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### - Art Gallery -

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor.

It can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite.

Cantor linked the Absolute Infinite with God,[1] and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.[2]

Cantor's view

Cantor said:

The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extraworldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or ordertype. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.[3]

Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original):[6]

A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a "sequence".

...

Now I envisage the system of all [ordinal] numbers and denote it Ω.

...

The system Ω in its natural ordering according to magnitude is a "sequence".
Now let us adjoin 0 as an additional element to this sequence, and place it, obviously, in the first position; then we obtain a sequence Ω′:

0, 1, 2, 3, ... ω0, ω0+1, ..., γ, ...
of which one can readily convince oneself that every number γ occurring in it is the type [i.e., order-type] of the sequence of all its preceding elements (including 0). (The sequence Ω has this property first for ω0+1. [ω0+1 should be ω0.])

Now Ω′ (and therefore also Ω) cannot be a consistent multiplicity. For if Ω′ were consistent, then as a well-ordered set, a number δ would correspond to it which would be greater than all numbers of the system Ω; the number δ, however, also belongs to the system Ω, because it comprises all numbers. Thus δ would be greater than δ, which is a contradiction. Therefore:

The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.

The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical to many. This is related to Cesare Burali-Forti's "paradox" which states that there can be no greatest ordinal number. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.

More generally, as noted by A. W. Moore, there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy. Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.

A standard solution to this problem is found in Zermelo's set theory, which does not allow the unrestricted formation of sets from arbitrary properties. Rather, we may form the set of all objects that have a given property and lie in some given set (Zermelo's Axiom of Separation). This allows for the formation of sets based on properties, in a limited sense, while (hopefully) preserving the consistency of the theory.

While this solves the logical problem, one could argue that the philosophical problem remains. It seems natural that a set of individuals ought to exist, so long as the individuals exist. Indeed, naive set theory might be said to be based on this notion. Although Zermelo's fix allows a class to describe arbitrary (possibly "large") entities, these predicates of the meta-language may have no formal existence (i.e., as a set) within the theory. For example, the class of all sets would be a proper class. This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.

Actual infinity
Limitation of size
Reflection principle
The Ultimate (philosophy)

Notes

§3.2, Ignacio Jané (May 1995). "The role of the absolute infinite in Cantor's conception of set". Erkenntnis. 42 (3): 375–402. doi:10.1007/BF01129011. "Cantor (1) took the absolute to be a manifestation of God [...] When the absolute is first introduced in Grundlagen, it is linked to God: "the true infinite or absolute, which is in God, admits no kind of determination" (Cantor 1883b, p. 175) This is not an incidental remark, for Cantor is very explicit and insistent about the relation between the absolute and God."
Infinity: New Research and Frontiers by Michael Heller and W. Hugh Woodin (2011), p. 11.
https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf
Translated quote from German:

Es wurde das Aktual-Unendliche (A-U.) nach drei Beziehungen unterschieden: erstens, sofern es in der höchsten Vollkommenheit, im völlig unabhängigen außerweltlichen Sein, in Deo realisiert ist, wo ich es Absolut Unendliches oder kurzweg Absolutes nenne; zweitens, sofern es in der abhängigen, kreatürlichen Welt vertreten ist; drittens, sofern es als mathematische Größe, Zahl oder Ordnungstypus vom Denken in abstracto aufgefaßt werden kann. In den beiden letzten Beziehungen, wo es offenbar als beschränktes, noch weiterer Vermehrung fähiges und insofern dem Endlichen verwandtes A.-U. sich darstellt, nenne ich es Transfinitum und setze es dem Absoluten strengstens entgegen.

[Ca-a, p. 378].
Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Georg Cantor, ed. Ernst Zermelo, with biography by Adolf Fraenkel; orig. pub. Berlin: Verlag von Julius Springer, 1932; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBN 3-540-09849-6.
The Rediscovery of the Cantor-Dedekind Correspondence, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974/75), pp. 104–139, at p. 126 ff.

Gesammelte Abhandlungen,[4] Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinness has discovered,[5] this is in fact an amalgamation by Cantor's editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.

Bibliography

The role of the absolute infinite in Cantor's conception of set
Infinity and the Mind, Rudy Rucker, Princeton, New Jersey: Princeton University Press, 1995, ISBN 0-691-00172-3; orig. pub. Boston: Birkhäuser, 1982, ISBN 3-7643-3034-1.
The Infinite, A. W. Moore, London, New York: Routledge, 1990, ISBN 0-415-03307-1.
Set Theory, Skolem's Paradox and the Tractatus, A. W. Moore, Analysis 45, #1 (January 1985), pp. 13–20.


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# Define every symbol as a command in LaTeX

I'm working on a large project involving multiple documents typeset in LaTeX. I want to be consistent in my use of symbols, so it might be a nice idea to define a command for every symbol that has a specific meaning throughout the project. Does anyone have any experience with this? Are there issues I should pay attention to?

A little more specific. Say that, throughout the document I would denote something called permability by a script P, would it be an idea to define

\providecommand{\permeability}{\mathscr{P}}

or would this be more like the case "defining a command for $n$"?

-

I have been doing this for anything that has a specific meaning and is longer than a single symbol, mostly to save typing:

\newcommand{\objId}{\mbox{$\mathit{objId}$}\xspace}
\newcommand{\insOp}[1]{#1\mbox{$^+$}\xspace}
\newcommand{\delOp}[1]{#1\mbox{$^-$}\xspace}

However then I noticed that I stopped making inconsistency errors (objId vs ObjId vs ObjID), so I agree that it is a nice idea.

However I am not sure if it is a good idea in case symbols in the output are, well, single Latin symbols, as in:

\newcommand{\numOfObjs}{$n$}

It is too easy to type a single symbol and forget about it even though a command was defined for it.

EDIT: using your example IMHO it'd be a good idea to define \permeability because it is more than a single P that you have to type in without the command. But it's a close call.

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Thanks. I think I'll use command, especially because I tend to change my notation sometimes. –  Martijn Jun 25 '09 at 6:25

A few tips:

• Using \providecommand will define that command only if it's not been previously defined. So if you're not getting the results you expected, you may be trying to define a command that's been defined elsewhere.
• If you wrap the math in your commands with \ensuremath, it will do the right thing regardless of whether you're in math mode when you issue the command:

\providecommand{\permeability}{\ensuremath{\mathscr{P}}}
Now I can easily use \permeability in text or $\permeability$ in math mode.

• Using your own commands allows you to easily change the typographical representation of something later. For instance:

\newcommand{\vect}[1]{\ensuremath{\mathbf{#1}}}

would print \vect{x} as a boldfaced x. If you later decide you prefer arrows above your vectors, you could change the command to:

\newcommand{\vect}[1]{\ensuremath{\vec{#1}}}

-

format: plain


text: The number $12 = 2^2 \times 3$ has 6 factors. What is the smallest natural number with exactly 36 factors?


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# The solution to the equation $- \Delta u = (\lambda - \log u)u$

$\Omega$ is a bounded domain in $\mathbb R^n$ with smooth boundary. Consider the Dirichlet problem:$$-\Delta u = (\lambda - \log u)u ~~~{\rm on}~~~ \Omega~~~ {\rm and}~~~u=0 ~~~{\rm on}~~\partial {\Omega}$$ where the solution $u\in {C^\infty }(\bar \Omega )$ and $u>0$ on $\Omega$ and $\lambda$ is a positive constant.

My question is: what conditions do we need for the existence and uniqueness of the solution $u$ ?

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Why do yout think that a solution of this problem is $C^\infty(\overline{\Omega})$? –  Tomás May 14 '13 at 12:25

Define $$f(t) = \left\{ \begin{array}{rl} -t\log{t} &\mbox{ if t\in[0,1]} \\ 0 &\mbox{ otherwise} \end{array} \right.$$

Note that $f$ is continuous. COnsider the problem

$$\tag{1} \left\{ \begin{array}{rl} -\Delta u-\lambda u=f(u) &\mbox{ in \Omega} \\ u=0 &\mbox{ in \partial\Omega} \end{array} \right.$$

1 - Existence

I - There is some methods avaliable to find a solution to $(1)$. For example, we can use the fact that $f$ is continuous with compact support, to prove that the energy functional $I:H_0^1\to\mathbb{R}$ associated with $(1)$ is (under some additional restriction) weakly lower semi continuous and coercive. Remember that if $F(t)=\int_0^t f(s)ds$, then $$Iu=\frac{1}{2}\int_\Omega |\nabla u|^2-\frac{\lambda}{2}\int_\Omega u^2-\int_\Omega F(u)$$

The additional restriction necessary here is $\lambda<\lambda_1$ (to guarantee coercivity).

II - (This is just a guess - I dont have verified the calculations): Also, you can prove existence, by noting that if $\lambda$ is not a eigenvalue of $-\Delta$, then the operator $-\Delta-\lambda I$ is invertible, and you can use arguments similar to this one.

Remark 1: The original function $t\log{t}$ is not monotone, so I dont know if this problem has uniqueness.

Remark 2: The function $f$ being continuous, we can conclude that $u\in C^1(\overline{\Omega})$.

Now, let's find conditions such that the solution $u$ of $(1)$ satisfies $0<u\leq 1$. Because $f\geq 0$ and there is a open set where $f>0$, you can conclude by the maximum principle that $u>0$. On the other hand, by regularity theory, you have that the solution $u$ of $(1)$ (See Breziz chapter 9) satisfies $(1)$ almost everywhere, i.e. $u$ is a Strong solution. If we suppose that the set where $u>1$ is non0empty, then we would have $-\Delta u-\lambda u=0$ almost everywhere in this set. If $\lambda$ is not a eingevalue, then this is a absurd, which implies that $u\geq 1$.

From the last paragraph, we can conclude that in fact $u$ is a solution to the original problem and satisfies $0<u\leq 1$. Hence, by using the first method proposed, we can conclude that if $\lambda<\lambda_1$, then you problem has a solution $u\in C^{2,\alpha}(\overline{\Omega})$ satisfying $0<u\leq 1$. I dont know if we can get more regularity, because the functions $-t\log{t}$ is only Holder continuous. A argument of bootstrap can be applied in the interior of $\Omega$, but in the boudary I dont know, because $-t\log{t}$ is not even Lipschitz for $t=0$.

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# Electronic – Equations of Power Flow Between Buses

power-flow-analysis

I'm working in a MATLAB algorithm to compute the state estimation of a power system given some measures. The classical method computes a Jacobian matrix of all the measures, so I have to program all the derivatives of all types of measures. The problem lies in the equations of the power flow between buses. I've searched several books and papers and I've found two distinct sets of equations (\$P_{ij}\$ and \$Q_{ij}\$ are the real and reactive power flow from node \$i\$ to node \$j\$, respectively):
$$P_{ij} = \left| {V_i } \right|^2 G_{ij} – \left| {V_i } \right|\left| {V_j } \right|\left( {G_{ij} \cos \delta _{ij} + B_{ij} \sin \delta _{ij} } \right)$$
$$Q_{ij} = – \left| {V_i } \right|^2 B_{ij} – \left| {V_i } \right|\left| {V_j } \right|\left( {G_{ij} \sin \delta _{ij} – B_{ij} \cos \delta _{ij} } \right)$$
Where \$V_i\$ and \$V_j\$ are the complex voltages of buses \$i\$ and \$j\$. \$G_{ij}\$ and \$B_{ij}\$ are the real and imaginary parts of \$Y_{ij}\$, the \$ij\$ element of the admitance matrix of the system. And \$\delta_{ij}\$ is the angle difference betwwen \$V_i\$ and \$V_j\$. These two equations are derived from the book of Abur and Expósito (2004), neglecting the lines' shunt admitance. However, from the paper of Larson, Tinney and Peschon (1970), the resulting equations are:
$$P_{ij} = -\left| {V_i } \right|^2 G_{ij} + \left| {V_i } \right|\left| {V_j } \right|\left( {G_{ij} \cos \delta _{ij} + B_{ij} \sin \delta _{ij} } \right)$$
$$Q_{ij} = \left| {V_i } \right|^2 B_{ij} + \left| {V_i } \right|\left| {V_j } \right|\left( {G_{ij} \sin \delta _{ij} – B_{ij} \cos \delta _{ij} } \right)$$
There are other documents that present either the first or the second set of equations, and I'm really bewildered, I absolutely don't know which set I must use. Can someone explain to me the reason of this discrepancy, and which is the correct set of equations? My most sincere thanks to whoever answers me.


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# 2-approximation edge-cover algorithm using primal-dual method

## The problem

Given an undirected graph $G=\left(V, E\right)$ and positive edge weights $w_e$, design a 2-approximation algorithm based on the primal-dual principle.

So far I managed to represent the problem with a linear program:

## Linear Program

Primal

$\min \underset{e\in E}{\sum} w_e x_e$

s.t.

• $\underset{e|v\in e}{\sum}x_e\geq 1$ for $\forall v\in V$
• $x_e \geq 0$ for $\forall e \in E$

Dual

$\max \underset{v\in V}{\sum} y_v$

s.t.

1. $y_u + y_v \leq w_e$ for $\forall e=\left(u, v \right) \in E$
2. $y_v \geq 0$ for $\forall v \in V$

## Tentative algorithm

1. $C \leftarrow \emptyset$

2. while $C$ is not a cover

2.1. Pick a node $v$ which is not covered

2.2. Increase $y_v$ until a contrains $y_u + y_v \leq w_e$ becomes tight for some $e$

2.3. $C \leftarrow C \bigcup \left\{ e \right\}$

Initial analysis

Since for every $e \in C$ the corresponding constraint is tight, then $\underset{e\in C}{\sum}w_e = \underset{e=\left(u, v \right) \in C}{\sum}y_u+y_v$. Also, since the dual solution is feasible, it must be that $\underset{v\in V}{\sum}y_v \leq \left| OPT \right|$.

So far I've managed to bound it by $d$ which is the highest degree in $G$, since $\underset{e=\left(u, v \right) \in C}{\sum}y_u+y_v \leq \underset{v|e=\left(u,v \right)\in C}{\sum}d \cdot y_v$. This is very similar to the classic version of set-cover using the primal-dual method.

I wonder if and how the analysis can be improved so it provides a 2-approximation ratio.

• The standard analysis (using only the primal) takes all edges with $x_e \geq 1/2$. Perhaps you can modify this analysis to use the primal-dual approach. Jun 30 '17 at 14:21


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# The relation y^3+y^2=x^3+x^2

Up a level : Differentiation, derivatives
Previous page : The circle and implicit derivatives
Next page : y^3+y^3=x^2+x

Let us say we should graph

${y^3} + {y^2} = {x^3} + {x^2}$

Technically we could make y the subject of it, but the solution is horrid (try on say Mathematica or Nspire CAS and you will see) and not very useful, so let us try to analyze it instead. To start with we can see that all points such that x=y are on the graph.

We can also see that x and y are used in a totally symmetric way, so we now that if a point (x, y) is on the graph then the point (y, x) is on the graph too.

Zeroes

Next we look for x and y intercepts.  If x=0 we get

${y^3} + {y^2} = 0$

Or

${y^2}(y + 1) = 0$

So y=0 or y=−1. If y=0 we get that

$0 = {x^3} + {x^2}$

This gives x=0 or x=−1.

For y=−1 we get

${( - 1)^3} + {1^2} = 0 = {x^3} + {x^2}$

That we have already dealt with. So we now have a line y=x and two points (0, −1) and (−1, 0)

Critical ponits – Horizontal tangents

To find a point with a horizontal tangent we should find points where y´=0.

We thus need to find the implicit derivative of our relation. We get

$3{y^3}y' + 2yy' = 3{x^2} + 2x$

or

$y'(3{y^3} + 2y) = 3{x^2} + 2x$

So if y´=0 we get

$0 = 3{x^2} + 2x = x(3x + 2)$

And thus x=0 or x=−2/3. Next we can find the y-values by substituting in these values in the original equation For  x=0 we have already found y=0 and y=−1.

The solution (0, 0) does not work since that is on the already established line y=x, that clearly does not have the slope 0.  And if you let x=0 and y=0 in the equation for the derivative that we can get from the result we got after the implicit derivative we get that y´=0/0, i.e. undefined. So what could happen at that point? Asking my student this one student replied after a short moment of thought that the graph might cross itself there, which indeed is the case. Nota bene that this was my introductory problem for implicit derivative.

So we have a point with the slope 0 at (0 ,1).

For x=−2/3 we get

$\begin{gathered} {y^3} + {y^2} = {\left( { - \frac{2}{3}} \right)^3} + {\left( { - \frac{2}{3}} \right)^2} = - \frac{8}{{27}} + \frac{4}{9} \hfill \\ \quad \quad \quad = - \frac{8}{{27}} + \frac{{12}}{{27}} = \frac{4}{{27}} \hfill \\ \end{gathered}$

So we need to solve the equation

${y^3} + {y^2} - \frac{4}{{27}} = 0$

Since we have x=−2/3 we myst, by symmetry we must have have y=−2/3 as one root. With a bit of algbra we can find that  y=1/3 is another solution – but that is not what we should concentrate on now, so, say we have just fond the roots in some way.

We can discard the solution y=−2/3 since it is on the y=x line. And as before we can suspect that the graph cross itself at that point.  That leaves us with the point (−2/3,  1/3).

Vertical tangents

Of symmetry reasons these must be at (1, 0) and (1/3,  −2/3). This gives us:

How to connect these points in some reasonable way? It looks like it could be some kind of ellipse there, and indeed, if we graph it we get:

Rather cool I must say.  Next thing is to prove that the ellipse-like figure actually is an ellipse, but I leave that to you to figure out, at least for the moment.

Up a level : Differentiation, derivatives
Previous page : The circle and implicit derivatives


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# Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.$r(t) = \langle \cos 2t, \cos 3t, \cos 4t \rangle$

## GeoGebra command:Curve $(\cos (2 t), \cos (3 t), \cos (4 t), t, 0,2$ pi)

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### Video Transcript

the problem. These used a computer to graph the curve we've given locked in creation. Make sure you choose a permitted on Wayne and will wants that reveal the true nature of the curve. My tears have to confine twenty side thirty on Science forty now, by using some graphing wise, we can graft the clear off follows here. This is a curve off, given Parametric equations and that shoes he's from zero to two parts.

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text:
# Naming of an [ordered-groups] tag

It seems like there are a decent number of questions about placing some sort of order on a group. I would like to tag these questions to better organize them, but I'm not sure what to call the tag.

The phrase "ordered group" to me implies that we have a total order $\leq$ which is preserved under both left and right multiplication. But people could be asking questions about partial or lattice orders, or the ordering could be preserved only under right multiplication, etc.

Creating separate tags for each of these niche subjects seems like overkill to me, so is the standard to just call it [ordered-groups] and assume people will be able to figure out the ambiguity? One alternative would be to call it [partially-ordered-groups] and hope that people realize that even if they have a stronger restriction, they can still use the tag.

Examples:

(These were taken by searching for questions tagged as both [group-theory] and [order-theory]. I filtered out the ones asking about the order of a group.)

• There is also a possibility to create two tags and make them synonyms. – Martin Sleziak Sep 19 '14 at 17:46
• Maybe provide a list of questions (with links) that you feel should use this tag. It would certainly help in ascertaining the viability of the tag. (Also, what would fall under the tag? Clearly not every question about the real line would, but what would the boundaries of the tag be?) – user642796 Sep 19 '14 at 18:46
• Thanks, I added some examples. @Arthur: it seems to me that if it's appropriate to add the [group-theory] tag and you care about an ordering, the [ordered-group] tag would apply. – Xodarap Sep 19 '14 at 20:16
• "I filtered out the ones asking about the order of a group." -- if [ordered-groups] exists, it will be applied to such questions too. Just a warning. – user147263 Sep 19 '14 at 23:26
• Thanks for creating tag-excerpt and tag-wiki. (There are many users who do not bother to do that when creating a new tag.) If you plan to add the tag to some of the examples you listed in your post, keep in mind not to do too many retags in a short timespan. – Martin Sleziak Sep 21 '14 at 7:29


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# How is the VP=VNP question in char 2 different from other char? What is the current frontier in regards to this question?

What are the caveats one should be aware of when pursuing VP=VNP question in char 2 compared to other char? What is the current frontier in regards to this question?

The relationship of the VP vs VNP question in different characteristics to Boolean complexity classes depends on the characteristic, but in an obvious way (e.g. Boolean complexity classes defined in terms of a similar characteristic, like $\mathsf{Mod_n P}$).
(Another caveat to be aware of is the distinction between functions over $\mathbb{F}_2$ and formal polynomials over $\mathbb{F}_2$. As functions, $x^2 - x$ and $0$ are the same, but as formal polynomials over $\mathbb{F}_2$ these are distinct. VP and VNP are usually defined in terms of formal polynomials.)
• No, it involves division by 2 (as do the proofs of the #P-hardness of perm). In characteristic 2, perm=det, so if the VNP-completeness proof for perm worked in characteristic 2 then we'd have that VP=VNP over $\mathbb{F}_2$, which implies (by Burgisser) that $\oplus P/poly \subseteq FNC^2/poly$... – Joshua Grochow Jul 28 '16 at 18:01


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# When will the supporting hyperplane of a convex set coincide with the tangent?

Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I want to discuss the uniqueness of the supporting hyperplane.

A special situation where uniqueness holds is when the supporting hyperplane at $p\in\partial C$ coincides with the tangent pf $\partial C$ at $p$. For example, when the $\partial C$ is piecewise linear, then the supporting hyperplanes must coincide with tangent planes except at vertices of $C$.

There are sufficient conditions, say $C$ being strictly convex, which guarantees the uniqueness of the supporting hyperplane. But strict convexity does not lead to the conclusion that the supporting hyperplane coincide with tangents. Moreover. since the supporting hyperplane theorem is no more than Hahn-Banach theorem, I was wondering

(i)Is there a necessary condition (on $\partial C$, say some sort of algebraic regularity; conditions on $C$ are also fine but less interesting) about the boundary of convex set $C$ to make the supporting hyperplane unique?

(ii)Is there a necessary condition (on $\partial C$, say some sort of algebraic regularity) about the boundary of convex set $C$ such that the supporting hyperplane of $C$ at any point of $p\in\partial C$ coincide with the tangent plane of $\partial C$ at $p$?

For my purpose, I only want to know the case $C\subset\mathbb{R}^n$ with the usual topology. ($C$ does not have to be closed convex set but only convex, closedness is too strong for my purpose.)

Additionally, how will the situation changes if $C\subset H$ for a general (infinite dimensional) Hilbert space? Or even more general, in a Banach space $B$.

• Keyword: smooth point (= point de lissité). – Francois Ziegler Mar 28 '17 at 14:21
• The finite-dimensional case is answered in detail in Rockafellar's book Convex Analysis. Basically, it is the case that the supporting hyperplane is unique at a point $p \in \partial C$ if and only if it is the tangent hyperplane at $p$ if and only if $\partial C$ is differentiable at $p$. – Deane Yang Mar 28 '17 at 14:44
• @DeaneYang After a careful look into the relevant sections of Rockafellar's treatise, the relevant result claimed in the book seems to be Theorem 18.8 and those in Sec 25. However, I did not recognize where he proved "iff", so if would be more explanatory if you could add your citations. Moreover, in Sec 18, Rockafellar actually showed that closed convex set in R^n is always envelope of tangent half-spaces, which does not serve as a necessary condition to my OP since I did not put closedness in OP. Appreciated. – Henry.L Mar 28 '17 at 15:34
• The way I recall it is this: Rotating the body if needed, the boundary near a given point is the graph of a convex function. Each element of the subdifferential of $f$ at that point defines a supporting hyperplane. $f$ is differentiable at the point if and only if there is only one element in the subdifferential and therefore only one supporting hyperplane. I don't have a copy of the book handy, but this is what I recall. – Deane Yang Mar 28 '17 at 19:04
• @DeaneYang Thanks for the clarification,I need some time to figure out which result yo are referring to, you are also welcomed to compose an answer! – Henry.L Mar 28 '17 at 20:05

I think it goes like this: Assume $C$ has nonempty interior. Rotating the convex set if needed, the boundary $\partial C$ near any given point $p \in \partial C$ can be written as the graph of a convex function. More or less by definition, a hyperplane is a supporting hyperplane at $p$ if and only if it corresponds to an element of the subdifferential of $f$. Moreover, $f$ is differentiable at $p$ if and only if the subdifferential contains only one element. In that case, the unique supporting hyperplane is the tangent hyperplane of $\partial C$ at $p$.

In summary, a supporting hyperplane is the tangent hyperplane at $p \in\partial C$ if and only if $f$ is differentiable at $p$ if and only if the subdifferential of $f$ at $p$ contains only one element if and only if there is only one supporting hyperplane at $p$.

This, I believe, follows by results stated and proved in the book Convex Analysis by Rockafellar.

EDIT: As HenryL points out, this holds only if the supporting hyperplane actually touches the boundary. If $C$ is noncompact, then it is possible this does not happen.

• If any upvoters happen to find the results relevant to this answer please comment, I am still searching for it, thanks! – Henry.L Mar 29 '17 at 17:08
• The claim " a hyperplane is a supporting hyperplane at $p$ if and only if it corresponds to an element of the subdifferential of $f$" is in Theorem 23.2(p.215 of [Rockafellar]); The claim "$f$ is differentiable at $p$ if and only if the subdifferential (at $p$) contains only one element." correspond to Theorem 25.1(p.242 of [Rockafellar]), however, it seems to require $f$ to be finite to make this claim hold. Therefore your claim is a partial answer which excludes the possible boundary at $\infty$. – Henry.L Apr 5 '17 at 12:58
• That's a good point. – Deane Yang Apr 5 '17 at 14:07
• Let $H(C,p)$ = family of supporting hyperplanes of $C$ at $p$, then, for any $r > 0$, $H(C,p) = H(C\cap \overline{ B(p,r)}, p)$, where $B(p,r)$ is the ball of radius $r$ centered at $p$. So, without loss of generality, one may assume $C$ is bounded. – VictorZurkowski Apr 5 '17 at 16:11
• I have no doubt that locally, $\partial C$ can be represented as the graph of a function defined on a supporting hyperplane, but the supporting hyperplane has to be chosen with some care. For example, when $C=\{(x,y) | 0 \le x \text{ and } x^2 \le y \}$, one cannot represent $\partial C$ near $(0,0)$ as the graph of a function defined on either $y=0$ or $x=0$ (both of which are supporting hyperplanes of $C$ at $p=(0,0)$ ). – VictorZurkowski Apr 5 '17 at 16:24


text:
# Direct sum and intersection of more than two subspaces

In "Linear Algebra Done Right" Axler points out that if you have more then two subspaces, it is not enough to test that each pair intersects only at 0, in order to say that they have direct sum. As an example, he asks to consider the following subspaces:
$$U_1 = \{(x, y, 0)\}$$
$$U_2 = \{(0, 0, z)\}$$
$$U_3 = \{(0, y, y)\}$$

where, he says, $$U_1 \bigcap U_2 = U_1 \bigcap U_3 = U_2 \bigcap U_3 = \{0\}$$ but they do not have direct sum.

I agree that they do not have direct sum, but if I test pair intersection I see that
$$U_1 \bigcap U_3 = \{(0, y, 0)\} \ne \{0\}$$
$$U_2 \bigcap U_3 = \{(0, 0, d)\} \ne \{0\}$$

So, to me it looks like contradiction, and that it is enough to check pair intersection. Where am I wrong?

There is no $$(0,y,0)$$ in $$U_3$$ with $$y\neq 0$$. The elements of $$U_3$$ are of the form $$(0,y,y)$$. So if $$y=0$$, we will have $$(0,0,0)$$. Hence we cannot produce $$(0,y,0)$$ with $$y\neq 0$$.


text:
# What is the slope intercept form of the line with a slope of -1  that passes through  (-5,7) ?

Aug 13, 2016

$y = - x + 2$

#### Explanation:

The equation of a line passing through $\left({x}_{1} , {y}_{1}\right)$ and having a slope of $m$ is given by

$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$

Hence equation of of a line with slope $- 1$ and paint through $\left(- 5 , 7\right)$ is

y-7=(-1)×(x-(-5)) or

y-7=(-1)×x-(-1)(-5) or

$y - 7 = - x - 5$ or

$y = - x - 5 + 7$ or

$y = - x + 2$ in slope intercept form, where coefficient of $x$, which is $- 1$ is the slope and $2$ is intercept on $y$-axis.


text:
Sample Size Matters: Design and Experiments

Previously, I introduced the idea that samples do not look exactly like the populations that they are drawn from, and had a closer look at what impact sample size has on our ability to estimate population statistics like mean, proportion or Cpk from samples. Here, I will have a closer look at how this uncertainty impacts our engineering process. In the next post, I will tie in the engineering impacts and decisions to the business value and costs.

Difference to detect

When we are testing, we’re either testing to determine that the new product or process performs better than the old or, for cost reduction projects, that the cheaper product or process is at least as good as the existing one.

This means that we need to detect a difference between old and new values, such as a difference in the mean weight between the new and old parts. The larger the sample size, $n$ the smaller the difference, $\Delta$ that we can detect. The error, $\epsilon$, in our estimate of the differences gets smaller as sample size increases:

Given the uncertainties in our estimate of $\mu$ and $\sigma$, illustrated above, it should be clear now that with small sample sizes we can only detect large differences of many multiples of the sample standard deviation, $S$.

Mean

When trying to determine if a new product or process is better than an old one, we are usually interested in shifting the mean. We want a product to be lighter, provide more power, or a process to work faster. In such cases, we need to estimate the difference of the means, $\Delta = \mu_2 - \mu_1$ and ensure that it is different than 0 (or some other pre-determined value). The minimum difference that we can reliable detect is plotted below for different sample sizes.

Standard deviation

In many Six Sigma projects, and any time we want to shift the mean closer to a specification limit, we need to compare the new population standard deviation with the old. The simplest way of making this comparison is by taking the ratio $F = \sigma_{2}^{2} / \sigma_{1}^{2}$, where $\sigma_{2}^{2}$ is the larger of the two variances. The dependence on sample size is illustrated below.

You can see from the inset plot, which includes sample sizes of 2 and 3, that small sample sizes really hurt comparisons of variance, and that interesting differences in variance can’t be detected until we have more than 10 samples.

Proportions

Proportions, such as fraction of defective parts between a new and old design, can be compared by looking at the difference between the two proportions, $\Delta = \left| p_1 - p_0 \right|$.

You can see from this that proportions data provides much less information than variable data; we need much larger sample sizes to achieve usefully small $\Delta$.

Summary and look forward

When designing experiments, the goal is to detect some difference between two populations. The uncertainty in our measurements and the variation in the parts has a big impact on how many parts we need to test, or greatly limits what we can learn from an experiment.

Next time, I’ll show how these calculations of sample size and uncertainty impact the busines.


text:
# Is it safe to design board at fab houses minimum trace width?

I am working on my first board to have professionally fabricated. Right now I am planning on using seeed studio mainly because they are the cheapest I have found. 10 5cmx5cm boards for $10 is pretty awesome. Their site says they can handle 6 mil traces with 6 mil spacing. I have routed my board to those specifications, and it passes their drc file. I am still a bit worried that this could lead to errors in manufacturing, or will be easy for me to screw up when soldering. It is mostly surface mount components of pretty generous pitches. SOIC and 0805 parts mostly, with a few through hole headers. Is there any danger to making smaller traces? Is there anything to be gained by making them thicker? I am not expecting current over 100mA or frequencies over 1MHz. I am also planning on paying for 100% etest, so I don't think I need to worry about the fab house delivering non-working boards. Information: Seeed Studios Seeed's Fusion PCB Service Seeed authorised user review and guide to using their Fusion service • Use larger traces where you have room, but generally consider that it's not the money you risk, but the time it would take to re-order. So check really carefully for design errors - things like wrong footprints, pads that don't leave enough room for hand assembly, wrong size through holes, etc. Print out the design and put the components on it. – Chris Stratton Nov 19 '11 at 5:18 • Seeed studio is awesome, Eric and the guys really get it. You'll be fine. – zak Nov 19 '11 at 9:20 • Related: This excellent reference - TI Analog Engineer’s Pocket Reference - 4th edition provides some useful information on PCB track current/ voltage drop / heat / fusing issues. Especially pages 55-68. – Russell McMahon May 5 '16 at 5:32 ## 6 Answers A reputable PCB house will solidly deliver their minimum standard track & spacing. Some will let you try finer at your risk and some will reject your work outright if under spec. I assume that you are using a temperature controlled iron. If not, do. If you are not confident in your own soldering abilities and you have plenty of room then fattening up tracks and especially pads associated with through hole parts does no harm. I'd give the through hole headers as much copper as you reasonably can - especially if the board is NOT PTH (plated through hole). With PTH you get substantially more strength. If the board is single sided (probably not, but ...) then you want to take great care with through hole pad soldering and due care with everything else. SMD part pads should be suited to the part and you have to learn to accommodate them rather than them accommodating you. PS: Others feel free to contradict or improve anything I've said. I have very substantial soldering experience but there is always good stuff and ideas to learn. ________________________________ Related: This excellent reference - TI Analog Engineer’s Pocket Reference - 4th edition as well as a vast amount of other useful material it provides some useful information on PCB track current/ voltage drop / heat / fusing issues. Especially pages 55-68. Do you need to go that small? If so, go for it -- it's advertised, and they will deliver. If not, and especially if the assembler (you?) has little soldering experience, go bigger. Larger traces are easier to solder to, and are better laminated to the FR-4, so you can, but still shouldn't, overheat areas with less risk of traces lifting. I usually don't go below 10 mil (0.25 mm), as it conveniently matches the finer-pitch leads. Minimum features sizes are useful when necking down to stick a trace between other features (between leads, BGA pads, etc.). Assembling 20/20 mil boards makes me happy. If they give 6/6 as their specs, then yes, it's safe to design to that. However, I wouldn't unless I really need the space. I use 8/8 as my own personal design rules unless there is a good reason not to. That way, any board house in the world can produce the board without problems or extra cost. Note that you can get tighter design rules from a number of places, but as the dimensions get smaller the cost goes up. Just because your board house of the week can do 6/6 before charging more doesn't make that a good design decision unless you need the extra space. I didn't look up the board house you mention, but$10 for any amount of board is suspiciously cheap. Does that include solder mask, silk screen at least on top, electrical testing? These can save a lot of trouble and human screwups later. Maybe you're OK without these, but you should make sure you know what you're getting. Personally, I wouldn't get boards without solder mask and silk screen on at least one side.

• yeah, they do solder mask and silkscreen on two sides. I have been a bit confused why they are an order of magnitude cheaper than most other alternatives I have seen, but people have reported success with them. – captncraig Nov 18 '11 at 18:19
• I've ordered a board and they delivered my boards in working conditions, in the stated time. They did add an extra number to the silkscreen though, probably to find back the boards for each order after depanelizing. – drxzcl Nov 19 '11 at 0:31
• I got 2 set of boards, silkscreen and solder mask was included. It was a great job for that money. – abdullah kahraman Nov 19 '11 at 9:31

They usually give a minimum that's slightly over their actual capabilities so you will be fine. The e-test makes sure anyway.

Thicker traces have less resistance and can handle more current, so use as necessary. Obviously you don't want to use tiny 6 mil traces for things like power rails.

As mentioned if you don't need to go that small, don't. Generally you would stick with the maximum you can get away with and set your trace width rules up for different nets. For 0805 parts I would probably use at least 10 mils.

If you are soldering this manually then make it as easy as you can for yourself. The thinner traces and especially annular rings are, the easier it is to lift/damage them when assembling or reworking.

I've used ITead Studio, who use the same manufacturer as Seeed Studio, for three sets of boards. They don't recommend 6 mil tracks, although the factory can manage them. I'd use 8 mil, as a minimum. I wouldn't have any hesitation using the minimum with other suppliers, though.

• iTead Studio does great job! They manufactured a board with 7 mil spacing and no problems at all. – abdullah kahraman Nov 19 '11 at 9:27

No, it is safe. If they can't do that reliability, they have factored that into their price. Paying for 100% etest covers you.

6/6 mil isn't bleeding edge.


text:
# Show Console in Windows Application?

Is there a way to show the console in a Windows Application?

What i want to do is something like that:

static class Program
{
static void Main(string[] args) {
bool consoleMode = Boolean.Parse(args[0]);

if (consoleMode) {
Console.WriteLine("consolemode started");
// ...
} else {
Application.EnableVisualStyles();
Application.SetCompatibleTextRenderingDefault(false);
Application.Run(new Form1());
}
}
}

best regards

-

What you want to do is not possible in a sane way. There was a similar question so look at the answers.

Then there's also an insane approach (site down - backup available here.) written by Jeffrey Knight:

Question: How do I create an application that can run in either GUI (windows) mode or command line / console mode?

On the surface of it, this would seem easy: you create a Console application, add a windows form to it, and you're off and running. However, there's a problem:

Problem: If you run in GUI mode, you end up with both a window and a pesky console lurking in the background, and you don't have any way to hide it.

What people seem to want is a true amphibian application that can run smoothly in either mode.

If you break it down, there are actually four use cases here:

User starts application from existing cmd window, and runs in GUI mode
User double clicks to start application, and runs in GUI mode
User starts application from existing cmd window, and runs in command mode
User double clicks to start application, and runs in command mode.

I'm posting the code to do this, but with a caveat.

I actually think this sort of approach will run you into a lot more trouble down the road than it's worth. For example, you'll have to have two different UIs' -- one for the GUI and one for the command / shell. You're going to have to build some strange central logic engine that abstracts from GUI vs. command line, and it's just going to get weird. If it were me, I'd step back and think about how this will be used in practice, and whether this sort of mode-switching is worth the work. Thus, unless some special case called for it, I wouldn't use this code myself, because as soon as I run into situations where I need API calls to get something done, I tend to stop and ask myself "am I overcomplicating things?".

Output type=Windows Application

using System;
using System.Collections.Generic;
using System.Windows.Forms;
using System.Runtime.InteropServices;
using System.Diagnostics;
using Microsoft.Win32;

namespace WindowsApplication
{
static class Program
{
/*
DEMO CODE ONLY: In general, this approach calls for re-thinking
There are 4 possible ways this can run:
1) User starts application from existing cmd window, and runs in GUI mode
2) User double clicks to start application, and runs in GUI mode
3) User starts applicaiton from existing cmd window, and runs in command mode
4) User double clicks to start application, and runs in command mode.

To run in console mode, start a cmd shell and enter:
c:\path\to\Debug\dir\WindowsApplication.exe console
To run in gui mode,  EITHER just double click the exe, OR start it from the cmd prompt with:
c:\path\to\Debug\dir\WindowsApplication.exe (or pass the "gui" argument).
To start in command mode from a double click, change the default below to "console".
In practice, I'm not even sure how the console vs gui mode distinction would be made from a
double click...
string mode = args.Length > 0 ? args[0] : "console"; //default to console
*/

[DllImport("kernel32.dll", SetLastError = true)]
static extern bool AllocConsole();

[DllImport("kernel32.dll", SetLastError = true)]
static extern bool FreeConsole();

[DllImport("kernel32", SetLastError = true)]
static extern bool AttachConsole(int dwProcessId);

[DllImport("user32.dll")]
static extern IntPtr GetForegroundWindow();

[DllImport("user32.dll", SetLastError = true)]
static extern uint GetWindowThreadProcessId(IntPtr hWnd, out int lpdwProcessId);

static void Main(string[] args)
{
//TODO: better handling of command args, (handle help (--help /?) etc.)
string mode = args.Length > 0 ? args[0] : "gui"; //default to gui

if (mode == "gui")
{
MessageBox.Show("Welcome to GUI mode");

Application.EnableVisualStyles();

Application.SetCompatibleTextRenderingDefault(false);

Application.Run(new Form1());
}
else if (mode == "console")
{

//Get a pointer to the forground window.  The idea here is that
//IF the user is starting our application from an existing console
//shell, that shell will be the uppermost window.  We'll get it
//and attach to it
IntPtr ptr = GetForegroundWindow();

int  u;

Process process = Process.GetProcessById(u);

if (process.ProcessName == "cmd" )    //Is the uppermost window a cmd process?
{
AttachConsole(process.Id);

//we have a console to attach to ..
Console.WriteLine("hello. It looks like you started me from an existing console.");
}
else
{
//no console AND we're in console mode ... create a new console.

AllocConsole();

Console.WriteLine(@"hello. It looks like you double clicked me to start
AND you want console mode.  Here's a new console.");
Console.WriteLine("press any key to continue ...");
}

FreeConsole();
}
}
}
}

-
Wow, it's insane! But it works, thanks! –  lesderid Nov 13 '10 at 16:42
I find it ironic with Microsoft and how it wants to create C# interfaces for all of it's API, yet there is no C# way to perform such a simple task. –  Ramon Zarazua Apr 17 '11 at 8:57
Rather than depending on console being the foreground window, you could get parent process id of current process using winapi: stackoverflow.com/a/3346055/855432 –  ghord Sep 11 '12 at 8:06
As of the time of writing, backup copy of the article is available here web.archive.org/web/20111227234507/http://www.rootsilver.com/… –  zespri Apr 18 '13 at 2:19
Hi! I found that if I ran this solution from shell like Far, nc it creates new console. If I attache to Far Console like cmd it works wrong. I recommend to create ConsoleApplication and if GUI needed then do FreeConsole(); Excellent article! Thanks! –  Maxim Vasiliev May 16 '13 at 9:44

This is a tad old (OK, it's VERY old), but I'm doing the exact same thing right now. Here's a very simple solution that's working for me:

    public static void ShowConsoleWindow()
{
var handle = GetConsoleWindow();

if (handle == IntPtr.Zero)
{
AllocConsole();
}
else
{
ShowWindow(handle, SW_SHOW);
}
}

public static void HideConsoleWindow()
{
var handle = GetConsoleWindow();

ShowWindow(handle, SW_HIDE);
}

[DllImport("kernel32.dll", SetLastError = true)]
static extern bool AllocConsole();

[DllImport("kernel32.dll")]
static extern IntPtr GetConsoleWindow();

[DllImport("user32.dll")]
static extern bool ShowWindow(IntPtr hWnd, int nCmdShow);

const int SW_HIDE = 0;
const int SW_SHOW = 5;

-
Worked much better than the accepted answer. –  Pavel Chikulaev Jun 6 '13 at 6:29

# Disclaimer

There is a way to achieve this which is quite simple, but I wouldn't suggest it is a good approach for an app you are going to let other people see. But if you had some developer need to show the console and windows forms at the same time, it can be done quite easily.

This method also supports showing only the Console window, but does not support showing only the Windows Form - i.e. the Console will always be shown. You can only interact (i.e. receive data - Console.ReadLine(), Console.Read()) with the console window if you do not show the windows forms; output to Console - Console.WriteLine() - works in both modes.

This is provided as is; no guarantees this won't do something horrible later on, but it does work.

## Project steps

Start from a standard Console Application.

Mark the Main method as [STAThread]

Add the standard Windows start code to your Main method:

## End Result

You will have an application that shows the Console and optionally windows forms.

## Sample Code

### Program.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Windows.Forms;

namespace ConsoleApplication9 {
class Program {

static void Main(string[] args) {

if (args.Length > 0 && args[0] == "console") {
Console.WriteLine("Hello world!");
}
else {
Application.EnableVisualStyles();
Application.SetCompatibleTextRenderingDefault(false);
Application.Run(new Form1());
}
}
}
}

### Form1.cs

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Windows.Forms;

namespace ConsoleApplication9 {
public partial class Form1 : Form {
public Form1() {
InitializeComponent();
}

private void Form1_Click(object sender, EventArgs e) {
Console.WriteLine("Clicked");
}
}
}

-
nice code here. but how do you disable showing the console if you want to sell or deploy or something else.??? –  r4ccoon Sep 20 '09 at 21:50
@r4ccoon - you can't. But you can easily move all your code to a normal Windows app. –  Sam Meldrum Dec 3 '10 at 9:22
Well, IMHO a simpler way to achieve the same effect is to create a Windows Forms Project as usual, then right click it in the Solution Explorer -> Properties, and change Output Type to Console Application. (edit: now I've realised it's basically ICR's answer) –  kamilk Jan 2 '13 at 11:22
Nice Step by step Answer –  Jeson Park Jun 24 '13 at 9:15

Easiest way is to start a WinForms application, go to settings and change the type to a console application.

-
Application -> output type: Console Application. Did it for me! –  Lodewijk Mar 22 at 20:54

In wind32, console-mode applications are a completely different beast from the usual message-queue-receiving applications. They are declared and compile differently. You might create an application which has both a console part and normal window and hide one or the other. But suspect you will find the whole thing a bit more work than you thought.

-


text:
We have a Python web service where we store some key-val pairs in redis. Occasionally, I want to delete some of the keys matching a certain pattern. Current, we are using redis-py-cluster for redis-related operations.

We can use scan_iter() method to search such keys and delete them. The first parameter to scan_iter() is the matching pattern. The code looks roughly like this:

for k in redis_client.scan_iter("prefix:*"):
redis_client.delete(k)

The above code kinda works, but it is awfully slow. We can add the count option to accelerate deletion. Option count specify how many keys per scan will return. According to here, the default count is 10, which is rather small.

batch_size = 500
keys = []

for k in redis_client.scan_iter("prefix:*", count=batch_size):
keys.append(k)
if len(keys) >= batch_size:
redis_client.delete(*keys)
keys = []
if len(keys) > 0:
redis_client.delete(*keys)

Using a large count will speed up the deletion process significantly. I have benchmarked on about 20000 keys. Here is what I have found:

batch size  Time taken (seconds)
100         929
500         260
1000        175
2000        133
4000        107
5000        106
10000       93


text:
# Max/Min with constraint

Printable View

• November 14th 2009, 01:05 PM
storchfire1X
Max/Min with constraint
I need help on the following problem:

Find the absolute maximum and minimum of the function $f(x,y) = x^2 + y^2$ subject to the constraint $x^4 + y^4 = 625$.

I tried using Lagrange's multiplier to this problem but here is where I'm stuck.

$\frac{\partial f}{\partial x}= \lambda\frac{\partial g}{\partial x}$

$2x = 4x^3 \lambda$

$\frac{\partial f}{\partial y} = \lambda\frac{\partial g}{\partial y}$

$2y = 4y^3\lambda$

I just can't seem to be able to solve for $x,y, \lambda$,which I need for the critical points
• November 14th 2009, 04:04 PM
TKHunny
Where is your third equation?

$2x\;=\;4x^{3}\lambda$

$2y\;=\;4y^{3}\lambda$

$x^{4}\;+\;y^{4}\;=\;625$

That should do it.


text:
# 3.2 Grouping symbols and the order of operations

 Page 1 / 2
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.

## Section overview

• Grouping Symbols
• Multiple Grouping Symbols
• The Order of Operations
• Calculators

## Grouping symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:

## ( ), [ ], { },

Parentheses : ( )
Brackets : [ ]
Braces : { }
Bar :

In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.

## Sample set a

If possible, determine the value of each of the following.

$9+\left(3\cdot 8\right)$

Since 3 and 8 are within parentheses, they are to be combined first.

$\begin{array}{cc}\hfill 9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\left(3\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}8\right)& =\phantom{\rule{2px}{0ex}}9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{24}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{33}\hfill \end{array}$

Thus,

$9+\left(3\cdot 8\right)=\text{33}$

$\left(\text{10}÷0\right)\cdot 6$

Since $\text{10}÷0$ is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."

## Practice set a

If possible, determine the value of each of the following.

$\text{16}-\left(3\cdot 2\right)$

10

$5+\left(7\cdot 9\right)$

68

$\left(4+8\right)\cdot 2$

24

$\text{28}÷\left(\text{18}-\text{11}\right)$

4

$\left(\text{33}÷3\right)-\text{11}$

0

$4+\left(0÷0\right)$

not possible (indeterminant)

## Multiple grouping symbols

When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.

## Sample set b

Determine the value of each of the following.

$2+\left(8\cdot 3\right)-\left(5+6\right)$

Combine 8 and 3 first, then combine 5 and 6.

$\begin{array}{cc}2+\mathrm{24}-\mathrm{11}\hfill & \text{Now combine left to right.}\\ \mathrm{26}-\mathrm{11}\hfill & \\ \mathrm{15}\hfill & \end{array}$

$\text{10}+\left[\text{30}-\left(2\cdot 9\right)\right]$

Combine 2 and 9 since they occur in the innermost set of parentheses.

$\begin{array}{cc}\mathrm{10}+\left[\mathrm{30}-\mathrm{18}\right]\hfill & \text{Now combine 30 and 18.}\\ \mathrm{10}+\mathrm{12}\hfill & \\ \mathrm{22}\hfill & \end{array}$

## Practice set b

Determine the value of each of the following.

$\left(\text{17}+8\right)+\left(9+\text{20}\right)$

54

$\left(\text{55}-6\right)-\left(\text{13}\cdot 2\right)$

23

$\text{23}+\left(\text{12}÷4\right)-\left(\text{11}\cdot 2\right)$

4

$\text{86}+\left[\text{14}÷\left(\text{10}-8\right)\right]$

93

$\text{31}+\left\{9+\left[1+\left(\text{35}-2\right)\right]\right\}$

74

${\left\{6-\left[\text{24}÷\left(4\cdot 2\right)\right]\right\}}^{3}$

27

## The order of operations

Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of $3+5\cdot 2$ . We could do either of two things:

Add 3 and 5, then multiply this sum by 2.

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}8\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}2\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{16}\hfill \end{array}$

Multiply 5 and 2, then add 3 to this product.

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}3\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{10}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{13}\hfill \end{array}$

We now have two values for one number. To determine the correct value, we must use the accepted order of operations .

## Order of operations

1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
2. Perform all exponential and root operations.
3. Perform all multiplications and divisions, moving left to right.
4. Perform all additions and subtractions, moving left to right.

## Sample set c

Determine the value of each of the following.

$\begin{array}{cc}\mathrm{21}+3\cdot \mathrm{12}\hfill & \text{Multiply first.}\hfill \\ \mathrm{21}+\mathrm{36}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$

$\begin{array}{cc}\left(\mathrm{15}-8\right)+5\cdot \left(6+4\right).\hfill & \text{Simplify inside parentheses first.}\hfill \\ 7+5\cdot \mathrm{10}\hfill & \text{Multiply.}\hfill \\ 7+\mathrm{50}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$

$\begin{array}{cc}\mathrm{63}-\left(4+6\cdot 3\right)+\mathrm{76}-4\hfill & \text{Simplify first within the parenthesis by multiplying, then adding.}\hfill \\ \mathrm{63}-\left(4+\mathrm{18}\right)+\mathrm{76}-4\hfill & \hfill \\ \mathrm{63}-\mathrm{22}+\mathrm{76}-4\hfill & \text{Now perform the additions and subtractions, moving left to right.}\hfill \\ \mathrm{41}+\mathrm{76}-4\hfill & \text{Add 41 and 76:}\phantom{\rule{8px}{0ex}}\mathrm{41}+\mathrm{76}=\mathrm{117}.\hfill \\ \mathrm{117}-4\hfill & \text{Subtract 4 from 117:}\phantom{\rule{8px}{0ex}}\mathrm{117}-4=\mathrm{113}.\hfill \\ \mathrm{113}\hfill & \end{array}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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text:
# In the given figure, P is a point in the interior of a parallelogram ABCD.

Question. In the given figure, P is a point in the interior of a parallelogram ABCD. Show that

(i) $\operatorname{ar}(A P B)+\operatorname{ar}(P C D)=\frac{1}{2} \operatorname{ar}(A B C D)$

(ii) $\operatorname{ar}(A P D)+\operatorname{ar}(P B C)=\operatorname{ar}(A P B)+\operatorname{ar}(P C D)$

Solution:

(i) Let us draw a line segment EF, passing through point $P$ and parallel to line segment $A B$.

In parallelogram $A B C D$,

$A B \| E F$ (By construction) ... (1)

ABCD is a parallelogram.

$\therefore A D \| B C$ (Opposite sides of a parallelogram)

$\Rightarrow \mathrm{AE} \| \mathrm{BF} \ldots(2)$

From equations (1) and (2), we obtain

$A B \| E F$ and $A E \| B F$

Therefore, quadrilateral ABFE is a parallelogram.

It can be observed that $\triangle A P B$ and parallelogram $A B F E$ are lying on the same base $A B$ and between the same parallel lines $A B$ and $E F$.

$\therefore$ Area $(\triangle \mathrm{APB})=\frac{1}{2}$ Area $(\mathrm{ABFE})$..(3)

Similarly, for $\triangle P C D$ and parallelogram EFCD,

Area $(\triangle \mathrm{PCD})=\frac{1}{2}$ Area $(\mathrm{EFCD})$..(4)

Adding equations $(3)$ and $(4)$, we obtain

Area $(\Delta \mathrm{APB})+$ Area $(\Delta \mathrm{PCD})=\frac{1}{2}[$ Area $(\mathrm{ABFE})+$ Area $(\mathrm{EFCD})]$

Area $(\Delta \mathrm{APB})+$ Area $(\Delta \mathrm{PCD})=\frac{1}{2}$ Area $(\mathrm{ABCD})$..(5)

Let us draw a line segment MN, passing through point $P$ and parallel to line segment $A D$.

In parallelogram $A B C D$,

MN $\| A D$ (By construction) $\ldots$ (6)

ABCD is a parallelogram.

$\therefore A B \| D C$ (Opposite sides of a parallelogram)

$\Rightarrow \mathrm{AM} \| \mathrm{DN} \ldots(7)$

From equations $(6)$ and $(7)$, we obtain

MN $\| A D$ and $A M \| D N$

Therefore, quadrilateral AMND is a parallelogram.

It can be observed that $\triangle A P D$ and parallelogram AMND are lying on the same base AD and between the same parallel lines AD and MN.

$\therefore$ Area $(\triangle \mathrm{APD})=\frac{1}{2}$ Area (AMND) ...(8)

Similarly, for $\triangle \mathrm{PCB}$ and parallelogram MNCB,

Area $(\triangle \mathrm{PCB})=\frac{1}{2}$ Area $(\mathrm{MNCB})$..(9)

Adding equations (8) and (9), we obtain

Area $(\Delta \mathrm{APD})+$ Area $(\Delta \mathrm{PCB})=\frac{1}{2}[$ Area $(\mathrm{AMND})+$ Area $(\mathrm{MNCB})]$

Area $(\Delta \mathrm{APD})+$ Area $(\Delta \mathrm{PCB})=\frac{1}{2}$ Area $(\mathrm{ABCD})$..(10)

On comparing equations $(5)$ and $(10)$, we obtain

Area $(\Delta \mathrm{APD})+$ Area $(\Delta \mathrm{PBC})=$ Area $(\Delta \mathrm{APB})+$ Area $(\Delta \mathrm{PCD})$


text:
how to use predictions on a single value?

I am comfortable using Machine learning on my train data and test data and validate it. But the question here is if I want to predict a single variable how do I do it? Let's suppose I have done feature selection using lasso or applied PCA or done feature engineering like transformation or scaling but when I want to utilize it for new data, maybe single data I am getting confused. Should I have to follow all the steps mentioned above even for the single row of data or how do I proceed with this. Please suggest.

Building a model pipeline is the way to go once you are done experimenting with your model. Following is a toy example -

from sklearn.svm import SVC
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import make_classification
from sklearn.model_selection import train_test_split
from sklearn.pipeline import Pipeline

#Generating random dataset
X, y = make_classification(random_state=0)

#Splitting test and tran sets
X_train, X_test, y_train, y_test = train_test_split(X, y,
random_state=0)

#Defining a pipeline with standardscaling transformer and then svc estimator
pipe = Pipeline([('scaler', StandardScaler()), ('svc', SVC())])

# The pipeline can be used as any other estimator
# and avoids leaking the test set into the train set
pipe.fit(X_train, y_train)

#During prediction on a test or on a fresh set of data, just use the pipe object to run the whole pipeline and get predictions.
pipe.score(X_test, y_test)

You can easily create a pipeline in Sklearn for almost every possible transform available in Sklearn. Check the user guide for more details.

Another example with PCA -

from sklearn.pipeline import Pipeline
from sklearn.svm import SVC
from sklearn.decomposition import PCA
estimators = [('reduce_dim', PCA()), ('clf', SVC())]
pipe = Pipeline(estimators)
pipe

Pipeline(steps=[('reduce_dim', PCA()), ('clf', SVC())])
#Now you can use the fit method of the pipeline object to perform feature engineering and train model, while the scoring method for predictions

For Sklearn Pipelines, you can have either a single or multiple transformation paths (which are combined together with a FeatureUnion) and then it has to end with an estimator (only a single estimator is allowed)

While all this is powerful enough for most purposes, sometimes, there are custom transformations that you will need for your data, and those transformations will not be a part of the Sklearn API. In that case you can actually use the base classes of sklearn to defined your own transformer which can easily become a part of the pipeline that you define. More information here.

You have to create what is known as a modeling pipeline this includes creating so-called transformers.

These transformers basically apply all necessary data transformations on your raw data to make the input "fit" for the trained model.

This might include things like One-Hot-Encoding, etc. It is important to "fit" these transformers on your training set only e.g. OHE only encodes values it knows from the training data set and treats everything else as NA.

Some of your transformers are a bit more involved though. If you have performed clustering / PCA for example and used the resulting dimensions in your model you cannot simply do the same steps on a new data. You have to create a suitable transformer which varies depending on your steps.

For example with a factor analysis you are usually able to extract a regression model which predicts the factors based on the raw input. You then apply this regression model fitted on the training data to the new unseen data.

There are certain points that one should keep practicing that always prepare data first properly and make it uniform shape by applying the transformation technique or any other feature engineering methodology. As u mentioned problem in question i assume that u suppose to do iterative training. In that case when new data comes it comes with new features so whatever parameter learned during the previous iteration it will not help.U might have to deal with such a scenario in iteration wise.At each iteration u will have to train new parameter(new model). Technique suggested by @Fnguyen is cool but not scalable in production.

If your new data come up with same feature set than u can use existing trained parameter of PCA or transformer. If new data is for training than load existing PCA and retrain it and transform new data and fit into model. PLease put feedback and your point of view as response.


text:
# Mesh for 3d dungeons game. [closed]

Hallo, I look for some F: R^2->R height function which would generate the Speleothem ceiling http://en.wikipedia.org/wiki/Speleothem for 3d game taking place in dungeons/caves. The function might be in any form usable for generating mesh.

-

## closed as off topic by David Roberts, Loop Space, Bill Johnson, Henry Cohn, Kevin WalkerJun 11 '12 at 14:42

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

This seems like an interesting question. However, perhaps depending on what preciely you look for, this is not the optimal site to ask it; there is AFAIK also a stackexchange site on game developpin for example. Still, I would like to see the question stay open here (at least for a while). Perhaps you could add some more details in the meantime. –  quid Jun 11 '12 at 11:22
I think this is off-topic. But for what it is worth, you'd probably be better off looking not for a fixed height function, but for something generated using something like context free contextfreeart.org/index.html or with similar design philosophy and tune your parameters. You may also want to track down the original paper mentioned in news.sciencemag.org/sciencenow/2005/01/18-03.html to see how cave formations are mathematically modeled. –  Willie Wong Jun 11 '12 at 14:18
I voted to reopen. IMO, this question on rocks is considerably more on-topic than the recent one on rooks. –  quid Jun 11 '12 at 17:46
Agree with quid. As probable answer I would try 2D LEVI flightts whatever it means. –  Alexander Chervov Jun 11 '12 at 17:50


text:
# A shopkeeper 2

A shopkeeper charged E17 for 4 textbooks of the same kind. What is the cost of one textbook?

x =  4.25 Eur

### Step-by-step explanation:

Did you find an error or inaccuracy? Feel free to write us. Thank you!

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text:
# Convert the polar equation to rectangular form. {eq}\theta=\pi {/eq}

## Question:

Convert the polar equation to rectangular form.

{eq}\theta=\pi {/eq}

## Polar to Rectangular form:

If we are given the value of {eq}\theta {/eq} in the polar form, then we can easily convert it in the rectangular form by using {eq}\tan \theta = \frac{y}{x}{/eq} . The term {eq}\tan \theta = \frac{y}{x}{/eq} is obtained by dividing the terms {eq}x = r\cos \theta {/eq} and {eq}y = r\sin \theta {/eq} . At last, substituting the value of {eq}\theta {/eq} into {eq}\tan \theta = \frac{y}{x}{/eq} will give us the required equation.

Become a Study.com member to unlock this answer!

Given:

• Consider the polar equation {eq}\theta = \pi {/eq} .

The objective is to convert the given polar equation to rectangular form.

Recal...


text:
2015
05-23

# Permutation

There is an arrangement of N numbers and a permutation relation that alter one arrangement into another.
For example, when N equals to 6 the arrangement is 123456 at first. The replacement relation is 312546 (indicate 1->2, 2->3, 3->1, 4->5, 5->4, 6->6, the relation is also an arrangement distinctly).
After the first permutation, the arrangement will be 312546. And then it will be 231456.
In this permutation relation (312546), the arrangement will be altered into the order 312546, 231456, 123546, 312456, 231546 and it will always go back to the beginning, so the length of the loop section of this permutation relation equals to 6.
Your task is to calculate how many kinds of the length of this loop section in any permutation relations.

Input contains multiple test cases. In each test cases the input only contains one integer indicates N. For all test cases, N<=1000.

Input contains multiple test cases. In each test cases the input only contains one integer indicates N. For all test cases, N<=1000.

1
2
3
10

1
2
3
16

#include<iostream>
#include<cstdlib>
#include<stdio.h>
#define ll __int64
using namespace std;
const int N=1000;
int prime[N]={0},num=1;
int isprime[N]={1,1};
ll dp[200][1010];
void sushu()
{
for(int i=2;i<N;i++)
{
if(!isprime[i])
prime[num++]=i;
for(int j=1;j<num&&i*prime[j]<N;j++)
{
isprime[i*prime[j]]=1;
if(!(i%prime[j]))
break;
}
}
}
int main()
{
sushu();
//cout<<num<<endl;
dp[0][0]=1;
num--;
for(int i=1;i<=num;i++)
{
for(int j=0;j<=1000;j++)
dp[i][j]=dp[i-1][j];//前i个质数和为j的不同最小公倍数个数
int res=prime[i];
while(res<=1000)
{
for(int j=0;j+res<=1000;j++)
if(dp[i-1][j]) dp[i][j+res]+=dp[i-1][j];
res*=prime[i];
}
}
int n;
while(scanf("%d",&n)!=EOF)
{
ll ans=0;
for(int j=1;j<=n;j++)
ans+=dp[num][j];
printf("%I64d\n",ans+1);
}
}


text:
Q

# Write the following in decimal form and say what kind of decimal expansion each has : (ii) 1/11

1. Write the following in decimal form and say what kind of decimal expansion each has :

(ii)     $\frac{1}{11}$

Views

We can rewrite  $\frac{1}{11}$  as

$\Rightarrow \frac{1}{11}= 0.09090909..... = 0.\overline{09}$
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion

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