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# estimator.rinse_and_repeat¶

estimator.rinse_and_repeat(f, n, alpha, q, success_probability=0.99, m=<Mock name='mock()' id='139938827128976'>, optimisation_target=u'red', decision=True, repeat_select=None, *args, **kwds)[source]

Find best trade-off between success probability and running time.

Parameters: f – a function returning a cost estimate n – LWE dimension n > 0 alpha – noise rate 0 ≤ α < 1, noise will have standard deviation αq/sqrt{2π} q – modulus 0 < q success_probability – targeted success probability < 1 optimisation_target – which value to minimise decision – set if f solves a decision problem, unset for search problems repeat_select – passed through to cost_repeat as parameter select samples – the number of available samples

format: plain


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# Statistics/Testing Data/Types of Tests

A statistical test is always about one or more parameters of the concerned population (distribution). The appropiate test depends on the type of null and alternative hypothesis about this (these) parameter(s) and the available information from the sample.

## Example

It is conjectured that British children gain more weight lately. Hence the population mean μ of the weight X of children of let's say 12 years of age is the parameter at stake. In the recent past the mean weight of this group of children turned out to be 45 kg. Hence the null hypothesis (of no change) is:

${\displaystyle \,H_{0}:\mu =45}$ .

As we suspect a gain in weight, the alternative hypothesis is:

${\displaystyle \,H_{1}:\mu >45}$ .

A random sample of 100 children shows an average weight of 47 kg with a standard deviation of 8 kg.

Because it is reasonable to assume that the weights are normally distributed, the appropriate test will be a t-test, with test statistic:

${\displaystyle T={\frac {{\bar {X}}-45}{S}}{\sqrt {100}}}$ .

Under the null hypothesis T will be Student distributed with 99 degrees of freedom, which means approximately standard normally distributed.

The null hypothesis will be rejected for large values of T. For this sample the value t of T is:

${\displaystyle t={\frac {47-45}{8}}{\sqrt {100}}=2.5}$ .

Is this a large value? That depends partly on our demands. The so called p-value of the observed value t is:

${\displaystyle p=P(T\geq t;H_{0})=P(T\geq 2.5;H_{0})\approx P(Z\geq 2.5)<0.01}$ ,

in which Z stands for a standard normally distributed random variable.

If we are not too critical this is small enough, so reason to reject the null hypothesis and to assume our conjecture to be true.

Now suppose we have lost the individual data, but still know that the maximum weight in the sample was 68 kg. It is not possible then to use the t-test, and instead we have to use a test based on the statistic max(X).

It might also be the case that our assumption on the distribution of the weight is questionable. To avoid discussion we may use a distribution free test instead of a t-test.

A statistical test begins with a hypothesis; the form of that hypothesis determines the type(s) of test(s) that can be used. In some cases, only one is appropriate; in others, one may have some choice.

For example: if the hypothesis concerns the value of a single population mean (${\displaystyle \mu }$ ), then a one sample test for mean is indicated. Whether the z-test or t-test should be used depends on other factors (each test has its own requirements).

A complete listing of the conditions under which each type of test is indicated is probably beyond the scope of this work; refer to the sections for the various types of tests for more information about the indications and requirements for each test.


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# CODEEN¶

The Euclid code integration platform (CODEEN) is hosted by the APC laboratory as a Linux virtual machine.
The needed softwares are installed regularly.
The TIPS prototype for NISP pixel simulations is one of the first codes installed and integrated under this platform.

We create a dedicated branche for the CODEEN integration developments.

svn checkout http://svn.oamp.fr/repos/tips/branches/tips_codeen

## Tests¶

Test was written with pyUnit and TIPS was integrated in the CODEEN/Jenkins:
https://apceucliddev.in2p3.fr/jenkins/job/TIPS_dev/

### Level 0 : unitary tests¶

Test every single part of the code (functions, methods ...).
Allow to be sure that every brick of the software is functioning correctly independently of the other parts.

• test sky
• test instrument

• test axesim

### Level 1 : integration tests¶

Allow testing the small interactions between the different units of the software.

• test observation

### Level 2 : full system tests¶

Allow testing the software in a global way, by assembling all the bricks that make it.
Not implemented at this stage because to heavy...
The example provide with TIPS could be convert to a level 2 test.

### Level 3 : non regression tests¶

Key tests one keeps during the whole development process.
Not implemented at this stage.


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# Sinusoidal transverse wave

#### The Transverse Wave

A sinusoidal transverse wave is traveling along a string in the negative direction of an x axis. The figure below shows a plot of the displacement as a function of position at time t = 0. The x axis is marked in increments of 20 cm and the y axis is marked in increments of 0.5 cm. The string tension is 4.4 N, and its linear density is 25 g/m.

(a) Find the amplitude.  m

(b) Find the wavelength.

(c) Find the wave speed. m/s

(d) Find the period of the wave. s

(e) Find the maximum speed of a particle in the string.  m/s

(f) Complete the equation describing the traveling sinusoidal transverse wave, in which x and y are in meters and t is in seconds.

y(xt) =  sin(  ?x +  ?t + ? )

The amplitude of the sinusoidal transverse wave is the maximum elongation from the x axis: $A = 5*0.5 cm = 2.5 cm$

and its wavelength is the distance between two consecutive maxima $lambda = 4*20 cm =80 cm$

and the wave speed is $v = sqrt{T/mu} =13.266 m/s$

The equation that relates the wavelength to the wave speed is:

$lambda = v*T$   and thus

$T = 0.0603 seconds$  and

$V max = omega*A =(2*pi/T)*A =(2*pi/0.0603)*0.025= 2.605 m/s$

The equation of the wave traveling in positive x direction is

$Y(x,t) = A*sin[2*pi*(x/lambda -t/T) +phi]$

The equation of wave travelling in the NEGATIVE x direction is

$Y(x,t) = A*sin(2*pi*(-x/lambda -t/T) +phi) = A*sin[2*pi(x/lambda +t/T) +phi]=$ $=0.025*sin(7.85*x +104.2*t +phi) meter$

From the initial conditions at $t=0$ (from the picture) we can determine the phase $phi$

$0.02 =0.025*sin(phi)$    $phi = 53.13 deg =0.927 rad$

Therefore the equation of this wave is

$Y(x,t)= 0.025*sin(7.85x +104.2t +0.927)$

You can continue reading a discussion on transversal waves at physics forums


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0th

Percentile

autoload creates a promise-to-evaluate autoloader and stores it with name name in .AutoloadEnv environment. When R attempts to evaluate name, autoloader is run, the package is loaded and name is re-evaluated in the new package's environment. The result is that R behaves as if package was loaded but it does not occupy memory.

.Autoloaded contains the names of the packages for which autoloading has been promised.

Keywords
programming, data
Usage
autoload(name, package, reset = FALSE, …)
.Autoloaded
Arguments
name

string giving the name of an object.

package

string giving the name of a package containing the object.

reset

logical: for internal use by autoloader.

other arguments to library.

Value

This function is invoked for its side-effect. It has no return value.

delayedAssign, library

Aliases
library(base) # NOT RUN { require(stats) autoload("interpSpline", "splines") search() ls("Autoloads") .Autoloaded x <- sort(stats::rnorm(12)) y <- x^2 is <- interpSpline(x, y) search() ## now has splines detach("package:splines") search() is2 <- interpSpline(x, y+x) search() ## and again detach("package:splines") # }


text:
Opened 11 years ago

# Ability to specify the author and creation time for ticket attachments

Reported by: Owned by: Thijs Triemstra osimons high XmlRpcPlugin major Thijs Triemstra 0.11

### Description

It's currently not possible to specify the creation time or author for a ticket attachment. The attached patch should fix this problem.

### comment:1 Changed 11 years ago by osimons

Hmm. Perhaps adding an attributes={} parameter to the end of the signature? That could then contain author and time_created keys? Ideally I would also like description to be in that dictionary, as unlike the other parameters it is just an optional extra.

Technically it would perhaps be possible to change description to attributes, and then (at least for a time) test the input - if input is a string then save as description, if it is an dictionary then update whatever attributes it contains or use any default values (like req.authname).

### comment:3 Changed 9 years ago by osimons

Priority: normal → high

With #5445 + #3988 completed, the same principles should be applied to ticket attachments too. Should add 'author' and 'when' as optional arguments at the end (no need to break the API), and also add test(s) to make sure it works.

### comment:4 Changed 6 years ago by Olemis Lang

See these tickets for a similar problem solved using a different approach (still WiP) .

### Modify Ticket

Change Properties


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VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
• On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets [Elektronski vir]
Eichfelder, Gabriele ; Povh, Janez, 1973-
In the paper we prove that any nonconvex quadratic problem over some set KRn with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of ... K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479495, 2009), which is actually a special instance of our result with K=Rn+ .
Vir: Optimization letters [Elektronski vir]. - ISSN 1862-4480 (Vol. 7, issue 6, 2013, str. 1373-1386)


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# Definition:Partial Function

## Definition

Let $S \subset \N^k$.

Let $f: S \to \N$ be a function.

Suppose that $\forall x \in \N^k \setminus S$, $f$ is undefined at $x$.

Then $f$ is known as a partial function from $\N^k$ to $\N$.

Thus we can specify a function that has values for some, but not all, elements of $\N$.

It can be seen that the definition of a partial function as given here is compatible with that of a partial mapping.


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# Consistency of PA: why other proofs?

Completeness theorem affirms that a formal first order system is consistent iff it has a model. The FOL number theory(PA) or First Order Arithmetic has a model, which is the natural numbers structure. So PA is consistent. But then,

why one needs proofs of the consistency of PA such (such those by Gentzen, Ackermann etc... obviously not in PA)? Is it only for the sake of proving the result "syntactically" and not by using semantic tools such as models, or what else?

-
The mathematical principles needed to prove the completeness theorem are considerably stronger than PA itself, and according to some, stronger than what is really needed to do mathematics. So in a philosophical sense, there's no real meaning to such a proof of the consistency of PA. – Zhen Lin Mar 14 '12 at 12:42
could you please cite a reference for these consistency proofs (by Gentzen, Ackermann, ect)? I am not sure I know what you are referring at – magma Mar 14 '12 at 15:43
– Martín-Blas Pérez Pinilla Jun 6 '14 at 8:48

One reason is that the construction of the natural number structure (and the proof that it is a model of PA) actually takes place in a much stronger system: (a fragment of) ZFC. Since you are using ZFC to construct a model of PA, this amounts to "ZFC $\vdash$ PA is consistent."

Now, what happens if (Xenu forbid!) (that fragment of) ZFC is itself inconsistent? In that case since ZFC proves everything, the fact that it proves the consistency of PA is rather worthless. What we can glean from this is that the consistency of ZFC implies the consistency of PA; an example of a relative consistency result, but does not give the absolute consistency of PA.

-
There is a little thing I don't understand completely. Usually, when treating with models, we don't want the structure to be exactly a structure described in a formal way. So it is immaterial whether the natural number structure take place in ZFC or not. It is just a non-formal structure we can talk about without referring to ZFC, isn't it? – Oo3 Apr 6 '12 at 9:34
The issue is that you need the machinery of sets and functions to even define what a model is. So, although you may not need to speak about sets and relations when you exhibit a model, the proof that the existence of a model implies consistence is dependent on the consistency of the "expressive" theory (the one doing the expressing) – nomen Jun 4 '14 at 0:03

One reason for investigating other consistency proofs (related to that given by Arthur Fischer) is philosophical/foundational. As Arthur Fischer pointed out, we can only prove consistency relative to some other, stronger system. Proving the consistency of $PA$ in $ZFC$ is rather like using a bazooka to open a can of beans; $ZFC$ is much, much stronger than $PA$ and accordingly, you might think, much more likely to be inconsistent. So from a philosophical/foundational perspective, one might want a consistency proof relative to a weaker theory than $ZFC$. Using less to prove consistency ought to improve ones faith that the theory is consistent. In the case of $PA$, Goedel showed that theory equiconsistent with the theory $T$, an extension in finite types of $PRA$ (primitive recursive arithmetic) and thus is in a certain sense finitistically justified.

There is a slightly more technical consideration lurking in the background here. By examining exactly how much one needs to prove a theory consistent, one gets an idea of the precise strength of the theory. Proving $CON(ZFC) \rightarrow CON(PA)$ tells you $ZFC$ is stronger than $PA$, but not by how much. Gentzen's consistency proof for $PA$ using transfinite induction up to $\epsilon_0$ yields an "ordinal analysis" of $PA$ ($\epsilon_0$ being the least ordinal $PA$ cannot prove recursive), and a correspondingly more refined conception of the strength of that theory.

-


text:
Proof. Let $A$ be a Noetherian complete local ring. By Lemma 15.49.2 it suffices to check that $B = A/\mathfrak q$ has geometrically regular formal fibres over the minimal prime $(0)$ of $B$. Thus we may assume that $A$ is a domain and it suffices to check the condition for the formal fibres over the minimal prime $(0)$ of $A$. Let $K$ be the fraction field of $A$.

We can choose a subring $A_0 \subset A$ which is a regular complete local ring such that $A$ is finite over $A_0$, see Algebra, Lemma 10.154.11. Moreover, we may assume that $A_0$ is a power series ring over a field or a Cohen ring. By Lemma 15.49.3 we see that it suffices to prove the result for $A_0$.

Assume that $A$ is a power series ring over a field or a Cohen ring. Since $A$ is regular the localizations $A_\mathfrak p$ are regular (see Algebra, Definition 10.109.7 and the discussion preceding it). Hence the completions $A_\mathfrak p^\wedge$ are regular, see Lemma 15.42.4. Hence the fibre $A_{\mathfrak p}^\wedge \otimes _ A K$ is, as a localization of $A_\mathfrak p^\wedge$, also regular. Thus we are done if the characteristic of $K$ is $0$. The positive characteristic case is the case $A = k[[x_1, \ldots , x_ d]]$ which is a special case of Lemma 15.49.5. $\square$

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).


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# How do you process historical sources with a Zettelkasten?

My working method is similar to what @pgrhowarth has described here: In short, I always have to refer back to the original sources, in addition to my own ideas and conclusions. And this is not a personal choice, but a requirement of my profession. But the "output" of a Zettelkasten, as far as I understand it, is a chain of associations of my own ideas, not some kind of source Wiki.

But I need both and more: on the one hand, I need to relate my sources to each other, i.e. "who quoted whom", "who learned from whom", "who wrote letters with whom", etc and all based on the originals. On the other hand, I want to process my ideas and thoughts on quotations, excerpts, themes, etc. in a Zettelkasten. But beyond that, I need to make sure that my Zettel always refer to the sources from which they are inspired.

The question I have is: what is the best way to do this? Do I need two "boxes", one for the verbatim quotes with sources and a proper Zettelkasten for my ideas? If so, how do I connect the two boxes? Or should I do "double bookkeeping" and always write the original quote and source on each Zettel?

To reiterate: I know how a wiki works, and I think I also know how to set up a Zettelkasten; what is not yet clear to me is how to connect these two systems.

• I think as a historian you just genererate an additional layer. Most other profession to not investigate the source itself (e.g. a chemist doesn't need to know about the connection of theology and alchemie and how the religious infrastrukture was a hub for alchemy; a mathematician doesn't need to know about beef between collegues as a motivator for their achievements). You also try to develop understanding about the exact events that happened.

That doesn't change anything about how you use the Zettelkasten Method. Just the content of the notes. You still use arguments, hypothesis, models etc. (https://zettelkasten.de/posts/reading-is-searching/)

I think you benefit a lot from explicit and conscious layering: https://zettelkasten.de/posts/layers-of-evidence/ (which is, by the way, heavily inspired by the rigor historians put to sources)

Up to this point I have two sets of items: The first one is knowledge items as layed out in the above article on reading. The second set is for stories: https://zettelkasten.de/posts/zettelkasten-fiction-writing-part-2-elements-of-story/ Perhaps, you might have to develop your own set of items specific to your need as a historian?

I am a Zettler

• Thank you for your answer, but to be honest, I am more of a hands-on person and understand such things better through concrete examples. But I'll try to use your terminology.

So you are saying that I need two different systems; some sort of Wiki in which I connect the "phenomena", i.e. the sources themselves, and a Zettelkasten for the "interpretation". Is that right?

If that is the case, then my question still remains: how to connect these two systems.

• @nurso said:
… But the "output" of a Zettelkasten, as far as I understand it, is a chain of associations of my own ideas, not some kind of source Wiki.

It is not a question of either/or but both. A ZK contains a spectrum of ideas and links. A spectrum of idea associations that call for links. From purely independent ideas to developing others' ideas to ideas that others have had about ideas. Some notes are cataclysmic and struggle to contain the burst of your ideas, and some spill out the history of Medieval knitting. All these may be important in the development of new and novel ideas. You impoverish your ZK by neglecting any part of the spectrum.

But I need both and more: on the one hand, I need to relate my sources to each other, i.e. "who quoted whom", "who learned from whom", "who wrote letters with whom", etc and all based on the originals. On the other hand, I want to process my ideas and thoughts on quotations, excerpts, themes, etc. in a Zettelkasten. But beyond that, I need to make sure that my Zettel always refer to the sources from which they are inspired.

Get in the habit of using a reference manager. This is a no-brainer. In the modern world, reference managers integrate with markdown and pandoc to make managing and tracking sources almost trivial. It is a habit that must be learned and ingrained into one's workflow.

The question I have is: what is the best way to do this? Do I need two "boxes", one for the verbatim quotes with sources and a proper Zettelkasten for my ideas? If so, how do I connect the two boxes? Or should I do "double bookkeeping" and always write the original quote and source on each Zettel?

No! You want to reduce redundancy and friction in your workflow as much as possible (but not too much).

To reiterate: I know how a wiki works, and I think I also know how to set up a Zettelkasten; what is not yet clear to me is how to connect these two systems.

At first, I was blind to this puzzle, but now I see how I evolved. The key is a personal map of connection types. Let me try and explain. Let's start with the easiest for me to explain.

For my links that are more WikiLink like, non-original idea to non-original idea, I use the format ›[[202202090838]]‹. These are inter-note links. These links are like WikiLinks, as you describe them, between notes without an intermediatary note.

Links to ideas that are a mix of my ideas and someone else ideas are formated [[202202090838]] like other links, but the note that I land on shows the mix of their ideas, in block quotes with a mix of quote levels, and some of my novel idea that is formatted in the note non-quoted. Here is a short example.

Opportunistic compression ›[[201901261934]]

01-26-2019 - 7:34 PM

> Progressive Summarization focuses, therefore on rebalancing the equation. It is a method for opportunistic compression — summarizing and condensing a piece of information in small spurts, spread across time, in the course of other work, and only doing as much or as little as the information deserves.

> Compression is a means to improve discoverability.

You summarize notes, not knowing what question might be asked in the future.

#thinking-skills
#zettelkasting

Progressive Summarization ......................................[[201901261654]]

———
References:
Progressive Summarization: A Practical Technique for Designing Discoverable Notes
evernote:///view/597091/s5/f0912e35-88e8-452b-a60d-aa0ef52360d8/f0912e35-88e8-452b-a60d-aa0ef52360d8/

Notes progress from mostly quoted ideas to less and less of other ideas to more and more of my own ideas. Who knows, maybe I'll rehash this note converting the ideas to my own.

Here is an example of a zettel that is all mine and has a Wiki-type link to an example that is someone else's idea. Notice that there is no block quoting in the note.

---
UUID:      ›[[202111071350]]
cdate:     11-07-2021 01:50 PM
tags:      #writing #thinking #dzogchen
---
# How Can Thoughts Have So Much Power?

**Use an existential question to start a riff in creative writing.**

What is a thought, and how can they have so much power? How can that thought "X" have so much influence on my life? In writing, rift on thought than say "How can this thought have so much influence on me?" Continue on!

- Your Environment Holds You In An Embrace [[202102210733]]
- Truth confirmation [[201905311111]]
- Shaping my writing environment [[202202060916]]

Super example by Carlo Rovelli ›[[202111060708]]‹

Will Simpson
“Read Poetry, Listen to Good Music, and Get Exercise”
kestrelcreek.com

• @nurso said:
Thank you for your answer, but to be honest, I am more of a hands-on person and understand such things better through concrete examples. But I'll try to use your terminology.

So you are saying that I need two different systems; some sort of Wiki in which I connect the "phenomena", i.e. the sources themselves, and a Zettelkasten for the "interpretation". Is that right?

If that is the case, then my question still remains: how to connect these two systems.

Everything is in your Zettelkasten. Aside from the sources themselves. And even then they could be included as notes (as I do with the bible).

I am a Zettler

• @Will said:
No! You want to reduce redundancy and friction in your workflow as much as possible (but not too much).

@Sascha said:
Everything is in your Zettelkasten. Aside from the sources themselves. And even then they could be included as notes (as I do with the bible).

Thanks to both of you.

So if I understood you correctly, you are saying that short quotations can certainly be verzettelt, but always with reference back to the source, which exists untouched outside the Zettelkasten. Like so:

┌───────┐    ┌──────────┐   ┌──────┐
│Sources├────► Zettel w.├───►Zettel│
└────┬──┘    │ quotes   │   └──────┘
│       └──────────┘
│
│        ┌──────┐
└────────►Zettel│
└──────┘

• Ignore the layer of Zettels.

It is Source -> Phenomenon (what you are describing) -> Interpretation -> Synthesis.

It is not important if everything is on one note or you divide it up. Just use the rule of thumb: Everything that you want to refer to specifically needs to have its own ID (and then needs to be captures by on an atomic note).

I am a Zettler

• @nurso

I know how a wiki works, and I think I also know how to set up a Zettelkasten; what is not yet clear to me is how to connect these two systems.

I have done quite a lot of research and writing in history (and published some of it). Personally, I just have a database that is something like a Zettelkasten in some ways, and a bit like a wiki in others. I don't see any need for methodological purity if the method gets in the way or is not perfectly suited to the task at hand. Indeed, I would say that some of the worth of a good method is that it can be adapted to suit differing needs. As has been pointed out before, Luhmann was a social scientist, and I can well imagine that he might have adopted a somewhat different method if he had been a chemist or a historian. In my own work I find that it is best to plunder methods for whatever is useful in them, rather than to adopt them wholesale, as they are rarely a perfect fit. But then I've always been a bit of a maverick.


text:
# What is the exact definition of a wave?

1. Aug 27, 2008

### granpa

I know it when I see it but what is the exact definition of a wave? what makes a wave a wave?

2. Aug 27, 2008

### kateman

3. Aug 27, 2008

### granpa

you call that exact? thats little better than 'I know it when I see it'.

4. Aug 27, 2008

### jaseh86

The textbook definition is something like "A propagating disturbance that carries energy".

What makes a wave really a wave would be its wave properties.
All waves experience the following under certain conditions:

Reflection, refraction, polarisation, interference and superposition and diffraction.

5. Aug 28, 2008

### Staff: Mentor

To me, a wave is a motion that satisfies the differential wave equation.

6. Aug 28, 2008

### pardesi

here is what i think of waves as

7. Aug 28, 2008

### atyy

Hmm, this won't work for nonlinear waves or the Schroedinger equation.

This doesn't work for standing waves.

Interference is my personal favourite - Young's double slit is so signature! But is it defined when superposition doesn't hold?

So maybe these definitions are still the best?

8. Aug 28, 2008

### jaseh86

A standing wave is still considered a disturbance in space-time right?

How about wave = "a disturbance....in the force"

9. Aug 28, 2008

### Phrak

My personal favorite is a cold snap, the inverse of a heat wave.

Odd that it's so hard to pin down. A string can support reflection if the mass density of the string changes, but as a one dimensional wave, diffraction and refraction are out.

10. Aug 29, 2008

### vanesch

Staff Emeritus
I think the word "wave" is heavily overloaded with meanings, definitions and all that, and it depends on the context what is exactly meant by it. However, within many contexts, an exact definition can be given, and there are some similarities to them over all these classes, which comes down to "stuff that wobbles in space and time".

11. Aug 29, 2008

### granpa

12. Aug 30, 2008

### atyy

13. Aug 30, 2008

### xCross

Wave is a simple harmonic motion which carries energy and information through an object but do not move object's particles..

14. Aug 30, 2008

### Antenna Guy

Within an antenna context, a wave describes the instantaneous energy passing through a closed surface containing the radiator. Usually, the closed surface is defined as a sphere for simplicity (since all points on the surface are equidistant from a single reference point - the phase center).

For extended objects (where the smallest sphere that completely contains the radiator is many wavelengths in diameter), the instantaneous energy distribution on that closed sperical surface can be anything but simple (although it is merely a superposition many simple waves).

Regards,

Bill

15. Aug 30, 2008

### Chaulesh

A wave is a disturbance that carries energy (not matter) from one point to another through a series of continuous movements.

There exists two types of waves : transverse and longitudinal waves.

The difference between the two is the way they carry the energy from one point to another.

A transverse wave transfers energy through series of upwards and downwards oscillations.

A longitudinal wave transfers energy through a series of compressions and rarefactions (expansions).

Some waves are a combination of both like water waves.

A good site to visit for clearer explanations on waves is:

http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

16. Aug 31, 2008

### Antenna Guy

Spherical waves have both tangential and radial components, but the radial component tends toward zero as the radius increases beyond $\frac{2 D^2}{\lambda}$.

Regards,

Bill

17. Sep 5, 2008

### richa0239

A wave is a disturbance that propagates through space and time, usually with transference of energy. Waves travel and transfer energy from one point to another, often with little or no permanent displacement of the particles of the medium instead there are oscillations around almost fixed locations.
Wave is
-A movement like that of a sudden occurrence or increase in a specified phenomenon.
-One of a series of ridges that moves across the surface of a liquid .

Examples of waves are:

Ocean surface waves, which are perturbations that propagate through water.
Sound — a mechanical wave that propagates through air, liquid or solids.
--------------
Richa

18. Sep 14, 2008

### skydancer46

A wave is not quite a thing (mass), but more than a thought (C squared)

Last edited: Sep 14, 2008
19. Sep 14, 2008

### skydancer46

A wave is not quite a thing (mass), but more than a thought (C squared)


text:
# Does Heisenberg apply to a collapsed wavefunction?

1. Feb 14, 2014

### T'Pau

Hi,

I'd like to argue Heisenberg doesn't apply to a collapsed wave...
I always interpretted the Heisenberg Uncertainty Principle (HUP) as follows:

1. HUP is *not* about measurement problems, it is fundamental
2. When (f.i.) an electron is a wave: HUP applies. The electron really *has* uncertain impulse/position
3. When the wave collapses, f.i. in a collission: the electron behaves like a particle and *does* have impulse/position.
4. *Immediately* after the collapse: the electron is a wave again, with uncertain impulse/position.

Point number 3, I learned, is controversial. I'd like to argue for it using an experiment.

I learned: HUP applies to photons as well (is that correct?).
Say, you perform a single-slit-experiment with a laser, say λ = 500 nm. (λ is well defined)
I use a very very narrow slit, λ << slit-width (no interference pattern behind the slit, only diffraction / bending of the laser light).

When you perform this experiment, on the screen you notice the same lightcolor as the laser emmited: is agreeable that behind the slit λ is still ≈ 500 nm? → Δλ < 1 nm?

I calculate the impulse-difference between λ = 500 nm and λ = 501 nm, → Δp ≈ 2.6 * 10^-30.

Δx Δp ≥ h/4π → Δx ≥ 20 μm.

But my slit-width is much smaller than that: it is < 0.5 μm....

So when you perform this experiment, for instance one photon at a time, everytime you see the screen light up, you know that at a very short instance just before that the photon:
a. *had* Δp ≈ 2.6 * 10^-30
b. *had* Δx < 0.5 μm.

This is better than what HUP allows for...

Notice: I wasn't able to predict where the screen would light up. My measurement doesn't allow me to make any predictions. But what I do know, is that for a very short moment in the past HUP has been broken...

I'm concerned with 'reality'. When something is a wave, it really *has* uncertain impulse/position.
But when this wave collapses, is that still true?

To put it otherwise: how can two electrons collide, when either position or impulse (or both) *is* uncertain?

I hope someone can help me out.

Paul

2. Feb 14, 2014

### bhobba

QM does not ascribe any property until its measured to have that property.

It is never a wave - its a quantum particle: Simply ask yourself - waves of what?
'In our ordinary world, “wave” and “particle” behavior are two different and opposite characteristics. It is difficult for us to think that they can be one of the same. Is light a particle or a wave? The simple, naïve answer to that is “both” or “neither”.

Light, or photon, was never defined as a “particle” the way we normally define a particle. Light is not defined to have a definite boundary in space like a ping-pong ball, or a grain of sand. Instead, light is defined as having quanta of energy. So the discreteness is not defined as discrete object in space, but rather in the energy it can carry. Already, this is not your regular “particle”, and should not be confused as such.

Secondly, in quantum mechanics, the description and properties of light has only ONE, single, consistent formulation, not two. This formulation (be it via the ordinary Schrodinger equation, or the more complex Quantum Electrodynamics or QED), describes ALL characteristics of light – both the wave-like behavior and the particle-like behavior. Unlike classical physics, quantum mechanics does not need to switch gears to describe the wave-like and particle-like observations. This is all accomplished by one consistent theory.

So there is no duality – at least not within quantum mechanics. We still use the “duality” description of light when we try to describe light to laymen because wave and particle are behavior most people are familiar with. However, it doesn’t mean that in physics, or in the working of physicists, such a duality has any significance.'

My gut tells me it might be wise for you to 'unlearn' the usual treatments found in beginning texts and popularizations and start with its actual conceptual core:
http://www.scottaaronson.com/democritus/lec9.html

If you are interested in the foundations of QM and seeing it developed in a proper axiomatic fashion while not skirting the interpretational issues THE book to get is Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/Quantum-Mechanics-A-Modern-Development/dp/9810241054

Thanks
Bill

Last edited by a moderator: May 6, 2017
3. Feb 14, 2014

### WannabeNewton

The HUP is a constraint on the variances of observables. It comes right out of the inability of certain observables to commute. As such it applies to any and all wavefunctions when you're taking the averages of observables for a given wavefunction.

4. Feb 14, 2014

### ZapperZ

Staff Emeritus
I measure Lz. Do you think the values for Lx and Ly are also determined?

Zz.

5. Feb 14, 2014

### Staff: Mentor

You know the exact moment that the screen flashed, but that does not tell you an exact moment that the photon passed through the slit, nor does it tell you how long the photon took to pass through the slit. Thus, you still haven't localized the photon.
(Note also that I can correctly say "X does not tell you Y" even if Y is something completely meaningless - I am not suggesting that "the moment the photon passed through the slit" is a meaningful concept when there was no detector there).

You may want to approach the uncertainty principle not as Heisenberg originally described it, but as the more modern statistical interpretations approach it (Bhobba recommends Ballentine, I second it). The traditional approach invites all sorts of confusion.

You'll also want to understand what's behind ZapperZ's terse but spot-on rhetorical question about the components of angular momentum. That's essential to understanding what is and is not "definite" after a "collapse ofthe wave function".

6. Feb 14, 2014

### atyy

Yes, the Heisenberg uncertainty principle applies to any wave function. For a wave to have a "wavelength", it must have many periods stretched out in space. If the wave is concentrated at a spot, then it doesn't have "a" wavelength, rather it is a superposition of waves of many wavelengths.

7. Feb 14, 2014

### Jilang

Hi Paul, it looks to me like you might confusing uncertainty in the wavelength (which is measured along the direction of motion) with the uncertainty in the position or width of the slit (which is at right angles to it). If you consider the components separately perhaps that will help.

Also I think you meant to say that for diffraction the wavelength is >> than the slit width.

8. Feb 14, 2014

### naima

I agree.
if Δx = 0.5 μm the wave function is a rectangular function rectf(x) which is null outside the slit
The momentum wave function will give you Δp:
the fourier transform of rectf(x) gives you the cardinal sine sinc(p) and you can take Δp as the distance between the first zeros of this function.
You will see that you does get what you wrote.

9. Feb 15, 2014

### T'Pau

Do I get it now?

Thanks everyone for helping me out.

I should have remembered: HUP is not about Δx times Δp_y, it's about Δx times Δp_x. So my experiment did not prove HUP wrong at all.

If I understand it correctly now, one of my miconceptions was that I thougt that a collapsed wave is not a wave. Is this true? → a collapsed wave is also a wave?

I'd like to check if I got it right this time.

Say, there is a totally free electron with zero uncertainty in impulse. The wavefunction of this electron is stretched out from minus infinity to plus infinity (it is as long as the universe).

Now, I trie to measure the position of this electron by shooting photons at it. Which is of course practically impossible: what are my chances of finding this electron if it's wave is that long? But say, after thousands of years, I get lucky: one of my photons strikes this electron.

The moment the photon strikes, the wave of the electron collapses. It becomes a new wave, say a rectf-function with Δx = 1 μm (I don't know the cross section of photons, is 1 μm agreeable?). This new wave does not have definite impuls anymore.

Just before I struck it: there was wave 1. Wave 1 has definite impulse, zero certainty in position.
I strike it: it becomes a new wave, wave 2. Wave 2 has uncertainty in impulse and uncertainty in position. The uncertainty in position is (say) Δx = 1 μm.
Just after I struck it: it becomes wave 3, or does it stay wave 2? Does wave 3 'keep' uncertainty in position of Δx = 1 μm? Or is there another wave 3, which just satisfies HUP?

(I know: I'm writing everywhere: 'it *is* a wave', I should write: 'I describe it as a wavefunction'.)

Thanks for all the help!

Paul

10. Feb 15, 2014

### bhobba

Precisely why do you think waves are involved at all?

A wavefunction is a state expanded in terms of position eigenfunctions. Sometimes it's mathematically like waves in other areas of physics - but it's NOT a wave.

Thanks
Bill

11. Feb 15, 2014

### naima

Before the photon strikes the atom, the uncertainty of the atom's momentum is described py a probability function P1(k). We have not to think that a photon is something with a well known momentum it has also a probability function P2(k).
After the interaction the atom has a new probability function P3(k)
How can we get P3?
let us find the probability P3(k0).
if the photon had a k1 momentum we would get k0 if the atom had k0 - k1 as momentum
this has a P1(k0 - k1) P(k1) probability.
this is also true for other values of the atom's momentum
We must sum all these probabilities to find P3(k0)
$$P3(k0) = \int P2(k0 -k) P1(k) dk$$
the result is the convolution of P1 and P2

12. Feb 15, 2014

### ZapperZ

Staff Emeritus
Actually, your biggest misconception is MORE fundamental than that. It appears that you do not understand at all the concept of commuting and non-commuting observables, and you do not understand the significance of it in terms of what can be known in a single measurement. It is why I think you didn't get the significance of my response earlier, and why I think you might not have been able to answer the question I posed.

In other words, you are focused on the shadow of the animal, without realizing what the actual animal looks like. The HUP and this so-called "collapse" are all consequences of the QM formulation. You need to go back to your QM lessons and trace the SOURCE of all this!

Zz.

13. Feb 15, 2014

### atyy

Yes, it is important to remember the uncertainty primciple applies to "canonically conjugate observables". Ballentine, author of a quantum mechanics text, makes a severe mistake by similarly forgetting this in his 1970 review. (Just in case you are thinking I'm criticising your question - I'm not - asking it on a forum and trying to learn from the discussion is different from publishing the claim in a top journal.)

Yes the collapsed wave function is a wave. It is a wave before collapse and after collapse. Before a measurement and after a measurement, the evolution of the wave function is governed by Schroedinger's equation. When a measurement is performed, the evolution is not governed by Schroedinger's equation, but collapses probabilistically according to the Born Rule.

The tricky part here is 'What is a measurement?", and quantum mechanics does not have a strict answer on this, but relies on our common sense ability to understand that a "measurement" is a "macroscopically irreversible reading".

But that aside, the wave function is a wave, before measurement and after measurement, ie. before and after collapse.

Last edited: Feb 15, 2014
14. Feb 15, 2014

### bhobba

Is a Dirac Delta function a wave?

What I am about to say is not directed at Atyy - I know he knows this stuff.

What its directed at is sloppy use of terminology that leads to confusion

Its never a wave - ever. Its a state. Sometimes when expanded in terms of a certain basis it mathematically looks like wave equations found in other areas of physics - but it's not a wave - its a state.

A quantum system is described by a state at all times. Sometimes, as a result of an observation, it changes state. This is actually the same as a state preparation procedure and is generally how its viewed these days. But often whats being observed is destroyed by the observation so always looking at it that way is not correct. The correct way to look at it is via the formalism of QM - not through the prism of simplifications found in popularization's and beginning level texts.

For that you need to study the proper formalism that is found in texts like Ballentine - QM - A Modern Development:
https://www.amazon.com/Quantum-Mechanics-A-Modern-Development/dp/9810241054

Here you will find QM developed from just 2 axioms with nothing mentioned about waves etc etc. It is regrettable that most beginning level texts speak about the wave particle duality and such - but strictly speaking its WRONG and leads to misunderstandings.

To illustrate it I will state the two axioms so you see nothing is mentioned at all about a wave or waves.

The first axiom is associated with any observation is a hermitian operator, O, whose eigenvalues give the possible outcomes of the observation.

The second axiom (also called Born's Rule) is there exists a positive operator of unit trace P such that the expected outcome of the observation E(O) = Trace (PO). By definition P is called the state of the system.

Immediately we see it has nothing to do with waves. A state, just like probabilities, is simply an aid in calculating the expected outcomes of observation. Saying a state is 'waves' is nonsense in a general sense. First, generally it's an operator, and secondly its not 'real' like a wave - its simply an aid in calculating the outcomes of observations. Sometimes, with so called pure states, the operator can be associated with elements of a vector space, and sometimes it makes sense to expand that element is terms of position eigenfunctions, and sometimes it mathematically looks like a wave - but that's it - that's all - only if those sometimes are satisfied does it even make sense to use the term wave-function - but its never a wave.

To answer the OP's original question the HUP is simply a theorem about non-commuting observable's. Its got nothing to do with waves, collapse or anything like that.

Thanks
Bill

Last edited by a moderator: May 6, 2017
15. Feb 15, 2014

### atyy

Why isn't it a wave?

Also, is the Dirac delta a physical wave function?

16. Feb 15, 2014

### WannabeNewton

"Wavefunction" generally refers to the position-space representation of a given state vector, $\langle x | \psi \rangle$ and the terminology is set in place for good reason. Even though Maxwell's equations in vacuum lead to equations that are formally wave equations i.e. second order in both time and space, Schrodinger's equation, while not formally a wave equation, also contains wave-like solutions i.e. solutions of the form $f(\vec{x} \pm \vec{v}t)$ which will of course just be Fourier decomposed into plane waves. Schrodinger's equation, just like Maxwell's equations, is linear so there's no problem having distributional solutions to the equation such as Dirac delta functions which can themselves be decomposed into Fourier modes. Distributional solutions are a problem for non-linear equations like Einstein's equation. So yes there certainly are solutions to Schrodinger's equation which are wave-like hence the term "wavefunction".

For example the general solution to Schrodinger's equation for a free particle is quite literally just a continuous superposition of Fourier modes which are individually of course sinusoidal waves acting as "stationary states" of the free particle. Whether or not the individual modes qualify as "physical" (hence the scare quotes in "stationary states") depends on whether you would be willing to call the non-normailzable generalized wavefunctions as "physical". Obviously the behavior of the localized solutions to the vacuum Maxwell equations will differ significantly from localized solutions to Schrodinger's equation for a free particle as the two are not in the same class of equations. For example an initial localized free Gaussian wave packet fed into Schrodinger's equation will tend to disperse whereas a vacuum EM wave packet will not disperse.

Last edited: Feb 15, 2014
17. Feb 15, 2014

### atyy

Could I get some commentary on this: the part that is perhaps misleading about thinking about the uncertainty principle as due to the wave-nature of the wave function is that the analogy to sound waves really comes from signal processing, which is the wave at a fixed position, and so may not have to do with a "wave" at all, but just a "signal". However, the part that I don't think is misleading is that in signal processing there is the conceptually analogous time-frequency uncertainty, and things like Cohen's class distributions, which are related to the Wigner function of quantum mechanics (known as the Wigner-Ville distribution in signal processing). Certainly one needs more than the time-frequency uncertainty, one needs that these correspond to the canonically conjugate observables of position and momentum. Also, once one goes to two particles, it becomes clear that the wave function is a wave in Hilbert space, not ordinary space. But basically, I think there is an uncertainty principle in a domain we are all familiar with - ordinary sound impinging on our ears - and it is related.

Last edited: Feb 15, 2014
18. Feb 15, 2014

### WannabeNewton

Exactly. So the antiquated wave-mechanics picture that strongly associates the wave-function with the physical space of a particle and describes a particle quite literally by a spatially extended wave in physical space will conceptually break down if one starts considering systems of more than one particle or non-coherent states in contrast with the coherent states allowed by, for example, the quantum harmonic oscillator. This is why one has to be careful when using physical space wavefunctions to provide intuition for the uncertainty relations associated with observables and quantum states. Furthermore QM allows for uncertainty relations between any pair of non-commuting observables such as the components of the spin operator so the intuitive descriptions making use of the antiquated wave-mechanics picture have limited scope.

19. Feb 16, 2014

### bhobba

Waves generally have peaks and troughs.

Its a wave-function by the definition of wave-function which is the state expanded in eigenstates of position. Thinking in terms of wave-functions is generally not a good idea because you are singling out position eigenvectors as somehow special - they aren't.

Thanks
Bill

Last edited: Feb 16, 2014
20. Feb 16, 2014

### Jilang

Exactly so. If you measure the position is becomes a delta function for the position (not so wavy), but in the momentum basis it is still is expressed a superposition of many eigenvectors with different momenta (very wavy).


text:
Number of unique determinants for an NxN (0,1)-matrix.

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore won't have a determinant. While it might also be interesting to ask what number of NxN (0,1)-matrices are singular or non-singular, I'd like to ignore singular matrices altogether in this question.

To get a better grasp on the problem I wrote a computer program to search for the values given an input N. The output is below:
1x1: 2 possible determinants
2x2: 3 ...
3x3: 5 ...
4x4: 9 ...
5x5: 19 ...

Because the program is simply designed to just a brute force over every possible matrix the computation time grows with respect to $O(2^{N^2})$. Computing 6x6 looks like it is going to take me close to a month and 7x7 is beyond hope without access to a cluster. I don't feel like this limited amount of output is enough to make a solid conjecture.

I have a practical application in mind, but I'd also like to get the bounds to satiate my curiosity.

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Singular matrices certainly do have a determinant; it's zero... –  Qiaochu Yuan Mar 18 '10 at 6:26
I'm sorry, I was a bit careless above; the 10 x 10 number is 269, but the 11 x 11 is unknown. See A013588 in the Online Encyclopedia of Integer Sequences, oeis.org/A013588 –  Gerry Myerson Mar 18 '10 at 6:30
I remember doing the 7x7 spectrum by hand years ago after some computer searching with a single desktop computer of 1990's vintage. It feels good to be compared to a computing cluster. Gerhard "Ask Me About System Design" Paseman, 2010.03.18 –  Gerhard Paseman Mar 18 '10 at 7:27
I also did the smaller cases by hand. For 4x4 I got 7, for 5x5 I got 11 and 6x6 I got 19. You might check your computer program. Gerhard "Ask Me About System Design" Paseman, 2010.03.18 –  Gerhard Paseman Mar 18 '10 at 14:51
The 12 x 12 case has recently been settled by a calculation of Richard Brent, Judy-anne Osborn, Paul Zimmermann, and myself: the conjecture at indiana.edu/~maxdet/spectrum13.html is correct. (The link is to the n=13 case of {-1,1} matrices, which corresponds to the 12 x 12 case of {0,1} matrices.) We hope to write up these and other results soon. The answer for 11 x 11 is still unknown, as our code works only for odd-sized {-1,1} matrices. –  Will Orrick Apr 15 '10 at 3:27

I have given some detail in a comment to another answer. I have a proof that the number of determinants is greater than 4 times the nth Fibonacci number for (n+1)x(n+1) (0,1) matrices, and I conjecture that for large n the number of distinct determinants approaches a constant times n^(n/2). Math Overflow has some hints of the proof in answers I made on other questions.

I am interested in your idea for an application, and am willing to share more information on this subject.

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Of course, the proof is for n greater than 5. The actual proof at grpmath.prado.com/Lemmas.html shows that there are (0,1)matrices of size n with determinant values 0 through 2*Fib(n-1) - 1, for all positive n. For n > 6, I use in addition a matrix with maximal determinant to get 4* fib(n-1) + 1 distinct values, both positive and negative and including 0. (Yes, I know about the faulty timestamp.) Gerhard "Ask Me About System Design" Paseman, 2010.03.18 –  Gerhard Paseman Mar 18 '10 at 15:06
The application isn't fully fleshed out. I was looking into the growth of the number of determinants to see asymptotic behaviour. I am looking to see if comparing determinants of adjacency matrices of simple graphs (after being processed by various transforms) can be used to solve the isomorphism problem probabilistically. I just started looking into the idea - no great progress or insight to be had yet. –  Ross Snider Mar 18 '10 at 18:40
Another possibility is to use a characteristic polynomial, det( A- Ix ) for A the adjacency matrix and x an indeterminate. Unfortunately there are a couple of nonisomorphic trees on something like 10 vertices which have the same polynomial. However, the determinant in combination with something else might be useful in making the equivalence classes almost as small as the isomorphism classes. –  Gerhard Paseman Mar 18 '10 at 22:24
I am accepting this answer because it provides an interesting lower bound, even if you haven't proved it rigorously here, and because it seems like the flurry of activity on this question has died out. –  Ross Snider Mar 19 '10 at 15:52
«Math Overflow has some hints of the proof in answers I made on other questions» is as helpful as the proverbial «It is in Euler's works somewhere»! –  Mariano Suárez-Alvarez Apr 16 '10 at 4:07

There are some answers in W P Orrick, The maximal {-1,1}-determinant of order 15, http://arXiv.org/pdf/math/0401179v1. Despite the title, toward the end of the paper the author does look at {0,1} matrices, and at the entire range, not just the maximal. The author also refers to an older paper on the topic, R Craigen, The range of the determinant function on the set of $n\times n$ (0,1) matrices, J Combin Math Combin Comput 8 (1990) 161-171.

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This subject is a current research interest of mine. Orrick maintains a web site which contains conjectured determinant spectra, Miodrag Zivkovic has done exhaustive reseearch involving Smith Normal forms up to 9x9, and I am searching for another class of matrices to push the bound above 4 times the nth Fibonacci number. A search for "Paseman determinant" will give you some of the internet links. Post more questions here and I will share what I know and believe. Gerhard "Ask Me About System Design" Paseman, 2010.03.19 –  Gerhard Paseman Mar 18 '10 at 7:04
I am also interested in applications. I may be induced to help you with your application. Gerhard "Ask Me About System Design" Paseman, 2010.03.19 –  Gerhard Paseman Mar 18 '10 at 7:08
Interestingly, the problem was studied by Metropolis, Stein, and Wells (see N. Metropolis. Spectra of determinant values in (0, 1) matrices. In A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory: Proceedings of the Science Research Atlas Symposium No. 2 held at Oxford, from 18-23 August, 1969, pages 271–276, London, 1971. Academic Press.) who had computed the answers up to n=7 back in 1969 (and claimed to be able to handle n=8). I wasn't aware of this work when I wrote the aforementioned paper. –  Will Orrick Apr 15 '10 at 3:37

From Hadamard's bound the largest possible determinant of an $n\times n$ (0,1) matrix is $h_n=2^{-n}(n+1)^{(n+1)/2}$. The data at http://www.indiana.edu/~maxdet/spectrum.html suggest several conjectures:

1. The spectrum is "dense" up to a certain point, after which it becomes "sparse". An integer in the "dense" part is almost certain to be the determinant of some $n\times n$ (0,1) matrix; and integer in the "sparse" part is almost certain not to be. The point at which the spectrum becomes sparse is asymptotically some constant times $h_n$. The data suggest that the constant is near 0.5. I think this is basically the conjecture made by Gerhard Paseman in his answer.
2. A stronger statement is as follows: let $g_n$ be the position of the first "gap", that is, the first positive integer that is not the determinant of some $n\times n$ (0,1) matrix. Then asymptotically $g_n$ is some constant times $h_n$, and again the constant appears to be close to 0.5. If $D_n$ denotes the set of positive $n\times n$ determinants, and if the above ideas are on the right track, then it seems likely that asymptotically $g_n/|D_n|$ is 1.

I don't have even a heuristic explanation as to why such conjectures ought to be true, and one can always worry about how much one should try to conclude from data that, in fact, go only as high as $n=21$. As mentioned in some of the comments to other answers, $D_n$ is really only known for $n\le 10$ and $n=12$. The sets $D_n$ for these $n$ are given at the above link, as are conjectures for all $n$ up to 16. Data for $17\le n\le21$ have not yet been added to the site. (Note that the $n$ on the web site refers to $(-1,1)$ matrices, so one should subtract 1 from it if one is talking about $(0,1)$ matrices.) I do have high confidence that the listed sets $D_n$ up to $n=16$ are not missing any values, even the ones that have not been proved complete.

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text:
# Variational problem - continuity exercise

Let $\Omega$ a bounded connected open regular set, and let $f \in L^2(\Omega)$. We consider the variational problem: find $u \in H^1(\Omega)$ such: $$\displaystyle\int_{\Omega} A \nabla u \cdot \nabla v dx + \left(\displaystyle\int_{\Omega} u dx\right)\left(\displaystyle\int_{\Omega} v dx\right) = \displaystyle\int_{\Omega} f v dx, \forall v \in H^1(\Omega)$$ with: $$\exists \alpha > 0, A(x) \xi \xi \geq \alpha |\xi|^2, \forall \xi \in \mathbb{R}^n, \exists \beta > 0, |A(x) \xi| \leq \beta |\xi|, \forall \xi \in \mathbb{R}^n$$

The question is: prove that this variational problem admits a unique solution $u \in H^1(\Omega)$.

My solution is: we put $$a(u,v) = \displaystyle\int_{\Omega} A \nabla u \cdot \nabla v dx + \left(\displaystyle\int_{\Omega} u dx\right)\left(\int_{\Omega} v dx\right)$$ $$L(v) = \displaystyle\int_{\Omega} f v dx.$$

To prove continuity of $a$, we have to prove that $$\exists M > 0, |a(u,v)| \leq ||u||_{H^1(\Omega)} ||v||_{H^1(\Omega}$$ Let $u \in H^1(\Omega)$, and let $v \in H^1(\Omega)$. $|a(u,v)| = |\displaystyle\int_{\Omega} A \nabla u \cdot \nabla v dx + (\displaystyle\int_{\Omega} u dx)(\displaystyle\int_{\Omega} v dx)|$

$|a(u,v)| \leq \displaystyle\int_{\Omega} |A \nabla u \cdot \nabla v| dx + (\displaystyle\int_{\Omega} |u|dx) (\displaystyle\int_{\Omega} |v|dx)$

we have by Cauchy-Schwarz inequality that: $\displaystyle\int_{\Omega}|A \nabla u \cdot \nabla v| dx \leq (\displaystyle\int_{\Omega} |A \nabla u|^2 dx)^{1/2} (\displaystyle\int_{\Omega} |\nabla v|^2 dx)^{1/2}$ $\leq \beta (\displaystyle\int_{\Omega} |\nabla u|^2 dx)^{1/2} (\displaystyle\int_{\Omega} |\nabla v|^2 dx)^{1/2} \leq \beta ||\nabla u||_{L^2} ||\nabla v||_{L^2}$

so $$\displaystyle\int_{\Omega} |A \nabla u \cdot \nabla v| dx \leq \beta ||u||_{H^1} ||v||_{H^1}$$

and we have by Cauchy-Schwarz inequality: $(\displaystyle\int_{\Omega} |u|dx)(\displaystyle\int_{\Omega} |v|dx) \leq ||u||_{L^2} ||v||_{L^2} \leq ||u||_{H^1} ||v||_{H^1}$ then, we conclude that $|a(u,v)| \leq \beta ||u||_{H^1} ||v||_{H^1} + ||u||_{H^1} ||v||_{H^1} = (\beta + 1) ||u||_{H^1} ||v||_{H^1}$ then, there exists $M = \beta + 1$ such $|a(u,v)| \leq M ||u||_{H^1} ||v||_{H^1}$.

How can I write a perfect proof of continuity of $a$?

Who can help me please, in my last question.

• Why registering a new account? math.stackexchange.com/questions/434702/… – Shuhao Cao Jul 3 '13 at 19:33
• And your proof of continuity is ok, but you might wanna add a factor of $|\Omega|$ in front of the estimate for $(\int_{\Omega} u)(\int_{\Omega} v)$, and absolute value is not necessary. – Shuhao Cao Jul 3 '13 at 19:35
• beacause i can't open my old acount – jijii Jul 3 '13 at 21:47
• how i can add a factor $|\Omega|$? and thoma tell me in a precedent topic that my proof is not complete beacause there are some constants missing and some details you have to improve. What's an perfect and complete proof please. – jijii Jul 3 '13 at 21:49
• You can always use "forgot password" option. Also when you use the Cauchy-Schwarz: $$\int_{\Omega} v\,dx \leq \left(\int_{\Omega} v^2\,dx\right)^{1/2} \left(\int_{\Omega} 1\,dx\right)^{1/2},$$ and the second term is the square root of the measure (area, volume,etc) of $\Omega$, together with the integral term for $u$ you have that factor. – Shuhao Cao Jul 3 '13 at 22:19


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# Circle a example of tangent

23.08.2019 |

Tangent Circle Formula With Solved Example Byjus Formulas

Equation of a Tangent to a Circle (solutions examples. Therefore to find this angle (angle k in the examples below), in other words, is point d tangent to the circle? show answer. since  \frac{1}{2}, this book contains interactive activities demonstrating circle properties and angle of the circle (tangent example of proofs involving circle properties..

Circles Tangents Worksheets Free Math Worksheets. The following example involves a common external tangent (where the tangent lies on the same side of both circles). from the center of the smaller circle,, how to use tangent in a sentence. example sentences with the word tangent. tangent example sentences..

Definition and examples of secant define secant

Tangent and Friends Examples Shmoop. 24/05/2011в в· this video shows how to solve for unknown values using the properties of tangent segments to a circle from a given point. complete video list: http://www, definition of tangent of a circle with its representation. example to learn how a straight line becomes tangent to the circle by passing through a point.); i'm trying to create an algorithm to find the tangent on a circle so that i can calculate the angle of reflection for that circle when it collides with an object. i, video: point of tangency: definition & example. to visualize what a line tangent to a circle looks like, imagine that you balance a ruler on a ball,.

Equation of a tangent to a circle вЂ“ Variation Theory

Tangent to a Circle Formulas Properties Theorems. A step by step example is included. use excel's tan function to find the tangent of an angle radians are related to the radius of the circle with one radian, what is a tangent to a circle? the word tangent comes from the latin word meaning "touching." this notion of touching is used in the geometrical meaning of tangent..

Tangents and normal to a curve. a tangent meets or touches a circle only at example 2: find the equation of the tangent and the normal to the curve y = x, video: point of tangency: definition & example. to visualize what a line tangent to a circle looks like, imagine that you balance a ruler on a ball,).

Definition and examples of tangent define tangent

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Top

Electrolyte Solution

In order to understand the complicated ideas about the study of electrolyte solutions we need to explore the kinetic entities in solution, the introduction of a dissolved electrolyte opens up the avenue of many solute entities like ions, solvated or un-solvated electrostatically associated groups of ions and covalently bound molecules and complex ions.

The concept of weak and strong electrolyte is more relevant if the associated and non-associated electrolytes are discussed as well.

 Related Calculators Calculate the Molality of the Solution Equation Solution Calculator Ionic Strength of a Solution Mole Fraction of a Solution

What is an Electrolyte?

Electrolyte solutions are solutions that can conduct electricity and the colligative properties like lowering of vapour pressure, depression of freezing point, elevation of the boiling point and osmotic pressure all depend on the number of individual particles present in solution. For some of the solutes it is found that the number of particles actually present in solution is higher than what is expected from the formula given in compound.

The self-ionisation of water which result in $H_{3}O^{+}$ (aq) and $OH^{-}$ (aq) always observed in aqueous electrolyte solution. The measured resistance or conductance is a composite quantity made up from the contribution of the ions of electrolyte and the $H_{3}O^{+}$ (aq) and $OH^{-}$ (aq). These can be obtained by measuring the resistance or conductance of $H_{2}O$ (l) used while making the electrolyte solutions.

The conductivity is subtracted from the conductivities found for each electrolyte solution.
• In the study of electrolyte solutions two types of solute can be distinguished
• The number of particles present is an integral number of times the number of particles expected on the basis of stoichiometric unit. This ratio does not change with change in concentration
• The number is greater than that corresponding to the stoichiometric unit which are much less than the values found in category ( ratio of actual number of particles to the stoichiometric number of stoichiometric units increase dramatically with decrease in concentration.

How to Make Electrolyte Solution?

Based on the basic characteristics of the solutes and solvent some basic factors are used to create the ground rules which helps in make electrolyte solutions.

The electrolytic solution can be made based on the following basic characteristics which are as follows:
• Molecular units: non conducting normal colligative properties, x ray structures showing discrete molecular units in the solid
• Molecular units plus ions: weakly conducting, colligative properties showing slightly more than the expected numbers of particles present, x ray structures showing discrete molecular units in the solid
• Ions: highly conducting, colligative properties, considerably greater than expected, x ray structure showing a giant ionic lattice.
Solutes in solution:
Molecular units are called non-electrolyte
Molecular units plus ions are called weak electrolytes
Ions only are called strong electrolytes \ygyk

Electrolyte Solution for Batteries

Batteries like the conventional lithium ion based the Lithium ions are formed at one electrode ion the considered battery and injected into the electrolyte at electrode / electrolyte interface. These Li+ ions are removed from the electrolyte at the other electrode and allowed to enter the electrode through the electrolyte / electrode interface.

Both the ions are transported inside the electrolyte under the influence of an electrical field through migration. The influence of convection is neglected while the ionic conductivity of the non-aqueous electrolyte used in Li ion batteries is relatively low as opposed to the electrolyte used in other forms like Nickel cadmium batteries. The total current that flows through electrolyte is the sum of the diffusion and migration contribution of both ion types as defined for chemical domain.

The electrical field inside the electrolyte cannot be neglected in Lithium ion model as it could be done in Nickel cadmium model. The electrolyte solutions are essential for all kinds of batteries as they help in transferring the respective ions from one electrode to another and also help in the working of these units.

In case we are using zinc electrodes in copper zinc batteries, the zinc ions move into the solution and at the same time anions from the other copper cell into the zinc unit to compensate the charge and finally neutralize the effects of these ions.

$Zn(s) \rightarrow Zn^{2+}(aq) + 2 e^{-}$

Solution of electrolyte is always required in batteries, even in dry cells. The simplest battery consists of two electrodes. The figure here illustrates a copper-zinc battery. The left hand is a zinc electrode. The zinc atoms have a tendency to become ions leaving the electrons behind.
$Cu^{2+}(aq) + 2 e^{-} \rightarrow Cu(s)$

In case of dry cells, these electrolytic solutions are replaced by fused paste in order to prevent the leakage from the dry cell units.

Electrolyte Solution Examples

For any electrolytic solution the two main categories are strong and weak electrolytes. For a strong electrolyte the resultants are strong acids, strong bases and corresponding salts. For any weak electrolyte the resultants are weak acids and weak bases.

Amongst strong electrolytes, the strong acids could be hydrogen chloride (HCl), Hydrogen bromide (HBr), Hydrogen sulphate $(H_{2}SO_{4})$, Nitric acid $(HNO_{3})$, $HClO_{3}$ etc.

The strong bases are sodium hydroxide (NaOH), potassium hydroxide (KOH), lithium hydroxide (LiOH), barium hydroxide Ba $(OH)_{2}$ etc.

Amongst the weak electrolytes, the weak acids are hydrogen fluoride (HF), acetic acid $(CH_{3}COOH)$, carbonic acid $(H_{2}CO_{3})$, phosphoric acid $(H_{3}PO_{4})$ etc. The weak bases are ammonia solution, pyridine etc.

For any weak electrolyte the partially dissociation into ions at moderate concentration increase the fraction of ions as the concentration decrease. For any strong electrolyte the ions present show no significant amount of molecular species present and the molar conductivity is independent of concentration.

Electrolyte Solution Experiment

Prior to the electrolyte experiment the mass M and inner volume V of the empty column are determined. The sample amount $m_{(AR)}$ of counter ion matric AR form the ion exchanger (A is the counter ion and R is the matrix). The ion exchanger is covered with distilled water and water is periodically run through to total swelling of the ion exchanger while the operation was carried out at a constant volume of the system or rather the ion exchanger solution.
For each of these experiments the effective volume is always equal to $\frac{Q}{c}$ and is calculated for the electrolyte in case its concentration is equal to concentration ‘c’ of the external solution, while the volume is basically the similitude of the retention volume in the liquid.


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# Finite models for torsion-free lattices

Let $$G$$ be a real, connected, semisimple Lie group and $$\Gamma < G$$ a torsion-free lattice. Then does there exist a finite $$CW$$-model for $$B\Gamma$$?

I know this to be true in many instances (e.g. when $$\Gamma$$ is uniform, when $$G=\mathrm{SO}_0(n,1)$$ or when $$\mathrm{rank}_{\mathbb R}(G) \geq 2$$ and $$\Gamma$$ is irreducible). In these instances, there always exist certain "canonical" finite CW-models for $$B\Gamma$$.

However, I am unaware of the situation for general such lattices, hence the question.

• I guess many (all?) cases were done in the 70's by Borel-Serre (at least when $G$ has finite center): there's a model that's a compact manifold (with boundary). I'd thought it was known in general, but I'm not sure. – YCor Feb 11 at 12:10
• Let $\Gamma$ be such a lattice, and assume $G$ has trivial center (if $G$ has finite center this is no restriction). Then there is a unique product decomposition $G=G_1\times \dots G_n$ such that $\Gamma$ meets $G_i$ in an irreducible lattice. Passing to the overgroup of finite index generated by projections, we can reduce to the irreducible case. Then rank $\ge 2$ is fine (probably by arithmeticity + Borel-Serre), remains rank 1. This case is certainly OK too (removing a disjoint family of horoballs and modding out directly yields a manifold with boundary). – YCor Feb 11 at 13:09
• There is a Ph.D. thesis, I don't remember now exactly by whom, whose main result is the existence of compactifications in the rank $1$ case (basically generalizing the situation for hyperbolic lattices in the way that you described). The Lie groups I am interested in are all the real points of an algebraic group, so they all have finite center anyways. – H1ghfiv3 Feb 11 at 13:16
• I think your argument shows only that each $\Gamma$ virtually has a finite CW-model, i.e that there exists a finite-index subgroup $\Gamma' < \Gamma$ possessing a finite CW-model. – H1ghfiv3 Feb 11 at 13:45
• "I think your argument shows only that each $\Gamma$ virtually..." No, I was careful on that point: I project the lattices on factors, so I embed $\Gamma$ into a larger torsion-free lattice which is a direct product (I said "overgroup"). – YCor Feb 11 at 16:17

I'll try to summarize YCor's comments into an answer (using big guns): Let $$G$$ be the real points of an algebraic group (a restriction by the OP in the comments) and assume $$\Gamma$$ irreducible.
Then Raghunathan shows that the answer is "yes" if $$\Gamma$$ is arithmetic. Margulis (Discrete subgroups of semisimple Lie groups) says that $$\Gamma$$ will be arithmetic if $$G$$ has rank at least $$2$$ (this is the sum over the ranks of the almost-factors). This leaves the case where $$G$$ is a single rank-1 factor. In that case $$\Gamma\backslash G/K$$ is a finite volume hyperbolic manifold from which one can cut off the cusps, see Theorem 12.7.2 in Ratcliffe, "Foundations of hyperbolic manifolds".
• The reduction on the irreducible case that YCor employed might be correct, I am not yet convinced (but that doesn't mean anything, I am not an expert in this area and all of what I've been asking might be very obvious to experts on Lie groups). However, the question on whether a general torsion-free discrete group $\Gamma$ has a finite CW-model for $B\Gamma$ whenever this virtually is the case is highly non-trivial, and as far as I know still open. A statement conforming this would indeed be of some profoundness (at least to me) – H1ghfiv3 Feb 12 at 18:30
• @H1ghfiv3 My attempt was to embed $\Gamma$ as subgroup of finite index in a product, so I didn't try to get finiteness of $B\Gamma$ from a finite index subgroup, but from a finite index overgroup. The problem with my approach is that the overgroup can have torsion even if the small group doesn't. – YCor Feb 14 at 16:45


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# $\lambda$-terms equal modulo $\alpha$-renaming, is this an equivalence relation?

Want to clarify few things.

It is said that two $\lambda$-terms are equal up to renaming of bound variables, such as $\lambda x.x$ equals $\lambda y.y$, so I think it is a relation actually, about how two terms are equal.

Is that (such equality up to renaming) is same as equivalence relation?

To show a relation is an equivalence relation on a set, have to prove it is reflexive, symmetric, or transitive. I know, but have not done any practice on it.

Let's say, I want to prove following

1. $\frac{}{x = x}$ \two variables
2. $\frac{ t_1 = t_3 \,\,\, t_2=t_4 } {t_1 \, t_2 \, = \,t_3 \, t_4}$ \applications
3. $\frac{ t_2=[y / x]t_2 }{\lambda x.t_1 \, = \, \lambda y.t_2}$ \abstractions

where $[y / x]t_2$ replace $x$ by $y$ in $t_2$ and $y$ is not occur in $t_2$.

I think above 3 rules are true.

How can I show proof of they holds? does that proof related to the proof of $\alpha$-equivalence is an equivalence relation on $\lambda$-terms?

I am bit fonfused, it would be great if anyone help me clear things for me.

• It certainly is an equivalence relation: terms that are equal up to an alpha conversion are colloquially referred as "equal". Let S be a set and let r be a subset of S x S. How do you prove that r is an equivalence relation? It seems you already understand that. Now, let S be "the set of lambda terms" and let r(x, y) be "there exists an alpha conversion that turns x into y", and prove each property individually. – quicksort Dec 5 '16 at 18:10
• You don't need to prove those rules. Those rules are the definition of $\alpha$-equality (you can write $=_{\alpha}$ instead of $=$). To prove it is an equivalence relation you have to prove reflexivity, symmetry, and transitivity, as you already stated. – Euge Dec 5 '16 at 18:28


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# American Institute of Mathematical Sciences

February  2007, 1(1): 77-93. doi: 10.3934/ipi.2007.1.77

## Approximation errors in nonstationary inverse problems

 1 Department of Physics, University of Kuopio, P.O. Box 1627, 70211 Kuopio, Finland 2 Department of Physics, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland

Received  September 2006 Published  January 2007

Inverse problems are known to be very intolerant to both data errors and errors in the forward model. With several inverse problems the adequately accurate forward model can turn out to be computationally excessively complex. The Bayesian framework for inverse problems has recently been shown to enable the adoption of highly approximate forward models. This approach is based on the modelling of the associated approximation errors that are incorporated in the construction of the computational model. In this paper we investigate the extension of the approximation error theory to nonstationary inverse problems. We develop the basic framework for linear nonstationary inverse problems that allows one to use both highly reduced states and extended time steps. As an example we study the one dimensional heat equation.
Citation: Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
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2020 Impact Factor: 1.639


text:
Tags
#analyst-notes #cfa-level-1 #corporate-finance #reading-35-capital-budgeting #study-session-10 #subject-3-investment-criteria
Question
An investment with an initial cost of $16,000 produces cash flows of$5000 annually. If the cash flow is evenly spread out over the year and the firm can borrow at 10%, the payback period is ______ years.

A. 3.20
B. 2.91
C. 3.33

PP = 16000/5000 = 3.20

Tags
#analyst-notes #cfa-level-1 #corporate-finance #reading-35-capital-budgeting #study-session-10 #subject-3-investment-criteria
Question
An investment with an initial cost of $16,000 produces cash flows of$5000 annually. If the cash flow is evenly spread out over the year and the firm can borrow at 10%, the payback period is ______ years.

A. 3.20
B. 2.91
C. 3.33
?

Tags
#analyst-notes #cfa-level-1 #corporate-finance #reading-35-capital-budgeting #study-session-10 #subject-3-investment-criteria
Question
An investment with an initial cost of $16,000 produces cash flows of$5000 annually. If the cash flow is evenly spread out over the year and the firm can borrow at 10%, the payback period is ______ years.

A. 3.20
B. 2.91
C. 3.33

PP = 16000/5000 = 3.20
If you want to change selection, open original toplevel document below and click on "Move attachment"

#### Parent (intermediate) annotation

Open it
dy>NPV measures the dollar benefit of the project to shareholders. However, it does not measure the rate of return of the project, and thus cannot provide "safety margin" information. Safety margin refers to how much the project return could fall in percentage terms before the invested capital is at risk.<body><html>

#### Original toplevel document

Subject 3. Investment Decision Criteria
on that capital. If a firm takes on a project with a positive NPV, the position of the stockholders is improved. Decision rules: The higher the NPV, the better. Reject if NPV is less than or equal to 0. <span>NPV measures the dollar benefit of the project to shareholders. However, it does not measure the rate of return of the project, and thus cannot provide "safety margin" information. Safety margin refers to how much the project return could fall in percentage terms before the invested capital is at risk. Assuming the cost of capital for the firm is 10%, calculate each cash flow by dividing the cash flow by (1 + k) t where k is the cost of capital and t is the year number.

#### Summary

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text:
# TeXnicCenter and svg

I'm wondering why I can't use the package svg with teXnicCenter. I've seen that one must use shell escape to make it functions. I've also downloaded the latest version of inkscape and of ImageMagick. So I've written that in my build profile :

--shell-escape -synctex=-1 -max-print-line=120 -interaction=nonstopmode "%wm"

I've put the shell first beacause I've seen that sometimes it would work better.

Here is my minimal code :

\documentclass{article}
\usepackage{svg}
\usepackage{amsmath}

\title{Inkscape package on Overleaf }
\begin{document}
\begin{figure}[htbp]
\centering
\includesvg[inkscapelatex=false,width=\linewidth]{test}
\caption{svg image}
\end{figure}

\end{document}

And I've checked my log file. Here are the parts concerned by escape :

Package svg Info: Calling Inkscape on input line 9. runsystem(inkscape -z -D --file="test.svg" --export-pdf="test_svg-raw.pdf" )...executed.

Package svg Warning: The export with Inkscape failed for file
(svg)                test.svg'
(svg)                Troubleshooting: Please check in the log file how
(svg)                the invocation of Inkscape took place and try to
(svg)                execute it yourself in the terminal on input line 9.

! Package svg Error: File test_svg-raw.pdf' is missing.

See the svg package documentation for explanation.
Type  H <return>  for immediate help.
...

l.9 ...inkscapelatex=false,width=\linewidth]{test}

Did you run the export with Inkscape? There's no file
./svg-inkscape/test_svg-raw.pdf'
although test.svg' was found.

[1

So, I don't really understand what is faulty here. Thanks if you can direct me in the right direction to make it work.

• it is likely that inkscape is not an active command e.g. from a cmd line enter inkscape --version if you get the response 'inkscape' is not recognized as an internal or external command, operable program or batch file. then it did not install itself on your path so you need to do it manually. If you are not sure which folder it is in and on windows the cmd is> where /r \ inkscape If you do get the version shown any where from your system then the --shell-escape did not work for a similar answer see tex.stackexchange.com/a/474119/170109
– user170109
Apr 1, 2019 at 19:09
• If I understand, the problem is that he can't find it. The fix should be to add the path to the folder of inskcape in the "variable environnement". I've added the path to the folder inkscape on my "user variables" and on my "system variables" under the variable "Path". To check if it would function, I've tried on the cmd "inskcape -version". I've alose added on miktex the path, by going to preferences>edit>typesetting. He still can't find it. I'm a bit confused now. Apr 2, 2019 at 5:05

1) enable shell escape, you have done that, otherwise you would be reporting a log error of Package svg Warning: You didn't enable 'shell escape' (or 'write18'), ! Package svg Error: File test_svg-raw.pdf is missing.
2) have a valid svg file in a directory you can write a subdirectory to, you don't appear to have that problem otherwise if missing you would see ! Package svg Error: File 'test.svg' is missing. However if the file was found and produced an invalid result or the subdirectory ./svg-inkscape could not be written in the folder with both tex and svg then the message in that case could be the one you see. Lets presume your test.svg file is valid and you can add a folder below the main tex folder (ensuring that main folder has no spaces or accent's etc.)


text:
# Summing infinite series that converge only for some parameter values

The input

Sum[d^t,{t,0,Infinity}]

produces output

1/(1-d)

which is correct for $|d|<1$. But for $|d|\geq1$ the sum does not converge so the output is wrong.

Why does Mathematica not produce any warning or indication that this is the case?

For more complicated cases where the non-convergence might not be obvious upon cursory inspection, is there a way for Mathematica to output both a sum and a condition for convergence?

• You can check first with SumConvergence, SumConvergence[d^t, t] gives Abs[d] < 1 – Jason B. Mar 11 '16 at 9:55

Sum[d^t, {t, 0, Infinity}, GenerateConditions -> True]