That does indeed make sense. Thanks for clarifying. --Jshflynn (talk) 23:53, 10 December 2012 (UTC)
It would be interesting to see which concepts are local to ProofWiki, which are too old to admit a citation, which are too general etc. ProofWiki is both a large and structurally regular body of information. It is only natural that I want to see patterns and statistics behind it. --Jshflynn (talk) 22:51, 10 December 2012 (UTC)
Most interesting. I didn't know you could do that with a rectangle. DHEI is half the area of ABCD. I wonder what would happen if you plotted all the "potential" E points such that that relationship holds. Is E somewhere special on the curve you obtain? --Jshflynn (talk) 23:53, 10 December 2012 (UTC)
Yes it is. That's the whole point. It's the incentre of that triangle. Unique for a given rectangle. --prime mover (talk) 06:11, 11 December 2012 (UTC)
Beg pardon, see what you mean. The locus of all the points $E$ such that $\Box DHEI = \Box ABCD / 2$. Should be straightforward to work out algebraically. My guess is that it's a quarter of an ellipse. --prime mover (talk) 23:07, 11 December 2012 (UTC)

D'oh! It is of course a hyperbola. Letting $(x, y)$ be the corner of the rect in question and letting the total area of rect be $ab$ the locus is of course described as $xy = ab/2$ which is a trivially simple example of $xy = c$. --prime mover (talk) 20:23, 28 December 2012 (UTC)

## Coretraction

Really? Definition:Section (Category Theory) referred to as coretraction? What source? (The use of "section" and "retraction" easily predates the entirety of CT.) --Lord_Farin (talk) 08:35, 18 January 2013 (UTC)

Cf. Barry Mitchell: Theory of Categories. Sorry L_F its such a pain setting up a source work I just assumed it was used throughout CT. You may remove at will. --Jshflynn (talk) 09:28, 18 January 2013 (UTC)
Well, as long as there exists a source, I won't remove it. I understand you're not going to bother writing up the whole source at this point; we all do what we do best. --Lord_Farin (talk) 09:35, 18 January 2013 (UTC)
I have taken the liberty of adding the source work, as well as the mathematician. Note that I can't find out when or where Barry Mitchell was born, so I'm guessing America 1930. He needs more of a page than an "Author" as there's at least one theorem named after him. --prime mover (talk) 17:01, 8 March 2013 (UTC)
... this wrk has now been invoked on the page in question.

## Spot the flaw or salvage the group

So the other day I thought I had found something isomorphic to the Klein 4 group.

My idea was to take all the statements generated by the grammar of propositional calculus and partition them into equivalence classes using the relation of logical equivalence.

Then, because every statement can be written in disjunctive normal form it can also be written with implication as its main connective (by the equivalent definition of disjunction).

So perhaps I could have functions on these equivalence classes that are based on the notions of converse, inverse and contrapositive.

It came from seeing the statement 'the inverse of the converse of a statement is its contrapositive'.

Naturally, we get the table below:

$\begin{array}{|c||c||c||c||c|} \hline \text{Composition} & \text{Inaction} & \text{Conversion} & \text{Inversion} & \text{Contraposition} \\ \hline \text{Inaction} & \text{Inaction} & \text{Conversion} & \text{Inversion} & \text{Contraposition} \\ \hline \text{Converse} & \text{Conversion} & \text{Inaction} & \text{Contraposition} & \text{Inversion} \\ \hline \text{Inversion} & \text{Inversion} & \text{Contraposition} & \text{Inaction} & \text{Conversion} \\ \hline \text{Contraposition} & \text{Contraposition} & \text{Inversion} & \text{Conversion} & \text{Inaction} \\ \hline \end{array}$

As much as one would like the table to be true and for this to be a rather quirky result to add to the database there is one massive and obvious flaw to it.

See if you can spot it or better yet salvage the group. --Jshflynn (talk) 22:17, 25 January 2013 (UTC)

Nope, can't spot the flaw. Looks like a Klein 4 group to me. --prime mover (talk) 22:40, 25 January 2013 (UTC)
That's what I thought, but the functions aren't well defined. For example, for inaction composed with inaction I could put inaction or contraposition. It's to do with how I've defined the underlying set.
You could possibly salvage the group by paying attention to the number of $\lnot$ signs in the antecedent and consequent and changing the definition of the underlying set accordingly but I would prefer to not have something so contrived. --Jshflynn (talk) 22:53, 25 January 2013 (UTC)
There are no flaws. You just need to define the entities appropriately. It's just an instance of composition of mappings. --prime mover (talk) 23:03, 25 January 2013 (UTC)
Contraposition $\circ$ Contraposition = Contraposition
Contraposition $\circ$ Contraposition = Inaction
Understand that I want to salvage a group example out of this as much as you do. --Jshflynn (talk) 23:15, 25 January 2013 (UTC)
Sorry, just noticed this is unresolved: how is Contraposition $\circ$ Contraposition = Contraposition? No comprendo. --prime mover (talk) 12:31, 2 February 2013 (UTC)

## Jordan Curve Theorem

I've seen that you've put up definitions of the Jordan Arc and Curve. Do you intend to upload the Jordan Curve Theorem as well? I'm going to need that in the Complex Analysis section. It would be nice if we could work together on a formulation of this theorem. --Anghel (talk) 11:59, 2 February 2013 (UTC)

## Metamath

As I'm a big fan of people learning much better when given the opportunity to find out for themselves, I will do no more here than make the observation that there has been a metamath citation template in existence for some time. You are encouraged to hunt down the appropriate page in the Help suite, in particularly that pertaining to House Style. --prime mover (talk) 07:37, 8 February 2013 (UTC)

## \restriction

The intended use of $\restriction$ is as on Definition:Restriction/Operation. Unfortunately, Definition:Subsemigroup wasn't a good source to copy-paste from. — Lord_Farin (talk) 22:00, 6 March 2013 (UTC)

## *ping* Subband of Induced Operation is Set of Subbands

Please complete said page. — Lord_Farin (talk) 13:31, 7 March 2013 (UTC)

## Equation template and all that

When putting an equation up in the equation template, you need to be careful to avoid including {{, }} and | symbols. The reason is apparent if you think about it.

In such cases, you put a space between the braces, and instead of | you use \vert.

This is documented in the Template:Eqn page but I think I need to put it in the FAQs too, because this keeps coming up. --prime mover (talk) 08:38, 8 March 2013 (UTC)

In the unlikely case a pipe (|) is needed outside math environment, use Template:!, i.e. {{!}}. — Lord_Farin (talk) 09:42, 8 March 2013 (UTC)

## Schröder Rule

I cleaned up what I believe were mistakes in the partial proof. We still need to finish it. Also, there's a claim right at the beginning that $A \circ B \subseteq C$ means, by the definitions of composition of relations and subset, that:

$\forall x, y, z \in S: \left({ (y, z) \in A \land (x, y) \in B \implies (x, z) \in C }\right)$

However, the definitions only get us to

$\forall x, z \in S: ((\exists y: (y, z) \in A \land (x, y) \in B) \implies (x,z) \in C)$

So we need some sort of theorem of predicate logic to bridge that gap. Want to give it a go?

Finally, I personally don't much care for truth table proofs escaping from the propositional logic category. Could that part be factored out as a proplog lemma? --Dfeuer (talk) 21:37, 28 May 2013 (UTC)

## What is the point?

What is the point in adding a same proof in 1+1 = 2/Proof 2? --Kc kennylau (talk) 07:11, 6 July 2013 (UTC)

## Colon

Don't forget the colons between the Template:BookReference and the relevant section when adding references. I take it that you will get round to creating the book and mathematician page in due time? — Lord_Farin (talk) 17:33, 23 July 2013 (UTC)

Also, you need to be careful when you link to a page with subpages - you need to go back and add the subpage into the link from the previous page or the thread gets lost.
Also, that book is styled as by "Thomas W. Hungerford", not "Thomas William Hungerford", so these pages will need to be revisited on that account as well. --prime mover (talk) 19:14, 23 July 2013 (UTC)

## Prev next work

When you come back after breaking off, don't forget to go back and add the "Next" tag to the last page you left with just a "Prev" tag. The best approach is to include the next tag along with the prev tag on the last page of an editing session, rather than put just a "prev" tag up, so you don't forget stuff like this. --prime mover (talk) 20:12, 25 July 2013 (UTC)

And like I said the other day, don't forget to go back and add the subpage to the "next" link on a page which has subpages, like I just did on the "next" link on Integers Modulo m under Multiplication form Commutative Monoid. I appreciate what you're doing here, but this stuff needs care to be taken.
Please take note of this point, I keep finding I have to go back and fill the "next" in for you. I'd rather not have to do that tedious job. --prime mover (talk) 19:14, 27 July 2013 (UTC)

## Extra blank line

Help:FAQ#Why the extra blank line?

I specifically mention it because it appears to be a detail of the house style that has passed you over. Please note that, in order to improve the presentational quality of pages, there is an extra blank line between all sections. --prime mover (talk) 06:10, 30 July 2013 (UTC)

And while I'm about it, there's no blank line above the $\blacksquare$ at the bottom of the proof. Please put the {{qed}} tag on the line immediately below the end of the proof. Again, it's just house style, it improves the look. --prime mover (talk) 06:18, 30 July 2013 (UTC)
Okay. --Jshflynn (talk) 08:31, 30 July 2013 (UTC)

## Hungerford book

Also, the contents of the source work which you are building up as a list in your user space is not in the format in which books are presented in the Books section. I mention this in case you are planning on putting it up in the "Books" section as the forthcoming Book:Thomas W. Hungerford/Algebra page.

This page needs to be structured as the contents list of the book as it is published. Please see examples of this. I appreciate it may look like a lot of work, which is tedious. You're right, it is. Get an OCR and scan the pages in, then all you need to do is tidy them up. --prime mover (talk) 06:24, 30 July 2013 (UTC)

The page User:Jshflynn/Hungerford Algebra is for myself. I have made the page Book:Thomas W. Hungerford/Algebra with the skeleton of the contents. I will fill the contents out in more than one session. --Jshflynn (talk) 08:31, 30 July 2013 (UTC)
Good stuff. Note the use of "text" form of section headings: $\text{I}$, $\text{II}$ etc., and also, in books contents we don't use bullets (they look untidy), just indents, which are also easier to manage. --prime mover (talk) 08:58, 30 July 2013 (UTC)


text:
•   OUTLINE Introduction General Definitions When can any weight function be turned into a strictly monotone weight function? Weighted Directed Graphs in Applications Autonomous Driving Quantum Walks Artificial Neural Networks   SUMMARY: We define a directed graph as a space with any binary relation, and we define a strictly monotone weight function on any directed graph. We prove when no weight function on a directed graph can be turned into a strictly monotone weight function, and we give a characterization of any strictly monotone weight function. Finally, we mention the use of weighted directed graphs in autonomous driving research, quantum information, and deep learning.    Introduction Here is an example…

•   OUTLINE Introduction Definitions Directed Graphs Infinite Chains (Def. 1 & Def. 2) When are Def. 1 and Def. 2 equivalent? A fix with the axiom of countable choice Directed Graphs in Applications Softwares Graph Neural Networks Quantum Information   SUMMARY: We introduce two notions of an infinite chain in a directed graph, and we show when these two notions are equivalent. We then mention the use of directed graphs in applications, such as artificial intelligence and quantum information. Introduction Consider this diagram $\begin{array}{ccccc} \bullet & \rightarrow & \bullet\\ \downarrow & & \downarrow\\ \bullet & \rightarrow & \bullet & \rightarrow & \bullet \end{array}$ which consists of vertices (the dots)…

•   OUTLINE Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence Quantum Optics   SUMMARY: We show why the tensor product is not the same as the Cartesian product, and we extend that result to categories. We then mention the use of the tensor product in applications, such as artificial intelligence and quantum optics. Introduction For any vectors $\left\langle x_{1},x_{2}\right\rangle$ of $\mathbb{R}^{2}$ and $\left\langle y_{1},y_{2},y_{3}\right\rangle$ of $\mathbb{R}^{3}$, a product of these two vectors, which is denoted as $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle$, is defined as the matrix \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} &…

• You may have had bad memories from mathematics at school, or maybe you liked it and stayed curious about it or even work in a related field. But in any case, you may ask yourselves: why research in mathematics is important? You may think that we already know all we need to know in mathematics as we are already able, for instance, to make the sophisticated computations required to successfully send a rocket into space to explore other planets. You might be surprised, but continuing research in mathematics can and will change our lives. And even save lots of human beings. How could mathematics save lives? A few years ago,…


text:
## blog.awill.me

You must have seen the original Jawbone around:

It was the first bluetooth headset that really tried to look cool, instead of some E.T. device on your ear. Unfortunately, it was a little big, and very expensive ($129). It was also only bluetooth v1.2 (but so was everything back then). So, Aliph went back and added bluetooth v2.0 and made it smaller (at the unfortunately expense of decreased battery life). They also made it more comfortable, whilst maintaining the high$129 price.

The only downside from the hardware side is the STUPID idea to use a proprietary charger.

Unlike every other gadget you have (obviously excluding Apple and they over proprietary-ness) most devices now connect to your computer, and get charged through USB, or a derivative. This means, I could share my phone charger with my headset charger. What a novel concept. Does Aliph really think I want to carry around 5 separate chargers with me?

Still, this might not be a huge deal to some people.

Since my job pays for my phone expenses, I figured there was no reason for the price tag to put me off.

So, time for a review:

The improved noise cancellation technology dubbed ‘Noise Assassin’ is very impressive. I walked outside, and stood next to sprinklers, and the person on the other end couldn’t tell. Impressive right? You’d think that would be the most important part of a headset right?

Well, technically, yes, but only if the rest of the stuff is at least up to par.

The problem? It’s somewhat surprising, but I couldn’t really hear them!

The speaker on the Jawbone is downright awful (and trust me, I’ve not got it configured incorrectly, and I know how to turn up the volume). The Jawbone’s interface of just two buttons is stupid, and causes it not to have its own volume interface. Increasing the volume on the Jawbone only increases the volume on your phone (volume is shared). Even the max volume setting required me to be in a silent environment. So, lets get this straight Aliph:

You have amazing noise elimination technology so that they can here me in a thunderstorm, but, I can’t hear them if I have the AC on in my car.

one word: Ridiculous.

It really would be like designing a car with a 500bhp V10 engine only to put a speed limiter on it for 50Mph. What’s the point Aliph?

No, seriously, I’d really like to know. It’s a wonder how things like this manage to get through testing, and QA.

Categories


text:
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• 12
• 9
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• 13

# DirectX Freeform Camera problem

This topic is 2809 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

This is my first post on these forums, but have been on #gamedev on the efnet IRC so howdy from Texas :)

To my problem. Please accept my apologies if I am not following some sort of forum etiquette. I read some initial rules so I hope nothing I do is out of line or looked down upon.

I am a Game and Simulation programming student who is working on a big project for our course. The 8 of us in the class on a team and we are to construct a demonstration/game that utilizes all the stuff we've learned from previous classes (AI, Graphics, physics, etc) - I promise that me asking for outside help is not against the class rules and is actually encouraged by our professor, so that's why I am stopping here.

Anyway, enough of the shenanigans. I have a freeform camera class in directX that I have structured and occasionally works quite well. The reason I say occasionally is because my camera often has fits depending on where I put it's starting position and it's target. Sometimes I can set a position and target, and the camera performs flawlessly, other times, I end up loosing my model completely and can never be found, or the 3d model (like a terrain that uses shaders) is skewed and leans randomly depending on the way the camera is facing. I am about 99.999% certain it's a mathematics problem. I have attempted several times in the past to really sit down and try to solve the issue, and I have a hunch specifically what it might be. I think that when I manually set a target and position, I am getting bad Right and Up vectors, which I have tried to correct but with no avail.

The code I have for the camera's structure is fairly complex and would possibly take several pages of a post to explain. Without stepping on anyone's toes, I would love for a kind soul to possibly get with me on Ventrilo/Teamspeak or heck even over the phone to help me hash out this problem. It would be much easier to explain over some sort of voice coms what the problem is and how it works, rather than type up the explanation here.

If someone would be willing to do this I would greatly appreciate it. I am not looking for someone to do this for me. I want to learn what my mistake is and I want to learn the solution that is required to fix it.

My code for most of the advanced directX users is going to be very basic structure. I am doing nothing that is cutting edge. I am actually using the book: "Introduction to 3D Game Programming w/ DirectX 9.0c - A Shader Approach" by Frank D. Luna so most of my code is based off of his. Surprisingly his code has similar issues for his FreeForm camera.

Thanks again,
Cameron

##### Share on other sites
If you're looking for someone to help you off of the forums, you might have better luck asking in Help Wanted or the Lounge. Since you're asking in the technical forums though, it might be better to keep the discussion here in the thread so that other readers can benefit. (IMO, at least.)

Anyway, even if there's a lot of code, I imagine the problem itself might not be as complex as it seems and could be solved here on the forums in the usual fashion.

I think the first step will be to define precisely what you mean by 'freeform'. Can you describe in detail the movement scheme for the camera? That is, what keys or other input devices control it, and what do the controls do?

##### Share on other sites
By FreeForm I mean "First Person" camera.. you use your normal "W and S" keys to move forward and backward in the direction you are looking, you use your "A and D" keys to strafe side to side. "Space" moves you up in the Y direction, like jumping. And last but not least holding down your shift key moves you faster, like a sprint. The mouse looks around like a normal FPS camera does to get you a new target.

My camera basically flies all over the Game World, no restrictions on where it can go.

Here is the good news. I figured out 90% of my problem with the camera's starting position and target!!! At least it seems to be working correctly now. But there is still a small issue that causes some odd behavior that I believe is related so let me explain.

My problem before is that I was not correctly setting my Up and Right vector depending on where I was and where I was looking. I've solved that initial issue now where I believe I am calculating them properly.

Here is my remaining problem. I have 2 methods in my camera class, one to set the position, and the other to set the target. Basically if I wanted to have the camera's target jump to a specific location in the middle the game without manually moving it with the mouse, then I could. Or if I wanted to all of a sudden change the camera's position by just one line of could, then I have that ability.
myCamera->setTarget(D3DXVECTOR3(0, 0, 0));

However, I need to be able to recalculate my UP and RIGHT vectors when I do that. I use the same code that I use in my camera's constructor to calculate the UP and RIGHT vectors. But when I do something like set a target or position, but it throws my my axes off.

Let me try and describe what happens. I have a flat basic terrain that is completely level. If I myCamera.setTarget(...) to a new position, the terrain will now lean and look slanted even though I am recalculating the UP and RIGHT vector.

If you need more explanation let me know. Here is my Camera's source file. I added the function "recalcVectors();" to help recalculate the UP and RIGHT vector at the beginning and possibly when I end up setting setTarget(...) or setPosition(...);

I am about 99.99% certain that it's still revolving around the calculation of the RIGHT and UP vector. I might be missing a key step in recalcVec()'s algorithm.

FYI, in the source code, my camera is a singleton class. The "adjust()" function is what's called every frame. Using normal DirectInput for the key commands. If you need anymore info on this then let me know!

Everything that has GS:: is our global settings static class. Target and Position are d3dxvectors. Speed and Sensitivity are floats. Mode is an int.

#include "DXCamera.h"#define DXCAMERA_FREEFORM 0;DXCamera::DXCamera() {	D3DXMatrixIdentity(&mView);	mPosition		= GS::CameraInitPosition;			//the position of the camera	mRight			= D3DXVECTOR3(1.0f, 0.0f, 0.0f);	// the right axis from the camera's position	mUpVec			= D3DXVECTOR3(0.0f, 1.0f, 0.0f);	//the up vector, default to Y	mTarget			= GS::CameraInitTarget;				//target that we want to look at	recalcVectors();									//calculate the vectors accordingly		mSpeed			= GS::CameraInitSpeed;				//camera's movement speed	mSensitivity	= GS::CameraInitSensitivity;		//camera's mouse sensativity	mCameraMode		= GS::CameraInitMode;				//camera mode}DXCamera::~DXCamera() {}DXCamera* DXCamera::Instance() {	static DXCamera *ptr = new DXCamera();		return ptr;}void DXCamera::adjust() {	switch(mCameraMode) 	{		case 0:			rotateFreeForm();			adjustFreeForm();					break;	}	buildView();}//mouse rotationvoid DXCamera::rotateFreeForm() {	if(gDInput->mouseButtonDown(1)) //if right mouse button are down	{ 				//grab angles		float pitch = gDInput->mouseDY() / 360; //get the change in the angle for moving the mouse up and down		float yAngle = gDInput->mouseDX() / 360; //get the change in the angle for moving the mouse left and right				//set sensativity		pitch *= mSensitivity;		yAngle *= mSensitivity;		//rotate Up and Down		D3DXMATRIX matRotAxis;		D3DXMatrixRotationAxis(&matRotAxis, &mRight, pitch);		D3DXVec3TransformCoord(&mTarget, &mTarget, &matRotAxis);		D3DXVec3TransformCoord(&mUpVec, &mUpVec, &matRotAxis);		//rotate left and right		D3DXMATRIX matRotYAxis;		D3DXMatrixRotationY(&matRotYAxis, yAngle);		D3DXVec3TransformCoord(&mRight, &mRight, &matRotYAxis);		D3DXVec3TransformCoord(&mUpVec, &mUpVec, &matRotYAxis);		D3DXVec3TransformCoord(&mTarget, &mTarget, &matRotYAxis);			}}void DXCamera::adjustFreeForm() {	//check speed modifier	float speedSet = mSpeed;	if(gDInput->keyDown(DIK_LSHIFT)) 	{		speedSet *= 5;	}	D3DXVECTOR3 dir(0.0f, 0.0f, 0.0f);	if(gDInput->keyDown(DIK_W))  //move forward	{		dir += mTarget;	}	if(gDInput->keyDown(DIK_S)) //move backward	{ 		dir -= mTarget;	}	if(gDInput->keyDown(DIK_D))  //strafe right	{		dir += mRight;	}	if(gDInput->keyDown(DIK_A)) //strafe left	{ 		dir -= mRight;	}	if(gDInput->keyDown(DIK_SPACE)) //move up in the y direction	{		dir.y += 1;	}	D3DXVec3Normalize(&dir, &dir);	mPosition += dir * speedSet;}//build and return the view matrix for the game worldvoid DXCamera::buildView() {	//Keep camera's axes orthogonal to each other and of unit length.	D3DXVec3Normalize(&mTarget, &mTarget);	//recalculate the up vector	D3DXVec3Cross(&mUpVec, &mTarget, &mRight);	D3DXVec3Normalize(&mUpVec, &mUpVec);	//D3DXVec3Cross(&mRight, &mUpVec, &mTarget);  // seems pointless	D3DXVec3Normalize(&mRight, &mRight);	//Fill in the view matrix entries.	float x = -D3DXVec3Dot(&mPosition, &mRight);	float y = -D3DXVec3Dot(&mPosition, &mUpVec);	float z = -D3DXVec3Dot(&mPosition, &mTarget);	mView(0,0) = mRight.x; 	mView(1,0) = mRight.y; 	mView(2,0) = mRight.z; 	mView(3,0) = x;   	mView(0,1) = mUpVec.x;	mView(1,1) = mUpVec.y;	mView(2,1) = mUpVec.z;	mView(3,1) = y;  	mView(0,2) = mTarget.x; 	mView(1,2) = mTarget.y; 	mView(2,2) = mTarget.z; 	mView(3,2) = z;   	mView(0,3) = 0.0f;	mView(1,3) = 0.0f;	mView(2,3) = 0.0f;	mView(3,3) = 1.0f;}//set the camera's movement speedvoid DXCamera::setSpeed(float speed) {	mSpeed = speed;}//get the camera's movement speedfloat DXCamera::getSpeed() {	return mSpeed;}//set the camera's mouse sensitivityvoid DXCamera::setSensitivity(float sens) {	mSensitivity = sens;}//get the camera's mouse sensitivityfloat DXCamera::getSensitivity() {	return mSensitivity;}//set the camera's position manuallyvoid DXCamera::setPosition(D3DXVECTOR3 pos) {	mPosition = pos;	//recalcVectors();}//get the camera's current positionD3DXVECTOR3 DXCamera::getPosition() {	return mPosition;}//set the camera's targetvoid DXCamera::setTarget(D3DXVECTOR3 target) {	mTarget = target;	//recalcVectors();}//get the camera's current targetD3DXVECTOR3 DXCamera::getTarget() {	return mTarget;}//set the camera's UP vector (default to 0, 1, 0)void DXCamera::setUpVec(D3DXVECTOR3 upVec) {	mUpVec = upVec;}//get the camera's current up vectorD3DXVECTOR3 DXCamera::getUpVec() {	return mUpVec;}D3DXMATRIX DXCamera::getView() {	return mView;}void DXCamera::recalcVectors(){	mTarget = mTarget - mPosition;	D3DXVec3Normalize(&mTarget, &mTarget);	D3DXVec3Cross(&mRight, &mUpVec, &mTarget);	D3DXVec3Cross(&mUpVec, &mTarget, &mRight);}

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As far as I see, you are overcomplicating the thing (and that is an understatement).

The usual way to make a first person view:
Rotate(ViewAngle.x,1,0,0);Rotate(ViewAngle.y,0,0,1);Translate(ViewPos[0],ViewPos[1],ViewPos[2]);

or
Translate(ViewPos.x,ViewPos.y,ViewPos.z);Rotate(ViewAngle_Z,0,0,1);Rotate(ViewAngle_X,1,0,0);
Depending on the matrix convention (row/column major)

ViewAngle is controlled by the mouse, ViewPos is controlled by the keyboard.

If you have targets, look into "look at".

Look at is basically this: (I haven't implemented it myself, I'm thinking it through by typing here):

For a transformation, you need a base (e1: forward,e2:left,e3:up). UP is the world's up vector.

You need to apply the inverse of the camera transform to the world.

I think the image tells everything you need.

If you have more camera types (fps, 6DOF, arcball whatever), I suggest to use separate functions for them, instead of one god function, that does everything.

I have to go now, if something isn't clear, let me know.

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In both of the places where you 're-build' the axes from the forward vector, I would try re-computing the axes as follows (pseudocode):
target = normalize(target);right = normalize(cross(vector3(0, 1, 0), target));up = cross(target, right);
The only place this might fail is if the camera looks more or less straight up or down. For this reason and others I recommend handling FPS motion differently than what you've shown here, but it would require significant revision to your code so I won't go into that further.

I would however recommend renaming your 'target' variable. 'Target' more or less implies a target position (a point), where what you really have here is a 'forward' vector. (As such, I'd just name it 'forward'.)

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Haven't posted something:

so, the final transfor will be:
[e1\e2\e3]^T * [-x, 0, 0                 0,-y, 0                 0, 0,-z]
or
[-x, 0, 0  0,-y, 0  0, 0,-z]  * [e1\e2\e3]^T
Depending on the convention. (note that the inverse of an orthogonal matrix is the same as its transpose)

e1,e2,e3 can be calculated in any ways. Maybe they are the base vectors of an object: this way you can 'bind' the camera to an object with arbitrary orientation and position.

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Um... I'm confused about something. Why 'right' vector and not 'left?'

The basis isn't a right handed stuff? (Or I'm into another flamewar of handedness?)

I used the world's UP = {0,0,1} convention.
Now I'm totally confused.

Sorry.

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Quote:
 The usual way to make a first person view:Rotate(ViewAngle.x,1,0,0);Rotate(ViewAngle.y,0,0,1);Translate(ViewPos[0],ViewPos[1],ViewPos[2]);orTranslate(ViewPos.x,ViewPos.y,ViewPos.z);Rotate(ViewAngle_Z,0,0,1);Rotate(ViewAngle_X,1,0,0);Depending on the matrix convention (row/column major)
Just FYI, transformation order doesn't have anything to do with whether the matrices are row-major or column-major. Rather, it's a function of vector notation convention (that is, whether row or column vectors are being used).

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Quote:
 Um... I'm confused about something. Why 'right' vector and not 'left?'The basis isn't a right handed stuff? (Or I'm into another flamewar of handedness?)I used the world's UP = {0,0,1} convention.Now I'm totally confused.
It's all just a matter of conventions. D3D can be set up to be either left-handed or right-handed, but in D3D use of a left-handed coordinate system is fairly common.

As for which direction is up, that's also pretty much just a matter of preference.

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*facepalm*

I don't know what I'm doing.

Anyway:
Rotate(ViewAngle.x,1,0,0);Rotate(ViewAngle.y,0,0,1);Translate(ViewPos[0],ViewPos[1],ViewPos[2]);
is in FPL openGL, but forget everything I said.


text:
# An algebra problem by Shriram Lokhande

Algebra Level 3

Define $$\varrho(n)$$ be a function which gives product of roots of the equation $\sqrt{n}^x+\frac{1}{\sqrt{n}^x}=2$ where $$n$$ is a gaussian integer .

Find $\prod_{gaussian integers}{\varrho(n)}$

Details

gaussian integer is an complex number $$a+bi$$ where a and b as integers

×


text:
Public Group

# Help with speed to platform distance scaling?

This topic is 671 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I'm in the process of creating an endless runner, and one of the features that I've been trying to implement effectively is a distance between platforms that scales with the speed you're running (-hspeed of the platforms). Essentially, the faster you run, the further you can jump. The problem is that I really don't know how to go about doing it. Here is my code for alarm[0] in the controller object that generates the platforms:

var len =irandom_range(global.playerspeed*10,global.playerspeed*16);
var rany = irandom_range(-20,20);
var floor_number = 0;
xx = 16
repeat (len){
new_floor = instance_create(room_width+xx,160+rany,obj_floor)
var yy2 = 0;
repeat(20){
yy2 += 16
instance_create(new_floor.x, new_floor.y + yy2, obj_floor);
}
floor_number ++
if (floor_number < len - 4 and floor_number > 4 ){
//if irandom(100) < 10 instance_create(room_width+xx,new_floor.y - 1, obj_enemy);
}

xx += 16;
}
alarm[0] = ????

I'm essentially trying to get the alarm to set to a scaling value at the end. Does anyone have any ideas? Am I going about the platform generation the wrong way? Any help or advice would be much appreciated!

Oh, and I also found that if the len == 1 global.playerspeed == 3 and alarm[0] == 10, the floors are seemingly spaced apart by their own width like this:

I've been playing around with those ratios but they don't seem to scale up properly. Maybe I'm missing something, though...

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In a similar project I used this:

float gap_limit = SLOW_GAP_LIMIT * speedRatio;
std::uniform_real_distribution<float> gapRand(0.0f, gap_limit);

Where speedRatio = speed / startingSpeed and SLOW_GAP_LIMIT was a tuned value.

You'll probably have to play with it a bit. That worked for the project it was in because the maximum speed kept it in a sane range. If we had uncapped the speed and somehow a player was able to continue accelerating for long enough it may have stopped working at some point.

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• ### Forum Statistics

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# Input file overview¶

This chapter introduces XML as it is used in the QMCPACK input file. The focus is on the XML file format itself and the general structure of the input file rather than an exhaustive discussion of all keywords and structure elements.

QMCPACK uses XML to represent structured data in its input file. Instead of text blocks like

begin project
id     = vmc
series = 0
end project

begin vmc
move     = pbyp
blocks   = 200
steps    =  10
timestep = 0.4
end vmc

QMCPACK input looks like

<project id="vmc" series="0">
</project>

<qmc method="vmc" move="pbyp">
<parameter name="blocks"  >  200 </parameter>
<parameter name="steps"   >   10 </parameter>
<parameter name="timestep">  0.4 </parameter>
</qmc>

XML elements start with <element_name>, end with </element_name>}, and can be nested within each other to denote substructure (the trial wavefunction is composed of a Slater determinant and a Jastrow factor, which are each further composed of $$...$$). id and series are attributes of the <project/> element. XML attributes are generally used to represent simple values, like names, integers, or real values. Similar functionality is also commonly provided by <parameter/> elements like those previously shown.

The overall structure of the input file reflects different aspects of the QMC simulation: the simulation cell, particles, trial wavefunction, Hamiltonian, and QMC run parameters. A condensed version of the actual input file is shown as follows:

<?xml version="1.0"?>
<simulation>

<project id="vmc" series="0">
...
</project>

<qmcsystem>

<simulationcell>
...
</simulationcell>

<particleset name="e">
...
</particleset>

<particleset name="ion0">
...
</particleset>

<wavefunction name="psi0" ... >
...
<determinantset>
<slaterdeterminant>
..
</slaterdeterminant>
</determinantset>
<jastrow type="One-Body" ... >
...
</jastrow>
<jastrow type="Two-Body" ... >
...
</jastrow>
</wavefunction>

<hamiltonian name="h0" ... >
<pairpot type="coulomb" name="ElecElec" ... />
<pairpot type="coulomb" name="IonIon"   ... />
<pairpot type="pseudo" name="PseudoPot" ... >
...
</pairpot>
</hamiltonian>

</qmcsystem>

<qmc method="vmc" move="pbyp">
<parameter name="warmupSteps">   20 </parameter>
<parameter name="blocks"     >  200 </parameter>
<parameter name="steps"      >   10 </parameter>
<parameter name="timestep"   >  0.4 </parameter>
</qmc>

</simulation>

The omitted portions ... are more fine-grained inputs such as the axes of the simulation cell, the number of up and down electrons, positions of atomic species, external orbital files, starting Jastrow parameters, and external pseudopotential files.

## Project¶

The <project> tag uses the id and series attributes. The value of id is the first part of the prefix for output file names.

Output file names also contain the series number, starting at the value given by the series tag. After every <qmc> section, the series value will increment, giving each section a unique prefix.

For the input file shown previously, the output files will start with vmc.s000, for example, vmc.s000.scalar.dat. If there were another <qmc> section in the input file, the corresponding output files would use the prefix vmc.s001.

<project> tag accepts additional control parameters <parameters/>. Batched drivers check against max_seconds and make efforts to stop the execution cleanly at the end of a block before reaching the maximum time. Classic drivers can also take the now-deprecated maxcpusecs parameter for the same effect in the per driver XML section.

In addition, a file named id plus .STOP, in this case vmc.STOP, stops QMCPACK execution on the fly cleanly once being found in the working directory.

## Random number initialization¶

The random number generator state is initialized from the random element using the seed attribute:

<random seed="1000"/>

If the random element is not present, or the seed value is negative, the seed will be generated from the current time.

To initialize the many independent random number generators (one per thread and MPI process), the seed value is used (modulo 1024) as a starting index into a list of prime numbers. Entries in this offset list of prime numbers are then used as the seed for the random generator on each thread and process.

If checkpointing is enabled, the random number state is written to an HDF file at the end of each block (suffix: .random.h5). This file will be read if the mcwalkerset tag is present to perform a restart. For more information, see the checkpoint element in the QMC methods Quantum Monte Carlo Methods and Checkpoint and restart files on checkpoint and restart files.


text:
# Reproduction Capabilities of Penalized Hyperbolic-polynomial Splines

This paper investigates two important analytical properties of hyperbolic-polynomial penalized splines, HP-splines for short. HP-splines, obtained by combining a special type of difference penalty with hyperbolic-polynomial B-splines (HB-splines), were recently introduced by the authors as a generalization of P-splines. HB-splines are bell-shaped basis functions consisting of segments made of real exponentials e^α x, e^-α x and linear functions multiplied by these exponentials, xe^+α x and xe^-α x. Here, we show that these type of penalized splines reproduce function in the space {e^-α x, x e^-α x}, that is they fit exponential data exactly. Moreover, we show that they conserve the first and second 'exponential' moments.

## Authors

• 3 publications
• 2 publications
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### A sufficient condition for penalized polynomial regression to be invariant to translations of the predictor variables

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Hyperbolic ordinal embedding (HOE) represents entities as points in hype...
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Aircraft failures alter the aircraft dynamics and cause maneuvering flig...
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## 1 Introduction

This short paper investigates the reproduction capabilities of hyperbolic-polynomial penalized splines. HP-splines, were recently introduced in [1] as a generalization of the better known P-splines (see [2, 3]), and combine a finite difference penalty with HB-splines that piecewise consist of real exponentials and monomials multiplied by these exponentials. Numerical examples show that the exponential nature of HP-splines may turn out to be useful in applications when the data show an exponential trend [4].

The HP-splines we consider have segments in the four-dimensional space

 E4,α:=span{eαx, xeαx, e−αx, xe−αx}, α∈R, (1)

with the frequency being an extra parameter to tune the smoother effects. Even though all details concerning their definition and construction can be already find in [1], the analysis of their reproduction capability is there missing. To fill the gap, here we show that these type of penalized splines reproduce function in the space , that is fit exponential data of the latter type exactly. We also show that they conserve the first and second ‘exponential’ moments, showing that HP-splines are the natural generalization of P-splines even with respect to reproduction and moment preservation.

Given the data points , , the uniform knot partition with knots distance , and denoting by a HB-splines basis of the spline space with segments in , the HP-spline approximating the given data is obtained by solving the minimization problem

 mina0,…,an+1m∑i=1wi(yi−n+1∑j=0ajBαj(xi))2+λn+1∑j=2((Δh,α2a)j)2, (2)

where the minimum is with respect to the HB-splines coefficients . The values are non-zero weights, is the difference operator acting on functions and on sequences respectively as (see [5] for this type of operators),

 Δh,α2u=u(x)−2e−αhu(x−h)+e−2αhu(x−2h),h>0 the knots distance
 (Δh,α2a)j=aj−2e−αhaj−1+e−2αhaj−2,j∈Z,

and is a regularization parameter that can be set in several different ways, e.g. with the discrepancy principle, the generalized cross-validation, or the L-curve method (see [6], for example)
It is not difficult to see that the HP-spline can be written as

 shp(x)=n+1∑j=0^ajBαj(x), (3)

where is the solution of the linear system

 (BαTWBα+λDh,α2TDh,α2)a=BαTWy, (4)

where , is the banded collocation matrix , is the diagonal matrix

 W:=(diag(wi))i=1,…,m∈Rm×m

and

 Dh,α2=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1−2e−αhe−2αh0⋯001−2e−αhe−2αh⋯0⋮⋱⋱⋱⋯⋮⋮⋮⋮1−2e−αhe−2αh⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦∈Rn×(n+2).

Note that for the space reduces to , HB-splines reduce to cubic B-splines and the difference operator reduces to the standard forward second order difference operator acting on a sequence as . Therefore, for HP-splines coincide with P-splines based on classical cubic B-splines proposed by Eilers and Max (see their recent monograph [7]). From now on, without loss of generality, we continue by assuming that

is the identity matrix.

P-splines are known to have a number of useful properties, essentially inherited from B-splines and from the special type of penalty: they can fit polynomial data exactly, they can conserve the first two moments of the data and show no boundary effects. The aim of this paper is to investigate similar reproduction properties of HP-splines. As in the polynomial case, the HP-spline properties are essentially inherited from HB-splines and this is why in Section 2 we first prove that HB-splines reproduce . Then, in Section 3, we show that, whatever the value of the smoothing parameter , HP-splines fit exponential data exactly as they reproduces . Moreover, we show that HP-splines conserve the first and second ‘exponential’ moments. Section 4 draws conclusion and highlights future works.

## 2 HB-splines and their reproduction properties

As shown in [8], in the ‘cardinal’  situation –corresponding to integer spline knots–, HB-splines can be defined through convolution. For the spline space with segment in , starting with the first order B-spline

 B1α(x)=eαxχ[0,1](x), (5)

supported in , the four order cardinal HB-spline supported on is obtained as

 B1α=(B1α∗B1α∗B1−α∗B1−α)withα=(α,α,−α,−α),

hence, for any integer , the HB-spline supported in is obtained by translation as . In case the knots are uniform but with a distance , the corresponding HB-splines are defined by dilation,

 Bhα=B1α(⋅h)=(B1α∗B1α∗B1−α∗B1−α)(⋅h), (6)

and then translation. Alternatively, we directly start with the scaled order-one HB-spline and use repeated convolution. In that case we see that when dealing with grid spacing , the frequency is scaled into , a fact that will also enter into the exponential reproduction discussion we are going to make.

Concerning the HB-spline reproduction, we prove the following result.

###### Proposition 2.1.

Let

be the vector of frequencies, and

the HB-spline function with uniform knots defined in (6) having support . Then,

 f=∑k∈ZcfkBhαh(⋅−hk),for allf∈E4,α.
###### Proof.

The starting point is [8, Proposition 2] that yields the particularly simple reproduction formulas for the order-two HB-spline

 ∑k∈ZeαkB1α,α(x−k)=eαx,∑k∈Z(k+1)eαkB1α,α(x−k)=xeαk,

providing, for , the reproduction formula for . In fact, from we arrive at

 eαx=∑k∈ZeαtkBhαh,αh(x−tk). (7)

Similarly, gives

 xheαhx=∑k∈Z(k+1)eαhkhBhα,α(x−hk)

and hence the reproduction formula for :

 xeαx=∑k∈Z(tk+h)eαtkBhαh,αh(x−tk). (8)

Next, we use the fact that the convolution product of with the functions or yields another exponential polynomial of the same type, that is

 Bh−αh,hαh∗f1=a0f1,andBh−αh,αhastf2=b0f1+b1f2,witha0,b0,b1∈R.

Therefore, since and , if we convolve both side of (7) and (8) with , we arrive at

 a0f1=∑k∈ZeαtkhBhαh(⋅−tk),andb0f1+b1f2=∑k∈Z(tk+h)eαtkhBhαh(⋅−tk),

that are the reproduction formulas for of a function in . Similarly we prove the reproduction of and therefore the reproduction of .∎

## 3 HP-splines reproduction of E2,−α and moment preservation

Based on the HB-splines reproduction properties shown in Proposition 2.1 in this section we show the reproduction capabilities of HP-splines, independently to the value of the smoothing parameter .

###### Proposition 3.1.

Let the data points , , be given together with the uniform knots partition () extended with the uniform left and right extra knots where . Let be the spline basis with segments in consisting of the uniform HB-splines and , with as in (6). Then, if the data are taken form a function , i.e.,  with , the HP-splines defined in (3) satisfies

 shp(xi)=yi,i=1,⋯,m.
###### Proof.

Form Proposition 2.1 we know that HB-splines reproduce meaning that there exists a sequence of coefficients satisfying

 n+1∑j=0cfjBαj(x)=f(x),x∈[a,b],f∈E4,α. (9)

With the notation assuming that the data are taken form a function , it is not difficult to see that the solution of the linear system (4) is exactly , since from (9) we have

 yi=n+1∑j=0^ajBαj(xi), i=1,⋯,m, (10)

and, for ,

 0 = Δh,α2f(x)=n+1∑j=0^ajΔh,α2Bαj(x)= (11) = n+1∑j=2(Δh,α2^a)jBαj(x)+^a0Bα0(x)+(^a1−2e−αh^a0)Bα1(x),

that, in combination with the linear independence of HB-splines, shows that for . The latter means that the model acts like non penalized regression and that the reproduction capabilities of the HB-splines transfer to the HP-splines. ∎

###### Remark 3.2.

It is important to remark that, even though HB-splines reproduces , Proposition 3.1 shows that HP-splines reproduces only. This limitation is due to the specific definition of the difference operator acting as shown in (11). A model based on rather than would reproduce .

Next, we show that HP-splines preserve the two ‘exponential’ moments

###### Proposition 3.3.

In the notation of Proposition (3.1), denoting by the vector of predicted values with elements , we have

 m∑i=1e−αxi^yi=m∑i=1e−αxiyi,andm∑i=1xie−αxi^yi=m∑i=1xie−αxiyi. (12)
###### Proof.

To see (12), we start from the two equations defining the reproduction of

 n+1∑j=0cjBαj(x)=e−αx,andn+1∑j=0djBαj(x)=xe−αx,withcj,dj∈R, (13)

and evaluate them at . Hence, for and we have the equivalence

 n+1∑j=0cjBαj(xi)=e−αxi, i=1,⋯,m  ⇔  Bαc=e, (14)

and, for and , the equivalence

 n+1∑j=0djBαj(xi)=xie−αxi, i=1,⋯,m   ⇔  Bαd=xe. (15)

Now, in consideration of the linear independence of the HB-splines, with the same reasoning done in Proposition 3.1, from

 0 = Δh,α2(e−αx)=Δh,α2(n+1∑j=0cjBαj(x))= = n+1∑j=2(Δh,α2c)jBαj(x)+c0Bα0(x)+(c1−2e−αhc0)Bα1(x),

we can write

 (Δh,α2c)j=0, j=2,…,n+1,

from which we conclude Similarly,

 0 = Δh,α2(xe−αx)=Δh,α2(n+1∑j=0djBαj(x))= = n+1∑j=2(Δh,α2d)jBαj(x)+d0Bα0(x)+(d1−2e−αhd0)

implies

 (Δh,α2d)j=0, j=2,…,n+1, (16)

and therefore Next, for the predicted values we have

 (17)

Multiplication of both sides of (17) respectively by and yields

and

Now, since and using (14) and (15) we arrive at

 eT(y−^y)=0,andxeT(y−^y)=0,

which are the vector versions of (12). ∎

###### Remark 3.4.

Note that the exponential moments (12), reduce to the classical moments preservation whenever that is to

 m∑i=1^yi=m∑i=1yi,andm∑i=1xi^yi=m∑i=1xiyi.

We conclude the paper with some figures showing the exponential-reproduction capabilities of HP-splines. Figures 1-2 refers to data taken from the exponential functions while Figure 2 to data from the function . They display the graph of the HP-spline (black ‘’ ) approximating the data for different selections of

combined with different level of absolute Gaussian noise with zero mean and standard deviation

, both specified in the figure captions. The data sites (red ‘’ ) and the spline knots location (blue ‘’ ) are also given in the figures. The smoothing parameter is always since not relevant to our discussion. The exact exponential fit is evident in absence of noise (left) while it is almost attained in case of a moderate noise (right) as well in case of an uncorrect selection of the frequency. For comparison, the graph of the P-splines approximating the data is also given (magenta ‘’ ) together with the graph of the function (blue ‘’ ).

## 4 Conclusions

This short paper enriches the study of HP-splines, penalized hyperbolic - splines with segments in the space consisting in the exponential polynomials , where is a real frequency. In particular, it investigates two important analytical properties of HP-splines: that they exactly fit function in and that they conserve the first and second ‘exponential’ moments, independently to the value of the smoothing parameter . A few numerical examples of reproduction are shown. A dynamic selection strategy of the parameter , that certainly deserve more attention, is presently under investigation.

Acknowledgement
The authors are members of INdAM-GNCS, partially supporting this work. They are also member of RItA (Rete ITaliana di Approssimazione) and UMI-T.A.A. group.

## References

• [1] R. Campagna, C. Conti, Penalized hyperbolic-polynomial splines, Applied Mathematics Letters 118 (2021) 107159.
• [2] P. H. C. Eilers, B. D. Marx, Flexible smoothing with B-splines and penalties, Statistical Science 11 (2) (1996) 89–121.
• [3] P. H. Eilers, B. D. Marx, M. Durbn, Twenty years of P-splines, SORT-Statistics and Operations Research Transactions 39 (2) (2015) 149–186.
• [4] R. Campagna, C. Conti, S. Cuomo, Computational error bounds for Laplace transform inversion based on smoothing splines, Applied Mathematics and Computation 383 (2020) 125376.
• [5] C. Conti, Sergio López-Ureña and Lucia Romani, Annihilation operators for exponential spaces in subdivision journal, Applied Mathematics and Computation, 418, 126796, 2022
• [6] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Society for Industrial and Applied Mathematics, 1998.
• [7] P. H. Eilers, B. D. Marx, Practical Smoothing: The Joys of P-splines, Cambridge University Press, 2021.
• [8] M. Unser, T. Blu, Cardinal exponential splines: part I - theory and filtering algorithms, IEEE Transactions on Signal Processing 53 (4) (2005) 1425–1438 Comp. Math. 11 (1999), 41-54.


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$2i\sqrt 2$
To express in terms of $i$ simply divide the number under the radical by -1 and write i as a multiplier. $\sqrt 8$=$2\sqrt 2$. Thus the answer is $2\sqrt 2i$.


text:
# Phase constant of waves

1. Mar 6, 2014

### grassstrip1

1. The problem statement, all variables and given/known data
I've attached the question where the graph can be found.
Essentially I have no problem determining A=0.04M K= 10π rad/m λ=0.2M ω=π/5 rad/s
I'm having trouble choosing what ø should be.

2. Relevant equations
y(x,t)= 0.04sin(10πx - π/5t +ø)

3. The attempt at a solution
Since the graph is for the particle at x=0 and t=0 and y=0
0=0.04sin(ø) solving for ø gives π, -π and 0

To try and determine the right phase constant I took the partial derivative to find the transverse velocity.
v(x,t)= (0.04)(-π/5)cos(10πx - π/5t +ø)
Since the particle has a positive velocity at t=1 , plugging in t=1 should give a positive velocity
Therefore ø cant = 0 since using 0 as a phase constant gives a negative velocity for t=1 and x=0
Problem is both plus and minus pi give the right velocity and I'm not sure how to pick the correct one. The correct answer is +π
I'd like to know in general as well how to pick the correct value. Thanks!
http://imageshack.com/a/img547/3680/2tjw.png [Broken]

Last edited by a moderator: May 6, 2017
2. Mar 6, 2014

### BvU

You did it just right. Many (me too) intuitively fill in $\phi$=0. But if you draw a sin(x), and pull it to the right, you see x=0 is moving down.

3. Mar 6, 2014

### grassstrip1

Okay thank you! I'm just wondering how you can tell the answer is positive pi and not negative since both would give the same sign for velocity

4. Mar 6, 2014

### nasu

It does not make any difference if you pick ∏ or -∏.
cos(a+∏)=cos(a-∏)=-cos(a)


text:
## Dynamical Systems

•  Boris, Kalinin, Penn State University
•  Local rigidity for partially hyperbolic toral automorphisms
•  11/26/2019
•  3:50 PM - 4:50 PM
•  C304 Wells Hall

We study perturbations of an irreducible ergodic toral automorphism $L$ with non-trivial center. For a small perturbation $f$ of $L$ with smooth center foliation we obtain results on regularity of the leaf conjugacy to $L$ and of a Holder topological conjugacy to $L$, when it exists. As a corollary, we show that for any symplectic perturbation a Holder conjugacy to $L$ must be smooth. For a totally irreducible $L$ with two-dimensional center, we establish various equivalent conditions that ensure smooth conjugacy between $L$ and $f$. This is joint work with Andrey Gogolev and Victoria Sadovskaya

## Contact

Department of Mathematics
Michigan State University