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# Can $C^\infty(\mathbb{T})$ become a Banach space?

Let $T$ be the unit circle and $C^\infty(\mathbb{T})$ the set of functions defined on $\mathbb{T}$ which have derivatives of every order. I know that $C^\infty(\mathbb{T})$ with the metric induced by the seminorms $$\sup_{t\in\mathbb{R}}|f^{(l)}(e^{it})|,l\geq 0$$ is complete (but not a Banach space with the seminorms themselves i.e. its locally convex structure cannot be defined by one norm).

Is there any chance that we can define some kind of norm on $C^\infty(\mathbb{T})$ in order to become a Banach space?

• Don't you mean "with the metric induced by the (semi)norms"? Completeness is a metric property, not a topological property (i.e. there exists a homeomorphism between a complete space and an incomplete space). – Nap D. Lover Apr 11 '16 at 19:04
• Do you want to demand some kind of additional properties on this norm? Otherwise you could just take a linear isomorphism between $C^\infty(\mathbb{T})$ and your favorite separable infinite-dimensional Banach space and use the induced norm... – Eric Wofsey Apr 11 '16 at 19:14
• For instance, the new norm might induce a completely different topology than the original metric. Operations like multiplication, differentiation, integration, might not be continuous under the new norm. Is that really what you want? – Nate Eldredge Apr 11 '16 at 19:31
• At first I was thinking if $C^p(T)$ for $p<\infty$ with its usual metric could be isomorphic to $C^\infty(T)$ with another metric. So I think that atleast these operations should be continuous. – Chris Apostol Apr 11 '16 at 19:38
• I added a precision in your question, hope this does not alter your intent. – Duchamp Gérard H. E. Jul 22 '17 at 9:22

No, it is impossible to define a topology of $C^\infty(\mathbb{T})$ with a single norm if you want to preserve derivation (w.r.t. any non zero vector of the tangent space) and the product as a continuous operators (otherwise why take $C^\infty(\mathbb{T})$ ?) because - exponential - Taylor formula is not true on all $C^\infty(\mathbb{R})$.
So, for example, call $D$ the derivation of $C^\infty(\mathbb{T})$ such that $$D(f)[e^{it}]=-ie^{-it}\frac{d}{dt}(t\to f(e^{it}))$$ (all other invariant derivations are proportional).
Then, if $D$ were continuous (i.e. bounded within the Banach structure), one would have, for all $t,h\in \mathbb{R}$ $$\sum_{n\geq 0}\frac{h^n}{n!}D^n(f)[e^{it}]=f(e^{i(t+h)})$$
which is not true for all $f\in C^\infty(\mathbb{T})$ (take e.g. any $f$ with support $\not= \mathbb{T}$). See a discussion on Taylor's formula here).
A shorter proof Apply $zD$ (Euler operator) above to the family $z^n$ (i.e. $e^{nit}$ through the parametrization) and see that it has an unbounded spectrum which is impossible for a bounded operator.


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# Guest Review by Josh Young – DragonBox and DragonBox+

29 Aug

Hi again!

If you’re wondering who I am, my name’s Josh, and I’ve come across from Mathematical Mischief (again), to bring you another maths app review! 🙂

Today, I’m covering DragonBox and DragonBox+, two apps developed by We Want To Know. They’re a small, French-Norwegian startup who are aiming to develop a new generation of digital learning games. Considering that the App Store is massive, it’s a high target to set in such a diverse environment.

Now, one might think upon opening DragonBox, that it looks nothing like maths. In actuality, it is a very, very, very clever con act. I mean, if it was just a maths app, do you sincerely think that any 8yr old would play it? Instead, the underlying motives of the game are hidden within the mechanics of the game, 🙂

So what is DragonBox?

DragonBox Title Screen – App Screenshot

DragonBox (and it’s enhanced app, DragonBox+) starts off as a simple title screen with a dog in a box. Once you start, you select a character and end up at a level selection screen.

The early levels show tutorials and instructions upon what to do. They’re artsy, clever, and intuitive, and most kids will be able to understand how the levels work. Ideally, to sit with a parent or older sibling that understands basic algebra would be best – having that extra bit of assistance would be useful.

The trick to DragonBox is its intuitive controls and manner of teaching the topic material.

In DragonBox, there’s 100 levels to try out (separated into five different level packs of varying difficulty), and there’s also an extra 100 levels (making that 200 levels) in DragonBox+. Unfortunately, there isn’t a section that tells you what each section is about, which is a bummer, but the level packs themselves tend to alternate between easy and more difficult problems, on about a ten level basis. I found this incredibly beneficial – in Level 1, you may learn what a zero is, then as you proceed, you might apply it with multiplication.

## How on earth is it maths?

See, the clever part about DragonBox is that it encourages students to apply maths visually.

Let’s take, for example, this equation:

$a\times y=b+c$

If we would like to isolate $y$ in this equation, we simply divide both sides of the equation by $a$. Then we obtain the solution:

$y=\frac{b}{a}+\frac{c}{a}$

Now, answer this: Do you honestly think any 8 year old wants to play that on an app? The answer to that is, indicatively, no.
What the designers of DragonBox have done is to mask this by using visual aids and cards as a device to practice these skills.

Now, the image below is of a level in-app.

The clever thing about DragonBox is that the maths is hidden within the app itself. – App Screenshot

Now, in this level, you can see that this level is represented by separate objects. Separate objects represent addition, whereas linked items represent multiplication. In this case, to solve the puzzle, you have to put a tomato under the linked objects (which represents division).

That is exactly the same as the equation above – but as it’s tackled in a visual aspect, no kid is going to think that they’re doing math. By hiding the maths in app, the child is learning about maths, but they aren’t being subjected to the archaic teaching styles that you see in the classroom these days. That’s not to say that the old styles don’t have their purpose – but in a modern digital environment, they aren’t nearly as useful in an adaptive, fun environment.

## Pros:

• Intuitive design, very kid friendly, fun and engaging. The music and design of the game invites the player to hatch and grow a monster for each level played in the pack – something that is pretty cool.
• Actually confronts algebra in a roundabout way.
• Develops critical thinking and analytical skills – critical for understanding basic algebra and higher forms of mathematics.
• Ability to save multiple profiles – beneficial for schools or families.

## Cons:

• Lack of directional outcome for some levels – ok, so in Level 1.1 I’m told that I want to get the box by itself. Thirty levels later, an x replaces the box and I’m not told any different. A bit of division based on outcomes (i.e. multiplication, addition, subtraction, division, working with zeroes) would be a far better way of separating the levels distinctively.
• The difference between DragonBox+ and DragonBox – Ok, so I’ve completed DragonBox, and I buy DragonBox+… only to find out all of the original levels are exactly the same as the original app. The only thing that really is + about it is that there’s a hundred extra levels – and if you’ve bought the first game, you’re paying twice as much for the same quantity of levels!
• Advice: If you’re sure you want to get the app, get DragonBox+. It’s a little bit more at the time of purchase, but you save the frustration about having to buy another app to get the rest of the levels.

## Verdict

All in all, DragonBox is a well developed, cleverly designed app that is sure to engage (and conspire) children of all ages involved with basic algebra.

 Title: DragonBox Publisher: We Want To Know AS Price: $2.99 Universal – will work on both iPhone and iPad. Title: DragonBox+ Publisher: We Want To Know AS Price:$6.49 Universal – will work on both iPhone and iPad.

Also available on the Mac App Store and Android.

You can find me at Mathematical Mischief, on Facebook, or Twitter. 🙂

All the best,
Josh

### 3 Responses to “Guest Review by Josh Young – DragonBox and DragonBox+”

1. jean 28/04/2013 at 5:01 am #

dragonbox 12+ just released, much more comprehensive. Starts off the same on the 2 first chapters, and then signs, parenthesis, fractions, factoring etc… FYI

• Josh Young 06/05/2013 at 2:31 pm #

Ooh – I’ll have to take a look. 🙂

format: plain


text:
to generate a flowchart or a drawing with annotations. It inherits the text properties (font, font size…) of the document. Just as some programming languages, TikZ also requires you to put a ; at the end of each line. will put the b-node to the right of the a-node, at a default distance of 2.0cm: The \draw[->] (a) -- (b); command creates a line joining the two nodes (using the node labels) with an arrow at the end of the line. The last one becomes the new current point.

To place nodes on a line or a curve use the "pos=" option, where fraction is a floating point number between 0 representing the previous coordinate and 1 representing the current coordinate. See the Inserting Images article for more information about this subject. User-defined paths can be created using the "to" operation. Using this option, you can specify that the picture should be raised or lowered such that the height ⟨dimension⟩ is on the baseline.

Sometimes however, it can be useful to generate images on their own, e.g. Once included, you can add a diagram or flowchart anywhere in your document by putting it in a tikzpicture environment: Let’s start with a minimal example LaTeX-file: If you save this as a .tex-file and then compile this with pdflatex, the resulting .pdf-document should contain a single, non-impressive line: The command to draw this line is reasonably self-explanatory: we want to draw a line, from coordinate (0.0, 0.0) to (1.0, 1.0), and this line has to be straight (--). Sometimes however, the required figures are not plots. To add a line the command \draw[gray, thick] defines a graphic element whose colour is gray and with a thick stroke. To load further libraries use. 1cm in 30 degree direction. To add a line the command \draw[gray, thick] defines a graphic element whose colour is gray and with a thick stroke. You can make the document a bit more tidy by putting the TikZ code into a separate .tex-file and including that using \input. Graphviz] that can do this for you. Another option to scale the entire picture is, Coordinates are specified in round brackets in an arbitrary TEX dimension either using Cartesian coordinates (comma separated), e.g. Two useful constants are e, which is equal to 2.718281828, and pi, which is equal to 3.141592654. Let me just mention some important aspects that might point you in the right direction for specific use cases: To conclude, here is the full source of the cover image of this post, that includes a lot of useful commands and properties: % tikzception: a TikZ picture within a node! Examples for libraries are "arrows", "automata", "backgrounds", "calendar", "chains", "matrix", "mindmap", "patterns", "petri", "shadows", "shapes.geometric", "shapes.misc", "spy", "trees". Line width options: "line width=", and abbreviations "ultra thin" for 0.1pt, "very thin" for 0.2pt, "thin" for 0.4pt (the default width), "semithick" for 0.6pt, "thick" for 0.8pt, "very thick" for 1.2pt, "ultra thick" for 1.6pt. For rectangles a special syntax exists. Here: Move to (0,0) straight line to (2,0), move to (0,1) straight line to (2,1): Two points can be connected by straight lines that are only horizontal and vertical.

TikZ commands are prevalently similar to Metafont, the option mechanism is similar to PsTricks syntax. (The syntax for a bend to the right may seem a little counter-intuitive. One could say that TikZ is to picture as LaTeX is to TeX. pdflatex will then generate a .png-image containing your image (this is exactly how the example images in this post were made). its colour, size, bounding box, position… I will show some of these properties at work later. The instruction has to end with a semicolon. Often it would be better to have the node placed beside the actual coordinate: Right ("right" or "anchor=west"), left ("left" or "anchor=east"), above ("above" or "anchor=south"), below ("below" or "anchor=north"). First of all, \node is the command you want TikZ to perform: defining a node.

instead of the usual pdflatex command. Open the file and comment out this line. Line pattern options: "dash pattern=" (e.g. E.g. at (0.0, 0.0) defines the location of the node (in the TikZ internal length units). standalone will automatically make sure the size of the document is adapted to that of the visible content. It comes with very good documentation and an extensive collection of examples: http://www.texample.net/tikz/. The positive x and y directions refer to right and up on a diagram respectively, and an increase in an angle is a counter-clockwise change, with 0 degrees pointing directly right. Inside a path operation use the following syntax after a given coordinate: The "()" is a name for later reference and it is optional. This situation is pretty much equivalent to including a figure using \includegraphics without any size options. This is very helpful if you e.g. Use a move-to operation to one corner and after "rectangle" the coordinates of the diagonal corner. While the previous systems (picture, epic, pstricks or metapost) focus on the how to draw, TikZ focuses more on the what to draw. PGF ("portable graphics format") is the basic layer, providing a set of basic commands for producing graphics, and TikZ ("TikZ ist kein Zeichenprogramm") is the frontend layer with a special syntax, making the use of PGF easier. A node is typically a rectangle or circle or another simple shape with some text on it. TikZ is not really a tool, but is a language package that is part of LaTeX, and that can be used to generate vector graphics that can be directly included in .pdf-documents generated with LaTeX. Let's start with circles, ellipses and arcs. as a .png-file. The line is actually defined by it's two endpoints, (-1,2) and (2,-4), joined by --. To load further libraries useExamples for libraries are \"arrows\", \"automata\", \"backgrounds\", \"calendar\", \"chains\", \"matrix\", \"mindmap\", \"patterns\", \"petri\", \"shadows\", \"shapes.geometric\", \"shapes.misc\", \"spy\", \"trees\".Drawing commands have to be enclosed in an tikzpicture environment or alternatively One possible option useful for inlined graphics is… Again, you might need to install standalone using your LaTeX package manager first. Possible color and thickness parameters in the tikz package: More colours may be available in your LaTeX distribution. Multiline text can be included inside a node. Here an example: Path construction operations try to be clever, such that the path starts at the border of the node's shape and not from the node's center.

Straight-edged ones use a similar syntax: The nodes are probably the most versatile elements in Tikz. You can change this default by providing an additional argument to the tikzpicture environment, e.g. While traditional plotting tools (certainly matplotlib) can be used for this purpose when you tweak them enough (a painful and slow process), there are much better tools available. A node is centered at the current coordinate by default. With "++" the last point of the path becomes the current position, with "+" the previous point stays the current path position.

(label) is what it says: this defines a label that will be attached to this node. If you only want to use nodes and you only care about their relative position, you can position the nodes using their properties, e.g.

One additional thing you might need to know is how to use the -- in the \draw command: Notice how the various \draw commands result in different orientations for the node connections: Note that while TikZ is very powerful in creating flowcharts, it is not very good at helping you figure out how to optimally position your nodes. Straight lines are given by coordinates separated by a double minus. In this section is explained how to create basic graphic elements.

The fill color "green!20!white" means 20% green and 80% white mixed together. Combinations are also possible, like "anchor=north east" or "below left". The point is actually a circle drawn by \filldraw[black], this command will not only draw the circle but fill it with black colour. Curved paths using a Bezier curve can be created using the "..controls() ..()" command, with one or two control points. PGF also has a math engine which enables you to plot functions: Many functions are possible, including factorial(\x), sqrt(\x), pow(\x,y), exp(\x), ln(\x), log10(\x), log2(\x), abs(\x), mod(\x,y), round(\x), floor(\x), ceil(\x), sin(\x), cos(\x), tan(\x), min(\x,y,), and max(\x,y). Line end, line join options: "line cap=", "arrows=-", "rounded corners", "rounded corners=", "line join=".

{This is a node} is the caption of the node, this is the text that will be displayed in the image. This is the general way commands in TikZ work. TikZ has a large number of additional libraries that help you create more powerful figures. The point is actually a circle drawn …

To do this, add a, similarly, you define your own colours using a. to reuse a specific drawing command, you can define your own LaTeX commands in the preamble of your document and then use them inside your TikZ code.

There are essentially three commands in this figure: A node definition, a node declaration and lines that join two nodes.

This example however did not give you any control over the size and position of the drawing: it was simply placed at the current position in the document, and the size was determined by the size of the drawing. Without specifying a unit (1,2), the standard one is cm (1cm,2cm). Drawing commands have to be enclosed in an tikzpicture environment, One possible option useful for inlined graphics is. Example: 2 standard units to the right of the last point used: A path is a series of straight and curved line segments (in a simplified explanation).

It's important to notice the semicolon ; at the end of each draw command. One instruction can spread over several lines, or several instructions can be put on one line. There are other tools (e.g. There exist some abbreviations: "at start" for "pos=0", "very near start" for "pos=0.125", "near start" for "pos=0.25", "midway" for "pos=0.5", "near end" for "pos=0.75", "very near end" for "pos=0.875", "at end" for "pos=1". However, much of the power of TikZ comes from its close ties with LaTeX, allowing you to embed LaTeX text into images, e.g. This will automatically load the pgf package. This will automatically load the pgf package.

.


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# Halemaumau Lava Lake

## Detailed Description

Movie of the lava lake in Kīlauea's summit vent in Halema‘uma‘u crater minutes after a slice of the vent wall had plunged into the lava. The lava lake is roiling violently as the recently added wall debris is digested by the lava. Slabs of rock can be seen breaking loose from the base of the overhanging vent wall and falling into the lava. The length of time it takes the rocks to reach the lava surface provides an indication of the size of the hole-it's about 150 m (492 ft) across and the rocks are falling about 60 m (197 ft).

## Details

Image Dimensions: 640 x 360

Date Taken:

Length: 00:00:17

Location Taken: HI, US

n/a


text:
# Expectation of an event

Let $S[4]$ be a binary array with elements of $S$ are taken uniformly and independently from $\{0,1\}$. Also take $k$ uniformly from $\{0,1\}$. Take $i=1$. Now run the following process:

1. Take $a,b$ uniformly and independently from $[1,4]$
2. Change $k=(k+S[a]) \bmod 2$
3. $S[b]=(k+S[i])\bmod 2$
4. $i=(i+1) \bmod 4$

Run the whole process (1,2,3,4) until all elements of S are filled with 0. What is expected number of runs for this event?

-
Yes. However, steps are differet. –  diamond kukur Jul 31 '11 at 8:59
In your step 1, I assume you mean take them uniformly and independently from $\{1,2,3,4\}$ and not $[1,4]$? –  cardinal Jul 31 '11 at 13:01
Exactly. I mean $a,b$ are taken uniformly from $\{1,2,3,4\}$. –  diamond kukur Jul 31 '11 at 14:20
I probably won't have time to spend actually solving this, but it shouldn't be hard to build the transition matrix for this Markov chain. Naively, the state space would be $\{0,1,2,3\}\times\{0,1\}^4$, but you can eliminate the variable $i$ from the state by replacing step 4 with "shift all elements of $S$ down by one step". That leaves you with 15 transient states and one absorbing one. –  Ilmari Karonen Aug 1 '11 at 2:50
This is very similar to your previous question. You should mention what's different, and explain what aspect of this you're having difficulties with despite the very detailed solution Ilmari provided in his answer there; otherwise a lot of effort might unnecessarily be duplicated. Also there's some similarity with this series of questions: 1, 2, 3, including the titles. Are you user12290? If so, please have your accounts merged. –  joriki Aug 1 '11 at 5:05


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Blog

Using ISO URNs

thammond – 2007 October 01

(Update - 2007.10.02: Just realized that there were some serious flaws in the post below regarding publication and form of namespace URIs which I’ve now addressed in a subsequent post here.)

By way of experimenting with a use case for ISO URNs, below is a listing of the document metadata for an arbitrary PDF. (You can judge for yourselves whether the metadata disclosed here is sufficient to describe the document.) Here, the metadata is taken from the information dictionary and from the document metadata stream (XMP packet).

The metadata is expressed in RDF/N3. That may not be a surprise for the XMP packet which is serialized in RDF/XML, as it’s just a hop, skip and a jump to render it as RDF/N3 with properties taken from schema whose namespaces are identified by URI. What may be more unusual is to see the document information dictionary metadata (the “normal” metadata in a PDF) rendered as RDF/N3 since the information dictionary is not nodelled on RDF, not expressed in XML, and not namespaced. Here, in addition to the trusty HTTP URI scheme, I’ve made use of two particular URI schemes: “iso:” URN namespaces, and “data:” URIs.

(Continues.)

As far as I am aware, there is no formal identifier for entries in the document information dictionary as specified by the PDF Reference from Adobe Systems, so it may be appropriate to use the HTTP URI for the Adobe homepage for the PDF Reference manual, from which specific editions are available.

For the PDF/X keys which are specified in the ISO standard ISO 15930-1 2001, I have used an ISO URN. (I don’t expect this to be correct in all details but it should give some idea of how it might be used. It may be that the URI should express the term itself, rather than the document from which the term was defined.) And finally, for the one additional user-supplied key here I have made use of a “data:” URI with no body (i.e. I’m speechless). One could have provided some text within the body of the “data:” URI if one wanted to differentiate between alternate user keys or to otherwise annotate these keys.

Note that the prefixes used in the information dictionary and in the metadata stream are unrelated, as are the mappings of property elements to schemas.

Well, that’s all really just for fun but it may show two things: 1) how a general description might be described with RDF and how general properties can be mapped to URIs (with possibly limited machine utility), and 2) how an ISO URN might be used.

# document infodict (object 58: 476983):
@prefix: pdfx: <urn:iso:std:iso-iec:15930:-1:2001> .
@prefix: _usr: <data:,> .
<>   _usr:Apag_PDFX_Checkup "1.3";
pdf:Author "Scott B. Tully";
pdf:CreationDate "D:20020320135641Z";
pdf:Creator "Unknown";
pdfx:GTS_PDFXConformance "PDF/X-1a:2001";
pdfx:GTS_PDFXVersion "PDF/X-1:2001";
pdf:Keywords "PDF/X-1";
pdf:ModDate "D:20041014121049+10'00'";
pdf:Producer "Acrobat Distiller 4.05 for Macintosh";
pdf:Subject "A document from our PDF archive. ";
pdf:Title "Tully Talk November 2001";
pdf:Trapped "False" .
# document metadata stream (object 41: 472418):
@prefix dc: <http://purl.org/dc/elements/1.1/> .
@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
<>     pdf:Keywords "PDF/X-1";
pdf:Producer "Acrobat Distiller 4.05 for Macintosh";
pdfx:Apag_PDFX_Checkup "1.3";
pdfx:GTS_PDFXConformance "PDF/X-1a:2001";
pdfx:GTS_PDFXVersion "PDF/X-1:2001";
xmp:CreateDate "2002-03-20T13:56:41Z";
xmp:CreatorTool "Unknown";
xmp:ModifyDate "2004-10-14T12:10:49+10:00";
xmpMM:DocumentID "uuid:bd7ae9a1-1110-43c0-8e84-632f2dbb55ab";
dc:creator  [
a rdf:Seq;
rdf:_1 "Scott B. Tully" ];
dc:description  [
a rdf:Alt;
rdf:_1 "A document from our PDF archive. "@x-default ];
dc:format "application/pdf";
dc:title  [
a rdf:Alt;
rdf:_1 "Tully Talk November 2001"@x-default ] .


text:
# zbMATH — the first resource for mathematics

## Bruss, Franz Thomas

Compute Distance To:
 Author ID: bruss.f-thomas Published as: Bruss, F. Thomas; Bruss, F. T. Homepage: http://homepages.ulb.ac.be/~tbruss/ External Links: MGP · Wikidata · GND
 Documents Indexed: 69 Publications since 1978, including 2 Books Reviewing Activity: 49 Reviews Biographic References: 2 Publications
all top 5

#### Co-Authors

 33 single-authored 7 Louchard, Guy 6 Ferguson, Thomas S. 3 Samuels, Stephen M. 3 Swan, Yvik C. 2 Delbaen, Freddy 2 Robertson, James B. 2 Rogers, L. C. G. 2 Yor, Marc 1 Drmota, Michael 1 Duerinckx, Mitia 1 Grübel, Rudolf 1 Heyvaert, Marie 1 Jammalamadaka, Sreenivasa Rao 1 Le Cam, Lucien Marie 1 Mahiat, Hervé 1 O’Cinneide, Colm Art 1 Paindaveine, Davy 1 Pierard, Martine 1 Rüschendorf, Ludger 1 Slavtchova-Bojkova, Maroussia Nikiforova 1 Turner, John Wm. 1 Ward, Mark Daniel 1 Zhou, Xian
all top 5

#### Serials

 14 Journal of Applied Probability 6 The Mathematical Scientist 6 Stochastic Processes and their Applications 5 The Annals of Probability 5 Annales de la Société Scientifique de Bruxelles. Série I 4 European Mathematical Society Newsletter 3 Advances in Applied Probability 3 The Annals of Applied Probability 3 Spektrum der Wissenschaft 2 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 2 Discrete Mathematics and Theoretical Computer Science. DMTCS 1 Mitteilungen der Deutschen Mathematiker-Vereinigung (DMV) 1 Cahiers du Centre d’Études de Recherche Opérationnelle 1 Gazette des Mathématiciens 1 Statistics & Probability Letters 1 Communications in Statistics. Stochastic Models 1 Algorithmica 1 Bulletin of the Belgian Mathematical Society - Simon Stevin 1 Contemporary Mathematics 1 Institute of Mathematical Statistics Lecture Notes - Monograph Series 1 Probability Surveys 1 ACM Transactions on Algorithms 1 Mathematica Applicanda
all top 5

#### Fields

 57 Probability theory and stochastic processes (60-XX) 18 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 14 Statistics (62-XX) 10 Operations research, mathematical programming (90-XX) 9 General and overarching topics; collections (00-XX) 4 History and biography (01-XX) 2 Computer science (68-XX) 2 Biology and other natural sciences (92-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Functions of a complex variable (30-XX) 1 Approximations and expansions (41-XX) 1 Numerical analysis (65-XX) 1 Systems theory; control (93-XX) 1 Mathematics education (97-XX)

#### Citations contained in zbMATH Open

46 Publications have been cited 300 times in 164 Documents Cited by Year
Sum the odds to one and stop. Zbl 1005.60055
Bruss, F. Thomas
2000
A note on extinction criteria for bisexual Galton-Watson processes. Zbl 0568.60073
Bruss, F. Thomas
1984
A counterpart of the Borel-Cantelli lemma. Zbl 0443.60002
Bruss, F. Thomas
1980
Optimal rules for the sequential selection of monotone subsequences of maximum expected length. Zbl 1005.60056
Bruss, F. Thomas; Delbaen, Freddy
2001
A note on bounds for the odds theorem of optimal stopping. Zbl 1059.60056
Bruss, F. Thomas
2003
‘Wald’s lemma’ for sums of order statistics of i.i.d. random variables. Zbl 0752.62057
Bruss, F. Thomas; Robertson, James B.
1991
On the multiplicity of the maximum in a discrete random sample. Zbl 1038.62047
Bruss, F. Thomas; Grübel, Rudolf
2003
On an optimal selection problem of Cowan and Zabczyk. Zbl 0596.60046
Bruss, F. Thomas
1987
A unified approach to a class of best choice problems with an unknown number of options. Zbl 0553.60047
Bruss, F. Thomas
1984
Selecting a sequence of last successes in independent trials. Zbl 0967.60047
Bruss, F. Thomas; Paindaveine, Davy
2000
A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length. Zbl 1071.60021
Bruss, F. Thomas; Delbaen, Freddy
2004
Multiple buying or selling with vector offers. Zbl 0905.60028
Bruss, F. Thomas; Ferguson, Thomas S.
1997
Pascal processes and their characterization. Zbl 0743.60043
Bruss, F. T.; Rogers, L. C. G.
1991
Embedding optimal selection problems in a Poisson process. Zbl 0745.60040
Bruss, F. Thomas; Rogers, L. C. G.
1991
On waiting times to populate an environment and a question of statistical inference. Zbl 0937.62120
Bruss, F. Thomas; Slavtchova-Bojkova, M.
1999
Minimizing the expected rank with full information. Zbl 0781.60035
Bruss, F. Thomas; Ferguson, Thomas S.
1993
Invariant record processes and applications to best choice modelling. Zbl 0665.60049
Bruss, F. Thomas
1988
What is known about Robbins’ problem? Zbl 1081.62059
Bruss, F. Thomas
2005
On the maximum and its uniqueness for geometric random samples. Zbl 0723.60011
Bruss, F. Thomas; O’Cinneide, Colm Art
1990
Half-prophets and Robbins’ problem of minimizing the expected rank. Zbl 0858.60045
Bruss, F. Thomas; Ferguson, Thomas S.
1996
The odds algorithm based on sequential updating and its performance. Zbl 1169.60006
Bruss, F. Thomas; Louchard, Guy
2009
On the $$N$$-tower problem and related problems. Zbl 1077.60051
Bruss, F. Thomas; Louchard, Guy; Turner, John W.
2003
Branching processes with random absorbing processes. Zbl 0381.60066
Bruss, F. Thomas
1978
A continuous-time approach to Robbins’ problem of minimizing the expected rank. Zbl 1200.62093
Bruss, F. Thomas; Swan, Yvik C.
2009
A unified approach to a class of optimal selection problems with an unknown number of options. Zbl 0592.60034
Bruss, F. Thomas; Samuels, Stephen M.
1987
Conditions for quasi-stationarity of the Bayes rule in selection problems with an unknown number of rankable options. Zbl 0704.62067
Bruss, F. Thomas; Samuels, Stephen M.
1990
On an interval splitting problem. Zbl 0707.60023
Bruss, F. Thomas; Jammalamadaka, S. Rao; Zhou, Xian
1990
Resource dependent branching processes and the envelope of societies. Zbl 1308.60103
Bruss, F. Thomas; Duerinckx, Mitia
2015
Optimal stopping on patterns in strings generated by independent random variables. Zbl 1023.60045
Bruss, F. Thomas; Louchard, Guy
2003
Patterns of relative maxima in random permutations. Zbl 0562.60041
Bruss, F. Thomas
1984
Stochastic processes with proportional increments and the last-arrival problem. Zbl 1255.60166
Bruss, F. Thomas; Yor, Marc
2012
Limits of any society. (Grenzen einer jeden Gesellschaft.) Zbl 1306.91108
Bruss, F. Thomas
2014
Sequential selection of the $$k$$ best out of $$n$$ rankable objects. Zbl 1397.60080
Bruss, F. Thomas; Louchard, Guy
2016
A survey of the linear-search problem. Zbl 0662.90042
Bruss, F. Thomas; Robertson, James B.
1988
The complete solution of the competitive rank selection problem. Zbl 0918.90002
Bruss, F. T.; Drmota, M.; Louchard, G.
1998
High-risk and competitive investment models. Zbl 1005.60054
Bruss, F. Thomas; Ferguson, Thomas S.
2002
Playing a trick on uncertainty. Zbl 1060.90659
Bruss, F. Thomas
2003
Die Kunst der richtigen Entscheidung. Zbl 1060.90661
Bruss, F. Thomas
2005
Advances in best choice modelling. Zbl 1062.90520
Bruss, F. Thomas
1987
The Schwarz-Christoffel transformation as a tool in applied probability. Zbl 1076.60070
Swan, Yvik; Bruss, F. Thomas
2004
Unexpected strategies. (Unerwartete Strategien.) Zbl 1288.91008
Bruss, F. Thomas
1998
Inverse auctions, injecting unique minima into random sets. Zbl 1300.91032
Bruss, F. Thomas; Louchard, Guy; Ward, Mark Daniel
2009
A mathematical approach to comply with ethical constraints in compassionate use treatments. Zbl 1414.60027
Bruss, F. Thomas
2018
A matrix-analytic approach to the $$N$$-player ruin problem. Zbl 1137.60035
Swan, Yvik C.; Bruss, F. Thomas
2006
On a class of optimal stopping problems with mixed constraints. Zbl 1296.60099
Bruss, F. Thomas
2010
Odds-theorem and monotonicity. Zbl 1463.60063
Bruss, F. Thomas
2019
Odds-theorem and monotonicity. Zbl 1463.60063
Bruss, F. Thomas
2019
A mathematical approach to comply with ethical constraints in compassionate use treatments. Zbl 1414.60027
Bruss, F. Thomas
2018
Sequential selection of the $$k$$ best out of $$n$$ rankable objects. Zbl 1397.60080
Bruss, F. Thomas; Louchard, Guy
2016
Resource dependent branching processes and the envelope of societies. Zbl 1308.60103
Bruss, F. Thomas; Duerinckx, Mitia
2015
Limits of any society. (Grenzen einer jeden Gesellschaft.) Zbl 1306.91108
Bruss, F. Thomas
2014
Stochastic processes with proportional increments and the last-arrival problem. Zbl 1255.60166
Bruss, F. Thomas; Yor, Marc
2012
On a class of optimal stopping problems with mixed constraints. Zbl 1296.60099
Bruss, F. Thomas
2010
The odds algorithm based on sequential updating and its performance. Zbl 1169.60006
Bruss, F. Thomas; Louchard, Guy
2009
A continuous-time approach to Robbins’ problem of minimizing the expected rank. Zbl 1200.62093
Bruss, F. Thomas; Swan, Yvik C.
2009
Inverse auctions, injecting unique minima into random sets. Zbl 1300.91032
Bruss, F. Thomas; Louchard, Guy; Ward, Mark Daniel
2009
A matrix-analytic approach to the $$N$$-player ruin problem. Zbl 1137.60035
Swan, Yvik C.; Bruss, F. Thomas
2006
What is known about Robbins’ problem? Zbl 1081.62059
Bruss, F. Thomas
2005
Die Kunst der richtigen Entscheidung. Zbl 1060.90661
Bruss, F. Thomas
2005
A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length. Zbl 1071.60021
Bruss, F. Thomas; Delbaen, Freddy
2004
The Schwarz-Christoffel transformation as a tool in applied probability. Zbl 1076.60070
Swan, Yvik; Bruss, F. Thomas
2004
A note on bounds for the odds theorem of optimal stopping. Zbl 1059.60056
Bruss, F. Thomas
2003
On the multiplicity of the maximum in a discrete random sample. Zbl 1038.62047
Bruss, F. Thomas; Grübel, Rudolf
2003
On the $$N$$-tower problem and related problems. Zbl 1077.60051
Bruss, F. Thomas; Louchard, Guy; Turner, John W.
2003
Optimal stopping on patterns in strings generated by independent random variables. Zbl 1023.60045
Bruss, F. Thomas; Louchard, Guy
2003
Playing a trick on uncertainty. Zbl 1060.90659
Bruss, F. Thomas
2003
High-risk and competitive investment models. Zbl 1005.60054
Bruss, F. Thomas; Ferguson, Thomas S.
2002
Optimal rules for the sequential selection of monotone subsequences of maximum expected length. Zbl 1005.60056
Bruss, F. Thomas; Delbaen, Freddy
2001
Sum the odds to one and stop. Zbl 1005.60055
Bruss, F. Thomas
2000
Selecting a sequence of last successes in independent trials. Zbl 0967.60047
Bruss, F. Thomas; Paindaveine, Davy
2000
On waiting times to populate an environment and a question of statistical inference. Zbl 0937.62120
Bruss, F. Thomas; Slavtchova-Bojkova, M.
1999
The complete solution of the competitive rank selection problem. Zbl 0918.90002
Bruss, F. T.; Drmota, M.; Louchard, G.
1998
Unexpected strategies. (Unerwartete Strategien.) Zbl 1288.91008
Bruss, F. Thomas
1998
Multiple buying or selling with vector offers. Zbl 0905.60028
Bruss, F. Thomas; Ferguson, Thomas S.
1997
Half-prophets and Robbins’ problem of minimizing the expected rank. Zbl 0858.60045
Bruss, F. Thomas; Ferguson, Thomas S.
1996
Minimizing the expected rank with full information. Zbl 0781.60035
Bruss, F. Thomas; Ferguson, Thomas S.
1993
‘Wald’s lemma’ for sums of order statistics of i.i.d. random variables. Zbl 0752.62057
Bruss, F. Thomas; Robertson, James B.
1991
Pascal processes and their characterization. Zbl 0743.60043
Bruss, F. T.; Rogers, L. C. G.
1991
Embedding optimal selection problems in a Poisson process. Zbl 0745.60040
Bruss, F. Thomas; Rogers, L. C. G.
1991
On the maximum and its uniqueness for geometric random samples. Zbl 0723.60011
Bruss, F. Thomas; O’Cinneide, Colm Art
1990
Conditions for quasi-stationarity of the Bayes rule in selection problems with an unknown number of rankable options. Zbl 0704.62067
Bruss, F. Thomas; Samuels, Stephen M.
1990
On an interval splitting problem. Zbl 0707.60023
Bruss, F. Thomas; Jammalamadaka, S. Rao; Zhou, Xian
1990
Invariant record processes and applications to best choice modelling. Zbl 0665.60049
Bruss, F. Thomas
1988
A survey of the linear-search problem. Zbl 0662.90042
Bruss, F. Thomas; Robertson, James B.
1988
On an optimal selection problem of Cowan and Zabczyk. Zbl 0596.60046
Bruss, F. Thomas
1987
A unified approach to a class of optimal selection problems with an unknown number of options. Zbl 0592.60034
Bruss, F. Thomas; Samuels, Stephen M.
1987
Advances in best choice modelling. Zbl 1062.90520
Bruss, F. Thomas
1987
A note on extinction criteria for bisexual Galton-Watson processes. Zbl 0568.60073
Bruss, F. Thomas
1984
A unified approach to a class of best choice problems with an unknown number of options. Zbl 0553.60047
Bruss, F. Thomas
1984
Patterns of relative maxima in random permutations. Zbl 0562.60041
Bruss, F. Thomas
1984
A counterpart of the Borel-Cantelli lemma. Zbl 0443.60002
Bruss, F. Thomas
1980
Branching processes with random absorbing processes. Zbl 0381.60066
Bruss, F. Thomas
1978
all top 5

#### Cited by 173 Authors

 23 Molina, Manuel 19 Mota, Manuel 18 Bruss, Franz Thomas 14 Gonzalez-Velasco, Miguel 13 Ramos, Alfonso 12 del Puerto, Inés Maria 8 Arlotto, Alessandro 7 Steele, J. Michael 7 Tamaki, Mitsushi 6 Gnedin, Alexander V. 5 Ano, Katsunori 4 Alsmeyer, Gerold 4 Dembińska, Anna 4 Grau, José María 4 Grübel, Rudolf 4 Louchard, Guy 4 Martínez, Rodrigo 3 Krasnosielska-Kobos, Anna 3 Mazalov, Vladimir Viktorovich 3 Pitman, Jim William 3 Rüschendorf, Ludger 2 Allaart, Pieter C. 2 Corbacho, Casimiro 2 Delbaen, Freddy 2 Faller, Andreas 2 Hitczenko, Paweł 2 Itoh, Yoshiaki 2 Jasiński, Krzysztof 2 Kabluchko, Zakhar A. 2 Kurushima, Aiko 2 Lytvynov, Eugene W. 2 Ma, Shixia 2 Mahmoud, Hosam M. 2 Matsui, Tomomi 2 Minuesa, Carmen 2 Rogers, L. C. G. 2 Rösler, Uwe 2 Seksenbayev, Amirlan 2 Sofronov, Georgy Yu. 2 Stadje, Wolfgang 2 Swan, Yvik C. 1 Acosta, A. Pascual 1 Alpern, Steve 1 Anagnostopoulos, Aris 1 Baker, David M. 1 Barranco-Chamorro, Inmaculada 1 Baryshnikov, Yuliy M. 1 Bayón, Luis 1 Bayraktar, Erhan 1 Beck, Anatole 1 Beckmann, Martin J. 1 Ben Abdelaziz, Fouad 1 Bíró, Csaba 1 Browne, Sid 1 Bunge, J. A. 1 Bunge, John 1 Cain, Nicholas 1 Chen, Robert W. 1 Christensen, Soren 1 Conceição Serra, Maria 1 Darji, Udayan B. 1 David, Israel 1 Deheuvels, Paul 1 Dendievel, Rémi 1 Dennert, Florian 1 Dizon-Ross, Rebecca 1 Doumerc, Yan 1 Drmota, Michael 1 Duerinckx, Mitia 1 Eaton, Julia 1 Egozcue, Martin 1 Eisenberg, Bennett 1 Fal’ko, I. A. 1 Feldheim, Naomi Dvora 1 Feldheim, Ohad Noy 1 Feldman, William M. 1 Ferenstein, Elżbieta Z. 1 Ferguson, Thomas S. 1 Fiat, Amos 1 Flesch, Janos 1 Fortuny, P. 1 Fuchs, Michael 1 Gans, Noah 1 Glickman, Hagit 1 Godbole, Anant P. 1 Goldenshluger, Alexander 1 Gorelik, Ilia 1 Gouet, Raúl 1 Grandoni, Fabrizio 1 Gurvich, Itai 1 Gutiérrez, Cristina 1 Haslegrave, John 1 Hsiau, Shoou-Ren 1 Hull, David M. 1 Hwang, Hsien-Kuei 1 Irle, Albrecht 1 Islas, José A. 1 Iung, Benoît 1 Ivashko, Anna Antonovna 1 Jacob, Christine ...and 73 more Authors
all top 5

#### Cited in 67 Serials

 24 Journal of Applied Probability 15 Stochastic Processes and their Applications 13 Statistics & Probability Letters 7 Advances in Applied Probability 7 The Annals of Applied Probability 4 Mathematical Biosciences 4 Journal of Statistical Planning and Inference 4 European Journal of Operational Research 4 Bernoulli 3 The Annals of Probability 3 Stochastic Analysis and Applications 3 Statistics 3 Test 3 Methodology and Computing in Applied Probability 2 Optimization 2 Sequential Analysis 2 Communications in Statistics. Theory and Methods 2 Combinatorics, Probability and Computing 2 Journal of Combinatorial Optimization 2 Stochastics 2 Probability Surveys 2 Journal of Theoretical Biology 2 Stochastic Systems 2 Mathematica Applicanda 1 Computers & Mathematics with Applications 1 Discrete Mathematics 1 Israel Journal of Mathematics 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Journal of Mathematical Biology 1 Metrika 1 Journal of Functional Analysis 1 Journal of Soviet Mathematics 1 Mathematics of Operations Research 1 Operations Research 1 Proceedings of the American Mathematical Society 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Semigroup Forum 1 Theoretical Population Biology 1 Transactions of the American Mathematical Society 1 Journal of Information & Optimization Sciences 1 International Journal of Production Research 1 Acta Mathematicae Applicatae Sinica. English Series 1 Probability Theory and Related Fields 1 Algorithmica 1 Annals of Operations Research 1 Random Structures & Algorithms 1 Neural Computation 1 Godishnik na Sofiĭskiya Universitet “Sv. Kliment Okhridski”. Fakultet po Matematika i Informatika 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Computational Statistics and Data Analysis 1 ZOR. Zeitschrift für Operations Research 1 Journal of Computer and Systems Sciences International 1 Theory of Probability and Mathematical Statistics 1 Statistical Papers 1 Journal of Mathematical Sciences (New York) 1 Mathematical Finance 1 Annals of Combinatorics 1 Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 Australian & New Zealand Journal of Statistics 1 Probability in the Engineering and Informational Sciences 1 Stochastic Models 1 Journal of Statistical Mechanics: Theory and Experiment 1 Proceedings of the Steklov Institute of Mathematics 1 ALEA. Latin American Journal of Probability and Mathematical Statistics 1 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 Dynamic Games and Applications 1 Journal of Statistical Distributions and Applications
all top 5

#### Cited in 18 Fields

 144 Probability theory and stochastic processes (60-XX) 52 Statistics (62-XX) 23 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 22 Operations research, mathematical programming (90-XX) 12 Biology and other natural sciences (92-XX) 8 Combinatorics (05-XX) 6 Computer science (68-XX) 3 Numerical analysis (65-XX) 3 Systems theory; control (93-XX) 2 Group theory and generalizations (20-XX) 2 Special functions (33-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Geometry (51-XX) 1 Convex and discrete geometry (52-XX) 1 Fluid mechanics (76-XX)

#### Wikidata Timeline

The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.


text:
Browse Questions

# Which of the following statements is correct for a galvanic cell,written in a cell diagram?

$\begin{array}{1 1}\text{Left electrode is cathode}\\\text{Right electrode is anode}\\\text{Left electrode is positive terminal}\\\text{Right electrode is positive terminal}\end{array}$

Answer : Right electrode is positive terminal
In galvanic cell,the right electrode is cathode and carries positive sign.


text:
# Pile of Crabs

Criterion 2021 Round 12
Limits 1s, 512 MB

There is a pile of crabs standing on one top of another.
There are total $N$ crabs numbered $1$ to $N$. The crab numbered $1$ is at the bottom of the pile. Crab numbered $2$ is on top of crab $1$ and so on. The crab numbered $N$ is at the top of the pile.
Each crab has an initial health value represented with an integer number.

You have to observe $Q$ events.
For each event, a crab numbered $X$ will start climbing from his position to the top of the pile. Every crab he will cross on his way, he will decrease their health value by $1$. If one’s health is equal to 0, the crab is dead and his value will not get decreased further. If the crab numbered $X$ is already on the top of the pile, he will not move.
If the crab numbered $X$ is already dead, he will not climb the pile. You have to report the crab is dead.

After observing all the events, you’ve to print the health value of each crab.

## Input

The first line on input consists of two integers $N$ and $Q$ $(3 ≤ N, Q ≤ 5 × 10^5)$ represents the number of crabs and the number of events respectively.
Each of the next $N$ lines consists of an integer value $H_i$ $(1 ≤ H_i ≤ 10^9)$ represents the health of $i^{th}$crab for each $i$ from $1$ to $N$.

Each of the next $Q$ lines consists of an integer $X$ represents the crab that will climb up the pile.

## Output

For each of the $Q$ events, if the crab numbered $X$ is already dead, print $Dead$ in separate lines. Otherwise, do not print anything.

After observing all the events, print a single line with $N$ integers $H_i$for each $i$ from $1$ to $N$seperated by space. $H_i$ represent the health of $i^{th}$ crab after all the events.

## Samples

InputOutput
4 3
2 3 1 4
2
3
1

Dead
2 2 0 2

After the first event, the health of crab 1, 2, 3, and 4 is equal to 2, 3, 0, 3 respectively.

In the second event, we can see crab 3 is already dead. So, it won’t move.

After the third event, the health of crabs 1, 2, 3, and 4 is equal to 2, 2, 0, 2 respectively.

InputOutput
10 16
19 5 26 71 16 2 22 8 85 24
8
1
2
1
4
4
9
9
9
5
8
2
6
10
6
9

Dead
13 0 24 65 10 0 18 1 78 17


text:
Asymptotics for the matrix-valued Keller-Rubinstein-Sternberg problem

We will talk about the asymptotical behaviour of a matrix-valued Allen-Cahn equation when a small parameter tends to zero. Precisely, we show that the limit system is a two-phases flow system: the phase interface evolves according to the mean curvature flow; in two bulk phase regions, the solution obeys the harmonic map heat flow into two different manifolds; on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition.

The proof follows the roadmap developed by de Mottoni-Schatzman and Alikakos-Bates-Chen: we first construct an approximate solution solving the regularized system up to arbitrary small terms, and then we prove a spectral lower bound for the linearized operator around it, and finally we estimate the difference between the true solution and the approximate solution. The main difficulties come from the inherent (partially minimal pairing) property of this problem.

This is joint work with Mingwen Fei, Fanghua Lin and Zhifei Zhang.


text:
# Periodic BC for Multigrid in MD

I know that this question might be very specific and maybe nobody will know the answer, but this is probably the only community where I could find an answer:

So, as part of my master's project, I am currently writing a multigrid code for Poisson's equation in order to calculate particle interactions in a molecular dynamcis setup. I'd like to use periodic BCs, but here arises one problem:

According to the Multigrid book by Trottenberg, periodic BCs only have a solution if $\sum_{x, y, z} f_h(x,y,z) = 0$, where $f_h$ is the right hand side and the sum is over all gridpoints.

The algorithm I found in some publications samples the charged particles via a density function to the grid. So in general, even if the total charge of the system is neutral, this condition will not be fulfilled.

Unfortunately, all papers I found only deal with this PBC problem on a quick sidenote and I wasn't able to get any valuable information out of it.

The code I wrote so far produced fine results when using different functions which fulfill this condition, but fails when going to the real data, i.e., sampled particle charge densities. My question now: Does anybody know this problem and the solution, or any idea which could help me?

• Hi @chris, and welcome to Scicomp! I'm curious... Are you using purely periodic boundary conditions? – Paul Mar 3 '13 at 19:46
• Hey and thank you. Yes I am. It is a conventional way when using FFT based methods such as SPME or P3M, so I also want to have periodic BCs to be able to compare the results – chris Mar 3 '13 at 20:27
• What kind of function are you using to map your particles to the grid? Do they preserve the sum of the quantity you are mapping? – Pedro Mar 3 '13 at 21:07

This isn't a multigrid issue, it's a problem formulation issue. Consider the symmetric system $A x = b$ and suppose that $A e = 0$ for some nonzero vector $e$ (the constant when $A$ is the Laplacian with periodic or Neumann boundary conditions). This system is singular and has no solution if you choose $b$ such that $e^T b \ne 0$. Indeed, the exact periodic problem is ill-posed because the integral of the Coulomb potential with some periodic change density (not averaging to zero) over all space is infinite. That is, if $\sigma(r')$ is periodic with nonzero integral, then
$$\Phi (r) = \int_{r'\in R^3} \frac{\sigma(r')}{4 \pi \vert r - r'\rvert} \mathrm{d} r' = \pm \infty .$$
$$A x = (I - e e^T) b$$
where $e$ is the normalized null space vector.
• Thank you, this works. Just for my personal understanding: Why can we introduce this shift of $b$ without affecting the field? – chris Mar 3 '13 at 23:13
• The field $F = \nabla \Phi = \nabla (\Phi + C)$ for any constant $C$. – Jed Brown Mar 3 '13 at 23:19


text:
# Find the centripetal acceleration of the space shuttle

1. Nov 7, 2009

### ambitiousgirl

1. Suppose a space shuttle is in orbit 400 km from the Earth's surface, and circles the eart about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g, the gravitational acceleration at the Earth's surface.

2. Ac=v2/r

3. I tried converting 400km/1.5 hr to km/h...then to m/s and divided by 90 min to find the centripetal acceleration, then divinding this answer by 9.8 to find the final answer in g's.

The answer is supposed to be .9 g's.

I have tried this probelm a million times, and I honestly think that my mistake is in converting 400km. I'm not sure though...

Last edited: Nov 7, 2009
2. Nov 7, 2009

### Delphi51

You haven't used 400 km as the radius, have you? The radius is the distance to the center of the Earth, not to the surface. Recommend you show your actual calc if you need more help.


text:
## Algebra 2 Common Core

$6\sqrt5$
Recall: For any real numbers $a, b,$ and $c$, $$a(b+c)=ab+ac$$ Use the rule above to obtain: \begin{align*} 5\sqrt5+\sqrt5&=(5+1)\sqrt5\\ &=6\sqrt5 \end{align*}


text:
# Place TikZ coordinate or \zlabel at every (base)line

I'm currently adjusting the code shown in Test if a paragraph has a page break in it? to support underlining and highlighting using TikZ similar as shown in Cool Text Highlighting in LaTeX, but without the need of soul or any other package which influences the typesetting of the text.

For this I'm placing a TikZ coordinate (using overlay,remember picture) at the begin and end and use this to draw the lines. At the moment I simply calculate the baselines of the intermediate lines by adding \baselineskip to the Y-coordinate of the first marker (or subtracting it from the second of there is a page break between them). This works fine for normal text but leads to wrong positions if the line distance is not equal to \baselineskip, e.g. caused by an overly high element on the line.

Would it be possible to automatically place such a TikZ marker on every line TeX produces? I also might use \zlabel from the zref package in addition to that to determine the page. I searched a little through The TeXBook and had a look at the source code of lineno but both without success. It would be enough to add the marker only when the line is higher than normal. (My current funny ideaTM to solve this is to make some frequent letter active, but I try to avoid such trickery.)

-

I don't think this is possible in TeX, because of its way of building paragraphs. There's no \everyline parameter, unfortunately.

One might try to typeset the paragraph in a \vbox and then dismantling it, adding to each line the code, see this answer.

I believe this is solvable with LuaTeX.

-

I implemented something similar with in the gmdoc-enhance package to have leading % signs on each comment line. It makes the space character active and expand in principle to \discretionary{}{\gmde@kommentarStart}{}~, where \gmde@kommentarStart puts a \hbox with the contents I want to show.

The problem here is that \gmde@kommentarStart will be evaluated for every space, not only for those that create a line break - but only for those the box will actually occur in the output. And also, I have to make sure that no other line breaks occur, which destroys most of TeX's line breaking algorithm (i.e. I essentially need \raggedright and no word-breaking). This is acceptable for my use case, but might not be for your one.

I'm not sure how this works together with TikZ (never used it).

(The package's documentation is in German, if someone needs more details, please comment.)

-


text:
The Community for Technology Leaders
Subscribe
Issue No.11 - November (2010 vol.32)
pp: 2006-2021
Zhaoshui He , RIKEN Brain Science Institute, Saitama and South China University of Technology, Guangzhou
Andrzej Cichocki , RIKEN Brain Science Institute, Saitama, Polish Academy of Sciences, Warsaw and Warsaw University of Technology, Warsaw
Shengli Xie , South China University of Technology, Guangzhou
Kyuwan Choi , ATR Computational Neuroscience Laboratories, Kyoto
ABSTRACT
Recently, there has been a growing interest in multiway probabilistic clustering. Some efficient algorithms have been developed for this problem. However, not much attention has been paid on how to detect the number of clusters for the general n-way clustering (n\ge 2). To fill this gap, this problem is investigated based on n-way algebraic theory in this paper. A simple, yet efficient, detection method is proposed by eigenvalue decomposition (EVD), which is easy to implement. We justify this method. In addition, its effectiveness is demonstrated by the experiments on both simulated and real-world data sets.
INDEX TERMS
Multiway clustering, probabilistic clustering, hypergraph, parallel factor analysis (PARAFAC), model order selection, multiway array, higher order tensor, supersymmetric tensors, affinity arrays, enumeration of clusters, estimation of PARAFAC components, principal components enumeration.
CITATION
Zhaoshui He, Andrzej Cichocki, Shengli Xie, Kyuwan Choi, "Detecting the Number of Clusters in n-Way Probabilistic Clustering", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 11, pp. 2006-2021, November 2010, doi:10.1109/TPAMI.2010.15
REFERENCES
 [1] D. Zhou, J. Huang, and B. Schölkopf, "Learning with Hypergraphs: Clustering, and Classification, Embedding," Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T. Hoffman, eds., vol. 19, pp. 1601-1608, MIT Press, 2007. [2] C. Berge, Hypergraphs, first ed. North Holland, Aug. 1989. [3] A. Shashua, R. Zass, and T. Hazan, "Multi-Way Clustering, Using Super-Symmetric Non-Negative Tensor Factorization," Lecture Notes in Computer Science, vol. 3954, pp. 595-608, Springer, July 2006. [4] A. Banerjee, S. Basu, and S. Merugu, "Multi-Way Clustering on Relation Graphs," Proc. SIAM Conf. Data Mining, 2007. [5] J.M. Buhmann and T. Hofmann, "A Maximum Entropy Approach to Pairwise Data Clustering," Proc. 12th IAPR Int'l Conf. Pattern Recognition, pp. 207-212, Oct. 1994. [6] T. Hofmann and J. Buhmann, "Hierarchical Pairwise Data Clustering by Mean-Field Annealing," Proc. Int'l Conf. Artificial Neural Networks, pp. 197-202, 1995. [7] T. Hofmann and J.M. Buhmann, "Pairwise Data Clustering by Deterministic Annealing," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 1, pp. 1-14, Jan. 1997. [8] D.J.C. MacKay, Information Theory, Inference and Learning Algorithms. Cambridge Univ. Press, Sept. 2003. [9] L. Kaufman and P.J. Rousseeuw, Finding Groups in Data: An Introduction to Cluster Analysis. Wiley Interscience, Mar. 2005. [10] R. Zass and A. Shashua, "A Unifying Approach to Hard Probabilistic Clustering," Proc. 10th IEEE Int'l Conf. Computer Vision, vol. 1, pp. 294-301, 2005. [11] H.A. Kiers, "Towards a Standardized Notation and Terminology in Multiway Analysis," J. Chemometrics, vol. 14, no. 3, pp. 105-122, 2000. [12] L. De Lathauwer, "A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization," SIAM J. Matrix Analysis and Applications, vol. 28, no. 3, pp. 642-666, http://publi-etis.ensea.fr/2006Del06, 2006. [13] T.G. Kolda and B.W. Bader, "Tensor Decompositions and Applications," SIAM Rev., vol. 51, no. 3, Sept. 2009. [14] R.A. Harshman, "Foundations of the PARAFAC Procedure: Models and Conditions for an 'Explanatory'," UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970. [15] R. Bro, "PARAFAC. Tutorial and Applications," Chemometrics and Intelligent Laboratory Systems, vol. 38, no. 2, pp. 149-171, Oct. 1997. [16] A. Cichocki, R. Zdunek, A.H. Phan, and S.I. Amari, Nonnegative Matrix and Tensor Factorizations. Wiley, Nov. 2009. [17] J.D. Carroll and J.-J. Chang, "Analysis of Individual Differences in Multidimensional Scaling via an $n$ -Way Generalization of 'Eckart-Young' Decomposition," Psychometrika, vol. 35, no. 3, pp. 283-319, http://ideas.repec.org/a/spr/psychov35y1970i3p283-319.html , Sept. 1970. [18] R.A. Harshman, "Determination and Proof of Minimum Uniqueness Conditions for PARAFAC1," UCLA Working Papers in Phonetics, vol. 22, pp. 111-117, 1972. [19] R.A. Harshman, "PARAFAC2: Mathematical and Technical Notes," UCLA Working Papers in Phonetics, vol. 22, pp. 30-47, 1972. [20] R. Zass and A. Shashua, "Doubly Stochastic Normalization for Spectral Clustering," Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T. Hoffman, eds., vol. 19, pp. 1569-1576, MIT Press, 2007. [21] P.M. Kroonenberg and T.H.A. van der Voort, "Multiplicatieve Decompositie van Interacties bij Oordelen over de Werkelijkheidswaarde van Televisiefilms [Multiplicative Decomposition of Interactions for Judgements of Realism of Television Films]," Kwantitatieve Methoden, vol. 8, no. 23, pp. 117-144, 1987. [22] J. Håstad, "Tensor Rank Is NP-Complete," J. Algorithms, vol. 11, no. 4, pp. 644-654, 1990. [23] M.E. Timmerman and H.A.L. Kiers, "Three-Mode Principal Components Analysis: Choosing the Numbers of Components and Sensitivity to Local Optima," British J. Math. and Statistical Psychology, vol. 53, no. 1, pp. 1-16, 2000. [24] R. Bro and H.A.L. Kiers, "A New Efficient Method for Determining the Number of Components in PARAFAC Models," J. Chemometrics, vol. 17, no. 5, pp. 274-286, 2003. [25] H.A.L. Kiers and A. der Kinderen, "A Fast Method for Choosing the Numbers of Components in Tucker3 Analysis," British J. Math. and Statistical Psychology, vol. 56, no. 1, pp. 119-125, May 2003. [26] E. Ceulemans and H.A.L. Kiers, "Selecting among Three-Mode Principal Component Models of Different Types and Complexities: A Numerical Convex Hull Based Method," British J. Math. and Statistical Psychology, vol. 59, no. 1, pp. 133-150, May 2006. [27] J.P.C.L. da Costa, M. Haardt, F. Römer, and G. Del Galdo, "Enhanced Model Order Estimation Using Higher-Order Arrays," Proc. 41st Asilomar Conf. Signals, Systems, and Computers, pp. 412-416, Nov. 2007. [28] J.P.C.L. da Costa, M. Haardt, and F. Römer, "Robust Methods Based on the HOSVD for Estimating the Model Order in PARAFAC Models," Proc. Fifth IEEE Sensor Array and Multichannel Signal Processing Workshop, pp. 510-514, July 2008. [29] J.B. Kruskal, Rank, Decomposition, and Uniqueness for 3-Way and N-Way Arrays. North-Holland Publishing Co., 1989. [30] P. Comon and J. ten Berge, "Generic and Typical Ranks of Three-Way Arrays," Proc. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing, pp. 3313-3316, Apr. 2008. [31] E. Kofidis and P.A. Regalia, "On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors," SIAM J. Matrix Analysis and Applications, vol. 23, no. 3, pp. 863-884, 2002. [32] T.P. Minka, "Automatic Choice of Dimensionality for PCA," Advances in Neural Information Processing Systems, T.K. Leen, T.G. Dietterich, and V. Tresp, eds, pp. 556-562, MIT Press, 2001. [33] M.O. Ulfarsson and V. Solo, "Dimension Estimation in Noisy PCA with SURE and Random Matrix Theory," IEEE Trans. Signal Processing, vol. 56, no. 12, pp. 5804-5816, Dec. 2008. [34] E. Radoi and A. Quinquis, "A New Method for Estimating the Number of Harmonic Components in Noise with Application in High Resolution Radar," EURASIP J. Applied Signal Processing, vol. 2004, no. 8, pp. 1177-1188, 2004. [35] J. Grouffaud, P. Larzabal, and H. Clergeot, "Some Properties of Ordered Eigenvalues of a Wishart Matrix: Application in Detection Test and Model Order Selection," Proc. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing,, vol. 5, pp. 2463-2466, 1996. [36] A. Quinlan, J.-P. Barbot, P. Larzabal, and M. Haardt, "Model Order Selection for Short Data: An Exponential Fitting Test (EFT)," EURASIP J. Advances in Signal Processing, vol. 2007, pp. 1-11, 2007. [37] A. Smilde, R. Bro, and P. Geladi, Multi-Way Analysis: Applications in the Chemical Sciences. John Wiley & Sons, Aug. 2005. [38] G. Milligan and M. Cooper, "An Examination of Procedures for Determining the Number of Clusters in a Data Set," Psychometrika, vol. 50, no. 2, pp. 159-179, June 1985. [39] W.J. Krzanowski and Y.T. Lai, "A Criterion for Determining the Number of Groups in a Data Set Using Sum-of-Squares Clustering," Biometrics, vol. 44, no. 1, pp. 23-34, Mar. 1988. [40] M.F. Antonio Cuevas and R. Fraiman, "Estimating the Number of Clusters," The Canadian J. Statistics, vol. 28, no. 2, pp. 367-382, June 2000. [41] W. Feller, An Introduction to Probability Theory and Its Applications, second ed., vol. 2. John Wiley & Sons, Jan. 1991. [42] R. Tibshirani, G. Walther, and T. Hastie, "Estimating the Number of Clusters in a Data Set via the Gap Statistic," J. Royal Statistics Soc. (Series B), vol. 63, no. 2, pp. 411-423, 2001. [43] M.A.F. Figueiredo and A.K. Jain, "Unsupervised Learning of Finite Mixture Models," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 3, pp. 381-396, Mar. 2002. [44] C.A. Sugar and G.M. James, "Finding the Number of Clusters in a Dataset: An Information-Theoretic Approach," J. Am. Statistical Assoc., vol. 98, no. 463, pp. 750-763, Sept. 2003. [45] M. Yan and K. Ye, "Determining the Number of Clusters Using the Weighted Gap Statistic," Biometrics, vol. 63, no. 4, pp. 1031-1037, Apr. 2007. [46] P. Guo, P. Chen, and M. Lyu, "Cluster Number Selection for a Small Set of Samples Using the Bayesian Ying-Yang Model," IEEE Trans. Neural Networks, vol. 13, no. 3, pp. 757-763, Apr. 2002. [47] X. Hu and L. Xu, "Investigation on Several Model Selection Criteria for Determining the Number of Cluster," Neural Information Processing—Letters and Rev., vol. 4, no. 1, pp. 1-10, 2004. [48] I.O. Kyrgyzov, O.O. Kyrgyzov, H. Maître, and M. Campede, "Kernel MDL to Determine the Number of Clusters," Lecture Notes in Computer Science, vol. 4571, pp. 203-217, Springer, 2007. [49] A. Dempster, N. Laird, and D. Rubin, "Maximum Likelihood from Incomplete Data via the EM Algorithm," J. Royal Statistical Soc., Series B, vol. 39, no. 1, pp. 1-38, 1977. [50] H. Akaike, "A New Look at the Statistical Model Identification," IEEE Trans. Automatic Control, vol. 19, no. 6, pp. 716-723, Dec. 1974. [51] H. Bozdogan, "Model Selection and Akaike's Information Criterion (AIC): The General Theory and its Analytical Extensions," Psychometrika, vol. 52, no. 3, pp. 345-370, Sept. 1987. [52] J. Rissanen, "Modelling by the Shortest Data Description," Automatica, vol. 14, pp. 465-471, 1978. [53] A. Barron, J. Rissanen, and B. Yu, "The Minimum Description Length Principle in Coding and Modeling," IEEE Trans. Information Theory, vol. 44, no. 6, pp. 2743-2760, Oct. 1998. [54] S.L. Sclove, "Some Aspects of Model-Selection Criterion," Proc. First US/Japan Conf. Frontiers of Statistical Modeling: An Informational Approach, H. Bozdogan, ed., vol. 2, pp. 37-67, 1994. [55] T. Calinski and J. Harabasz, "A Dendrite Method for Cluster Analysis," Comm. Statistics, vol. 3, pp. 1-27, 1974. [56] J.A. Hartigan, Clustering Algorithms. John Wiley & Sons, Apr. 1975. [57] C.A. Andersson and R. Bro, "The $n$ -Way Toolbox for MATLAB," Chemometrics and Intelligent Laboratory Systems, vol. 52, no. 1, pp. 1-4, 2000. [58] D.D. Lee and H.S. Seung, "Learning the Parts of Objects by Nonnegative Matrix Factorization," Nature, vol. 401, no. 6755, pp. 788-791, 1999. [59] D.D. Lee and H.S. Seung, "Algorithms for Non-Negative Matrix Factorization," Advances in Neural Information Processing Systems, T.K. Leen, T.G. Dietterich, and V. Tresp, eds., pp. 556-562, MIT Press, 2001. [60] A. Georghiades, P. Belhumeur, and D. Kriegman, "From Few to Many: Illumination Cone Models for Face Recognition under Variable Lighting and Pose," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 6, pp. 643-660, July 2001. [61] J. Ho, M.-H. Yang, J. Lim, K.-C. Lee, and D. Kriegman, "Clustering Appearances of Objects under Varying Illumination Conditions," Proc. 2003 IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 11-18, 2003. [62] E. Donchin, K.M. Spencer, and R. Wijesinghe, "The Mental Prosthesis: Assessing the Speed of a P300-Based Brain-Computer Interface," IEEE Trans. Rehabilitation Eng., vol. 8, no. 2, pp. 174-179, June 2000. [63] F. Piccione, F. Giorgi, P. Tonin, K. Priftis, S. Giove, S. Silvoni, G. Palmas, and F. Beverina, "P300-Based Brain Computer Interface: Reliability and Performance in Healthy and Paralysed Participants," Clinical Neurophysiology, vol. 117, no. 3, pp. 531-537, Mar. 2006.


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# 32 Straws Thingys! Aaah!

[This is post #26 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first or previous post.]

There’s one more modification we can make that changes the arrangement of straws without significantly affecting the overall composition: swap the roles of long and short ends! We’ve been sliding the short ends into the long ones to make the seam rest nicely along the flexy bits, but we should at least consider the opposite. In fact, my very first Straws Thingy made this choice instead, because I thought it nicely complemented the (less symmetric) 4-color color scheme.

But you know what this means: Each of yesterday’s 16 Straws Thingys can be reversed in this way, resulting in a whopping total of 32 alternative, distinguishable-but-just-barely Straws Thingys! And with 5 independent binary choices to make, we’ve built ourselves a lovely, colorful 5-dimensional hypercube of Straws Thingys.

Or maybe just an amorphous pile. Whatever you prefer.

Anyway, I’m sorry to say that this is the last day of exponential growth / straw Tribble invasion. But never fear—there are a few posts yet to come before we wrap up the series!

# 4D Hypercube of Straws Thingys!

[This is post #25 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

We’ve been accumulating lots of variant Straws Thingys, all subtly different, but all just as symmetrical as the others. We saw two ways of messing with the 5 macroscopic tetrahedra themselves, and then we zoomed into the molecular level and rearranged the 20 triangles that build these tetrahedra. Today, we finally venture to atomic scales and manipulate the 60 straws directly.

Let’s interrogate another scaffold feature. The arrows on the scaffold told us which way each straw should point: long end goes here, short end points this way. But what’s special about the arrows’ placement?

As with yesterday, there are two answers. Firstly, if you disregard the arrows and insert straws randomly (still following the dashed lines, of course), then the straws end up inconsistently oriented: you’ll get two long or two short ends trying to come together, when you actually need one of each.

But secondly, why not reverse all the arrows at once? Each straw gets turned around in place, so the triangles remain intact. This is a fine thing to do!

The difference is rather subtle (especially to photograph!), so here’s another way to see it: the location of the seam between successive straws. I’ve changed one straw from yellow to blue to highlight the effect:

In the original (left), the seam is neatly tucked away underneath another straw. After reversing straw orientation (right), the seam jumps outside the weave, a tad more visible. Only a tad. But for this reason (and only this reason), this arrow decision was deliberate, unlike the three prior differences where I broke ties arbitrarily.

Finally, as before, we can make this change on all 8 of our prior versions. This gives us 16 different Straws Thingys! Will it never end?! Will Target never run out of straws?!

By the way, these 16 things are determined by 4 independent binary axes, so this is a four-dimensional hypercube of Straws Thingys!

# Triangle Jitterbug: Up to Eight Versions

[This is post #24 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

We began describing Straws Thingy’s shape as 5 intersecting tetrahedra, so it is fitting that the two asymmetries we’ve identified so far (color scheme and mirror image) depend only on these tetrahedra.

Yesterday’s interloper, by contrast, requires us to look more closely at the 20 triangles, not just the 5 tetrahedra. Indeed, in yesterday’s Spot the Difference image,

the tetrahedra are in the same locations with the same color scheme, but each tetrahedron is woven differently from its four triangles.

Said differently, the spiral at each vertex rotates in the opposite direction. So this interloper occupies a middle ground between original and mirror: each tetrahedron is mirrored in place, but the tetrahedron arrangement isn’t affected.

And of course, this can be done to all four of our prior variants, so there are really 8 different Straws Thingys! (I still can’t resist making a full set!)

This also answers a question you may have considered while building your Straws Thingy:

Why are the elbows arranged as they are on the scaffold? What goes wrong if I don’t follow the elbows and weave the corners the other way around?

The practical answer is that the scaffold gets upset. Indeed, this alternate weaving needs the scaffold holes in different locations. (In fact, this is a completely different arrangement of cylinders, and the inflation optimization even returns a different ideal length! [28.42 instead of 27.43, which, thankfully, fits comfortably in our error margins.])

But more fundamentally, nothing goes wrong! I equally could have settled on the other scaffold instead, and I don’t have a strong preference between the two. So we now have 8 different Straws Thingys, all distinguishable, but none especially distinguished from the crowd.

# But Wait, There’s More!

[This is post #23 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

To finish yesterday’s train of thought, how should you proceed if you wanted to make some mirror image Straws Thingys?

You you could pull some trickery and print the scaffold in mirror image (do printers make this easy to do?), but it’s equivalent to just assemble the regular scaffold inside-out and look at the ink from behind. Follow the arrows and elbows as normal, just from the other side of the paper.

That way, you can make a nice, complete collection of all four Stra…

PSST!

What?! Can it wait? I’m trying to finish a blog post here!

Well, it’s just, uh, there’s another one here to see you.

Another what?

Um, another Straws Thingy.

That’s the same as before! What does this Thingy think it is, claiming to be new and different…

Well, it is, um, very insistent. Maybe you could ask them? (Reduce to conspiratorial whisper.) The readers? (Cue strings.)

Yes, perhaps you’re right. Readers? It looks like we have another game of Spot the Difference on our hands! What makes this Straws Thingy different from the other Straws Thingys?

(Original on the left, insistent interloper on the right.)

# Mirror Thingy on the Wall

[This is post #22 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

If you step in front of a mirror, don’t be alarmed if your Thingy looks a bit strange—your Straws Thingy, that is.

The Straws Thingy is not identical to its mirror image! In other words, it is chiral. Here’s a physical copy of this mirror image for closer inspection. (This is two real thingys side-by-side. I didn’t just flip the picture.)

One way to tell them apart is to look at their driving conduct. If we imagine cars driving toward and past each other along edges, starting at neighboring vertices (in this case, blue and green), the cars will keep to the right side of the road. In the mirror images, they keep left.

Seen a different way, as the straws spiral outward from the center of a face, they spiral counterclockwise for our original, but clockwise in the mirror image.

And as we saw yesterday, the original and the mirror image each have two distinct color schemes. So there are really four different Straws Thingys! I couldn’t resist building all of them.

# About Faces

[This is post #21 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Knowing the color pattern around any one of the twelve faces (5-sided stars) of a Straws Thingy tells us exactly how the tetrahedra are colored, so it lets us reconstruct the entire color scheme. In other words, any one face determines all 12 faces. This is useful for quickly comparing color schemes. The “original” Straws Thingy from yesterday has these twelve faces, in no particular order:

For example, the face pictured in the top left has the colors Blue (B), Orange (O), Pink (P), Green (G), and Yellow (Y) in counterclockwise order around the face, so we’ll write it as BOPGY. It also doesn’t matter where we start—this is the same ordering as PGYBO, or YBOPG, or…—so to be consistent, let’s choose to always start at Blue. Then these twelve orderings are, respectively:

BOPGY, BPYGO, BGYOP, BPGOY, BGPYO, BYPGO, BOYPG, BYOGP, BPOYG, BOGYP, BYGPO, and BGOPY.

You can check that all 12 of these orderings are different, but remember that any one of them can be used to identify this original Straws Thingy.

So if we’re handed a new Straws Thingy and we want to know if it matches the original one, we only need to check a single face of the new one: if it’s in this list, the two Straws Thingys are identical, so the twelve faces of the new one will be exactly these twelve faces. If not, then there can be no face matches at all: the twelve orderings of the new faces can’t be in the above list.

So how many orderings aren’t in this list?

Well, how many orderings are there in total? We have to start with Blue, but then any of the four remaining colors can come next, then any of the three colors can come after that, then either of the two remaining colors follows, finishing with the only color remaining. So there are $$4\times 3\times 2\times 1 = 4! = 24$$ face orderings in total.[1]

With 12 of those orderings belonging to the “original” Straws Thingy, there are only 12 other faces to pick from. So any other Straws Thingy, like yesterday’s “odd one out,” has to use exactly the other 12 orderings. In other words, there’s only one other Straws Thingy color arrangement! Any Straws Thingy you build will match either the “original” or the “odd one out”; which did you build?

### Notes

1. That middle term is not an excited “4”. It’s pronounced 4 factorial, and in general, $$n! = n\times (n-1)\times\cdots\times 2\times 1$$ counts the number of ways to arrange n items in a line, or as we saw here, $$n+1$$ items in a circle. []

# Scheming Colors

[This is post #20 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first , previous, or next post.]

Yesterday I asked what differentiates one of these from the other two:

The answer: color scheme! They all have 5 tetrahedra, one in each color, but they’re not positioned in the same way every time. The two Straws Thingys on the edges are identical, though it’s not apparent until we rotate one to match the other:

To avoid motion sickness, here’s that final frame again. Check that the colors match perfectly in the two assemblies:

By contrast, when comparing one of these “original” Straws Thingys with the remaining “odd one out” (pictured above, center), the colors refuse to match up, try as you might to rotate them into alignment. We can get pretty close, though: below, only green and yellow have swapped places. (The “original” is on the right, with the “odd one out” on the left.)

Does this mean we get a third unique color scheme if we swap, say, green and blue instead? How many colorways are possible in total? Let’s investigate this tomorrow.

# Odd One Out?

[This is post #19 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Congrats again on finishing your Straws Thingy! (If you had or are having trouble, please let me know.)

During the next week we’ll prod more deeply into a few features of this sculpture’s symmetries, especially those that become apparent during construction.

To begin, a puzzle: all three of these Straws Thingys were built precisely according to the instructions outlined earlier, and yet, they’re not identical arrangements! What differs?

This is not meant to be a trick question: their dissimilarity is clearly visible in this image without zooming in or guessing. And in fact, two of these three are identical. Who’s the odd one out? Are there other ways this difference can manifest?

# Step 10: A Savage Ending

[This is post #18 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

The straws are locked and loaded, but how do we get the scaffold off?

Violently!

It’s finally complete. Congratulations on building your very own Straws Thingy!

Look forward to a return to mathematical musings tomorrow.

# Step 9: Quintuplicate!

[This is post #17 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Here’s where we left off yesterday:

Now just repeat four more times!


text:
# Prove that $\lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!} = e$

Define $$e, e'$$ by $$e: =\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{n} \quad \text{and} \quad e' := \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!}$$

Prove that $$e' \in \mathbb R$$ and $$e' = e$$.

Could you please verify whether my attempt is fine or contains logical gaps/errors? Any suggestion is greatly appreciated!

PS: I've not learned about derivative and logarithmic function yet.

My attempt:

To make the presentation easier to follow, I set $$e_n := \left(1+\frac{1}{n}\right)^{n} \quad e'_n := \sum_{k=0}^{n} \frac{1}{k!}$$ Clearly, $$e'_n + 1/((n+1)!) = e'_{n+1}$$ and thus the sequence $$(e'_n)$$ is increasing. On the other hand, $$e'_n = 2 + \sum_{k=2}^{n} \frac{1}{k!} \le 2 + \sum_{k=1}^{n} \frac{1}{k(k+1)} = 2+(1-\frac{1}{n+1}) = 3 - \frac{1}{n+1} < 3$$. So the sequence $$(e'_n)$$ is bounded from above and thus $$e'$$ is well defined.

By binomial theorem, $$e_n = \sum_{k=0}^n {n \choose k} \frac{1}{n^k}$$ where $${n \choose k} \frac{1}{n^{k}}=\frac{1}{k !} \frac{n \cdot(n-1) \cdot \cdots \cdot(n-k+1)}{n \cdot n \cdot \cdots \cdot n} \leq \frac{1}{k !}$$. As such, $$e_n \le \sum_{k=0}^n \frac{1}{k !} = e'_n$$ and so $$e \le e'$$. Next we prove that $$e \ge e'$$.

For $$n \ge m$$, we have

\begin{aligned} e_{n}=\left(1+\frac{1}{n}\right)^{n} &=\sum_{k=0}^{n} {n \choose k} \frac{1}{n^{k}} \geq \sum_{k=0}^{m} {n \choose k} \frac{1}{n^{k}} \\ &=1+\sum_{k=1}^{m} \frac{1}{k !} \frac{n (n-1) \cdots (n-k+1)}{n \cdots n} \\ &= 1+\sum_{k=1}^{m} \frac{1}{k !} \left[ 1 \cdot\left(1-\frac{1}{n}\right) \cdots\left(1-\frac{k-1}{n}\right) \right] \end{aligned}

For $$n \ge m$$, I set $$x_{n,m} = 1+\sum_{k=1}^{m} \frac{1}{k !} \left [ 1 \cdot\left(1-\frac{1}{n}\right) \cdots\left(1-\frac{k-1}{n}\right) \right]$$. Then $$e_n \ge x_{n,m}$$ and thus $$e = \lim _{n \rightarrow \infty} e_n \ge \lim _{n \rightarrow \infty} x_{n,m} = e'_m$$. As such, $$e \ge e'$$. This completes the proof.

• See here : math.stackexchange.com/questions/99016/… Aug 9, 2019 at 8:41
• Seems correct to me. Aug 9, 2019 at 8:45
• wow, this was perfectly written (and correct) Aug 9, 2019 at 8:47
• Thank you so much for all of your support ^^ Aug 9, 2019 at 8:50


text:
# bases

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2. ### Bases

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3. ### Bases, Exponents, and Digits

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6. ### Finding subspaces and bases of vector bases

Hi guys I am having trouble with the following question S= {(2,-1,4), (1,1,-1)}, T= {(x,y,x,-2y):x,y\in R} a)Show that S\subseteq T and write down a vector in R3 that does not belong to T b)Show that T is a subspace of R3 Is the first part of the question asking to prove that S is a...
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8. ### Desperately need help with three questions concerning images, kernels, bases, etc.

Please see this link: Linear Algebra Questions - TexPaste There are three questions. Shouldn't take longer than 10 minutes. Please give me answers for now. When I'm working it out, if there's anything I don't get, I'll come back for guidance. Thank you so much.
9. ### Basic concepts on bases

1. Can the basis be {(1, -1, 0, 0)T, (0, 0, 1, -1)T}? I wonder if the ans is right. Here x2, x4 are supposed to be the free variables, and x1, x3 as dependent variables; but the ans used x1, x3 as free variables. Or am I wrong???? Both can be bases?? 2. When asked to find a basis, we just...
10. ### Addition in Other Bases

When we add 2 + 4, the answer is 6. The addition above is called addition in base 10, right? How do we add in other bases? For example, how do we add 2 + 4 in base 2? How about in base 3?
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\text{If }a=\log_{12} 18, b=\log_{24}54\text{, then find the value of }ab+5(a-b) I am trying in different ways to equalize the bases. But I can't do anything as the bases are unequal.
12. ### Linear Algebra-Transition Matrices

Hey guys, I'm after some help on this problem. Help would be much appreciated.
13. ### Bases for R2

Course: Linear Algebra So I have the set \left\{(0,0) , (1,3)\right\} which I will call S. I'll have the system: c_2 = x_1 3c_2 = x_2 Which gives me c_2 = x_1 = \frac{1}{3}x_2 Can this mean that span(S) = R2? Is the zero vector redundant?
14. ### Determine the matrices that represent the following rotations of R^3

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15. ### Logarithmic equation with different bases

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16. ### Grobner Bases - Second question on D&F Proposition 24

I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases. I have a second problem (see previous post for first problem) understanding a step in the proof of Proposition 24, Page 322 of D&F Proposition 24 reads as follows: Proposition 24. Fix a...
17. ### Grobner Bases

I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases. I have a problem understanding a step in the proof of Proposition 24, Page 322 of D&F Proposition 24 reads as follows: Proposition 24. Fix a monomial ordering on R= F[x_1, ... , x_n]...
18. ### Find the Matrix for T Relative to the bases {1, t, t^2} and {1, t, t^2, t^3}.

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19. ### Need help finding one of the bases

I have a trapezoid with a top base (b1) of 10 and a height of 5. Considering the area, find b2 Do I need to find the area first? If so, can someone explain how I go about that. You don't need to give me the answer, but I'm not sure where to start. Thanks!
20. ### Solving Equation: Exponent with different bases

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