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# What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 4

For $s > 1$, Riemann’s famous zeta function is defined by

$\zeta(s) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$.

This is also called a p-series in calculus.

What I didn’t know (Gamma, page 41) is that, in 1748, Leonhard Euler exactly computed this infinite series for $s = 26$ without a calculator! Here’s the answer:

$\displaystyle 1 + \frac{1}{2^{26}} + \frac{1}{3^{26}} + \frac{1}{4^{26}} + \dots = \frac{1,315,862 \pi^{26}}{11,094,481,976,030,578,125}$.

I knew that Euler was an amazing human calculator, but I didn’t know he was that amazing.

When I researching for my series of posts on conditional convergence, especially examples related to the constant $\gamma$, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.


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# A reference to tell OPs to include in the original question the details added in comments

I often comment questions asking the OPs to provide more details, and I also suggest to add those details in the original question. However, this is frequently ignored, and most of the details remain in the comments. I don't think this helps improving the quality of the question, which should be the real goal of commenting. Is there a reference on the help pages or in this Meta site I could cite, which encourages OP to include details provided in comments, into the main question?

EDIT just to be sure I understood some of the comments below, if I don't want the question to be closed, but to be put on hold because it's unclear, I click on flag and select this option, right?

• I don't know, but I often start my comments with "please edit your question to give us more details on...", in many cases it seems to work better then simply asking for clarification. – Tim Nov 23 '17 at 9:08
• @Tim good, but it doesn't seem to be enough for some OPs. I could add a little note in the help center, stressing the importance of updating the main question, if someone shows me/points me to a link on how to do it. – DeltaIV Nov 23 '17 at 9:14
• Well, someone could provide such explanation in this thread and you can always send link to it, however I'm afraid that the people who do not read/understand your comments, would be exactly the same group that would not visit the link. – Tim Nov 23 '17 at 9:18
• @DeltaIV: True, but the way it usually seems to work out is that when someone overlooks the request to provide more information by editing the question their response is thin and incomplete in any case (at which point I usually give up rather than repeat myself in writing). Note that closing unclear questions in addition to commenting helps you to make the point. – Scortchi - Reinstate Monica Nov 23 '17 at 10:31
• @Scortchi how long should we give an OP before suggesting a close? I have seldom voted to close as I hold the good hope that someone does eventually edit the question or responds to the comments otherwise. Of course, any single arbitrary time cut-off will be flawed, but what kind of timeframe is considered reasonable? A couple of days, a week or a month even? – IWS Nov 23 '17 at 10:47
• IWS: For 'close' read 'place on hold pending improvement'. The sooner the better in my opinion. – Scortchi - Reinstate Monica Nov 23 '17 at 11:05
• @Tim if someone provides the explanation in this thread, I will for sure include the link - at least it shows that it's not just a pet peeve of mine, but it's something backed up by people with considerably more reputation and/or status (e.g., moderators) than me. You think it won't work? Maybe, but at least I'll know I tried my best to convince them. – DeltaIV Nov 23 '17 at 13:04
• @Scortchi please see my edited question :-) you mean I should flag the question as "unclear what you're asking", right? I do that often, but there are cases where I think the question is not bad (so I would feel a bit awkward flagging it), but the problem (prediction intervals for mixed models) is one of those where the answer depends even on "tiny" details . Anyway, if that helps the site quality, I can start flagging more questions. – DeltaIV Nov 23 '17 at 13:22
• @DeltaIV: I think you're misunderstanding something: to place a question on hold is to close it. The latter wording - users see '[on hold]' appended to the title of a closed question - was adopted to make it clearer that improvement followed by re-opening is the expected path. With over 3k rep., you can cast close votes yourself, with no need to raise flags. – Scortchi - Reinstate Monica Nov 23 '17 at 13:33
• @Scortchi yes, I was definitely misunderstanding that. I know I can cast close votes, but since I also see the flag option I thought there was some difference...I now think the only difference is that in the first case I don't need to get the moderators involved. Thanks for clarifying – DeltaIV Nov 23 '17 at 13:38

If a poster won't act on your comment alone, they likely won't act on anything else (less than closure).

It does sometimes help to be specific when asking, to mention the word edit specifically; some posters hang around for an hour or so and may fix their question if you catch them early.

If they do clarify in comments, you can always edit to include the details if need be.

In relation to whether to flag to close or whether to vote to close:

• When you're able to take an appropriate action to deal with or partly deal with something yourself (like vote to close) you should take that action.

• flagging notifies moderators that you think there is something they should do (above what ordinary high-rep users like yourself can). When you don't have relevant voting privileges it makes sense to use the flag option for a closure. If you voted for closure but a good while has passed (many hours at least, perhaps more like a day, say) and you don't think it's going to close from lack of attention (but should still close), in that case flagging also makes some sense.

• in cases where it's important for a moderator to see it or something that's urgent, then you could both vote on it and flag it appropriately

format: plain


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# LMIs in Control/Applications/Hinf LMI SatelliteAttitudeControl

LMIs in Control/Applications/Hinf LMI SatelliteAttitudeControl

This is a ${\displaystyle H_{\infty }}$ LMI for Satellite Attitude Control. Satellite attitude control is necessary to allow satellites in orbit accomplish their mission. Poor satellite attitude control results in poor pointing performance which can result in increased cost, delayed service, and reduced lifetime of the satellite.

## The System

The full derivation of the system from first principles is accomplished in the companion LMI for ${\displaystyle H_{2}}$  Satellite Attitude Control. The link to that page is at the bottom with the references.

Continuous Time:

{\displaystyle {\begin{aligned}I_{x}{\ddot {\phi }}+4(I_{y}-I_{z})\omega _{0}^{2}\phi +(I_{y}-I_{z}-I_{z})\omega _{0}{\dot {\psi }}&=T_{cx}+T_{dx}\\I_{y}{\ddot {\theta }}+3(I_{x}-I_{z})\omega _{0}^{2}\theta &=T_{cy}+T_{dy}\\I_{z}{\ddot {\psi }}+(I_{x}+I_{z}-I_{y})\omega _{0}{\dot {\psi }}&=T_{cz}+T_{dz}\\\end{aligned}}}

The above model was derived by substituting satellite attitude kinematics into the attitude dynamics of a satellite. The following are definitions of the variables above:

• Moments of inertia about the corresponding axis: ${\displaystyle I_{x},I_{y},I_{z}}$
• Euler Angles: ${\displaystyle \phi ,\theta ,\psi }$
• Disturbance Torques (flywheel, gravitational, and disturbance): ${\displaystyle T_{c},T_{g},T_{d}}$
• Rotational-angular velocity of the Earth: ${\displaystyle \omega _{0}=7.292115x10^{-5}{\text{ [rad/s]}}}$

The state-space representation of the system can be found by the following steps. Let

{\displaystyle {\begin{aligned}x&={\begin{bmatrix}q&{\dot {q}}\end{bmatrix}}^{T},&z_{\infty }=10^{-3}M{\ddot {q}},&z_{2}=q,\\\end{aligned}}}

Introduce the notations

{\displaystyle {\begin{aligned}I_{\alpha \beta }&=I_{\alpha }-I_{\beta },&I_{\alpha \beta \gamma }=I_{\alpha }-I_{\beta }-I_{\gamma }\\\end{aligned}}}

where ${\displaystyle \alpha ,\beta ,\gamma }$  stand for any element in ${\displaystyle {x,y,z}}$ . Then the state-space system is:

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}u+B_{2}d\\z_{\infty }&=C_{1}x+D_{1}u+D_{2}d\\z_{2}&=C_{2}x\\\end{aligned}}}

where the matrices in the above state-space representation are defined as follows:

{\displaystyle {\begin{aligned}A={\begin{bmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\{\frac {-4\omega _{0}^{2}I_{yz}}{I_{x}}}&0&0&0&0&{\frac {-\omega _{0}I_{yzx}}{I_{x}}}\\0&{\frac {-3\omega _{0}^{2}I_{xz}}{I_{y}}}&0&0&0&0\\0&0&{\frac {-\omega _{0}^{2}I_{yx}}{I_{z}}}&{\frac {\omega _{0}I_{yzx}}{I_{x}}}&0&0\end{bmatrix}}\\\\B_{1}=B_{2}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\{\frac {1}{I_{x}}}&0&0\\0&{\frac {1}{I_{y}}}&0\\0&0&{\frac {1}{I_{z}}}\\\end{bmatrix}}\\\\C_{1}=10^{-3}\times {\begin{bmatrix}-4\omega _{0}^{2}I_{yz}&0&0&0&-\omega _{0}I_{yxz}\\0&-3\omega _{0}^{2}I_{xz}&0&0&0&0\\0&0&-\omega _{0}^{2}I_{yx}&\omega _{0}I_{yxz}&0&0\end{bmatrix}}\\\\C_{2}={\begin{bmatrix}I_{3x3}&0_{3x3}\end{bmatrix}}\\\\D_{1}=10^{-3}\times L_{1},D_{2}=10^{-3}\times L_{2}\end{aligned}}}

## The Data

Data required for this LMI include moments of inertia of the satellite being controlled and the angular velocity of the earth. Any knowledge of the disturbance torques would also facilitate solution of the problem.

## The Optimization Problem

The idea is to design a state feedback control law for the previous satellite state-space system of the form

${\displaystyle u=Kx}$

This control law is designed so that the closed-loop system is stable and the transfer function matrix from disturbance to output

${\displaystyle G_{z_{\infty }d}(s)=(C_{1}+D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2}+D_{2}}$

satisfies

${\displaystyle ||G_{z_{\infty }d}(s)||_{\infty }=(C_{1}+D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2}+D_{2}}$

for a minimal positive scalar ${\displaystyle \gamma _{\infty }}$  which represents the minimum attenuation level.

The idea here is to attenuate the disturbances as much as possible while still maintaining the ability of the satellite to track. This minimum attenuation level is found from the LMI in the following section.

• Objective: Hinf norm
• Variables: Controller Gains
• Constraints: Satellite Attitude Dynamics and Kinematics. Maximum safe rotational rate of Satellite, maximum jet pulse thrust

## The LMI: ${\displaystyle H_{\infty }}$ Feedback Control of the Satellite System

Duan and Yu approach the ${\displaystyle H_{\infty }}$  satellite system as follows. The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.

{\displaystyle {\begin{aligned}\\&min\gamma _{\infty }\\\\&{\text{s.t. }}X>0\\\\&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(C_{1}X+D_{1}W)^{T}\\B_{2}^{T}&-\gamma _{H\infty }I&D_{2}^{T}\\C_{1}X+D_{1}W&D_{2}&-\gamma _{H\infty }I\end{bmatrix}}<0\\\end{aligned}}}

which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the ${\displaystyle H_{\infty }}$  problem.

## Conclusion:

The Duan and Yu textbook takes as typical values of the satellite moment of inertias as:

{\displaystyle {\begin{aligned}I_{x}=1030.17kg\cdot m^{2},I_{y}=3015.65kg\cdot m^{2},I_{Z}=3030.43kg\cdot m^{2}\\\end{aligned}}}

They then proceed to solve the optimization problem to find a controller gain that yields an attenuation level of 0.0010. Though this value is very small and represents very good attenuation the optimized controller pushes the poles of the closed loop system very close to the imaginary axis, resulting in slow oscillatory behavior with a very long settling time.

To address this a second approach was used by the authors which involves modifying the final LMI in the expression above and requiring that it be constrained as follows

{\displaystyle {\begin{aligned}&-I<{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(C_{1}X+D_{1}W)^{T}\\B_{2}^{T}&-\gamma _{H\infty }I&D_{2}^{T}\\C_{1}X+D_{1}W&D_{2}&-\gamma _{H\infty }I\end{bmatrix}}<0\\\end{aligned}}}

These results are planned verified in the linked code implementation using YALMIP, whereas the authors took advantage of the MATLAB LMI Toolbox to achieve their results.

## Implementation

A link to CodeOcean or other online implementation of the LMI


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# Zoran Skoda seminar Nerves and Grothendieck's homotopy hypothesis

Here I will sketch the outline and literature from my expositional seminar

• Nerves and Grothendieck’s homotopy hypothesis, Seminar for topology, Department of Mathematics, University of Zagreb, 31 Jan 2011, 11:15-12:45

### General ideas on higher groupoids and homotopy types

Introduction about Grothendieck’s Pursuing stacks (1983) and the idea of sheaves of higher groupoids; generalizing nonabelian cohomology to higher coefficients.

A sketch of the definition of weak 2-categories (bicategories).

Fundamental groupoid, bigroupoid,…infinity groupoid. Insufficency of strict infinity-groupoids. Toward weak infinity categories in order to model homotopy types and present coefficients for higher nonabelian cohomology. Homotopy conjecture: weak infinity groupoids model weak homotopy types. It has many variants, some almost trivial, soem very hard, depending how precisely one models homotopy types and which model for infinity groupodis one takes. Now people consider models involving simplicial techniques, what makes it easier, but Grothendieck wanted to consider the true algebaric generalizations of groupoids and bigroupoids.

### Nerves

General nonsense on nerve and realization (see the link). Main examples: Grothendieck’s nerve $Cat\to sSet$, Duskin’s nerve $Bicat\to sSet$, homotopy coherent nerve $sSet-Cat\to sSet$, $\omega$-nerve $\omega Cat\to sSet$, Moore’s normalization functor (in Dold-Kan correspondence), cyclic nerve of cyclic sets… The corresponding geometric realization is in all these situations defined as certain left Kan extension. In all these situations, which lead to pairs of a nerve functor and a geometric realization, the geometric realization is a left adjoint to the nerve functor. Characterization of the essential image of the Grothendieck’s nerve by Grothendieck via the unique horn filler conditions. Kan complexes are characterized by horn filler conditions without uniqueness. The subcategory $Kan \subset sSet$ of Kan complexes should be the image of a nerve functor from $\infty$-groupoids (of any kind) to $sSet$. In particular the Street’s $\omega$-nerve is such; Duskin’s “hypergroupoids” are such by the definition.

### Simplicial homotopy theory

Simplicial homotopy theory. Kan fibrations, inner Kan fibrations, Kan complexes, inner Kan complexes aka quasicategories. Higher fundamental groupoids and the homotopy conjecture via Kan complexes.

### Other models of $(\infty,1)$-categories.

Categories enriched over monoidal categories. Closed monoidal categories are enriched over itself. Usually it is useful to enrich over close monoidal categories. Topological categories - categories enriched over $Top$ (in this case, $Top$ is a convenient category of topological spaces). Simplicially enriched categories – categories enriched and over $sSet$ ($Top$ and $sSet$ are closed symmetric monoidal categories with respect to categorical product). $Top-Cat$ and $sSet-Cat$ can be treated as the models of the $(\infty,1)$-categories of $(\infinity,1)$-categories. Many $\infty$-categorical notions in this model differ from the default notions of enriched category theory (e.g. limits are not quite the same as enriched limits…). It is difficult to define the infinity category of infinity functors in that approach, even among (co)fibrant objects; this is an important point where it is easier to work with quasicategories.

### Quillen model categories as presentations

Quillen model categories: axioms, via the notion of weak factorization system?s. Outline of general tools and rough passage through basic terminology (e.g. fibrant objects, left homotopy, homotopy category…). Main examples: Quillen model structure on $sSet$. Strøm model structure on the category of all topological spaces. Model categories on categories of simplicial presheaves. Quillen adjunctions, Quillen equivalences.

Idea of model structures as presentations of bicomplete locally presentable $(\infinity,1)$-categories.

### Topos $\mathbb{U}$ and strong shape

Coherent nerve $\mathbb{U}$ of the large category of Kan complexes is important in many situations; in particular it is the terminal $(\infty,1)$-topos.

$(\infty,1)$-topoi are special $(\infty,1)$-categories which are analogues of usual Grothendieck topoi in ordinary category theory. The usual Grothendieck topos is a category which is equivalent to a category of sheaves on some Grothendieck site (which is a generalization of the category of open subsets of a topological space); a Grotehndieck topos can be internally characterized by abstract properties which are called Giraud’s axioms. A Grothendieck $(\infty,1)$-topos is an $(\infty,1)$-category satisfying the infinite/categorical analogues of Giraud’s axioms. In the quasicategory language this is expanded in Lurie’s book Higher Topos Theory; in an another model (“Segal topoi”) it appeared earlier in a work of Toen and Vezzosi; there is also some earlier treatment in the works of Rezk, Joyal and others.

For a locally presentable infinity category $D$, the $(\infinity,1)$-category $Pro(D)= Ind(D^{op})^{op}$ is the full $(\infty,1)$-subcategory of $Fun(D,D)^{op}$ whose objects are pro objects which are defined as left exact accessible $(\infty,1)$-functors $D\to \mathbb{U}$. In particular, for $D = \mathbb{U}$, the $(\infty,1)$-topos $Pro(\mathbb{U})$ is the $(\infinity,1)$-category of shapes.

A geometric morphism $p_* : C\to \mathbb{U} : p^*$ for $C$ a $(\infty,1)$-topos, canonical up to homotopy. Strong shape of $(\infty,1)$-topos (after Lurie’s book) as $p_* p^*\in Pro(\mathbb{U})$. If $X$ is a paracompact space then the strong shape of $X$ is induced by the geometric morphism $p_*:Sh_\infty(X)\to Sh_\infty(Y):p^*$ induced by the map $p : X\to \star$ of spaces. The strong shape of $C$ is defined even for the infinity topoi which are not necessarily of the form $Sh_\infty(X)$ for a paracompact $X$; examples not strictly in classical topology abound.

I was asked if it is worthy to learn so many abstract infinite-categorical tools and other tools presented; of course it depends on wanted applications and personal inclinations, but the machinery actually gives cleaner and simpler perspective, it is easier to think in wide range of applications and (in the point of view of our nLab), it is the most natural way of doing things involving homotopy, structured (geometric) spaces and so on. Now there are very many results and fruitful new fields of mathematics, including the proof of cobordism hypothesis, derived algebraic geometry, “new brave worlds of homotopy theory” including motivic homotopy theory, higher gauge theory in physics etc. Lurie’s book Higher Topos Theory in chapter 7 gives some applications of $(\infty,1)$-topoi to the classical topology, for example to the dimension theory and proper base change theorems.


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## anonymous one year ago How do I calculate the absolute uncertainty when there is an exponent? (see question 2)

1. anonymous

1) when the number is (5.3322∗ 10^-5) would the uncertainty be (1*10^-9) or (0.0001)?

2. anonymous

2) how to calculate the absolute uncertainty when there is an exponent? For example $(-7.5899*10^{-6}) *(22.62 \pm 0.01)$.

3. anonymous

for example lets say u are given 10.1 you dont know if it was 10.05 to 10.15 so your uncertainty is 10.1 +/- 0.05

4. anonymous

We were told in class that when the uncertainty is implicit we will assume its the last digit and it will always be 1. For example: 0.0967 the uncertainty will be 0.0001.

5. anonymous

(5.3322∗ 10^-5) same thing this part 5.3322 you dont know if it was 5.33215 to 5.33225 so 5.3322 +/- 0.00005

6. anonymous

ok i guess that maybe fine too its like roudning up the uncertainty itself

7. anonymous

+/- 0.00005 = +/- 0.0001

8. anonymous

ooohhhh ok. so the uncertainty will be determined regardless of the exponent.

9. anonymous

yeah that is right

10. anonymous

(5.33220∗ 10^-5) if they want to go upto exponent

11. anonymous

you have to add that 0

12. anonymous

ok. thanks!!

13. anonymous

sure np

14. anonymous

for your question 2, absolute uncertainty is that the uncertainty that is based on proportion

15. anonymous

iwhen y=x^n, is the formula n* (relative uncertainty), relative uncertainty= (uncertainty/value)

16. anonymous

okay i see

17. anonymous

here is a more logical approach to calculating your absolute uncertainty

18. anonymous

suppose you have 10.0+/- 0.1 and 2.0+/- 0.2 you see 10.0*2.0 = 20.0 then you see the max and mix that happens with the uncertainty 10.1*2.2 = 20.2 + 2.02 = 22.22 and the min 9.9*1.8 = 18-0.18=17.82

19. anonymous

(22.22-17.82)/2 would be the uncertainty?

20. anonymous

ok

21. anonymous

let me check how u actually find absolute uncertainties hmm

22. anonymous

look at this http://web.uvic.ca/~jalexndr/192UncertRules.pdf

23. anonymous

that is probably best forget what i said so far

24. anonymous

ok. thank you so much!!


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If a set has 20 elements.How many ordered pairs having distinct elements can be formed? A. 150 B. 170 C. 190 D. 210

eg. (a,b), $a\epsilon A \and\ b\epsilon B$
The number of ordered pairs = $^{20}C_{2}$=190


text:
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• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

• Legal notice
• Personal data

## Section: Research Program

### Scientific foundations

The research activities undertaken by the team aim at developing innovative numerical methodologies putting the emphasis on several features:

• Accuracy. The foreseen numerical methods should rely on discretization techniques that best fit to the geometrical characteristics of the problems at hand. Methods based on unstructured, locally refined, even non-conforming, simplicial meshes are particularly attractive in this regard. In addition, the proposed numerical methods should also be capable to accurately describe the underlying physical phenomena that may involve highly variable space and time scales. Both objectives are generally addressed by studying so-called $hp$-adaptive solution strategies which combine $h$-adaptivity using local refinement/coarsening of the mesh and $p$-adaptivity using adaptive local variation of the interpolation order for approximating the solution variables. However, for physical problems involving strongly heterogeneous or high contrast propagation media, such a solution strategy may not be sufficient. Then, for dealing accurately with these situations, one has to design numerical methods that specifically address the multiscale nature of the underlying physical phenomena.

• Numerical efficiency. The simulation of unsteady problems most often relies on explicit time integration schemes. Such schemes are constrained by a stability criterion, linking some space and time discretization parameters, that can be very restrictive when the underlying mesh is highly non-uniform (especially for locally refined meshes). For realistic 3D problems, this can represent a severe limitation with regards to the overall computing time. One possible overcoming solution consists in resorting to an implicit time scheme in regions of the computational domain where the underlying mesh size is very small, while an explicit time scheme is applied elsewhere in the computational domain. The resulting hybrid explicit-implicit time integration strategy raises several challenging questions concerning both the mathematical analysis (stability and accuracy, especially for what concern numerical dispersion), and the computer implementation on modern high performance systems (data structures, parallel computing aspects). A second, often considered approach is to devise a local time stepping strategy. Beside, when considering time-harmonic (frequency-domain) wave propagation problems, numerical efficiency is mainly linked to the solution of the system of algebraic equations resulting from the discretization in space of the underlying PDE model. Various strategies exist ranging from the more robust and efficient sparse direct solvers to the more flexible and cheaper (in terms of memory resources) iterative methods. Current trends tend to show that the ideal candidate will be a judicious mix of both approaches by relying on domain decomposition principles.

• Computational efficiency. Realistic 3D wave propagation problems involve the processing of very large volumes of data. The latter results from two combined parameters: the size of the mesh i.e the number of mesh elements, and the number of degrees of freedom per mesh element which is itself linked to the degree of interpolation and to the number of physical variables (for systems of partial differential equations). Hence, numerical methods must be adapted to the characteristics of modern parallel computing platforms taking into account their hierarchical nature (e.g multiple processors and multiple core systems with complex cache and memory hierarchies). In addition, appropriate parallelization strategies need to be designed that combine SIMD and MIMD programming paradigms.

From the methodological point of view, the research activities of the team are concerned with four main topics: (1) high order finite element type methods on unstructured or hybrid structured/unstructured meshes for the discretization of the considered systems of PDEs, (2) efficient time integration strategies for dealing with grid induced stiffness when using non-uniform (locally refined) meshes, (3) numerical treatment of complex propagation media models (e.g. physical dispersion models), (4) algorithmic adaptation to modern high performance computing platforms.


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# Pierpont prime

A Pierpont prime is a prime number of the form

Named after James Pierpont Thousands Infinite Pierpont number 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889 3×210,829,346 + 1 A005109
${\displaystyle 2^{u}3^{v}+1\,}$

for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections.

A Pierpont prime with v = 0 is of the form ${\displaystyle 2^{u}+1}$, and is therefore a Fermat prime (unless u = 0). If v is positive then u must also be positive (because a number of the form ${\displaystyle 3^{v}+1}$ would be even and therefore non-prime, since 2 cannot be expressed as ${\displaystyle 3^{v}+1}$ when v is a positive integer), and therefore the non-Fermat Piermont primes all have the form 6k + 1, when k is a positive integer (except for 2, when u = v = 0).

The first few Pierpont primes are:

2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, ... (sequence A005109 in the OEIS)

## Distribution

 Unsolved problem in mathematics:Are there infinitely many Pierpont primes?(more unsolved problems in mathematics)

Distribution of the exponents for the smaller Pierpont primes

Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed. There are 36 Pierpont primes less than 106, 59 less than 109, 151 less than 1020, and 789 less than 10100. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among n-digit numbers of the correct form ${\displaystyle 2^{u}3^{v}+1}$ , the fraction of these that are prime should be proportional to 1/n, a similar proportion as the proportion of prime numbers among all n-digit numbers. As there are Θ(n2) numbers of the correct form in this range, there should be Θ(n) Pierpont primes.

Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately 9n Pierpont primes up to 10n.[1] According to Gleason's conjecture there are Θ(log N) Pierpont primes smaller than N, as opposed to the smaller conjectural number O(log log N) of Mersenne primes in that range.

## Primality testing

When ${\displaystyle 2^{u}>3^{v}}$ , the primality of ${\displaystyle 2^{u}3^{v}+1}$  can be tested by Proth's theorem. On the other hand, when ${\displaystyle 2^{u}<3^{v}}$  alternative primality tests for ${\displaystyle p=2^{u}3^{v}+1}$  are possible based on the factorization of ${\displaystyle p-1}$  as a small even number multiplied by a large power of three.[2]

## Pierpont primes found as factors of Fermat numbers

As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors. The following table[3] gives values of m, k, and n such that

${\displaystyle k\cdot 2^{n}+1{\text{ divides }}2^{2^{m}}+1.\,}$

The left-hand side is a Pierpont prime when k is a power of 3; the right-hand side is a Fermat number.

m k n Year Discoverer
38 3 41 1903 Cullen, Cunningham & Western
63 9 67 1956 Robinson
207 3 209 1956 Robinson
452 27 455 1956 Robinson
9428 9 9431 1983 Keller
12185 81 12189 1993 Dubner
28281 81 28285 1996 Taura
157167 3 157169 1995 Young
213319 3 213321 1996 Young
303088 3 303093 1998 Young
382447 3 382449 1999 Cosgrave & Gallot
461076 9 461081 2003 Nohara, Jobling, Woltman & Gallot
495728 243 495732 2007 Keiser, Jobling, Penné & Fougeron
672005 27 672007 2005 Cooper, Jobling, Woltman & Gallot
2145351 3 2145353 2003 Cosgrave, Jobling, Woltman & Gallot
2478782 3 2478785 2003 Cosgrave, Jobling, Woltman & Gallot
2543548 9 2543551 2011 Brown, Reynolds, Penné & Fougeron

As of 2018, the largest known Pierpont prime is 3 × 210829346 + 1,[4] whose primality was discovered by Sai Yik Tang and PrimeGrid in 2014.[5]

## Polygon construction

In the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation.[6] It follows that they allow any regular polygon of N sides to be formed, as long as N ≥ 3 and of the form 2m3nρ, where ρ is a product of distinct Pierpont primes. This is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector.[1] Regular polygons which can be constructed with only compass and straightedge (constructible polygons) are the special case where n = 0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes.

In 1895, James Pierpont studied the same class of regular polygons; his work is what gives the name to the Pierpont primes. Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, the primes N for which Pierpont's construction works are exactly the Pierpont primes. However, Pierpont did not describe the form of the composite numbers with 3-smooth totients.[7] As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ given above.[1]

The smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections). All other regular N-gons with 3 ≤ N ≤ 21 can be constructed with compass, straightedge and trisector.[1]

## Generalization

A Pierpont prime of the second kind is a prime number of the form 2u3v − 1. These numbers are

2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 131071, 139967, 165887, 294911, 314927, 442367, 472391, 497663, 524287, 786431, 995327, ... (sequence A005105 in the OEIS)

A generalized Pierpont prime is a prime of the form ${\displaystyle p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdot p_{3}^{n_{3}}\cdot ...\cdot p_{k}^{n_{k}}+1}$  with k fixed primes {p1, p2, p3, ..., pk}, pi < pj for i < j. A generalized Pierpont prime of the second kind is a prime of the form ${\displaystyle p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdot p_{3}^{n_{3}}\cdot ...\cdot p_{k}^{n_{k}}-1}$  with k fixed primes {p1, p2, p3, ..., pk}, pi < pj for i < j. Since all primes greater than 2 are odd, in both kinds p1 must be 2. The sequences of such primes in OEIS are:

 {p1, p2, p3, ..., pk} +1 −1 {2} {2, 3} {2, 5} {2, 3, 5} {2, 7} {2, 3, 5, 7} {2, 11} {2, 13}

• Safe prime, the primes for which p − 1 is as non-smooth as possible

## Notes

1. ^ a b c d Gleason, Andrew M. (1988), "Angle trisection, the heptagon, and the triskaidecagon", American Mathematical Monthly, 95 (3): 185–194, doi:10.2307/2323624, MR 0935432. Footnote 8, p. 191.
2. ^ Kirfel, Christoph; Rødseth, Øystein J. (2001), "On the primality of ${\displaystyle 2h\cdot 3^{n}+1}$ ", Discrete Mathematics, 241 (1–3): 395–406, doi:10.1016/S0012-365X(01)00125-X, MR 1861431.
3. ^ Wilfrid Keller, Fermat factoring status.
4. ^ Caldwell, Chris. "The largest known primes". The Prime Pages. Retrieved 14 March 2017.
5. ^ "PrimeGrid's 321 Prime Search" (PDF). PrimeGrid. Retrieved 14 March 2017.
6. ^ Hull, Thomas C. (2011), "Solving cubics with creases: the work of Beloch and Lill", American Mathematical Monthly, 118 (4): 307–315, doi:10.4169/amer.math.monthly.118.04.307, MR 2800341.
7. ^ Pierpont, James (1895), "On an undemonstrated theorem of the Disquisitiones Arithmeticæ", Bulletin of the American Mathematical Society, 2 (3): 77–83, doi:10.1090/S0002-9904-1895-00317-1, MR 1557414.


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# Transposition of a matrix

1. Jul 21, 2011

### matqkks

Why is the transpose of a matrix important?
To find the inverse by cofactors we need the transpose but I would never find the inverse of a matrix by using cofactors.

2. Jul 21, 2011

### Fredrik

Staff Emeritus
I think the main reason why the transpose is useful is that the standard inner product on the vector space of n×1 matrices is $\langle x,y\rangle=x^Ty$. This implies that a rotation R must satisfy $R^TR=I$.

I think that cofactor stuff is sometimes useful in proofs, but you're right that if you just want to find the inverse of a given matrix, there are better ways to do it.

3. Jul 21, 2011

### kdbnlin78

There are of course many ways to invert a matrix but thie is not the only use for the transpose.

Systems of linear equations can be reformulated into matrix systems by looking at the equation xAx^{T} = b where x is a n x 1 column vector with entries {x_{1},...,x_{n}} and Z is a square matrix n x n with entries (for real valued equations, say) in /mathbb{R}. The matrix b is then also an $n x 1$ column matrix of numbers in /mathbb{R} too.


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# Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

I do not understand how to go about completing this problem or even where to start.

• It may help to note that $|x| = \max(x,-x)$. Jun 26, 2013 at 3:09
• I would start by convincing yourself that the proof is meaningful with several concrete examples. Set x=1, y=-3, for example, and try to get an intuition that it's true (before you try to prove why it's true). Then many of the comments above should be helpful. Jun 26, 2013 at 4:23
• Just do a case-by-case analysis: Evaluate both sides once for the case $x\leq y$ and once for $y\leq x$. Jun 26, 2013 at 6:23
• possible duplicate of Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ Mar 7, 2014 at 13:43
• @TuckerRapu: IIRC, here and in a couple of other places you've marked an older question as a duplicate of a newer question. Mar 7, 2014 at 14:52

Here is another way of looking at it:

We have $|x| = \max(x,-x)$. Also, $\max(a,b)+c = \max(a+c,b+c)$, and if $c \geq 0$, then $c \max(a,b) = \max(ac,bc)$.

Hence \begin{eqnarray} \frac{1}{2}(x+y+|x-y|) &=& \frac{1}{2}(x+y+\max(x-y,y-x)) \\ &=& \frac{1}{2}(\max(x-y+x+y,y-x+x+y)) \\ &=& \frac{1}{2}(\max(2x,2y)) \\ &=& \max(x,y) \end{eqnarray}

• +1 it would be nice, if you noted the $\text{min}(x,y)$ for the OP as well. :) Jun 26, 2013 at 5:00
• @BabakS.: Thanks! The $\min$ result follows immediately from $\max(x,y)+\min(x,y) = x+y$. Jun 26, 2013 at 5:40
• Great! This should be accepted! Jun 26, 2013 at 12:57
• on the first line why is it max(x-y,y-x) and not max(x-y,-x-y)? Oct 21, 2021 at 0:27
• @AGarza Because $|z|=\max(z,-z)$. Now let $z=x-y$. Oct 21, 2021 at 0:32

This probably isn't as rigorous as it should be, but I think it's intuitive enough.

Hmm... We don't know which of $x$ or $y$ is bigger, but we do know one thing: their average. If we call the average $z$, then $z=\frac{x+y}{2}$. Now, the distance between $x$ and $y$ is $|x-y|$, so the distance from $z$ to both $x$ and $y$ is $\frac{|x-y|}{2}$.

So if we imagine a number line, the distance from $0$ to $z$ is $\frac{x+y}{2}$, and the distance from $z$ to max(x, y) is $\frac{|x-y|}{2}$. Thus, the total distance from $0$ to max(x, y) is $\frac{|x-y|}{2}$ + $\frac{x+y}{2}$, as desired.

• I like this answer. If not as rigorous, this is exactly how one would come up with the expression in the first place. Also, one can easily come up with similar expression for $\operatorname{min}$ in this manner. Oct 3, 2015 at 19:23

Hint: If $x\ge y$ then $|x-y|=x-y$. If $x\lt y$ then $|x-y|=-(x-y)=y-x$.

We have used the fact that in general $|w|=w$ if $w\ge 0$ and $|w|=-w$ if $w\lt 0$.

• +1 This is the most straightforward one here. Also @user71925 it should also be noted that $\min\{x,y\} = \frac{1}{2}(x+y-|x-y|)$. May 20, 2014 at 18:40

$$\max\{x,y\} =\frac{x+y+|x-y|}{2}$$ $$=> 2.\max\{x,y\} =x+y+|x-y|$$ there are two possible situation :
1. $y>x$, i.e $\max\{x,y\}=y$ then $y-x=|x-y|$, this equation is true because we assume that
$y>x$
2. $x>y$, or $\max\{x,y\}=x$, then $x-y=|x-y|$, which is true if $x>y$

Conceptually, focus on $|x-y|$ as the absolute difference of the two numbers.

Without loss of generality, assume $x > y$. Then $y + |x - y| = x$. This can be understood as representing that if we add the difference between two numbers to the smaller, we get the larger.

The specific equation will naturally fall out with this observation.

Without loss of generality, let $y=x+k$ for some nonnegative number $k$. Then,

$$\frac{x+(x+k)+|x-(x+k)|}{2} = \frac{2x+2k}{2} = x+k = y$$

which is equal to $\max(x,y)$ by the assumption.

For another approach we can do as follows:

$$max(x, y) - max(-x, -y) = x + y$$ and, $$max(x, y) + max(-x, -y) = |x - y|$$

Adding those together yeilds: $$2\times max(x, y) = x + y + |x - y|$$ Dividing by two, gives us what we want.


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### Session DE: Mini-symposium on Pentaquarks I

7:00 PM–9:45 PM, Tuesday, September 20, 2005
Ritz-Carlton Hotel Room: Amphitheater

Chair: Makoto Oka, Tokyo Institute of Technology

Abstract ID: BAPS.2005.HAW.DE.3

### Abstract: DE.00003 : Search for $\Theta^{+}$ via $\pi^-p \to K^{-}X$ reaction near production threshold

7:45 PM–8:00 PM

Preview Abstract MathJax On | Off   Abstract

#### Author:

Koji Miwa
(Kyoto University)

We have searched for $\Theta^{+}$ via $\pi^{-}p \to K^{-}X$ reaction using 1.87 and 1.92GeV/c $\pi^{-}$ beam at the K2 beamline of KEK-PS 12GeV Proton Synchrotron. We irradiated 2.9$\times 10^{9}$ and 3.0$\times 10^{9}$ $\pi^{-}$ beam to the SCIFI and the polyethylene targets respectively at beam momentum 1.87GeV/c. At 1.92GeV/$c$, 7.4$\times 10^{9}$ $\pi^{-}$ beam was irradiated to only polyethylene target. In the missing mass spectrum at beam momentum 1.92GeV/$c$, a bump was found at 1530MeV/$c^{2}$ which is consistent with the mass reported by several experiments. The statistical significance of this bump, however, was only 2.5$-$2.7$\sigma$(preliminary). Therefore we derived the upper limit of $\Theta^{+}$ production cross section via $\pi^{-}p \to K^{-}\Theta^{+}$ reaction and obtained 4.1$\mu$b(preliminary) at 90\% confidence level assuming that $\Theta^{+}$ was produced isotropically in the center of mass system.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2005.HAW.DE.3


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TheInfoList

thumb|200px|Geometry of the water molecule with values for O-H bond length and for H-O-H bond angle between two bonds Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom. Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of molecule, i.e. they can be understood as approximately local and hence transferable properties.

Determination

The molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas electron diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances, dihedral angles, angles, and connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries (conformational isomerism) that are close in energy on the potential energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas. The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking bond lengths of two joined atoms, bond angles of three connected atoms, and torsion angles (dihedral angles) of three consecutive bonds.

The influence of thermal excitation

Since the motions of the atoms in a molecule are determined by quantum mechanics, "motion" must be defined in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) In addition to translation and rotation, a third type of motion is molecular vibration, which corresponds to internal motions of the atoms such as bond stretching and bond angle variation. The molecular vibrations are harmonic (at least to good approximation), and the atoms oscillate about their equilibrium positions, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, so that the wavefunction of a single vibrational mode is not a sharp peak, but an exponential of finite width (the wavefunction for ''n'' = 0 depicted in the article on the quantum harmonic oscillator). At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that "the molecules will vibrate faster"), but they oscillate still around the recognizable geometry of the molecule. To get a feeling for the probability that the vibration of molecule may be thermally excited, we inspect the Boltzmann factor , where Δ''E'' is the excitation energy of the vibrational mode, ''k'' the Boltzmann constant and ''T'' the absolute temperature. At 298 K (25 °C), typical values for the Boltzmann factor β are: * ''β'' = 0.089 for Δ''E'' = 500 cm−1 * ''β'' = 0.008 for Δ''E'' = 1000 cm−1 * ''β'' = 0.0007 for Δ''E'' = 1500 cm−1. (The reciprocal centimeter is an energy unit that is commonly used in infrared spectroscopy; 1 cm−1 corresponds to ). When an excitation energy is 500 cm−1, then about 8.9 percent of the molecules are thermally excited at room temperature. To put this in perspective: the lowest excitation vibrational energy in water is the bending mode (about 1600 cm−1). Thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero. As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that at higher temperatures more molecules will rotate faster, which implies that they have higher angular velocity and angular momentum. In quantum mechanical language: more eigenstates of higher angular momentum become thermally populated with rising temperatures. Typical rotational excitation energies are on the order of a few cm−1. The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature. Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.

Bonding

Molecules, by definition, are most often held together with covalent bonds involving single, double, and/or triple bonds, where a "bond" is a shared pair of electrons (the other method of bonding between atoms is called ionic bonding and involves a positive cation and a negative anion). Molecular geometries can be specified in terms of bond lengths, bond angles and torsional angles. The bond length is defined to be the average distance between the nuclei of two atoms bonded together in any given molecule. A bond angle is the angle formed between three atoms across at least two bonds. For four atoms bonded together in a chain, the torsional angle is the angle between the plane formed by the first three atoms and the plane formed by the last three atoms. There exists a mathematical relationship among the bond angles for one central atom and four peripheral atoms (labeled 1 through 4) expressed by the following determinant. This constraint removes one degree of freedom from the choices of (originally) six free bond angles to leave only five choices of bond angles. (Note that the angles ''θ''11, ''θ''22, ''θ''33, and ''θ''44 are always zero and that this relationship can be modified for a different number of peripheral atoms by expanding/contracting the square matrix.) :$0 = \begin \cos \theta_ & \cos \theta_ & \cos \theta_ & \cos \theta_ \\ \cos \theta_ & \cos \theta_ & \cos \theta_ & \cos \theta_ \\ \cos \theta_ & \cos \theta_ & \cos \theta_ & \cos \theta_ \\ \cos \theta_ & \cos \theta_ & \cos \theta_ & \cos \theta_ \end$ Molecular geometry is determined by the quantum mechanical behavior of the electrons. Using the valence bond approximation this can be understood by the type of bonds between the atoms that make up the molecule. When atoms interact to form a chemical bond, the atomic orbitals of each atom are said to combine in a process called orbital hybridisation. The two most common types of bonds are sigma bonds (usually formed by hybrid orbitals) and pi bonds (formed by unhybridized p orbitals for atoms of main group elements). The geometry can also be understood by molecular orbital theory where the electrons are delocalised. An understanding of the wavelike behavior of electrons in atoms and molecules is the subject of quantum chemistry.

Isomers

Isomers are types of molecules that share a chemical formula but have difference geometries, resulting in different properties: * A pure substance is composed of only one type of isomer of a molecule (all have the same geometrical structure). * Structural isomers have the same chemical formula but different physical arrangements, often forming alternate molecular geometries with very different properties. The atoms are not bonded (connected) together in the same orders. ** Functional isomers are special kinds of structural isomers, where certain groups of atoms exhibit a special kind of behavior, such as an ether or an alcohol. * Stereoisomers may have many similar physicochemical properties (melting point, boiling point) and at the same time very different biochemical activities. This is because they exhibit a handedness that is commonly found in living systems. One manifestation of this chirality or handedness is that they have the ability to rotate polarized light in different directions. * Protein folding concerns the complex geometries and different isomers that proteins can take.

Types of molecular structure

A bond angle is the geometric angle between two adjacent bonds. Some common shapes of simple molecules include: * Linear: In a linear model, atoms are connected in a straight line. The bond angles are set at 180°. For example, carbon dioxide and nitric oxide have a linear molecular shape. * Trigonal planar: Molecules with the trigonal planar shape are somewhat triangular and in one plane (flat). Consequently, the bond angles are set at 120°. For example, boron trifluoride. * Angular: Angular molecules (also called ''bent'' or ''V-shaped'') have a non-linear shape. For example, water (H2O), which has an angle of about 105°. A water molecule has two pairs of bonded electrons and two unshared lone pairs. * Tetrahedral: ''Tetra-'' signifies four, and ''-hedral'' relates to a face of a solid, so "tetrahedral" literally means "having four faces". This shape is found when there are four bonds all on one central atom, with no extra unshared electron pairs. In accordance with the VSEPR (valence-shell electron pair repulsion theory), the bond angles between the electron bonds are arccos(−) = 109.47°. For example, methane (CH4) is a tetrahedral molecule. * Octahedral: ''Octa-'' signifies eight, and ''-hedral'' relates to a face of a solid, so "octahedral" means "having eight faces". The bond angle is 90 degrees. For example, sulfur hexafluoride (SF6) is an octahedral molecule. * Trigonal pyramidal: A trigonal pyramidal molecule has a pyramid-like shape with a triangular base. Unlike the linear and trigonal planar shapes but similar to the tetrahedral orientation, pyramidal shapes require three dimensions in order to fully separate the electrons. Here, there are only three pairs of bonded electrons, leaving one unshared lone pair. Lone pair – bond pair repulsions change the bond angle from the tetrahedral angle to a slightly lower value.Miessler G.L. and Tarr D.A. ''Inorganic Chemistry'' (2nd ed., Prentice-Hall 1999), pp.57-58 For example, ammonia (NH3).

VSEPR table

The bond angles in the table below are ideal angles from the simple VSEPR theory (pronounced "Vesper Theory"), followed by the actual angle for the example given in the following column where this differs. For many cases, such as trigonal pyramidal and bent, the actual angle for the example differs from the ideal angle, and examples differ by different amounts. For example, the angle in H2S (92°) differs from the tetrahedral angle by much more than the angle for H2O (104.48°) does.

3D representations

* Line or stick – atomic nuclei are not represented, just the bonds as sticks or lines. As in 2D molecular structures of this type, atoms are implied at each vertex. * Electron density plot – shows the electron density determined either crystallographically or using quantum mechanics rather than distinct atoms or bonds. * Ball and stick – atomic nuclei are represented by spheres (balls) and the bonds as sticks. * Spacefilling models or CPK models (also an atomic coloring scheme in representations) – the molecule is represented by overlapping spheres representing the atoms. * Cartoon – a representation used for proteins where loops, beta sheets, and alpha helices are represented diagrammatically and no atoms or bonds are explicitly represented (e.g. the protein backbone is represented as a smooth pipe). The greater the amount of lone pairs contained in a molecule, the smaller the angles between the atoms of that molecule. The VSEPR theory predicts that lone pairs repel each other, thus pushing the different atoms away from them.

* Jemmis mno rules *Lewis structure * Molecular design software * Molecular graphics * Molecular mechanics * Molecular modelling * Molecule editor * Polyhedral skeletal electron pair theory * Quantum chemistry * Topology (chemistry)

References


text:
## Chemistry: The Central Science (13th Edition)

Published by Prentice Hall

# Chapter 13 - Properties of Solutions - Exercises: 13.51a

#### Answer

$0.150\;mol\;SrBr_2$

#### Work Step by Step

Recall that molarity = moles solute / liter solution $\frac{600\;mL\;solution}{}(\frac{1\;L\;solution}{1000\;mL\;solution})(\frac{0.250\;mol\;SrBr_2}{1\;L\;solution})=0.150\;mol\;SrBr_2$

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 Ok! The equation you wrote (conservation of momentum) reads $$p_{initial}=p_{final}\Rightarrow m\cdot u+M\cdot 0=m\cdot \frac{u}{2}+M\cdot V_A$$ so you have for $V_A$ ...