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July 14, 2020

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# .

In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski.

Specifically, if (X,d) is a metric space, x0 is a point in X, and Cb(X) denotes the Banach space of all bounded continuous real valued functions on X with the supremum norm, then the map

$$\Phi : X \rarr C_b(X)$$

defined by

$$\Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox{for all}\quad x,y\in X$$

is an isometry.[1]

Note that this embedding depends on the chosen point x0 and is therefore not entirely canonical.

The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a closed subset of a convex subset of some Banach space.[2] (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry

$$\Psi : X \rarr C_b(X)$$

defined by

$$\Psi(x)(y) = d(x,y) \quad\mbox{for all}\quad x,y\in X$$

The convex set mentioned above is the convex hull of Ψ(X).

In both of these embedding theorems, we may replace Cb(X) by the Banach space  ∞(X) of all bounded functions XR, again with the supremum norm, since Cb(X) is a closed linear subspace of  ∞(X).

These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain X, it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing X.

History

Formally speaking, this embedding was first introduced by Kuratowski[3], but a very close variation of this embedding appears already in the paper of Fréchet[4] where he first introduces the notion of metric space.

Tight span, an embedding of any metric space into an injective metric space defined similarly to the Kuratowski embedding

References

^ Juha Heinonen (January 2003), Geometric embeddings of metric spaces, retrieved 6 January 2009
^ Karol Borsuk (1967), Theory of retracts, Warsaw. Theorem III.8.1
^ Kuratowski, C. Quelques problèmes concernant les espaces métriques non-separables. Fund. Math. 25 (1935), 534-545.
^ Fréchet M. Sur quelques points du calcul fonctionnel. — Rendiconti del Circolo Matematico di Palermo. — 1906. — 22. — pp. 1—74.

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# zbMATH — the first resource for mathematics

Algorithmic problems in Brunn-Minkowski theory. (English) Zbl 0847.52012
Trier: Univ., FB 4, 153 p. (1995).
This thesis is an extensive study of the computational complexity of several questions on convex polytopes or more general convex bodies, arising from Brunn-Minkowski theory. Main topics are volumes of Minkowski sums, the computation of mixed volumes, and Minkowski’s existence (or reconstruction) problem. The binary Turing machine model of computation is employed (thus rational inputs and outputs are required), and polytopes occur either as $${\mathcal V}$$-presentations (convex hulls of vertices) or $${\mathcal H}$$-presentations (intersections of halfspaces). For general convex bodies, different oracle concepts are used. Since it is known that computing the volume of a polytope given by an $${\mathcal H}$$- or by a $${\mathcal V}$$-presentation is $$\# \mathbb{P}$$-hard, one expects similar results for mixed volumes. The following are examples of the results obtained. Computing the volume of a zonotope generated by rational vectors is $$\# \mathbb{P}$$-hard. Approximately computing the volume of the Minkowski sum of ellipsoids is $$\# \mathbb{P}$$-hard. Computing the mixed volume of $$n$$ (axes-parallel) boxes of dimension $$n$$ with rational edge length is $$\# \mathbb{P}$$-hard. There are also corresponding easiness results for mixed volume computation. A further problem treated is how efficiently mixed volumes can be approximated in fixed dimension by means of deterministic algorithms. For variable dimension, under suitable restrictions, polynomial-time randomized algorithms for the relative approximation of (“few”) mixed volumes of well-presented convex bodies with given error are described.
Minkowski’s existence theorem for polytopes with given normals and volumes of facets is treated in the following adaptation to the model of computation, called MinkApp: [Instance: $$m,n \in \mathbb{N}$$; vectors $$a_1, \dots, a_m \in \mathbb{Q}^n \backslash \{0\}$$, mutually noncollinear, which span $$\mathbb{R}^n$$; a positive rational vector $$v = (\nu_1, \dots, \nu_m)$$ such that $$\sum^m_{i = 1} \nu_i a_i = 0$$; a positive rational error bound $$\varepsilon$$. Task: Determine a rational vector $$\widetilde b = (\widetilde \beta_1, \dots, \widetilde \beta_m )^T$$ such that the facet volumes $$\widetilde \mu_1, \dots, \widetilde \mu_m$$ of the polytope $\widetilde P = \bigl\{ x \in \mathbb{R}^n : \langle a_i, x \rangle \leq \widetilde \beta_i, \quad \text{for} \quad 1 = 1,2, \dots, m \bigr\}$ satisfy $$\max_{i = 1,\dots, m} \bigl |\widetilde \mu_i - \nu_i |a_i |\bigr |< \varepsilon.]$$ If the dimension $$n$$ is fixed, the corresponding problem is called $$n$$-MinkApp, and the exact version $$(\varepsilon = 0)$$ is MinkRecon. The problem MinkRecon is equivalent to a certain convex program (as already used by Minkowski). Approximating the solution of MinkRecon, using stability results, plays an important role. The following are the principal results obtained. For each fixed $$n\in\mathbb{N}$$, $$n$$-MinkApp can be solved in polynomial time. MinkApp can be solved in polynomial time whenever an oracle for approximating the volume of $${\mathcal H}$$-polytopes is available. The problem MinkApp is $$\# \mathbb{P}$$-hard and $$\# \mathbb{P}$$-easy.
##### MSC:
 52B55 Computational aspects related to convexity 68Q25 Analysis of algorithms and problem complexity 68-02 Research exposition (monographs, survey articles) pertaining to computer science 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry


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Faculté des sciences

## Valuations with Crofton formula and Finsler geometry

### In: Advances in Mathematics, 2007, vol. 210, no. 2, p. 733-753

Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of... Plus

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Summary
Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of Gelfand–Smirnov that Crofton densities are projective is derived. The Holmes–Thompson volumes in a projective Finsler space are studied. It is shown that they induce in a natural way valuations and that the Alesker product of the k-dimensional and the l-dimensional Holmes–Thompson valuation is the (k+l)-dimensional Holmes–Thompson valuation.


text:
Question: Error in reading sbml file
0
9 months ago by
tjal00010
tjal00010 wrote:

Hi,

I tried to read sbml model using the following commands:

file <- system.file("sbml", "msb2010108-s2.xml", package = "rsbml")

But I always get this error: Error: File unreadable.

I tried following commands and it works fine:

doc <- rsbml_read("msb2010108-s2.xml",dom=FALSE)

I even could see the graph nodes and everything is fine. The sbml model is mTOR signalling pathway and it contains a lot of nodes and edges. Verifying if the result of the second code is right or not is not possible. Hence, I appreciate if someone could assist me with the reason behind the error message in the first commands.

Kind Regards,

rsbml R • 139 views
modified 9 months ago by Michael Lawrence11k • written 9 months ago by tjal00010
2
9 months ago by
United States
Michael Lawrence11k wrote:

The system.file() call is just to get paths to example files installed with the rsbml package. If you have your own file, just pass the correct path, like you already did.


text:
# whileboolexpr with arrayjobx arrays generates unwanted whitespaces in output

While improving on a class file of mine, I found that I used the multido package in just a single line of code and naturally I set out to remove that dependency.

My class uses several counters and arrays to compute and typeset a lot of information generated by comparably few user input. Thus, in the following, I cannot simply change the variable structure of my minimal example. Also, I would like to keep the mathmode to output the numbers to stay consistent with the rest of the document.

Furthermore, the actual loop-body is a lot more complicated than this example.

\documentclass{article}

\usepackage{etoolbox}%
\usepackage{multido}%
\usepackage{arrayjobx}%

\newcounter{RUNNER}%
\newcounter{THISOFTEN}%
\setcounter{THISOFTEN}{3}%
\newarray{SHOWME}%
\readarray{SHOWME}{Item1&Item2&Item3}%
\newarray{SHOWMORE}%
\readarray{SHOWMORE}{-3&-10&55}%

\newcommand*\PrintOLD{%
\multido{\i=1+1}{\value{THISOFTEN}}{%
\SHOWME(\i)($\SHOWMORE(\i)$)~}}%

\newcommand*\PrintNEW{%
\setcounter{RUNNER}{0}%
\whileboolexpr{test{\ifnumless{\value{RUNNER}}{\value{THISOFTEN}}}}{%
\stepcounter{RUNNER}%
\SHOWME(\value{RUNNER})($\SHOWMORE(\value{RUNNER})$)~}}%

\begin{document}
Hello World: \PrintOLD

Hello World: \PrintNEW
\end{document}

# Generated Output

My solution using etoolbox adds white spaces I cannot seem to get rid of.

pdflatex and xelatex (which I normally use) generated the same output.

So far I tried to circumvent the if-clause inside the boolean expression of whileboolexpr by adding a new boolean or toggle to be used there and moved the if-clause at the end of the loop-body. I also tried unlessboolexpr as well as several other if… versions. Furthermore, I played with expansion order, but I am certainly not an expert with this.

And in a more desperate attempt I shrunk the macro to a single line without any spaces at all – just to make sure.

I am currently reluctant to try recursions as I find those harder to maintain for complicated, evolving loop-bodies. Furthermore, I have a couple of ideas on using the whileboolexpr construct to implement some features still on the TODO-list which so far have caused me headaches.

Thus I would be most grateful for Ideas on making whileboolexpr work in this case or any other solution that resembles a classical loop (preferably without introducing a new package dependency).

• Please correct your code so that it compiles without error and outputs something demonstrating the problem you are asking about. – cfr Jan 1 '15 at 2:34
• the problem doesn't appear to be with etoolbox. It seems that between the etoolbox code and the code from arrayjobx the whitespace is entering. Replace the \SHOWME.... with any string such as hello% and you'll see that no extra white space has been generated. – A.Ellett Jan 1 '15 at 3:37
• Why aren't you using etoolbox lists? I would never recommend arrayjobx. – egreg Jan 1 '15 at 10:24
• @A.Ellett Yes, I should have phrased my question more precisely. Since all other macros accessing araay-elements like this seem to work fine, I opted to use in this case in my question. – Valryne Jan 1 '15 at 11:13
• @egreg This is a project I started for learning more of Latex/Tex than using sections. I just removed ifthen and a lot of other ballast when I learned about etoolbox. arrayjobx is used so intesively, that I wanted to get rid of other packages I knew had problems before even thinking about switching to etoolbox lists. Whith what I know now, arrayjobx will have to go soon. – Valryne Jan 1 '15 at 11:32

## 1 Answer

The problem seems to be with the way that arrayjobx parses numbers such as in

  \SHOWME(\value{RUNNER})($\SHOWMORE(\value{RUNNER})$)~%%

\value{RUNNER} returns LaTeX's internal representationof a number and not a printable number, which is what arrayjobx is expecting

Try the following macro and everything works out fine:

\newcommand*\PrintNEWer{%
\def\aecnt{0}%\setcounter{RUNNER}{0}%
\whileboolexpr{
test { \ifnumcomp{\aecnt}{<}
{\value{THISOFTEN}}
}}
{%%
\edef\aecnt{\number\numexpr\aecnt+1\relax}%%
\SHOWME(\aecnt)($\SHOWMORE(\aecnt)$)~%%
}}%%

But also, just rewriting you \SHOWME slightly differently will suffice

either

\SHOWME(\number\value{RUNNER})

or

\SHOWME(\theRUNNER)

will work.

• Yes, a simple X\SHOWME(\value{RUNNER})X (with the counter set to 1) prints “X[space][space]Item1X”. The macros are too involved to propose a fix. – egreg Jan 1 '15 at 10:23
• I would have never guessed that arrayjobx does not expect a number as most other packages I used so far. I was aware of a difference between internal and printable numbers, but will certainly pay more attention to this in the future. I will use \theRUNNER until I have the time to switch to etoolbox lists as @egreg proposed. Just one more question: Is there a particular reason for having %% (two percentage signs) at some lines in your code? – Valryne Jan 1 '15 at 11:41
• @Valryne there's nothing special about %% other than I find it easier to see when writing a document. The habit is old and kind of entrenched. But, other than being a habit of mine, there is no deeper significance. – A.Ellett Jan 1 '15 at 15:16


text:
# Math2111: Chapter 3: Additional Material: For which curves and regions does Green’s theorem apply?

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

Here we discuss the conditions under which Green’s theorem (see also here) applies, more precisely for which curves and regions enclosed by the curve.

The formula in Green’s theorem states that

$\displaystyle \oint_{\mathcal{C}} \boldsymbol{F} \cdot \mathrm{d} \boldsymbol{s} = \iint_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm{d} A, \qquad\qquad\qquad (1)$

where $\boldsymbol{F}:\mathbb{R}^2\to\mathbb{R}^2,$ $\boldsymbol{F}=P\widehat{\boldsymbol{i}}+Q\widehat{\boldsymbol{j}},$ is a continuously differentiable vector field, $\mathcal{C}$ is a curve which is simple, closed, smooth and oriented positively, and ${}R$ is the region enclosed by the curve $\mathcal{C}.$

First, the conditions on the curve $\mathcal{C}$ and the region ${}R$ must be such that the line integral and double integral in (1) are well defined.

The curve $\mathcal{C}$

Let us discuss the line integral and curve $\mathcal{C}$ first. If $\mathcal{C}$ has a parameterisation $\boldsymbol{c}$ which is piecewise continuously differentiable, then the line integral is well defined. On the other hand, the line integral is also defined under weaker conditions: it is enough to assume that $\boldsymbol{c}:[a,b]\to\mathbb{R}^2$ is continuous and has bounded variation. Let $\boldsymbol{c}(t) = x(t) \widehat{\boldsymbol{i}} + y(t) \widehat{\boldsymbol{j}}.$ Then $\boldsymbol{c}$ has bounded variation if ${}x$ and ${}y$ each have bounded variation, which means that $V(x), V(y) \textless \infty,$ where ${}V$ stands for variation. The variation is defined by

$\displaystyle V(x)= \sup_{P \in \mathcal{P}} \sum_{i=1}^{N_P} |x(t_i)-x(t_{i-1})|,$

where the supremum is taken over all partitions ${}P=\{t_0,\ldots, t_{N_P}\}$ of the interval $[a,b],$ where $a=t_0 \textless t_1 \textless \cdots \textless t_{N_P-1}\textless t_{N_P}=b.$ Note that if ${}x$ is continuously differentiable, then $V(x) = \int_a^b |x^\prime(t)|\,\mathrm{d} t.$

It can be shown that if $\boldsymbol{c}:[a,b]\to\mathbb{R}^2$ is continuous and such that each of the components ${}x, {}y$ have bounded variation, then the parameterised curve $\boldsymbol{c}$ has finite length. We call such curves rectifiable.

Note that assuming that the parameterisation $\boldsymbol{c}$ is only continuous is not sufficient, since there are curves which are continuous everywhere but nowhere differentiable and which have infinite length (see the Koch curve).

Let now $\boldsymbol{F}(x,y)=P(x,y) \widehat{\boldsymbol{i}} + Q(x,y) \widehat{\boldsymbol{j}}$ be a continuous vector field and let $\boldsymbol{c}(t)= x(t) \widehat{\boldsymbol{i}} + y(t) \widehat{\boldsymbol{j}}$ be rectifiable. Then we define

$\displaystyle \int_{\boldsymbol{c}} \boldsymbol{F}\cdot\mathrm{d} \boldsymbol{s} = \int_a^b P(x(t),y(t)) \,\mathrm{d} x(t) + \int_a^b Q(x(t)),y(t)) \,\mathrm{d} y(t),$

where the last two integrals are Riemann-Stieltjes integrals.

Hence, the line integral is well-defined if $\boldsymbol{c}$ is continuous and each of its components have bounded variation.

The region ${}R$

We now discuss the region ${}R$ which is enclosed by the curve $\mathcal{C}$ (which is parameterised by $\boldsymbol{c}$). The double integral is certainly defined (since we assume that $Q_x - P_y$ are continuous) if ${}R$ is Jordan-measurable.

A non-empty bounded set $R \subseteq \mathbb{R}^2$ is called Jordan-measurable if the function

$\displaystyle 1_R(\boldsymbol{z})=\left\{\begin{array}{rl} 1 & \mbox{if } x \in R, \\ &\\ 0&\mbox{otherwise,} \end{array} \right.$

is Riemann-integrable.

There is a theorem which states that a set ${}R$ is Jordan-measurable if and only if its boundary $\partial R$ has Lebesgue measure zero. The boundary of ${}R$ is just the curve $\mathcal{C},$ since ${}R$ is defined as the region enclosed by $\mathcal{C}.$ (The last statement is well-defined by the Jordan-curve theorem.) Since $\boldsymbol{c}$ is rectifiable it has Lebesgue measure zero (the curve is Lebesgue measurable since it is the image of a continuous map defined on a compact set and hence the curve is compact). Therefore, the region ${}R$ enclosed by the simple closed curve which is parameterised by a continuous and rectifiable function $\boldsymbol{c}$ is Jordan-measurable (such curves are also called rectifiable Jordan curves).

Thus the assumptions on $\boldsymbol{c}$ guarantee that the region ${}R$ is Jordan-measurable and, in turn, the double integral is well defined.

Curves and regions for which Green’s theorem applies

So far we investigated curves and regions which ensure that the line integral and double integral in (1) are well defined. We now turn to Green’s theorem itself and investigate in more detail for which regions Green’s theorem holds.

Let $[a,b]$ be an interval with $a\textless b$ and let $\phi_1,\phi_2:[a,b]\to\mathbb{R}$ be two continuous functions such that $\phi_1(x) \textless \phi_2(x)$ for all $x \in [a,b].$ The set

$\displaystyle R_x=\{(x,y)\in\mathbb{R}^2: \phi_1(x) \le y \le \phi_2(x)\}$

is called ${}x$-simple. Analogously, let $[c,d]$ be an interval with $c \textless d$ and let $\psi_1,\psi_2: [c,d]\to\mathbb{R}$ be two continuous functions such that $\psi_1(y) \textless \psi_2(y)$ for all $y\in [c,d].$ We call the set

$\displaystyle R_y=\{(x,y) \in \mathbb{R}^2: \psi_1(y)\le x \le \psi_2(y)\}$

${}y$-simple.

For such regions the following two lemmas apply.

Lemma
Let $D$ be an ${}x$-simple region and let $P:D\to\mathbb{R}$ be continuously differentiable. Then

$\displaystyle \int_{\mathcal{C}^+} P \,\mathrm{d} x= -\int_D \frac{\partial P}{\partial y} \,\mathrm{d} x \,\mathrm{d}y,$

where $\mathcal{C}^+$ is the boundary curve of ${}D$ oriented counterclockwise.

Lemma
Let $D$ be an ${}y$-simple region and let $Q:D\to\mathbb{R}$ be continuously differentiable. Then

$\displaystyle \int_{\mathcal{C}^+} Q \,\mathrm{d} y= \int_D \frac{\partial Q}{\partial x} \,\mathrm{d} x \,\mathrm{d}y,$

where $\mathcal{C}^+$ is the boundary curve of ${}D$ oriented counterclockwise.

A proof of these results can be found here. Notice that the proof only requires that the functions $\phi_1,\phi_2,\psi_1,\psi_2$ are continuous – no further smoothness assumption is needed.

Green’s theorem can now be obtained by adding up the two equations from the above lemmas and by assuming that the region ${}R$ is simultaneously ${}x$-simple and ${}y$-simple. Regions which are both ${}x$-simple and ${}y$-simple are in the following called simple.

What is not yet clear however is the following. In the definition of ${}x$-simple and ${}y$-simple we assumed that the boundary is only continuous. However, it is known that there are simple closed curves (the Koch curve for instance) which are not rectifiable and hence for which line integrals over such curves are not well defined. The following result states that this cannot happen. That is, the boundary curve of a region which is simple (i.e. ${}x$-simple and ${}y$-simple) has bounded variation.

Lemma
Let $R\subset\mathbb{R}^2$ be a region which is ${}x$-simple and ${}y$-simple. Then the boundary curve $\partial R$ has bounded variation and is therefore rectifiable.

The proof is left as an exercise.

The last result now implies that the line integral is well-defined, and since the boundary curve is a simple closed curve which is continuous and rectifiable, the region ${}R$ is Jordan-measurable. Hence also the double integral is well defined.

More general regions

Green’s theorem also applies to regions ${}R$ enclosed by a simple closed curve, where ${}R$ can be divided into a finite number of simple regions.

Green’s theorem
Let $R\subset \mathbb{R}^2$ be a region whose boundary is made up of a finite number of simple closed curves. Suppose that by means of a finite number of line segments parallel to the coordinate axes, ${}R$ can be decomposed into a finite number of simple regions $R_1,\ldots, R_N$ with the boundary of each $R_k$ oriented counterclockwise. Then if ${}P$ and ${}Q$ are continuously differentiable on ${}R,$ then

$\displaystyle \int_{\partial R} P \,\mathrm{d} x + Q \,\mathrm{d} y = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\,\mathrm{d} x\,\mathrm{d}y,$

where the boundary curve is oriented counterclockwise.

To prove this result one applied Green’s theorem to each $R_k,$ $k=1,\ldots, N.$ The line integral over the curves inside the region cancel which then implies the result. The details are left as an exercise.


text:
Phys. Rev. Lett. (in the press); preprint at http://arxiv.org/abs/1405.6241 (2014)

Using data from CERN's Large Hadron Collider, the ATLAS Collaboration has collected evidence of a process known as vector boson scattering (VBS). The vector boson in question is the W boson, discovered at CERN in the early 1980s. Now, ATLAS has identified an excess of events — in a 20.3 fb−1 sample of proton–proton collisions at a centre-of-mass energy of 8 TeV — that is consistent with the scattering of two W bosons, each having been radiated from the quarks inside the colliding protons.

By seeking a particular signature of VBS — of two same-electric-charge W bosons decaying to a neutrino plus either an electron or a muon, and accompanied by two jets of particles — the physicists were able to home in on the 'electroweak' class of VBS processes, and compare their cross-sections with the standard-model predictions. The now-discovered Higgs boson is a vital part of the theory, keeping the VBS amplitude under control (lest unitarity break down at the TeV scale). However, VBS is still usefully sensitive to 'new physics' — but, almost inevitably, ATLAS conclude that all is in line with the standard model.


text:
# Spherical cap

(Redirected from Spherical dome)
An example of a spherical cap in blue (and another in red.)

In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

## Volume and surface area

If the radius of the sphere is ${\displaystyle r}$ , and the height of the cap is ${\displaystyle h}$ , then the volume of the spherical cap is[1]

${\displaystyle V={\frac {\pi h^{2}}{3}}(3r-h)\,.}$

and the curved surface area of the spherical cap is[1]

${\displaystyle A=2\pi rh}$

or

${\displaystyle A=2\pi r^{2}(1-\cos \theta )\,,}$

in which ${\displaystyle \theta }$  is the polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. (If it ${\displaystyle \phi }$  denotes the latitude in geographic coordinates, ${\displaystyle \theta +\phi =\pi /2=90^{\circ }\,}$ .)

The relationship between ${\displaystyle h}$  and ${\displaystyle r}$  is irrelevant as long as ${\displaystyle 0\leq h\leq 2r}$ . For example, the red section of the illustration is also a spherical cap for which ${\displaystyle h>r}$ .

These formulas can be rewritten to use the radius ${\displaystyle a}$  of the base of the cap instead of ${\displaystyle r}$ , using the relationship:

${\displaystyle r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2}\,,}$

so that

${\displaystyle r={\frac {a^{2}+h^{2}}{2h}}\,.}$

Substituting this into the formulas gives:

${\displaystyle V={\frac {1}{6}}\pi h(3a^{2}+h^{2})\,,}$
${\displaystyle A=2\pi {\frac {(a^{2}+h^{2})}{2h}}h=\pi (a^{2}+h^{2})\,.}$

The volume formula may be derived by integrating under a surface of rotation and factorizing as follows.

${\displaystyle V=\int _{r\cos \theta }^{r}\pi \left(r^{2}-x^{2}\right)dx={\frac {\pi }{3}}r^{3}(2-3\cos \theta +\cos ^{3}\theta )={\frac {\pi }{3}}r^{3}(2+\cos \theta )(1-\cos \theta )^{2}\,.}$

## Applications

### Volumes of union and intersection of two intersecting spheres

The volume of the union of two intersecting spheres of radii ${\displaystyle r_{1}}$  and ${\displaystyle r_{2}}$  is [2]

${\displaystyle V=V^{(1)}-V^{(2)}\,,}$

where

${\displaystyle V^{(1)}={\frac {4\pi }{3}}r_{1}^{3}+{\frac {4\pi }{3}}r_{2}^{3}}$

is the sum of the volumes of the two isolated spheres, and

${\displaystyle V^{(2)}={\frac {\pi h_{1}^{2}}{3}}(3r_{1}-h_{1})+{\frac {\pi h_{2}^{2}}{3}}(3r_{2}-h_{2})}$

the sum of the volumes of the two spherical caps forming their intersection. If ${\displaystyle d\leq r_{1}+r_{2}}$  is the distance between the two sphere centers, elimination of the variables ${\displaystyle h_{1}}$  and ${\displaystyle h_{2}}$  leads to[3][4]

${\displaystyle V^{(2)}={\frac {\pi }{12d}}(r_{1}+r_{2}-d)^{2}\left(d^{2}+2d(r_{1}+r_{2})-3(r_{1}-r_{2})^{2}\right)\,.}$

### Surface area bounded by parallel disks

The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius ${\displaystyle r}$ , and caps with heights ${\displaystyle h_{1}}$  and ${\displaystyle h_{2}}$ , the area is

${\displaystyle A=2\pi r|h_{1}-h_{2}|\,,}$

or, using geographic coordinates with latitudes ${\displaystyle \phi _{1}}$  and ${\displaystyle \phi _{2}}$ ,[5]

${\displaystyle A=2\pi r^{2}|\sin \phi _{1}-\sin \phi _{2}|\,,}$

For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016[6]) is 2π·6371²|sin 90° − sin 66.56°| = 21.04 million km², or 0.5·|sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth.

This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the Tropics.

## Generalizations

### Sections of other solids

The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

### Hyperspherical cap

Generally, the ${\displaystyle n}$ -dimensional volume of a hyperspherical cap of height ${\displaystyle h}$  and radius ${\displaystyle r}$  in ${\displaystyle n}$ -dimensional Euclidean space is given by [7]

${\displaystyle V={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(t)\,\mathrm {d} t}$

where ${\displaystyle \Gamma }$  (the gamma function) is given by ${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\mathrm {e} ^{-t}\,\mathrm {d} t}$ .

The formula for ${\displaystyle V}$  can be expressed in terms of the volume of the unit n-ball ${\displaystyle C_{n}={\scriptstyle \pi ^{n/2}/\Gamma [1+{\frac {n}{2}}]}}$  and the hypergeometric function ${\displaystyle {}_{2}F_{1}}$  or the regularized incomplete beta function ${\displaystyle I_{x}(a,b)}$  as

${\displaystyle V=C_{n}\,r^{n}\left({\frac {1}{2}}\,-\,{\frac {r-h}{r}}\,{\frac {\Gamma [1+{\frac {n}{2}}]}{{\sqrt {\pi }}\,\Gamma [{\frac {n+1}{2}}]}}{\,\,}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1-n}{2}};{\tfrac {3}{2}};\left({\tfrac {r-h}{r}}\right)^{2}\right)\right)={\frac {1}{2}}C_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right)}$  ,

and the area formula ${\displaystyle A}$  can be expressed in terms of the area of the unit n-ball ${\displaystyle A_{n}={\scriptstyle 2\pi ^{n/2}/\Gamma [{\frac {n}{2}}]}}$  as

${\displaystyle A={\frac {1}{2}}A_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right)}$  ,

where ${\displaystyle \scriptstyle 0\leq h\leq r}$ .

Earlier in [8] (1986, USSR Academ. Press) the following formulas were derived: ${\displaystyle A=A_{n}p_{n-2}(q),V=C_{n}p_{n}(q)}$ , where ${\displaystyle q=1-h/r(0\leq q\leq 1),p_{n}(q)=(1-G_{n}(q)/G_{n}(1))/2}$ ,

${\displaystyle G_{n}(q)=\int \limits _{0}^{q}(1-t^{2})^{(n-1)/2}dt}$ .

For odd ${\displaystyle n=2k+1:}$

${\displaystyle G_{n}(q)=\sum _{i=0}^{k}(-1)^{i}{\binom {k}{i}}{\frac {q^{2i+1}}{2i+1}}}$ .

#### Asymptotics

It is shown in [9] that, if ${\displaystyle n\to \infty }$  and ${\displaystyle q{\sqrt {n}}={\text{const.}}}$ , then ${\displaystyle p_{n}(q)\to 1-F({q{\sqrt {n}}})}$  where ${\displaystyle F()}$  is the integral of the standard normal distribution.

A more quantitive way of writing this, is in [10] where the bound ${\displaystyle A/A_{n}=n^{\Theta (1)}\cdot [(2-h/r)h/r]^{n/2}}$  is given. For large caps (that is when ${\displaystyle (1-h/r)^{4}\cdot n=O(1)}$  as ${\displaystyle n\to \infty }$ ), the bound simplifies to ${\displaystyle n^{\Theta (1)}\cdot e^{-(1-h/r)^{2}n/2}}$ .

## References

1. ^ a b Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023.
2. ^ Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc. 107: 1118–1124. doi:10.1021/ja00291a006.
3. ^ Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. 6: 133–135. doi:10.1016/0097-8485(82)80006-5.
4. ^ Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. 68: 441–451. doi:10.1021/j100785a001.
5. ^ Scott E. Donaldson, Stanley G. Siegel. "Successful Software Development". Retrieved 29 August 2016.
6. ^ "Obliquity of the Ecliptic (Eps Mean)". Neoprogrammics.com. Retrieved 2014-05-13.
7. ^ Li, S (2011). "Concise Formulas for the Area and Volume of a Hyperspherical Cap". Asian J. Math. Stat. 4 (1): 66–70. doi:10.3923/ajms.2011.66.70.
8. ^ Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission. 22 (4): 49–54.
9. ^ Chudnov, Alexander M (1991). "Game-theoretical problems of synthesis of signal generation and reception algorithms (rus.)". Problems of Information Transmission. 27 (3): 57–65.
10. ^ Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. 2016. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (SODA '16), Robert Kraughgamer (Ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 10-24.


text:
# Kunita–Watanabe inequality

In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes.

## Statement of the theorem

Let M, N be continuous local martingales and H, K measurable processes. Then

${\displaystyle \int _{0}^{t}\left|H_{s}\right|\left|K_{s}\right|\left|\mathrm {d} \langle M,N\rangle _{s}\right|\leq {\sqrt {\int _{0}^{t}H_{s}^{2}\,\mathrm {d} \langle M\rangle _{s}}}{\sqrt {\int _{0}^{t}K_{s}^{2}\,\mathrm {d} \langle N\rangle _{s}}}}$

where the brackets indicates the quadratic variation and quadratic covariation operators. The integrals are understood in the Lebesgue–Stieltjes sense.

## References

• Rogers, L. C. G.; Williams, D. (1987). Diffusions, Markov Processes and Martingales. II, Itô; Calculus. Cambridge University Press. p. 50. doi:10.1017/CBO9780511805141. ISBN 0-521-77593-0.


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# Insertion of a node in M-way search tree

I am currently studying data structures and i am looking to see if my implementation of the insertion method in an M-way search tree is good and i am looking forward to improving my coding approach.

There are some basic cases in inserting a node in an M-way search tree:-

Basic Case: If there is still space in the node for new key.

Otherwise,

Case 1: If the data is smaller than the first key of the node then we move towards the first child.

Case 2: If the data is greater than the last node then we move towards the last child.

Case 3: According to the property we check accordingly if the value ranges between the values of two keys.

Code:

#include<iostream>
using namespace std;

struct mWay{
int index=-1;
static const int m=3;
int k[m-1];
mWay *child[m]={NULL};
};

mWay* createNode(int value){
mWay *node=new mWay;
(node->index)++;
node->k[node->index]=value;
return node;
}

void sorting(mWay *s){      //using bubble sort..
int temp;
for(int i=0;i<(s->index);i++){
for(int j=0;j<(s->index);j++){
if((s->k[j])>(s->k[j+1])){
temp=(s->k[j]);
(s->k[j])=(s->k[j+1]);
(s->k[j+1])=temp;
}
}
}
}

mWay* Insert(int data,mWay *root){
if(root==NULL){         //creating a new mWay tree...
root=createNode(data);
return root;
}
else{                   //updating the mWay tree...
mWay *ptr=root;
mWay *par=NULL;
int store;          //we will store the index of next child node in this variable (in the last case)...
while(ptr!=NULL){
par=ptr;
if((ptr->index)<(ptr->m)-2){    //checking to see if we can insert more keys in the current node...
(ptr->index)++;
ptr->k[ptr->index]=data;
sorting(ptr);               //sorting keys..
return root;
}
else{       //we have to move to the next node..
if(data<ptr->k[0]){     //if the data is smaller than the first key of the node then we move towards the first child...
ptr=ptr->child[0];
}
else if(data>ptr->k[(ptr->m)-2]){       //if the data is greater than the last node then we move towards the last child...
ptr=ptr->child[(ptr->m)-1];
}
else{       //according to the property we check accordingly if the value ranges between the values of two keys ...
for(int i=1;i<(ptr->m)-1;i++){
if(data>(ptr->k[i-1]) && data<(ptr->k[i])){
store=i;
ptr=ptr->child[i];
break;
}
}
}
}
}   //end of while loop which means we just have to insert the node now...
if(data<(par->k[0])){
par->child[0]=createNode(data);
}
else if(data>(par->k[(par->m)-2])){
par->child[(par->m)-1]=createNode(data);
}
else{
par->child[store]=createNode(data);
}
return root;
}
}

• Do you mind adding a little more information about M-Way search trees for those of us not familiar with the data structure? – Dannnno Jul 13 '18 at 21:32
• @Dannnno I am so sorry for replying late. I forgot to reply you. Here is the detail for this data structure (although i think you have already studied about it ;) ) : webdocs.cs.ualberta.ca/~holte/T26/m-way-trees.html – sazharsha Aug 4 '18 at 7:06

This looks more like C than C++.

The only features of C++ I see used are #include <iostream>, using namespace std; and new.

Since this is flagged , I will review it as such, but if it is intended to use with C, a review for that language might give better insights.

# Don't reinvent the wheel

The C++ standard library provides many features that would help. Need a sorting algorithm? Use std::sort. Need to find something in a container? Use std::find, std::lower_bound, std::upper_bound or some other algorithm that suits your needs!

# Naming

Many variables have very short and cryptic names. This makes the code hard to read.

Example: What does mWay::k stand for?

# Issues

• Elements of <iostream> aren't used anywhere.

• using namespace std; is discouraged, especially in headers. If someone else includes that header, it might cause ambiguities in their code!

# Design

C++ has classes. Why not use them to define the tree, instead of a loose set of free standing functions?

For example, a rough outline could look like this:

#include <array>
#include <memory>
#include <algorithm>

template<typename ValueType, size_t Arity>
class nary_search_tree {
static_assert(Arity >= 2u, "Tree needs an arity greater than or equal to 2!");

class node {
size_t size = 0;
std::array<ValueType, Arity-1> values;
std::array<std::unique_ptr<node>, Arity> children;

void insert_in_order(ValueType value) {
auto insert_index = find_child_index(value);

if(insert_index == size) {
values[insert_index] = value;
++size;
return;
}

std::copy_backwards(&values[insert_index], &values[size-1], &values[size]);
values[insert_index] = value;
++size;
}

void find_child_index(ValueType value) const {
for(auto i = 0u; i < size; ++i) {
if(values[i] > value) return i;
}

return size;
}

public:
void insert(ValueType value) {
if(index < values.size()) {
insert_in_order(value);
return;
}

auto index = find_child_index(value);
if(!children[index]) {
children[index] = std::make_unique<node>();
}

children[index]->insert(value);
}
};

std::unique_ptr<node> root = nullptr;

public:
void insert(ValueType value) {
if(!root) root = std::make_unique<node>();

root->insert(value);
}
};

Makes reasoning much easier, doesn't it? And as a bonus, it's much more generic, too!

Note: This code requires C++11 or later to work.

Of course, this is just the beginning... This code could be improved, e.g. by using const ValueType& to prevent copies and by using overloads for ValueType&& to support movable types. Also, other search tree functionality is missing, like looking up an element

• It seems that there is much to learn. I am a beginner and a student who is taking this DS and A course. My aim was to implement this data structure from scratch all by myself :) maybe this is the reason it looks for like C. I should’ve done this in C then. oh well. Thank you :) – sazharsha Jul 14 '18 at 6:33
• @sazharsha: The intention behind data structures is structuring data in order to better reason about it. The intention behind classes is to structure behavior in order to better reason about it :) The C style implementation of the data structure works, but reasoning about what it does is a lot harder IMO. Often, it's easy to implement data structure as classes, since their behavior easily maps to methods. In any way, extracting helper functions nearly always improves readability and reusability, so try to do so if possible. – hoffmale Jul 14 '18 at 9:20

# Don’t write using namespace std;.

You can, however, in a CPP file (not H file) or inside a function put individual using std::string; etc. (See SF.7.)

Forget that the legacy NULL macro ever existed. Don’t use it — ever. ⧺ES.47: Use nullptr rather than 0 or NULL

Don't write explicit comparisons against nullptr. Use the contextual conversion to bool as "OK to use".

⧺C.149 — no naked new or delete.

You should probably make this a unique_ptr as a drop-in replacement without otherwise changing the architecture.

Bubble sort?! Just call std::sort.

Prefer prefix over postfix

The biggest problem is this part of the code.

        if((ptr->index)<(ptr->m)-2){    //checking to see if we can insert more keys in the current node...
(ptr->index)++;
ptr->k[ptr->index]=data;
sorting(ptr);               //sorting keys..
return root;
}

Lets start with the minor problems, the sorting is a bubble sort ie. its an O(n^2) sort, not only is it slow but unneeded as we know our array is already sorted before adding a new value. So only the new value needs to be placed, we can find it with a binary search O(log n) plus a insert O(n) which together is O(n), where n is the number of values in the node.

The other thing is that this works ... as long as you haven't deleted any of the values. If you want to support delete and new inserts you have to move the child pointers too.

• ohh. I have only implemented insertion method. i will implement this including deletion and searching in B tree. Half of this DSA course is still left and its giving me a hard time. – sazharsha Jul 14 '18 at 9:01
• i still didn’t get the sorting part.I want all the elements in the array of the node to be in sorted order after inserting a new one (which is not possible every time) so i implemented bubble sort for that ,in this multi way search tree. – sazharsha Jul 14 '18 at 9:11
• I'd use a linear search (see the snippet in my answer) instead of binary search. If the arity of the tree is big enough that binary search gives a measurable performance win, then reducing the arity of the tree would likely result in the same performance win, if not better. ($\log{2}(n)$ vs $\log{Arity}(n)$) – hoffmale Jul 14 '18 at 9:16
• After seeing the latest iteration of Alexandescu's Fastware from NDC I have to agree with you that linear search is better for small arities. – Surt Jul 19 '18 at 22:04


text:
# Relationship between reductio ad absurdum and modus tollens

### Math Forum Discussions

There is some instability in the terminology here. Many authors use Reductio Ad Absurdum (RAA) as meaning the same as proof by contradiction and indirect. lus, only one of them, modus ponens, is clearly indemonstrable. Never- theless, I try to show . Where '→' stands for conditional relationship. In my view, it is not only linked in logic to Reductio ad Absurdum, but also to the Ex. Contradictione. general form of argument known as reductio ad absurdum. is the same as modus tollens, and the only difference between those three is.

### Modus tollens - Wikipedia

Сердечный приступ. Беккер безучастно кивнул: - Так мне сказали. Лейтенант вздохнул и сочувственно помотал головой.


text:
# FiveThirtyEight

## Science

One key piece of the Obama administration’s latest proposal for reducing carbon emissions from power plants, announced on Monday, would have states start or join existing regional cap-and-trade programs. While the success of cap and trade in cutting power-plant emissions that cause acid rain is widely known, using the same approach to regulate carbon dioxide emissions has a shorter, and bumpier, track record. Since 2009, nine northeastern U.S. states have operated a cap-and-trade market for power plant carbon dioxide emissions, and my analysis indicates that this market had a minimal effect on emissions. Why? The “cap” part of cap and trade was, until recently, too high for the program be effective.

Known as the Regional Greenhouse Gas Initiative (RGGI), the program was designed and agreed upon by Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New Jersey, New York, Rhode Island and Vermont in 2005. (New Jersey Gov. Chris Christie pulled his state out of the program in 2011.) The RGGI set a cap on carbon dioxide emissions for the entire region, and specific allowances for each state, that were based upon actual emissions amounts in the early 2000s. In its original design, the program set a fixed cap for 2009 through 2014, with decreases in the cap of 2.5 percent per year scheduled to begin in 2015.

The basic concept of cap and trade is simple: The government sets an overall “cap” on the quantity of a particular compound that can be emitted into the atmosphere. Entities like power plants then purchase permits, known as “allowances,” in order to legally release that compound. The allowances are traded among entities on an open market, so that companies must purchase permits to pollute, giving them a financial incentive to invest in cleaner technologies that reduce their emissions. In most cap-and-trade schemes, the overall cap on the total emissions gets lower over time, with a corresponding increase in the market price of allowances.

To assess the effectiveness of the northeastern states’ program at reducing carbon dioxide emissions, I compared emissions data from the nine RGGI states with similar data from all the other states, which had no regulations on carbon dioxide emissions.1 If the rate of change in carbon dioxide emissions in RGGI states differed from non-RGGI states, that would be a good initial indication that the RGGI program was working.

The chart above shows that carbon dioxide emissions have declined considerably during recent years in both RGGI and non-RGGI states. This nationwide pattern cannot be attributed to a regional program like the RGGI. Instead, it results from two major factors: the recession, which hit hardest in 2009 just as the RGGI program began, and increasingly abundant natural gas, which emits less carbon dioxide than coal. If you compare the two lines from 2009 to 2012, it’s not clear whether the RGGI states have been lowering their carbon emissions any faster than the rest of the country. It’s possible that the large initial decline from 2008 to 2009 in RGGI states, corresponding with the start of the RGGI market, was in part driven by the program, but it’s a stretch to make that conclusion from this data alone.

We can look more closely, and perform a statistical test, by using data from individual states.

Here, each line represents a different state, and the carbon emissions of each state are normalized to their 2008 values. There are two charts: one for RGGI states and the other for non-RGGI states, each with an overall trend. Both show a decrease in carbon dioxide emissions over time, but there is no statistically significant difference between RGGI and non-RGGI states in the rate of decrease.2 Again, there is no evidence that the nine northeastern states, under a cap-and-trade program, saw their carbon emissions decline any faster than the rest of the country did.

The RGGI program, the first regulation of carbon dioxide emissions from power plants in U.S. history, had little initial effect because the cap turned out to be much too high. The cap amount was set in 2005, and was based upon actual emissions amounts during the early 2000s. The assumption was that without regulation carbon dioxide emissions would continue to rise and should therefore be capped and held at their current levels.

Instead, carbon dioxide emissions decreased considerably due to the recession and the natural gas boom, but the RGGI program wasn’t flexible enough for the cap to move along with them. In its first five years, the RGGI’s market-based cap-and-trade program gave power plants no incentive to reduce their emissions because the cap amount simply was irrelevant.

You can see in the chart above that the regional cap of 165 million tons of carbon dioxide was way above the actual emissions output for the region. A new regional cap of 91 million tons went into effect this year. This new cap is low enough to be relevant; in fact, emissions data from the first quarter of 2014 has the region on track to exceed the cap unless subsequent quarters see lower emissions.3

In case you’re still not convinced that RGGI used a too-high cap for carbon dioxide emissions from 2009 to 2013, let’s look at the market price of carbon under the program. Here’s what allowances to emit carbon dioxide traded for over the life of the program:

As you can see, the first sales of allowances, in late 2008, put the price of carbon dioxide emissions at around $3 a ton. Under a cap-and-trade program that’s working as intended, the price of allowances should increase over time, giving power plants an incentive to reduce emissions. In the case of the RGGI, however, the price plummeted in mid-2009 to around$2 a ton, and for most of 2010 through 2012 it remained below \$2. It likely would have stayed at that price but for the dramatically lower cap starting in 2014, which will decline by 2.5 percent each year until 2020. After that cap was announced, the market price of carbon allowances promptly began increasing; the regional cap was suddenly relevant.

We can take some lessons from the RGGI and apply them to the new federal program to regulate carbon emissions, which includes giving states the option of creating new cap-and-trade programs. The system must be able to adapt to unforeseen conditions. The RGGI’s big flaw was that there was no mechanism in place to lower the cap if something unexpected happened with energy markets.

The Obama administration’s new program has one huge advantage over the RGGI in this regard: Because it’s a federal program, it should in principle be more adaptable, since it’s being imposed from the top. The RGGI is a regulatory agreement among nine states, so any changes to the agreement have to be negotiated among nine different governor’s offices, state legislatures and other relevant entities. On some level, it’s no wonder it took the states four years to pull off lowering the cap.

An optimistic observer would hope that the U.S. Environmental Protection Agency, which will administer the new carbon emissions rule, will be in a better position to adjust its new program in response to unforeseen conditions than the RGGI initially was. Although the proposed rule will give states a number of different options to choose from in order to meet required emissions reductions, the standards themselves — the caps — will be determined by the EPA.

### Footnotes

1. California is the only non-RGGI state that now also regulates carbon dioxide emissions from power plants, but its cap-and-trade program didn’t begin until 2013. ^
2. I fit linear mixed-effects models, via maximum likelihood, to the full data set of all states, with state as a random effect. I fit a model with “Year” as a continuous fixed effect and RGGI/non-RGGI as a binary fixed effect. I tested this model against a null model in which only the fixed effect of Year was present. I tested the RGGI/non-RGGI model against the null with a likelihood ratio test, which resulted in a p-value of 0.09. I also tested a model that additionally included an interaction term between RGGI/non-RGGI and Year; the likelihood ratio test for this model against the null resulted in a p-value of 0.12. ^
3. The point shown in the chart for 2014 is based upon reported emissions for the first quarter of 2014, extrapolated to a full year of emissions at that level. ^

All Brooks Miner

### What Does Public Support in Obama’s Second Inaugural Speech? Jan 22

All Climate Change

All Science


text:
# Welding defect

A welding defect is any flaw that compromises the usefulness of a weldment. There is a great variety of welding defects. Welding imperfections are classified according to ISO 6520[1] while their acceptable limits are specified in ISO 5817 [2] and ISO 10042. [3]

## Major causes

According to the American Society of Mechanical Engineers (ASME), welding defect causes are broken down as follows: 45 percent poor process conditions, 32 percent operator error, 12 percent wrong technique, 10 percent incorrect consumables, and 5 percent bad weld grooves.[4]

### Residual stresses

Main article: Residual stress

The magnitude of stress that can be formed from welding can be roughly calculated using:[5]

$E \alpha \Delta T$

Where E is Young's modulus, α is the coefficient of thermal expansion, and ΔT is the temperature change. For steel this calculates out to be approximately 3.5 GPa (510,000 psi).

## Types

### Cracks

Defects related to fracture.

#### Arc strike cracking

Arc strike cracking occurs when the arc is struck but the spot is not welded. This occurs because the spot is heated above the material's upper critical temperature and then essentially quenched. This forms martensite, which is brittle and micro-cracks. Usually the arc is struck in the weld groove so this type of crack does not occur, but if the arc is struck outside of the weld groove then it must be welded over to prevent the cracking. If this is not an option then the arc spot can be postheated, that is, the area is heated with an oxy-acetylene torch, and then allowed to cool slowly.[6]

#### Cold cracking

Residual stresses can reduce the strength of the base material, and can lead to catastrophic failure through cold cracking, as was the case with several of the World War II Liberty ships' hulls. Cold cracking is limited to steels and is associated with the formation of martensite as the weld cools. The cracking occurs in the heat-affected zone of the base material. To reduce the amount of distortion and residual stresses, the amount of heat input should be limited, and the welding sequence used should not be from one end directly to the other, but rather in segments.[7]

Cold cracking only occurs when all the following preconditions are met:[citation needed]

• susceptible microstructure (e.g. martensite)
• hydrogen present in the microstructure (hydrogen embrittlement)
• service temperature environment (normal atmospheric pressure): -100 to +100 °F
• high restraint

Eliminating any one of these will eliminate this condition.

#### Crater crack

Crater cracks occur when a crater is not filled before the arc is broken. This causes the outer edges of the crater to cool more quickly than the crater, which creates sufficient stresses to form a crack. Longitudinal, transverse and/or multiple radial cracks may form.[8]

#### Fusion-line cracks

The following is an excerpt from a research on Damage Behavior of Welded Joints with Defects on Fusion Line

Two kinds of welded joints damage models,with notch and crack on fusion line,were applied to analyse the influences of the mechanical property heterogeneity and geometric heterogeneity such as intensity matching, base metal rupture strain,base metal strain-hardening exponential and welding line width on the damage distribution and the relationship between damage parameter and rupture paremater. In addition, in the presence of entire damage zone method, the crack extension behaviors of the welded joints were investigated.

Base metal mechanical property has great effects on the damage distribution of the welded joints. On condition of similar load,the welded joints damage value increases with the decreasing base metal rupture strain and the increasing base metal strain-hardening exponential and intensity matching.

For under-matched welded joints,the damage concentration zone of the welded joints near the notch will transfer from the welded metal side to the base metal side as the base metal rupture strain decreases and strain-hardening exponential increases. but for over-matched welded joints, the damage concentration zone will transfer from the base metal side to the welded metal side as the base metal rupture strain increases and strain-hardening exponential decreases.

On the other hand, the varied welded line width has little effect on the damage distribution of the welded joints.For the welded joints with crack on fusion line,when loaded and when the crack mouth opening distance reaches the same level,the maximum damage value in front of the crack increases as the base metal strain-hardening exponential and intensity matching increase,but decreases as the base metal rupture strain increases.

The establishment of the failure principle on the welded structure should take mechanical property heterogeneity into serious consideration. The numerical calculations of the crack extension behavior which is based on the entire damage zone viewpoint reveal that the crack extension path is parallel to the fusion line on condition of plane stress but on condition of plane strain the angle between the crack extension path and the fusion line is approximate 45°.The numerical calculations and analysis indicate that micro-damage parameter and macro -rupture parameter both can describe the stress-strain behaviors,and their relationship intensively relies on mechanical property heterogeneity. The investigation on the micro-damage parameter and macro -rupture parameter is undoubtedly supplement to the safety evaluation and life-span prediction of the welded joints,which is of great theory significance.

#### Hat crack

Hat cracks get their name from the shape of the cross-section of the weld, because the weld flares out at the face of the weld. The crack starts at the fusion line and extends up through the weld. They are usually caused by too much voltage or not enough speed.[8]

#### Hot cracking

Hot cracking, also known as solidification cracking, can occur with all metals, and happens in the fusion zone of a weld. To diminish the probability of this type of cracking, excess material restraint should be avoided, and a proper filler material should be utilized.[7] Other causes include too high welding current, poor joint design that does not diffuse heat, impurities (such as sulfur and phosphorus), preheating, speed is too fast, and long arcs.[9]

An undercut crack, also known as a heat-affected zone (HAZ) crack,[10] is a crack that forms a short distance away from the fusion line; it occurs in low alloy and high alloy steel. The exact causes of this type of crack are not completely understood, but it is known that dissolved hydrogen must be present. The other factor that affects this type of crack is internal stresses resulting from: unequal contraction between the base metal and the weld metal, restraint of the base metal, stresses from the formation of martensite, and stresses from the precipitation of hydrogen out of the metal.[11]

#### Longitudinal crack

Longitudinal cracks run along the length of a weld bead. There are three types: check cracks, root cracks, and full centerline cracks. Check cracks are visible from the surface and extend partially into weld. They are usually caused by high shrinkage stresses, especially on final passes, or by a hot cracking mechanism. Root cracks start at the root and extent part way into the weld. They are the most common type of longitudinal crack because of the small size of the first weld bead. If this type of crack is not addressed then it will usually propagate into subsequent weld passes, which is how full cracks (a crack from the root to the surface) usually form.[8]

#### Reheat cracking

Reheat cracking is a type of cracking that occurs in HSLA steels, particularly chromium, molybdenum and vanadium steels, during postheating. The phenomenon has also been observed in austenitic stainless steels. It is caused by the poor creep ductility of the heat affected zone. Any existing defects or notches aggravate crack formation. Things that help prevent reheat cracking include heat treating first with a low temperature soak and then with a rapid heating to high temperatures, grinding or peening the weld toes, and using a two layer welding technique to refine the HAZ grain structure.[12][13]

#### Root and toe cracks

A root crack is the crack formed by the short bead at the root(of edge preparation) beginning of the welding, low current at the beginning and due to improper filler material used for welding.Major reason for happening of these types of cracks is hydrogen embrittlement. These types of defects can be eliminated using high current at the starting and proper filler material. Toe crack occurs due to moisture content present in the welded area,it as a part of the surface crack so can be easily detected. Preheating and proper joint formation is must for eliminating these types of defects.

#### Transverse crack

A gap or break in the surface of a weld perpendicular to the weld axis that may be completely within the weld metal or may extend from the weld metal into the base metal. Excess hydrogen, an excessively strong weld metal, and high levels of residual stress result in transverse cracks.

### Distortion

Welding methods that involve the melting of metal at the site of the joint necessarily are prone to shrinkage as the heated metal cools. Shrinkage then introduces residual stresses and distortion. Distortion can pose a major problem, since the final product is not the desired shape. To alleviate certain types of distortion the workpieces can be offset so that after welding the product is the correct shape.[14] The following pictures describe various types of welding distortion:[15]

### Gas inclusion

Gas inclusions is a wide variety of defects that includes porosity, blow holes, and pipes (or wormholes). The underlying cause for gas inclusions is the entrapment of gas within the solidified weld. Gas formation can be from any of the following causes: high sulphur content in the workpiece or electrode, excessive moisture from the electrode or workpiece, too short of an arc, or wrong welding current or polarity.[10]

### Inclusions

There are two types of inclusions: linear inclusions and rounded inclusions. Inclusions can be either isolated or cumulative. Linear inclusions occur when there is slag or flux in the weld. Slag forms from the use of a flux, which is why this type of defect usually occurs in welding processes that use flux, such as shielded metal arc welding, flux-cored arc welding, and submerged arc welding, but it can also occur in gas metal arc welding. This defect usually occurs in welds that require multiple passes and there is poor overlap between the welds. The poor overlap does not allow the slag from the previous weld to melt out and rise to the top of the new weld bead. It can also occur if the previous weld left an undercut or an uneven surface profile. To prevent slag inclusions the slag should be cleaned from the weld bead between passes via grinding, wire brushing, or chipping.[16]

Isolated inclusions occur when rust or mill scale is present on the base metal.[17]

### Lack of fusion and incomplete penetration

Lack of fusion is the poor adhesion of the weld bead to the base metal; incomplete penetration is a weld bead that does not start at the root of the weld groove. Incomplete penetration forms channels and crevices in the root of the weld which can cause serious issues in pipes because corrosive substances can settle in these areas. These types of defects occur when the welding procedures are not adhered to; possible causes include the current setting, arc length, electrode angle, and electrode manipulation.[18] Defects can be varied and classified as critical or non critical. Porosity (bubbles) in the weld are usually acceptable to a certain degree. Slag inclusions, undercut, and cracks are usually non acceptable. Some porosity, cracks, and slag inclusions are visible and may not need further inspection to require their removal. Small defects such as these can be verified by Liquid Penetrant Testing (Dye check). Slag inclusions and cracks just below the surface can be discovered by Magnetic Particle Inspection. Deeper defects can be detected using the Radiographic (X-rays) and/or Ultrasound (sound waves) testing techniques.

### Lamellar tearing

Lamellar tearing is a type of welding defect that occurs in rolled steel plates that have been welded together due to shrinkage forces perpendicular to the faces of the plates.[19] Since the 1970s, changes in manufacturing practices limiting the amount of sulfur used have greatly reduced the incidence of this problem.[20]

Lamellar tearing is caused mainly by sulfurous inclusions in the material. Other causes include an excess of hydrogen in the alloy. This defect can be mitigated by keeping the amount of sulfur in the steel alloy below 0.005%.[20] Adding rare earth elements, zirconium, or calcium to the to control the configuration of sulfur inclusions throughout the metal lattice can also mitigate the problem.[21]

Modifying the construction process to use casted or forged parts in place of welded parts can eliminate this problem, as Lamellar tearing only occurs in welded parts.[19]

### Undercut

Undercutting is when the weld reduces the cross-sectional thickness of the base metal, which reduces the strength of the weld and workpieces. One reason for this type of defect is excessive current, causing the edges of the joint to melt and drain into the weld; this leaves a drain-like impression along the length of the weld. Another reason is if a poor technique is used that does not deposit enough filler metal along the edges of the weld. A third reason is using an incorrect filler metal, because it will create greater temperature gradients between the center of the weld and the edges. Other causes include too small of an electrode angle, a dampened electrode, excessive arc length, and slow speed.[22]

## References

1. ^ BS EN ISO 6520-1: "Welding and allied processes — Classification of geometric imperfections in metallic materials — Part 1: Fusion welding"(2007)
2. ^ BS EN ISO 5817: "Welding — Fusion-welded joints in steel, nickel, titanium and their alloys (beam welding excluded) — Quality levels for imperfections" (2007)
3. ^ BS EN ISO 10042: "Welding. Arc-welded joints in aluminium and its alloys. Quality levels for imperfections" (2005)
4. ^ Matthews, Clifford (2001), ASME engineer's data book, ASME Press, p. 211, ISBN 978-0-7918-0155-0.
5. ^ Bull, Steve (2000-03-16), Magnitude of stresses generated, University of Newcastle upon Tyne, archived from the original on 2009-12-06, retrieved 2009-12-06.
6. ^ Rampaul 2003, pp. 207–208.
7. ^ a b Cary & Helzer 2005, pp. 404–405.
8. ^ a b c Raj, Jayakumar & Thavasimuthu 2002, p. 128.
9. ^ Bull, Steve (2000-03-16), Factors promoting hot cracking, University of Newcastle upon Tyne, archived from the original on 2009-12-06, retrieved 2009-12-06.
10. ^ a b Raj, Jayakumar & Thavasimuthu 2002, p. 126.
11. ^ Rampaul 2003, p. 208.
12. ^ Bull, Steve (2000-03-16), Reheat cracking, University of Newcastle upon Tyne, archived from the original on 2009-12-06, retrieved 2009-12-06.
13. ^ Bull, Steve (2000-03-16), Reheat cracking, University of Newcastle upon Tyne, archived from the original on 2009-12-06, retrieved 2009-12-06.
14. ^ Weman 2003, pp. 7–8.
15. ^ Bull, Steve (2000-03-16), Welding Faults and Defects, University of Newcastle upon Tyne, archived from the original on 2009-12-06, retrieved 2009-12-06.
16. ^ Defects/imperfections in welds - slag inclusions, archived from the original on 2009-12-05, retrieved 2009-12-05.
17. ^ Bull, Steve (2000-03-16), Welding Faults and Defects, University of Newcastle upon Tyne, archived from the original on 2009-12-05.
18. ^ Rampaul 2003, p. 216.
19. ^ a b Bull, Steve (2000-03-16), Welding Faults and Defects, University of Newcastle upon Tyne, archived from the original on 2009-12-03.
20. ^ a b Still, J. R., Understanding Hydrogen Failures, retrieved 2009-12-03.
21. ^ Ginzburg, Vladimir B.; Ballas, Robert (2000), Flat rolling fundamentals, CRC Press, p. 142, ISBN 978-0-8247-8894-0.
22. ^ Rampaul 2003, pp. 211–212.


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## Girard’s Theorem

Here’s one of my favorite theorems from geometry, both because of the result itself, and because of the proof I’ll describe in this blog post. I suspect it was much better known when spherical trigonometry used to be part of the standard curriculum, but that was before my time. Now it seems to be forgotten, and if it’s mentioned at all it’s in a differential geometry course as an elementary consequence of the more general and powerful Gauss-Bonnet theorem.

One of the very old and well known results about triangles is that the sum of the angles is equal to $\pi$ (or $180^o$ if you measure angles in degrees instead of radians, which is almost never done in modern mathematics). Girard’s theorem can be viewed as the analogous result in the context of spherical geometry, where the analogue of line segments in the plane is great circle arcs on the sphere.

Girard’s Theorem For a spherical triangle on a unit sphere (i.e. radius equal to one) with interior angles $A,B,C$ the quantity $A+B+C-\pi$ is equal to the area of the triangle.

The way this is proved nowadays (which is not the proof I’ll discuss later in this post) is by first proving the Gauss-Bonnet theorem. To do that you need to work in the more general context of Riemannian geometry where lines in a plane generalize to geodesics in a surface (or an $n$ dimensional Riemannian manifold if you really want to generalize). And then you need to introduce the concept of curvature.

Once you’ve done all that, you can finally state and prove the Gauss-Bonnet theorem, which says that the quantity $A+B+C-\pi$ (where the curves that make up a triangle are geodesics) is equal to the integral of the curvature over the surface of the triangle. For a unit sphere the curvature is one, and for the plane the curvature is zero, and so the Gauss-Bonnet theorem can be used to prove both results mentioned above.

Just as there’s a much more direct and elementary proof of the result for planar triangles, there’s a much more direct and elementary proof of the result for spherical triangles, and that’s what I’ll write about in this post.

To warm up for the proof, I’d like to discuss the solution to a seemingly unrelated combinatorial problem. It turns out to be very closely related, since an approach to solving it is exactly the same as an approach to proving Girard’s theorem.

Problem Consider all words consisting of three characters $a, b,c$ How many words of length three are there where all three characters occur? Continue reading “Girard’s Theorem”

## Pythagoras’ Theorem

I often find myself thinking about two of the triangle results from antiquity. They’re Pythagoras’ theorem relating the sides of right angled triangles to each other, and Heron’s theorem relating the sides of an arbitrary triangle to the triangle’s area.

Of the two, Pythagoras’ theorem has been far more fruitful for the advancement of mathematics (in terms of implications and generalizations). In this blog post I’ll focus on Pythagoras’ theorem and describe some easily visualized (i.e. two and three dimensional) generalizations. Continue reading “Pythagoras’ Theorem”

## Volumes and Permutations

The factorial function $n! = n \cdot (n-1) \ldots 3 \cdot 2 \cdot 1$ is traditionally introduced as the number of permutations of $n$ items.

It also crops up as the reciprocal of the ($n$ dimensional) volume of the $n$ dimensional simplex defined by

$x_1 \textgreater 0, x_2 \textgreater 0, \ldots x_n \textgreater 0, x_1 + x_2 + \ldots + x_n \textless 1$

There’s a routine proof by induction that the volume of an $n$ simplex is $1/n!$, but it leaves the connection with the number of permutations of $n$ items a little mysterious.

Here’s one way to see why they’re related. Continue reading “Volumes and Permutations”


text:
Limited access

Researchers wanted to investigate how an individual's age relates to his or her political leanings. A random sample of $100$ individuals in each of three age categories was asked to classify themselves as being "Liberal" or "Conservative". The results are displayed in the side-by-side bar graph below.

Based on this display, what is the conditional distribution of Liberals by age category?

A

Liberals = $51.67\%$
Conservative = $48.33\%$

B

Age $25$ and Under = $70\%$
Age $26-55$ = $50\%$
Over Age $55$ = $35\%$

C

Age $25$ and Under = $33.33\%$
Age $26-55$ = $33.33\%$
Over Age $55$ = $33.33\%$

D

Age $25$ and Under = $45.16\%$
Age $26-55$ = $32.26\%$
Over Age $55$ = $22.58\%$

E

There is no way to determine the conditional distribution from the graph provided.

Select an assignment template


text:
How to implement level transitioning for 2D Platformer (tile-based)

I'm making a 2D tile-based platformer in C++/SDL and I want the player to move to the next level when they run off the side of the level, but I'm not sure how to implement this. I have tried putting colliders on the tiles at the edge of the map but it seems like there could be a better than than designating special tiles.

I wasn't sure what to keywords to search for when looking for a solution so if you can point me to a useful guide that would be great!

I'm looking for something very similar to how Super Metroid handles level changing.

Without knowing any of the mechanics of your code it can be difficult to suggest a solution that is efficient, simple to integrate, and extendable. Please consider that when asking future questions. It would be a better use of time for the both of us to tailor an answer to your needs than to give general advice.

Now with that said, given free reign of the problem, I would approach it with the idea in mind that you may want to end levels through different scenarios in the future.

First I would create an interface specifically for that purpose.

Interface LevelFinisher {

finishLevel()

}

Then you have the ability to end the level in numerous ways. You could assign the functionality to an item that you use, a platform you hit, an enemy you kill, etc.

Now your original question suggested you want to end levels after reaching a certain point. Using your new interface you can create an object that is fed the player's position and checks to see if they have passed the x or y coordinate that marks the level's end. Then simply call that object's finishLevel() method.


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# Solution to this limit

Can anyone tell me how to calculate this limit. It is a puzzle so I think there must be some trick.

$lim_{x \to \dfrac{\pi}{20}} \left( {(\sec x)}^{\cos x} + {(\csc x)}^{\sin x} \right)$

• I suspect that it is $\pi/2$, which makes it a bit more interesting. – Aravind Oct 18 '15 at 19:00

$\sin$ and $\cos$ are analytic, so they're continuous everywhere, and in particular,

$$\lim_{x \to \frac{\pi}{20}} \sin x = \sin \frac{\pi}{20}$$

and

$$\lim_{x \to \frac{\pi}{20}} \cos x = \cos \frac{\pi}{20}$$

Since $\sin \frac{\pi}{20} \neq 0$ and $\cos \frac{\pi}{20} \neq 0$, the same applies to $\sec$ and $\csc$, so you can just evaluate this limit by substitution: replace all the $x$ with $\frac{\pi}{20}$ and you're done.

Notice, both the limits exist separately so we can break the given limit as follows $$\lim_{x\to \frac{\pi}{20}}\left(\sec^{\cos x}(x)+\csc^{\sin x}(x)\right)$$ $$=\lim_{x\to \frac{\pi}{20}}\left(\sec x\right)^{\cos x}+\lim_{x\to \frac{\pi}{20}}\left(\csc x\right)^{\sin x}$$ $$=\left(\sec \frac{\pi}{20}\right)^{\cos \frac{\pi}{20}}+\left(\csc \frac{\pi}{20}\right)^{\sin \frac{\pi}{20}}$$

$$\lim_{x \to \frac{\pi}{20}} \left( {(\sec(x))}^{\cos(x)} + {(\csc(x))}^{\sin(x)} \right)=$$ $$\lim_{x \to \frac{\pi}{20}} \sec^{\cos(x)}(x)+\lim_{x \to \frac{\pi}{20}} \csc^{\sin(x)}(x)=$$ $$\sec^{\cos\left(\frac{\pi}{20}\right)}\left(\frac{\pi}{20}\right)+\csc^{\sin\left(\frac{\pi}{20}\right)}\left(\frac{\pi}{20}\right)\approx 2.34901$$