text:
×

# Happy Pi day!!

A very happy Pi day to all the Brilliant people out there!!!While most people complain about the kids and teens of this generation you people really give me hope for a better tomorrow..You people are awesome....Never stop learning!!And also the hardworking staff members of the website you people are equally awesome!!!....

2 years, 10 months ago

Sort by:

Just something to add, next year is going to be an epic Pi Day because it will be 3-14-15. We will be so happy at 9:26:53 next year. · 2 years, 10 months ago

That will be AWESOME · 2 years, 9 months ago

Wow didn't notice. · 2 years, 10 months ago

I want to wish everyone a happy Pi Day, and hopefully, you won't overindulge on pie. · 2 years, 10 months ago

My Pi Day website. Fun stuff. · 2 years, 10 months ago

Happy Einstein's birthday. · 2 years, 10 months ago

Who could forget the great physicist? · 2 years, 10 months ago

Physicist ! Not a physician.....physician means a doctor,,, · 2 years, 10 months ago

Haha · 2 years, 10 months ago

Our prodigy! · 2 years, 10 months ago

I invite you to join the movement to reestablish truth and beauty, celebrate $$\tau$$ day, June 28th. Staff · 2 years, 10 months ago

How about $$e$$ day on the 27th January ? Also there could be $$\phi$$ day on the 16th of January. · 2 years, 10 months ago

or 71st february? · 2 years, 10 months ago

Hahahehehehehahahehehahaehhehah · 2 years, 10 months ago

Isn't $$e=2.71...$$?So why the 21st of January instead of the 27th? · 2 years, 10 months ago

Typo!!! · 2 years, 10 months ago

Go on. Is this tied to another insurgent movement in mathematics? Staff · 2 years, 10 months ago

I calculated the algorithm to find the value of $$\pi$$ yesterday. I can calculate the value upto 31 correct decimal places(My calculator cant calculate beyond)............Its very interesting......... · 2 years, 10 months ago

Show how you would derive pi in https://brilliant.org/discussions/thread/pi-day/ · 2 years, 10 months ago

I am too stressed out to enjoy it..I have college decisions due today :( · 2 years, 10 months ago

Really, Are you from Ethiopia, me too -_- · 2 years, 10 months ago

Yeah I am..Its rare to find a fellow math enthusiast · 2 years, 10 months ago

Pi day!!! It sounds nice.. · 2 years, 10 months ago

Happy pi and Einstein day.Yes,the great man was born on 14th of this month too. · 2 years, 10 months ago

As of today, I identify as a Taoist - see the Half Tau Day page.

I highly recommend reading the Tao Manifesto and the Pi Manifesto together. I came to the overwhelming conclusion that Tau was the superior circle constant, but I'm open to input!

Happy 1/2 Tau Day!

The Linux User · 2 years, 10 months ago

Congratulations! Staff · 2 years, 10 months ago


text:
Question
Fusion bombs use neutrons from their fission trigger to create tritium fuel in the reaction $n + {}^{6}\textrm{Li} \to {}^{3}\textrm{H} + {}^{4}\textrm{He}$. What is the energy released by this reaction in MeV?
$4.781 \textrm{ MeV}$
Solution Video

OpenStax College Physics Solution, Chapter 32, Problem 53 (Problems & Exercises) (1:18)

View sample solution

Calculator Screenshots

Video Transcript

This is College Physics Answers with Shaun Dychko. We are going to calculate the energy released when neutrons from a fission trigger hit lithium that's in the fusion part of the fusion bomb then that will create tritium and this will be what fuses in the fusion part of the reaction. So to calculate the energy released, we'll find the total mass of the reactants; there's neutron plus the mass of lithium-6 and then we'll take away from that the total mass of tritium and helium-4. And so we look up each of these masses in our appendix A and we have equal numbers of electrons surrounding the lithium and around the tritium and helium so these atomic masses even though they include electron masses, we don't need to worry about it because we have as many added to here as we are taking away amongst these two. So when we take the difference in the total number of atomic mass units of the reactants and the products and then multiply that by 931.5 megaelectron volts per c squared for every atomic mass unit times c squared we get an energy released of 4.781 megaelectron volts.

format: plain


text:
# Harmonic oscillator with light damping

My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$x = Ae^{- \gamma t} \cos (\omega \, t)$$ for $\gamma = \dfrac b {2m}$.

It says this implies the amplitude follows the relationship: $$x_{max} = Ae^{-\gamma t}$$

I am not understanding where this second equation comes from. Why does the cosine term disappear if $\omega$ is a constant?

-

$\cos (\omega t)$ behaves like any other cosine term. $\cos (c)$ where $c$ is any constant is itself a constant. Since cosine oscillates between $[ -1 , 1]$ then when cosine is at it's maximum value it can be replaced with $1$.
Therefore: $$x_{max} = max(Ae^{- \gamma t} \cos (\omega t)) = Ae^{- \gamma t} \cdot max(\cos (\omega \, t)) = Ae^{- \gamma t} \cdot 1 = Ae^{- \gamma t}$$


text:
# Cut the Cake

Time Limit: 5000/5000 MS (Java/Others)

Memory Limit: 102400/102400 K (Java/Others)

## Description

Mehmet is a nut cake seller. As the whole cake is too large and too heavy, a customer generally buys a specified part of it. The whole cake is a rectangle, while the specified part is a convex polygon region inside the rectangle. To get such specified part, Mehmet uses his sword to cut the cake. The only operation he can do is splitting a piece of cake into two parts by a straight cutting. As the cake is very solid, it cost as much as 1 unit energy to cut each length of cake. What's the minimum energy Mehmet has to consume to satisfy the customer?

## Input

There are multiple test cases. Process to the End of File.

The first line of each test cases contains three integers 3≤N≤100, 0<W≤5000, 0<H≤5000, where W and H are the width and height of the nut cake. The next N lines are the vertices of the specified part in clockwise or counterclockwise order. Each line contains two integers 0<Xi<W and 0<Yi<H.

## Output

For each test case, output the minimum energy with six digits after decimal point.

## Sample Input

4 100 100
10 10
20 10
20 20
10 20
5 40 30
12 25
28 25
32 20
20 8
8 20

## Sample Output

150.000000
109.494748

zhuyuanchen520

## Source

2013 Multi-University Training Contest 9


text:
# Other Mathematics Commons™

## All Articles in Other Mathematics

1,792 full-text articles. Page 2 of 76.

Compilación De Procesos Investigativos En Educación Matemática, 2022 Universidad de Cundinamarca

#### Compilación De Procesos Investigativos En Educación Matemática, Martha Lidia Barreto Moreno, Yeferson Castellanos Novoa, María Alejandra Mayorga Henao, Diana Marcela Contento Sarmiento, Jesús Antonio Villarraga Palomino, Andrés Alberto Gutiérrez Morales, Juan David Firigua Bejarano, Yineth Marleidy Parra Ubaque, Lady Johanna Silva Marín

##### Educación

En el libro Compilación de procesos de investigación en educación matemática, consta de cuatro capítulos donde se presentan los procesos desarrollados en el marco de proyectos de investigación a nivel de pregrado y postgrado en Educación.

El primer capítulo consiste en la sistematización de la acción docente desarrollada en el marco de los Talleres Itinerantes de Alfabetización Computacional en la provincia de Sumapaz, propuesta de innovación para implementar procesos didácticos que contribuyan al desarrollo del pensamiento matemático computacional en educación básica primaria rural.

El segundo capítulo contiene el proceso investigativo que dio continuidad al trabajo realizado en la Fase1, sobre …

The Degree Gini Index Of Several Classes Of Random Trees And Their Poissonized Counterparts—Evidence For Duality, 2022 The George Washington University, Washington, DC 20052, USA

#### The Degree Gini Index Of Several Classes Of Random Trees And Their Poissonized Counterparts—Evidence For Duality, Carly Domicolo, Panpan Zhang, Hosam Mahmoud

##### Journal of Stochastic Analysis

No abstract provided.

2022 University of Southern California, Los Angeles, CA 90089, USA

#### A Sharp Rate Of Convergence In The Functional Central Limit Theorem With Gaussian Input, S.V. Lototsky

##### Journal of Stochastic Analysis

No abstract provided.

2022 Università di Bari, n.4, Via E. Orabona, 70125 Bari, Italy

#### Quantization Of The Free Poisson Central Limit Theorem, Yungang Lu

##### Journal of Stochastic Analysis

No abstract provided.

2022 Università di Bari, n.4, Via E. Orabona, 70125 Bari, Italy

#### Quantization Of The Boolean Poisson Central Limit Theorem And A Generalized Boolean Bernoulli Sequence, Yungang Lu

##### Journal of Stochastic Analysis

No abstract provided.

2022 Polytechnique Montréal, Montréal, Québec H3C 3A7, Canada

#### A First-Passage Problem For Exponential Integrated Diffusion Processes, Mario Lefebvre

##### Journal of Stochastic Analysis

No abstract provided.

Domain Of Exotic Laplacian Constructed By Wiener Integrals Of Exponential White Noise Distributions, 2022 Universitá di Roma Tor Vergata, Via di Torvergata, Roma, Italy

#### Domain Of Exotic Laplacian Constructed By Wiener Integrals Of Exponential White Noise Distributions, Luigi Accardi, Un Cig Ji, Kimiaki Saitô

##### Journal of Stochastic Analysis

No abstract provided.

Symmetric Generations And An Algorithm To Prove Relations, 2022 California State University, San Bernardino

#### Symmetric Generations And An Algorithm To Prove Relations, Diddier Andrade

##### Electronic Theses, Projects, and Dissertations

In this thesis we have discovered homomorphic images of several progenitors such as 3^(*56):(23:(3:7), 3^(*14):(23:(3:7)), 5^(∗24) : S5, 2^(∗10) : (10 : 2), 56^(∗24) : (A5 : 2), and 11^(∗12) :m L2(11). We give isomorphism types of each image that we have found.
We then create a monomial representation of L2(11) by lifting 5:11 onto it.
We manually perform Double Coset Enumeration of 3:(2×S5) over D12
to create its Cayley graph. This is achieved by solving many word problems. The
Cayley graph is used to find a permutation representation of 3:(2×S5). We also
perform Double Coset Enumeration S3 × A5 …

Academic Hats And Ice Cream: Two Optimization Problems, 2022 Moscow Power Engineering Institute (National Research University)

#### Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova

##### Journal of Humanistic Mathematics

This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!

The Construction And Estimation Of Hidden Semi-Markov Models, 2022 University of Sulaimani, Sulaymaniyah, Iraq

#### The Construction And Estimation Of Hidden Semi-Markov Models, Kurdstan Abdullah, John Van Der Hoek

##### Journal of Stochastic Analysis

No abstract provided.

#### Propuestas Y Resultados De Investigación Transmoderna, Translocal Y Digital Desde Jóvenes Semilleristas, Xiomara Gonzalez Gaitan

##### Institucional

En el presente libro intitulado Propuestas y resultados de investigación transmoderna, translocal y digital desde jóvenes semilleristas, se encuentran compilados las propuestas, avances y resultados de los proyectos en curso de los Semilleros de Investigación de la Universidad de Cundinamarca, Colombia, que se presentaron en el “II encuentro de semilleros de investigación: ciencia, tecnología e innovación en la era digital” en su versión 2020. Hacemos la labor de publicar estos proyectos con la intensión de difundir el conocimiento y como muestra del esfuerzo y alcance de la labor investigativa de los semilleristas de la Universidad de Cundinamarca. Esperamos que lo …

On Superoscillations And Supershifts In Several Variables, 2022 Chapman University

#### On Superoscillations And Supershifts In Several Variables, Yakir Aharonov, Fabrizio Colombo, Andrew N. Jordan, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen

##### Mathematics, Physics, and Computer Science Faculty Articles and Research

The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.

The Thermodynamics Of A Stochastic Geometry Model With Applications To Non-Extensive Statistics, 2022 University of Mazandaran, Babolsar, Iran

#### The Thermodynamics Of A Stochastic Geometry Model With Applications To Non-Extensive Statistics, O.K. Kazemi, A. Pourdarvish, J. Sadeghi

##### Journal of Stochastic Analysis

No abstract provided.

Artificial Neural Network Concepts And Examples, 2022 University of Missouri-St. Louis

#### Artificial Neural Network Concepts And Examples, Harcharan Kabbay

##### Theses

Artificial Neural Networks have gained much media attention in the last few years. Every day, numer- ous articles on Artificial Intelligence, Machine Learning, and Deep Learning exist. Both academics and business are becoming increasingly interested in deep learning. Deep learning has innumerable uses, in- cluding autonomous driving, computer vision, robotics, security and surveillance, and natural language processing. The recent development and focus have primarily been made possible by the convergence of related research efforts and the introduction of APIs like Keras. The availability of high-speed compute resources such as GPUs and TPUs has also been instrumental in developing deep learning …

2022 Università di Bari, n.4, Via E. Orabona, 70125 Bari, Italy

#### Quantization Of The Poisson Type Central Limit Theorem (1), Yungang Lu

##### Journal of Stochastic Analysis

No abstract provided.

A Closed Form Formula For The Stochastic Exponential Of A Matrix-Valued Semimartingale, 2022 Heinrich Heine University, Düsseldorf, Germany

#### A Closed Form Formula For The Stochastic Exponential Of A Matrix-Valued Semimartingale, Peter Kern, Christian Müller

##### Journal of Stochastic Analysis

No abstract provided.

Ultrametrics And Complete Multipartite Graphs, 2022 Institute of Applied Mathematic and Mechanics of NAS of Ukraine

#### Ultrametrics And Complete Multipartite Graphs, Viktoriia Viktorivna Bilet, Oleksiy Dovgoshey, Yuriy Nikitovich Kononov

##### Theory and Applications of Graphs

Let $$(X, d)$$ be a semimetric space and let $$G$$ be a graph. We say that $$G$$ is the diametrical graph of $$(X, d)$$ if $$X$$ is the vertex set of $$G$$ and the adjacency of vertices $$x$$ and $$y$$ is equivalent to the equality $$\diam X = d(x, y)$$. It is shown that a semimetric space $$(X, d)$$ with diameter $$d^*$$ is ultrametric iff the diametrical graph of $$(X, d_{\varepsilon})$$ with $$d_{\varepsilon}(x, y) = \min\{d(x, y), \varepsilon\}$$ is complete multipartite for every $$\varepsilon \in (0, d^*)$$. A refinement of the last result is given for totally bounded ultrametric spaces. …

2022 University of Massachusetts Amherst

#### General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul

##### Doctoral Dissertations

In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.

Self-Repelling Elastic Manifolds With Low Dimensional Range, 2022 University of Rochester, Rochester, NY 14627, USA

#### Self-Repelling Elastic Manifolds With Low Dimensional Range, Carl Mueller, Eyal Neumann

##### Journal of Stochastic Analysis

No abstract provided.

Induced Matrices: Recurrences And Markov Chains, 2022 Southern Illinois University, Carbondale, Illinois 62901, USA

#### Induced Matrices: Recurrences And Markov Chains, Philip Feinsilver

##### Journal of Stochastic Analysis

No abstract provided.


text:
# Gravity CharacterController on moving platform

I have a third person character witha character controller. In the C# Player script I set the move direction like this which works pretty nice to simulate gravity

private Vector3 _moveDirection = Vector3.zero;

private float Gravity = 20.0f;

void Update()
{
// ....
_moveDirection.y -= Gravity * Time.deltaTime;

_characterController.Move(_moveDirection * Time.deltaTime + mExternalMovement);
}

The mExternalMovement (also Vector3) I set when the player is on a platform that moves. This works as well for platforms the move e.g. from left to right. But when I use the same for platforms that move up and down the player is like trembling- because of the gravity that is always set for the y movement.

But I have to set the gravity as well when standing on the platform.... any ideas?

• Have you considered adding gravity only when you're not in contact with a ground surface, such as a moving platform? – DMGregory Mar 24 '20 at 19:46
• @DMGregory: Yes, but it hs not effect and it seems like that it isnet the gravity. It is doing this in one step: _characterController.Move(_moveDirection * Time.deltaTime + mExternalMovement); But when I move this _characterController.Move(mExternalMovement) to LateUpdate and keep _characterController.Move(_moveDirection * Time.deltaTime in Update it works smoothly. – Jay Mar 24 '20 at 20:49
• Sounds like you should update your question then to reflect your current understanding of the problem. – DMGregory Mar 24 '20 at 20:52

## 1 Answer

Add a Rigidbody and use the gravity there?

• Dont know what you mean. I have a Rigidbody but what does it had to do with the character controller? – Jay Mar 24 '20 at 19:24


text:
## Fanduekisses one year ago Graph Transformations: What does a negative sign do to an absolute value graph if it's in front of the || ?

1. Fanduekisses

I know that if it's in fron of the x it flips the graph. What about in this case: $-|-x-3|+2$

2. zepdrix

When the negative is on the x, it flips it horizontally. Example: $$\large\rm f(x)=\sqrt{x}$$ would flip across the y-axis if we did this: $$\large\rm g(x)=\sqrt{-x}$$ A negative played on the outside of everything, is really just a negative being placed on y. And it results in a flip vertically. Example: $$\large\rm j(x)=x^2$$ would flip across the x-axis if we did this: $$\large\rm k(x)=x^2$$ But with your problem, notice that we have neither of these things happening. We're not applying the negative to the 2. So normally I would say, we're reflecting the graph about the x-axis. But in this case, we're reflecting the graph about the line y=2.

3. zepdrix

Woops, $$\large\rm k(x)=-x^2$$

4. Fanduekisses

Ohh so in this case it flips both horizontally and vertically?

5. zepdrix

|dw:1442976962331:dw|The $$\rm \color{green}{green}$$ is the original function. The $$\rm \color{blue}{blue}$$ is the new one. The function is being reflected vertically across the $$\rm \color{#DD4747}{pink}$$ line.


text:
# GATE2002-11

16 votes
1.4k views

The following recursive function in C is a solution to the Towers of Hanoi problem.

void move(int n, char A, char B, char C)  {
if (......................) {
move (.............................);
printf("Move disk %d from pole %c to pole %c\n", n, A,C);
move (.....................);
}
}

Fill in the dotted parts of the solution.

recategorized
2
If(n==1) { Printf("\n %s %c %s %c","move disk1 from pole",A,"to pole",B); Return; } Towers(n-1,A,C,B); Printf.... Towers(n-1,C,B,A);

## 2 Answers

20 votes

Best answer
void move(int n, char A, char B, char C) {
if (n > 0) {
move (n-1, A, C, B);
printf("Move disk %d from pole %c to pole %c\n", n, A, C);
move (n-1, B, A, C);
}
}

edited by
12

The snippet tells that A must be a source and C the destination.That means function would be move(src,spare,dest).

Firstcall(A,B,C) A to C using B intermediate

If we go by this defination, first move(n-1,A,C,B) will move n-1 disks to B using C as spare.

Now the 1 that remains is moved from A(src) to C(dest).

Shouldn't the last call then be move(n-1,B,A,C) ,that is moving the n-1 disk that were previously moved to intermediate node B over C(the destination) ?

3
correct now?
1
yes
1
Yes it is correct !!
7 votes
If(n==1) { Printf("\n %s %c %s %c","move disk1 from pole",A,"to pole",B); Return; } Towers(n-1,A,C,B); Printf.... Towers(n-1,C,B,A);

## Related questions

26 votes
2 answers
1
3.7k views
In the C language: At most one activation record exists between the current activation record and the activation record for the main The number of activation records between the current activation record and the activation records from the main depends on the actual ... the activation record for the recursive function to be saved in a different stack before the recursive function can be called.
20 votes
3 answers
2
3.7k views
The C language is: A context free language A context sensitive language A regular language Parsable fully only by a Turing machine
10 votes
2 answers
3
3.1k views
The results returned by function under value-result and reference parameter passing conventions Do not differ Differ in the presence of loops Differ in all cases May differ in the presence of exception
21 votes
2 answers
4
2.8k views
Consider the following C program: void abc(char*s) { if(s[0]=='\0')return; abc(s+1); abc(s+1); printf("%c",s[0]); } main() { abc("123"); } What will be the output of the program? If $abc(s)$ is called with a null-terminated string $s$ of length $n$ characters (not counting the null ('\0') character), how many characters will be printed by $abc(s)$?


text:
EN FR
• Legal notice
• Accessibility - non conforme

## Section: Research Program

### Local Interactions and Transient Analysis in Adaptive Dynamic Systems

Participants : Nicolas Gast, Bruno Gaujal, Florence Perronnin, Jean-Marc Vincent, Panayotis Mertikopoulos.

Many systems can be effectively described by stochastic population models. These systems are composed of a set of $n$ entities interacting together and the resulting stochastic process can be seen as a continuous-time Markov chain with a finite state space. Many numerical techniques exist to study the behavior of Markov chains, to solve stochastic optimal control problems [89] or to perform model-checking [48]. These techniques, however, are limited in their applicability, as they suffer from the curse of dimensionality: the state-space grows exponentially with $n$.

This results in the need for approximation techniques. Mean field analysis offers a viable, and often very accurate, solution for large $n$. The basic idea of the mean field approximation is to count the number of entities that are in a given state. Hence, the fluctuations due to stochasticity become negligible as the number of entities grows. For large $n$, the system becomes essentially deterministic. This approximation has been originally developed in statistical mechanics for vary large systems composed of more than ${10}^{20}$ particles (called entities here). More recently, it has been claimed that, under some conditions, this approximation can be successfully used for stochastic systems composed of a few tens of entities. The claim is supported by various convergence results [70], [79], [102], and has been successfully applied in various domains: wireless networks [52], computer-based systems [73], [84], [97], epidemic or rumour propagation [62], [77] and bike-sharing systems [66]. It is also used to develop distributed control strategies [101], [83] or to construct approximate solutions of stochastic model checking problems [54], [55], [56].

Within the POLARIS project, we will continue developing both the theory behind these approximation techniques and their applications. Typically, these techniques require a homogeneous population of objects where the dynamics of the entities depend only on their state (the state space of each object must not scale with $n$ the number of objects) but neither on their identity nor on their spatial location. Continuing our work in [70], we would like to be able to handle heterogeneous or uncertain dynamics. Typical applications are caching mechanisms [73] or bike-sharing systems [67]. A second point of interest is the use of mean field or large deviation asymptotics to compute the time between two regimes [92] or to reach an equilibrium state. Last, mean-field methods are mostly descriptive and are used to analyse the performance of a given system. We wish to extend their use to solve optimal control problems. In particular, we would like to implement numerical algorithms that use the framework that we developed in [71] to build distributed control algorithms [64] and optimal pricing mechanisms [72].


text:
How do comets' nuclei stick together?

Comets are commonly called icy dust balls or dirty snowballs and as such rather light:

Known comets have been estimated to have an average density of $$0.6 {\rm g}/{\rm cm}^3$$. Because of their low mass, comet nuclei do not become spherical under their own gravity and therefore have irregular shapes.

I read about their live times, I learned that they can rotate, how they can be classified by their type. NASA summarizes the key facts as follows

Comets are frozen leftovers from the formation of the solar system composed of dust, rock and ices. They range from a few miles to tens of miles wide, but as they orbit closer to the sun, they heat up and spew gases and dust into a glowing head that can be larger than a planet. This material forms a tail that stretches millions of miles.

What interests me has to do with their formation. I know that comets are not spherical and rather small in size. I am now wondering how comets' nuclei stick together, more specifically if there is some empiral knowledge or even some theory I should know about. Obviously, gravitation is not strong enough, so it must be merely the structural stability of the material, but how can that be estimated?

References

• One word: "poorly".
– Mark
Feb 5 '21 at 23:35

Van-der-Waals forces and gravitation are the main answer.

The main property keeping together bodies smaller than a few $$100 {\rm km}$$ are their constituents surface forces between the grains (van-der-Waals) and chemical/ molecular bonding within the grains themselves. It's the same mechanic which makes the dust and grease in your flat stick to surfaces and the fluffy dirt clumps under the bed stick together, thus cohesion/ adhesion.

The typical internal strength of such dirty iceball as comets are, or rubble piles as some minor asteroids like Ruygu seem to be (e.g. Wada et al, 2018), does not need to be large - there is nothing in deep space which pulls it apart. And even the few µg of self-gravitation help to keep smaller fragments and ejecta to return to the parent body when hit by the inevitable micro-meteorites after their intial formation (Blum et al, 2017) (so yes, there is still self-gravitation involved which helps form them; they are just too small to achieve hydrostatic equilibrium and differentiation).

67P Churyumov-Gerasimenke provided a nice cometary physics lab which allowed in-situ investigation. The strength of the material could be analysed directy (Heinisch et al, 2018), (Grousin et al, 2018) with the lander Philae, even though it landed somewhat precariously (see also this overview by Blum et al (2018) focussing on planet formation and lessons learned from 67P in particular).

In summary: most measurement, both in-situ and lab experiments, indicate that the internal strength of comets is within the range of $$1 {\rm Pa} \ldots ... 10 {\rm kPa}$$ and most likely at the lower end of this range. Their strength is thus determined by their small gravity and the inter-particle forces, thus van-der-Waals which in part is possibly enhanced a bit by sintering together the different constituents grains.

• Wow, you are quick. The essential part I overlooked is that nothing is really pulling comets appart, usually, when being far away from e.g. Jupiter. Also, I like the comparison to the fluffy dust balls. Feb 5 '21 at 12:06
• I added two more wiki-links and fixed a typo and some typesetting, hope you are ok with it. Feb 5 '21 at 19:45
• Just to add to how little you need to hold comets together, the OSIRIS-Rex spacecraft that recently visited Bennu had to slow its orbital speed to a paltry 5 cm/sec to not escape the gravity. The real question here, to me, is what keeps them stuck together immediately after an impact, i.e., why don't they just crash and bounce off? Feb 5 '21 at 20:04
• @zephyr see aanda.org/articles/aa/pdf/2017/07/aa30345-16.pdf plot 13 (which an updated plot from some way earlier Weidenschilling work): relative collision velocities for particles of that size in a solar nebula are mm/s. So it's far from 'smashing'. Yet the 'bouncing barrier' at slightly larger (and faster) particles is something only overcome by turbulent disks and gas-solid interactions in things like eddies to create local over-densities of solids and enhance their self-gravity Sep 20 '21 at 9:10


text:
Dataset Restricted Access

# PAN19 Authorship Analysis: Style Change Detection

Zangerle, Eva; Tschuggnall, Michael; Specht, Günther; Potthast, Martin; Stein, Benno

This is the data set for the Style Change Detection task of PAN@CLEF 2019.

The goal of the style change detection task is to identify text positions within a given multi-author document at which the author switches. Detecting these positions is a crucial part of the authorship identification process, and for multi-author document analysis in general. Note that, for this task, we make the assumption that a change in writing style always signifies a change in author.

Tasks

Given a document, we ask participants to answer the following two questions:

• Was the given document written by multiple authors? (task 1)
• For each pair of consecutive paragraphs in the given document: is there a style change between these paragraphs? (task 2)

In other words, the goal is to determine whether the given document contains style changes and if it indeed does, we aim to find the position of the change in the document (between paragraphs).

All documents are provided in English and may contain zero up to ten style changes, resulting from at most three different authors. However, style changes may only occur between paragraphs (i.e., a single paragraph is always authored by a single author and does not contain any style changes).

Data

To develop and then test your algorithms, two data sets including ground truth information are provided. Those data sets differ in their topical breadth (i.e., the number of different topics that are covered in the documents contained). dataset-narrow contains texts from a relatively narrow set of subjects matters (all related to technology), whereas dataset-wide adds additional subject areas to that (travel, philosophy, economics, history, etc.).

Both of those data sets are split into three parts:

• training set: Contains 50% of the whole data set and includes ground truth data. Use this set to develop and train your models.
• validation set: Contains 25% of the whole data set and includes ground truth data. Use this set to evaluate and optimize your models.
• test set: Contains 25% of the whole data set. For the documents on the test set, you are not given ground truth data. This set is used for evaluation.

Input Format

Both dataset-narrow and dataset-wide are based on user posts from various sites of the StackExchange network, covering different topics. We refer to each input problem (i.e., the document for which to detect style changes) by an ID, which is subsequently also used to identify the submitted solution to this input problem.

The structure of the provided datasets is as follows:

train/
dataset-narrow/
dataset-wide/
validation/
dataset-narrow/
dataset-wide/
test/
dataset-narrow/
dataset-wide/

For each problem instance X (i.e., each input document), two files are provided:

problem-X.txt contains the actual text, where paragraphs are denoted by \n\n.
truth-problem-X.json contains the ground truth, i.e., the correct solution in JSON format:

{
"authors": NUMBER_OF_AUTHORS,
"structure": ORDER_OF_AUTHORS,
"site": SOURCE_SITE,
"multi-author": RESULT_TASK1,
"changes": RESULT_ARRAY_TASK2
}

The result for task 1 (key "multi-author") is a binary value (1 if the document is multi-authored, 0 if the document is single-authored). The result for task 2 (key "changes") is represented as an array, holding a binary for each pair of consecutive paragraphs within the document (0 if there was no style change, 1 if there was a style change). If the document is single-authored, the solution to task 2 is an array filled with 0s. Furthermore, we provide the order of authors contained in the document (e.g., [A1, A2, A1] for a two-author document), the total number of authors and the Stackoverflow site the texts were extracted from (i.e., topic).

An example of a multi-author document, where there was a style change between the third and fourth paragraph could look as follows (we only list the two relevant key/value pairs here):

{
"multi-author": 1,
"changes": [0,0,1,...]
}

A single-author document would have the following form (again, only listing the two relevant key/value pairs):

{
"multi-author": 0,
"changes": [0,0,0,...]
}

Version 2.0: added validation and test set
##### Restricted Access

You may request access to the files in this upload, provided that you fulfil the conditions below. The decision whether to grant/deny access is solely under the responsibility of the record owner.

Please request access to the data with a short statement on how you want to use it. Thanks!

We would like to point out that you can register on pan.webis.de to be part of the PAN community.

• Eva Zangerle, Michael Tschuggnall, Günther Specht, Martin Potthast, and Benno Stein. Overview of the Style Change Detection Task at PAN 2019. In Linda Cappellato, Nicola Ferro, David E. Losada, and Henning Müller, editors, CLEF 2019 Labs and Workshops, Notebook Papers, September 2019. CEUR-WS.org.

389
39
views
downloads
All versions This version
Views 38977
Downloads 3912
Data volume 251.7 MB65.4 MB
Unique views 28455
Unique downloads 228


text:
# Superposition and the Winning Jackpot Numbers

Let's say I buy myself a lottery ticket (Mega-Millions). I have $\frac{1}{175,711,536}$ chance of winning. Before I tune on the tv/radio and listen to the winning numbers (i.e. make an observation), is it correct to say that the winning numbers are in some kind of 'superposition' of states? And the act of watching the numbers come out of the drawing machine is somehow 'collapsing' the wave function? Could this possibly affect the way I think about quantum mechanics?

EDIT: I found the following article very similar to what I've proposed: http://iopscience.iop.org/0031-9120/38/2/606/pdf/pe32m6.pdf

-

Someone may in principle know the exact velocities etc. of the balls or whatever is used to pick the random numbers in advance. But if we do not know them, we may describe them by "probabilistic distributions" which means that we assign every possible velocity and position of all the balls, and therefore every possible outcome of the lottery, some probability (or probability density).

However, probabilities and probability densities may be used even in classical (non-quantum) physics, to quantify our uncertainty and partial knowledge about the state of the physical system (the balls). Quantum mechanics is more "novel" than just probability distributions because the probabilities in quantum mechanics are calculated from some complex "probability amplitudes" by squaring them and the probability amplitudes are matrix elements of linear operators on the Hilbert space, and so on, so your picture seems to be missing all this actual "beef" of quantum mechanics.

For all practical purposes, large balls such as those that decide the lottery may be described just by classical physics so even if we use probability distributions, they're classical probability distributions. In principle, however, all the particles in the system obey the laws of quantum mechanics (which are only relevant if we want to be really accurate in our description of the balls) and in that case, it's true that the wave function evolves to a general superposition and the act of the measurement or its perception chooses one outcome or another, the "collapse". But the "collapse" isn't any material process affecting physical objects; it is the change of the knowledge leading one to replace the original probability distributions by conditional ones, conditioned by the freshly observed fact.

Again, as I mentioned at the beginning, it is possible – in these macroscopic, nearly classical, situations – to imagine that someone has "measured" or "perceived" the outcome before the citizen who is waiting for the lottery results at home so the "collapse" effectively took place before some people realized it. Whenever there is a possibility that the different microstates or outcomes may "reinterfere again", we must however be very careful to avoid any collapse up to the moment of an actual measurement, an irreversible perception of some classical information (usually accompanied by "decoherence").

-


text:
Towers

 Towers Engineering & Build Event Forum Threads 2018 2017 2012 2011 There are no tests available for this event Images Image Gallery There are no question marathons for this event Division B Champion Tower Heights Middle School Division C Champion Grand Haven High School

Towers is a Division B and Division C event for the 2017 and 2018 seasons. In this event, given certain parameters of height and material, each team is to design, build and test the lightest and tallest tower to carry a maximum standard load. The tower must span an opening (in most years, a 20 cm x 20 cm square), and support a load at a minimum height.

How to Build a Tower

This section describes the process for building a tower. While the specific construction parameters change from year to year, this process generally remains the same, aside from specifics regarding the design.

The first issue of tower building is the wood selection. Many consider balsa wood to be the best, followed by bass. However, there are many different opinions on which to use. Here are some opinions:

• One opinion recommends an all balsa wood tower, but it may be pretty tricky to handle. To use any type of wood, an extremely sharp knife is needed for each cut to avoid smashing the wood fibers (A blade will have a lot of life left after you change it for tower building, and one way to save costs is to save the blades and use them later for other projects, etc.). Smashing the wood fibers damages the integrity of the wood and may result in a lower score.
• Another opinion states that heavy basswood should be used when first trying to understand the dynamics of the structure. Once a working design is completed using basswood, weight can then be removed by using smaller sizes or balsa. Using balsa can make it more difficult if it is your first time.
• Another opinion suggests a mix of balsa and bass. Bass is used where extra strength is needed. In all other places, balsa is fine. The legs are normally bass and bracings balsa. A mix of the two brings out the best of both in one tower.

Balsa vs. Bass:

• Balsa is on average lighter for a specific strength, although individual pieces vary in both woods.
• Balsa is easier to bend. This is generally not relevant in Towers, but may be important for some specialized designs.
• Bass has less variation between pieces than balsa with regard to the range of density for a certain size, and the range of strength for a specific density.
• Bass is generally stronger for a certain size, though usually heavier as well.

Step 1

• The next step is design. A four legged tower is easier to build but may not be as efficient. A 3-legged tower is extremely difficult to build, as the angles aren't nice and neat. Furthermore, a bracing strip needs to be angled 2 ways, whereas in a 4 legged design it only needs to be angled in one direction. Also, don't forget about laminating wood together- many good towers have been built by laminating very small wood together for the legs.
• Make sure to comply to the rules for dimensions. In 2018, for division B, the tower must be 50cm tall, and in division C, the tower must be 60cm tall. Also, the 2018 rules state that it must be able to pass through an 8cm diameter circle at 25cm from the ground for division B, and 20cm from the ground for division C. Therefore, a two-part design is essentially required in 2018, whereas in 2017, one-part designs were more common.
• After the number of legs is decided, the bracing is the part that towers are won and lost by. It depends on the type of leg design is used, but generally, balsa wood legs of dimensions 3/16x1/8cm need to be braced every 5 or 6cm all
the way up (by the way, a solid wood square is not the best way to go for legs). Also, with the bracing, equality is of high importance. If you brace at 5cm and then the next space is 8cm, you're testing the 8cm space. It would be better to brace at 6.5 cm and 6.5 cm in this example.
• In addition, when creating a four-legged tower, there is a choice between building a rectangular base or a square base. A rectangular base tower needs stiffer legs and is lighter, but has a high chance of tipping over. A way to avoid this (though will add weight) is to slant the wide side of the tower until it almost forms a pyramid.
• It is highly advised to test, test, and test some more. The more you build, the better your chances of getting a good design, and the better you get at building. Analyze any failed structures. Knowing which member broke first (which can be determined by the type of break, sheer, torque (twisting or bending) will tell you what piece of wood needs to be denser, or larger in cross section. It is a very good idea to understand how the stresses are flowing in your tower. Eventually, time spent on building individual towers will decrease, while the quality will increase. Also, never test towers over the maximum weight that the rules say. If it holds, set it aside and see where you can cut back. If you fail, then you still have a backup.

Step 2a: Materials

Get the Following Recommended Materials:

• Easy Cutter by Midwest (Easy, fast, and clean cuts, even angled) ~$15-$20 on eBay.
• A pack of extra blades may also be useful
• 1/8 and 1/16 BalsaWood. (Make sure the 1/16 isn't too flimsy and cheap)
• Zap-A-Gap glue (reasonably strong, and fast curing)

Step 2b: Design

So how should a tower be built? The best way is to really create a full-scale blueprint on graph paper. First of all, a more general design should be drawn out. Here is a basic design from 2006.

Once that is finished, the design should be drawn out on graph paper. It makes it a lot easier if you create a "jig" of your blueprint. In other words, use pieces of scrap wood to create a template shown below:

(picture to come)

The jig makes it possible for concise construction of the tower, ensuring it will not be slanted.

Step 2c: Bracings

Once the design of the tower fully planned out, it is time to consider what type of bracing to use.

• One idea suggests using an X Brace, where two pieces of wood cross over each other, creating an X. The advantage of this method is that it will prevent the legs from bowing outwards and slightly inwards, but at the cost of more wood.
• The second idea mentions using a Z Brace, where a diagonal piece of wood lies in between two horizontal members. The benefit to having this is that the side will be held together, but will start to torque in the direction of the diagonal brace.
• Another method is to use Z Bracings without the horizontal members, similar to this (Imagine the chimney on its side): |/\/\/\/\/\/\|. The benefit to having this is that it requires less wood and almost provides as much support as the Z Brace. However, since there is no horizontal brace to keep the tower from bowing inwards, you would need to angle the bracings at a smaller angle and have a medium-high density wood.

There are many other types of bracings, such as the V brace and the /\ brace, too. However, their scores in the previous years are not too high, so other types of bracings are much better to use.

Below are examples of what these braces would look like. Note that these are only examples of what the would look like, not how many of them you should have.

Step 3: Construction

• Construction is rather easy with an easy cutter and plans with measurements. All you have to do now is to cut all the pieces out that you have measured, although this can take up hours if you have never done it before. Remember to cut out 4x the wood on the blueprint, then connect the two halves with bracings that form the 3rd and 4th sides.
• The straightness of the tower is incredibly important. If the balance is off and the weight is not distributed equally it will fail when using lighter smaller wood.
• If you haven't drawn any plans, STOP! You must go back to 2a.

Tips and Tricks

• The Test Base opening is typically a square of 20 cm sides. This means that a tower with a base of 15 or 16 cm can be oriented so that each leg rests at the midpoint of the opening's sides, rather than building a tower with a base of 20 or 21 cm with the legs resting at the corners.
• It is far more difficult to increase the size of a stick than to decrease the size. Hence, cut sticks so that there is some extra length, making it easier to adjust - if a stick is cut too short, there's no way to correct it.
• Make sure to apply only the amount of glue needed - in most cases, significantly less than one drop. Too much glue will increase the mass of the tower and it will not significantly increase the strength - in an adequately constructed tower, the glue joints are among the strongest parts. The faces being glued to each other should be as flat as possible to minimize the amount of space-filling done with glue.
• Short, light bracings are better than long, heavy bracings.
• When attaching braces, make sure to glue them on top of the frame as opposed to adjacent to the frame. It makes the tower much stronger because it provides a greater contact area.
• Remember that using too much wood will weigh down a tower and result in a lower score. Most winning towers win because of their weight, so even if a tower holds all 15 kg, the score will still be more dependent of the weight of the tower itself.
• Ideally, the load should be equally distributed across each leg of the tower. To help the tower rest on the test base more evenly, sanding the bottom of the legs can be helpful. However, be careful not to end up with a tower that is shorter than the minimum allowed height.
• Keep in mind that if a tower's base is 29 cm wide, a load score bonus of 5000g will be rewarded (2018 season). Of course, this isn't always recommended due to the risk of a lower score. If you do decide to get the bonus, remember that your tower will be weaker. It is also advised to have a base slightly bigger than the requirement to avoid risking the potential bonus and weakening your tower for nothing.
• Build on a consistent basis and test regularly.
• It is recommended to record a test with a slow motion camera. Towers typically begin failing at a single point, which cascades into a failure of other members. A slow-motion video makes it easier to identify the point of failure and fix/improve it.
• NEVER GIVE UP! Towers is a frustrating event in which many towers have to be built in order to succeed. Don't give up and keep persevering. The reward is very satisfying.

Competition

Check in

Before device testing, teams will check in to verify that the tower meets the requirements. Teams measure the tower verify that the tower is tall enough, pass an 8-cm diameter ring gauge and measure where the gauge rests, and put the tower on the scale so the supervisor can determine the mass.

Scoring

A tower is scored primarily according to its efficiency - load held mass divided by the mass of the tower. For example, a tower that weighs 5 grams and holds 15 kilograms would earn a score of $\frac{15,000 \textrm{ g}}{5 \textrm{ g}}=3000$

The maximum "load held" is 15 kilograms.

In most years, there are additional factors that affect the score:

• In 2018, there is a bonus awarded if the tower spanned a 29cm circle centered on the 20cm by 20cm opening. The bonus adds 5000 grams to "mass held".
• In 2017, there was a bonus awarded if the tower spanned a 29cm circle centered on the 20cm by 20cm opening. The bonus added 2000 grams to "mass held".
• In 2012, the tower efficiency was weighted by the height of the tower. The specific weight depended on the tournament level (regional, state, national), such that at higher tournament levels a corresponding increase in height would have a greater impact on the score, encouraging taller towers at higher levels of competition.
• In 2011, the score was calculated using the square of the load held, in kilograms, divided by the mass of the tower in grams.

Teams who violate construction or competition rules (sections 3 and 4 of the rules, respectively), and teams whose towers are unable to be tested (e.g. the tower doesn't span the gap) are ranked in lower tiers.

Competitiveness of teams in Towers at different tournaments varies greatly. As of the 2017 season, in the majority of states, a score of 1000 is sufficient to finish near the top 4-6. In some larger states, high scores reach 2500+, or even 3000+. At the 2017 National Tournament, the medal-winning scores were approximately 2900-3500 in Division B, and 2700-3300 in Division C.


text:
## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

(a) $f = 1.5\times 10^{9}~Hz$ (b) $\lambda = 1.0\times 10^{-6}~m$
(a) We can find the frequency of an electromagnetic wave with a wavelength of 20 cm. $f = \frac{v}{\lambda}$ $f = \frac{3.0\times 10^8~m/s}{0.20~m}$ $f = 1.5\times 10^{9}~Hz$ (b) We can find the wavelength of a sound wave in water which has this frequency. $\lambda = \frac{v}{f}$ $\lambda = \frac{1500~m/s}{1.5\times 10^{9}~Hz}$ $\lambda = 1.0\times 10^{-6}~m$


text:
上海理工大学学报  2022, Vol. 44 Issue (4): 368-372 PDF

An hp-version Legendre spectral collocation method for nonlinear fractional differential equations
LI Shan, AN Xiao, SUN Guilei
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The hp-version Legendre spectral collocation method for solving nonlinear fractional differential equations was studied. At first, the multifractional differential equation was transformed into an equivalent Volterra integral equation. Then a numerical method to approximate the original equation was constructed. Finally, the correctness of the algorithm theory and the effectiveness of the proposed numerical method were demonstrated by numerical experiments.
Key words: nonlinear fractional differential equations     Legendre spectral collocation method     hp-version error bounds
1 问题的提出

 $\left\{ {\begin{array}{*{20}{l}} {{{\rm{D}}^{\alpha + 1}}u = f(t,u(t)),\quad t \in [0,T]} \\ {u(0) = 0,\quad {u'}(0) = 0} \end{array}} \right.$ (1)

 $\begin{split}& {{\rm{D}}^\alpha }u = \frac{1}{{\Gamma (n - \alpha )}}\int_0^t {{{(t - \tau )}^{n - \alpha - 1}}} {u^{(n)}}(\tau ){\text{d}}\tau, \\& \quad {{t}} > 0,\quad {{n}} - 1 < \alpha < {{n}}{} \end{split}$

Caputo导数 ${{\rm{D}}^\alpha }$ 不同于传统的微分算子，它是一种全局算子。目前分数阶导数主要分为3种类型：Riesz 意义下的分数阶导数、Riemann-Liouville 分数阶导数和Caputo 型分数阶导数。1832年Liouville采用级数的形式给出了分数阶导数，1853 年Riemann 采用定积分的形式给出了另一种分数阶微分，Grünwald 和Krug 将Riemann 和Liouville 的结论进行了统一，得到了Riemann-Liouville 分数阶导数的定义。1967 年Caputo 提出了Caputo 型分数阶导数。Caputo 型分数阶导数因其广泛的应用背景和相对简便的运算被广泛地应用于各个领域。目前在分数阶微分方程理论的研究中，关于其数值解进行了大量的研究。在Caputo 型分数阶定义下的初值条件具有明确的物理意义，因而在工程物理学领域该方程具有重要的应用价值。但由于解的复杂性，不能直接应用于实际工程中，所以，求解Caputo 型分数阶微分方程的数值解受到广泛关注。

2 预备知识

${I_h}$ 是区间 $I = \left[ {0,T} \right]$ 上的一个网格, ${I_h}: = \{ {t_n}: 0 = {t_0} < {t_1} < \cdots < {t_N} = T \}$ ，并且设 ${h_n} = {t_n} - {t_{n - 1}}$ ${I_n} = ({t_{n - 1}},{t_n}]$ 。通常情况下，对于一个给定区间 $\varLambda$ 和某一权函数 $\chi (x)$ ，定义

 ${L}_{\chi }^{2}(\varLambda )=\left\{v|v是可测的并且{\Vert v\Vert }_{{L}_{\chi }^{2}(\varLambda )} < \infty \right\}$

${L_k}(x),x \in ( - 1,1)$ 是标准的阶数为 $k$ 的Legendre多项式。Legendre 多项式集合构成一个完备的 ${L^2}( - 1,1)$ 正交系统，即

 $\int_{ - 1}^1 {{L_k}} (x){L_j}(x){\rm{d}}x = \frac{2}{{2k + 1}}{\delta _{k,j}}$ (2)

 ${L_{n,k}}(t) = {L_k}\left( {\frac{{2t - {t_{n - 1}} - {t_n}}}{{{h_n}}}} \right),\quad t \in {I_n},\quad k \geqslant 0$ (3)

${L_{n,k}}(t),k \geqslant 0$ 的集合构成一个完备的 ${L^2}\left( {{I_n}} \right)$ 正交系统，即

 $\int_{{I_n}} {{L_{n,k}}} (t){L_{n,j}}(t){\rm{d}}t = \frac{{{h_n}}}{{2k + 1}}{\delta _{k,j}}$ (4)

 ${t_{n,j}} = \frac{1}{2}\left( {{h_n}{x_{n,j}} + {t_{n - 1}} + {t_n}} \right),\quad 1 \leqslant n \leqslant N,\quad 0 \leqslant j \leqslant {M_n}$

 $\int_{{I_n}} \phi (t){\rm{d}}t = \frac{{{h_n}}}{2}\sum\limits_{j = 0}^{{M_n}} \phi \left( {{t_{n,j}}} \right){\omega _{n,j}},\quad \phi \in {\mathcal{P}_{2{M_n} + 1}}\left( {{I_n}} \right)$ (5)

 $\begin{split}& \sum\limits_{j = 0}^{{M_n}} {{L_{n,p}}} \left( {{t_{n,j}}} \right){L_{n,q}}\left( {{t_{n,j}}} \right){\omega _{n,j}} = \frac{2}{{2p + 1}}{\delta _{p,q}},\\& \qquad 0 \leqslant p + q \leqslant 2{M_n} + 1 \end{split}$ (6)

 ${\mathcal{I}_{s,{M_n}}}v\left( {{t_{n,j}}} \right) = v\left( {{t_{n,j}}} \right),\quad 0 \leqslant j \leqslant {M_n}$ (7)

 $\mathcal{I}_{\xi ,{M_n}}^tv\left( {{\xi _{n,j}}} \right) = v\left( {{\xi _{n,j}}} \right),\quad 0 \leqslant j \leqslant {M_n}$ (8)

 ${\xi _{n,j}} = \sigma \left( {{t_{n,j}},t} \right): = {t_{n - 1}} + \frac{{\left( {t - {t_{n - 1}}} \right)\left( {{t_{n,j}} - {t_{n - 1}}} \right)}}{{{h_n}}}$

 $\mathcal{I}_{\xi ,{M_n}}^tv(\xi ) = {\mathcal{I}_{\lambda ,{M_n}}}v(\sigma (\lambda ,t))$

 $\begin{split} & \int_{{I_k}} {{{(t - s)}^\nu }} \phi (s){\rm{d}}s = \sum\limits_{j = 0}^{{M_k}} \phi \left( {{t_{k,j}}} \right)\tilde \omega _{k,j}^v(t),\\&\qquad t \in {I_n},\quad k < n \end{split}$ (9)

 $\tilde \omega _{k,j}^v(t) = \int_{{I_k}} {{{(t - s)}^\nu }} {l_{k,j}}(s){\rm{d}}s$ (10)

 $\int_{{t_{n - 1}}}^t {{{(t - \xi )}^\nu }} \psi (\xi )d\xi = \sum\limits_{j = 0}^{{M_n}} \psi \left( {{\xi _{n,j}}} \right)\hat \omega _{n,j}^\nu (t),\quad t \in {I_n}$ (11)

 $\hat \omega _{n,j}^\nu (t) = \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\nu }} l_{{{n}},j}^t(\xi ){\rm{d}}\xi$ (12)

3 hp型谱配置法

 $u(t) = \frac{1}{{\Gamma (1 + \alpha )}}\int_0^t {{{(t - s)}^\alpha }} f(s,u(s)){\rm{d}}s$ (13)

${u^n}(t)$ 是第 $n$ 个区间上式(13)的解，即

 ${u^n}(t): = u(t),\quad t \in {I_n},\quad 1 \leqslant n \leqslant N$

 $\begin{split}& {u^n}(t) = \frac{1}{{\Gamma (1 + \alpha )}}\Bigg( \sum\limits_{k = 1}^{n - 1} {\int_{{I_k}} {{{(t - s)}^\alpha }} } f\left( {s,{u^k}(s)} \right){\rm{d}}s +\Bigg.\\&\quad \Bigg. \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} f\left( {\xi ,{u^n}(\xi )} \right){\rm{d}}\xi \Bigg) \end{split}$ (14)

 $\begin{split}& {U^n}(t) =\\& \quad{{\cal I}_{t,{M_n}}}\left[ {\frac{1}{{\Gamma (\alpha + 1)}}\left( {\sum\limits_{k = 1}^{n - 1} {\int_{{I_k}} {{{(t - s)}^\alpha }} } {{\cal I}_{s,{M_k}}}f\left( {s,{U^k}(s)} \right){\rm{d}}s} \right.} \right. +\\& \quad \left. {\left. { \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} {\cal I}_{\xi ,{M_n}}^tf\left( {\xi ,{U^n}(\xi )} \right){\rm{d}}\xi } \right)} \right]\\[-12pt] \end{split}$ (15)

 $U(t) = {U^k}(t),\quad t \in {I_k},\quad 1 \leqslant k \leqslant N$

 ${U^n}(t) = \sum\limits_{p = 0}^{{M_n}} {u_p^n} {L_{n,p}}(t)$ (16)

 $\begin{split}& {\mathcal{I}_{t,{M_n}}}\int_{{I_k}} {{{(t - s)}^\alpha }} {\mathcal{I}_{s,{M_k}}}f\left( {s,{U^k}(s)} \right){\rm{d}}s = \\& \quad {\mathcal{I}_{t,{M_n}}}\sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\tilde \omega _{k,{{\text{p}}'}}^\alpha (t) = \\& \quad \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right){\mathcal{I}_{t,{M_n}}}\tilde \omega _{k,{{\text{p}}'}}^\alpha (t) = \\& \quad \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\sum\limits_{p = 0}^{{M_n}} {\omega _{k,{p'},p}^{n,\alpha }} {L_{n,p}}(t) = \\& \quad \sum\limits_{p = 0}^{{M_n}} {a_{k,p}^n} {L_{n,p}}(t) \\[-8pt] \end{split}$ (17)

 $\begin{split}& a_{k,p}^n = \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\omega _{k,{p'},p}^{n,\alpha } , \omega _{k,{p^\prime },p}^{n,\alpha } = \\& \qquad \frac{{2p + 1}}{2}\sum\limits_{j = 0}^{{M_k}} {\tilde \omega _{k,{{\text{p}}'}}^\alpha } \left( {{t_{n,j}}} \right){L_{n,p}}\left( {{t_{n,j}}} \right){\omega _{n,j}} \end{split}$ (18)

 $\begin{split}& {\mathcal{I}_{t,{M_n}}}\left( {\int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} \mathcal{I}_{\xi ,{M_n}}^tf\left( {\xi ,{U^n}(\xi )} \right){\rm{d}}\xi } \right) = \\& \qquad {\mathcal{I}_{t,{M_n}}}\sum\limits_{j = 0}^{{M_n}} f \left( {{\xi _{n,j}},{U^n}\left( {{\xi _{n,j}}} \right)} \right)\hat \omega _{n,j}^\alpha (t) =\\& \qquad \sum\limits_{p = 0}^{{M_n}} {b_p^n} {L_{n,p}}(t) \\[-8pt] \end{split}$ (19)

 $\begin{split} b_p^n =&\frac{{2p + 1}}{2}\cdot\\& \sum\limits_{i,j = 0}^{{M_n}} f \left( {{\xi _{n,j}},{U^n}\left( {{\xi _{n,j}}} \right)} \right)\hat \omega _{n,j}^\alpha \left( {{t_{n,i}}} \right){L_{n,p}}\left( {{t_{n,i}}} \right){\omega _{n,i}} \end{split}$ (20)

 $\begin{split}& \sum\limits_{p = 0}^{{M_n}} {u_p^n} {L_{n,p}}(t) = \frac{1}{{\Gamma (\alpha + 1)}}\sum\limits_{p = 0}^{{M_n}} {\sum\limits_{k = 1}^{n - 1} {a_{k,p}^n} } {L_{n,p}}(t) +\\& \qquad \quad \frac{1}{{\Gamma (\alpha + 1)}}\sum\limits_{p = 0}^{{M_n}} {b_p^n} {L_{n,p}}(t) \end{split}$ (21)

 $u_p^n = \frac{1}{{\Gamma (\alpha + 1)}}\left( {\sum\limits_{k = 1}^{n - 1} {a_{k,p}^n} + b_p^n} \right)$ (22)

4 数值结果

 $\begin{gathered} {E_{{L^2}(0,T)}}: = {\left( {\sum\limits_{k = 1}^N {\frac{{{h_k}}}{2}} \sum\limits_{j = 0}^{{M_k}} {{{\left( {u\left( {{t_{k,j}}} \right) - U\left( {{t_{k,j}}} \right)} \right)}^2}} {\omega _{k,j}}} \right)^{\frac{1}{2}}} \\ {E_{{L^\infty }(0,T)}}: = \mathop {\max }\limits_{_{1 \leqslant k \leqslant N}} \left( {\mathop {\max }\limits_{_{0 \leqslant j \leqslant {M_k}}} \left| {u\left( {{t_{k,j}}} \right) - U\left( {{t_{k,j}}} \right)} \right|} \right) \\ \end{gathered}$

 $\left\{ {\begin{array}{*{20}{l}} {{{\rm{D}}^{1.5}}y(t) + y(t) = {t^\alpha } + \dfrac{{\Gamma (\alpha + 1)}}{{\Gamma (\alpha - 1/2)}}{t^{\alpha - 3/2}}} \\ {u(0) = 0,\quad {u^\prime }(0) = 0} \end{array}} \right.$ (23)

 图 1 例1在 ${L^2}$ 范数和 ${L^\infty }$ 范数下的收敛阶 Fig. 1 The convergence order under L2-norm and ${{\boldsymbol{L}}^\infty }$ -norm for example 1

 [1] HILFER R. Applications of fractional calculus in physics[M]. Singapore: World Scientific, 2000. [2] PODLUBNY I. Fractional differential equations[M]. San Diego: Academic Press, 1999. [3] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier, 2006. [4] CARRERAS B A, LYNCH V E, ZASLAVSKY G M. Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model[J]. Physics of Plasmas, 2001, 8(12): 5096-5103. DOI:10.1063/1.1416180 [5] TARASOV V E. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media[M]. New York: Springer, 2011. [6] MAINARDI F. Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models[M]. London: Imperial College Press, 2010. [7] SUGIMOTO N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves[J]. Journal of Fluid Mechanics, 1991, 225: 631-653. DOI:10.1017/S0022112091002203 [8] SRIVASTAVA V, RAI K N. A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues[J]. Mathematical and Computer Modelling, 2010, 51(5/6): 616-624. [9] GUO Y L, WANG Z Q. An hp-version Chebyshev collocation method for nonlinear fractional differential equations[J]. Applied Numerical Mathematics, 2020, 158: 194-211. DOI:10.1016/j.apnum.2020.08.003 [10] BRUNNER H. Collocation methods for Volterra integral and related functional differential equations[M]. New York: Cambridge University Press, 2004. [11] WANG C L, WANG Z Q, JIA H L. An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels[J]. Journal of Scientific Computing, 2017, 72(2): 647-678. DOI:10.1007/s10915-017-0373-3 [12] ZAKY M A, AMEEN I G. On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems[J]. Computational and Applied Mathematics, 2019, 38(3): 144. DOI:10.1007/s40314-019-0922-5


text:
# zbMATH — the first resource for mathematics

An introduction to the theory of point processes. Vol. II: General theory and structure. 2nd revised and extended ed. (English) Zbl 1159.60003
Probability and Its Applications. New York, NY: Springer (ISBN 978-0-387-21337-8/hbk; 978-0-387-49835-5/ebook). xvii, 573 p. (2008).
In the presented research monograph the authors have made a major reshaping of their work in their first edition of 1988 [An introduction to the theory of point processes. Springer (1988; Zbl 0657.60069)] and now present An Introduction to the Theory of Point Processes in two volumes with subtitles Volume I: Elementary Theory and Methods and Volume II: General Theory and Structure.
Volume I [New York, NY: Springer (2003; Zbl 1026.60061)] consists of 8 chapters, including standard material: Poisson process, stationary point processes on the line, renewal processes, finite point processes, Cox, cluster and marked point processes, conditional intensities and likelihood and second order properties of stationary point processes. In the three appendices basic concepts of topology and measure theory, measures of metric spaces and conditional expectations, stopping times and martingales are reviewed, where the latter underlies the discussion of predictability and conditional intensity of Volume II, Chapter 14.
In the second volume they set out a general unified framework for the theory of point processes by starting from their interpretation as random measures. The material represents a reorganized version of those parts of Chapter 6–14 of the first edition, not already covered in Volume I, continued with the more theoretical topics of the first edition, including limit theorems, ergodic theory, Palm theory and evolutionary behavior via martingales and conditional intensity and the structure of spatial point processes.
Readers of the second volume should be more familiar with aspects of measure theory and topology than those of the first volume. The second volume starts with Chapter 9: Basic theory of random measures and point processes. Then (Chapter 10) special classes of processes are discussed like infinite divisible point processes, point processes defined by Markov chains and marked point processes. Chapter 11 deals with convergence concepts and limit theorems, including superpositions, thinning and random transitions. Stationary point processes and random measures is the topic of Chapter 12: ergodic theorems, infinite divisible point processes, asymptotic stationarity and convergence to equilibrium, long range dependence, self similarity. The Palm theory is treated extensively in Chapter 13. In Chapter 14 a general introduction to the notions of compensators and point process martingales and their links to the Doob-Meyer theorems are given. Further extensions to random measures and marked point processes are given. Then conditional intensities are treated as a Radon-Nikodym derivative of Campbell measures are re-introduced. Finally various applications are given (likelihood and time change theorems, martingale-type central limit theorem, entropy). Spatial point processes (in $$\mathbb R^2$$ and $$\mathbb R^3$$) are dealt in the last Chapter 15: In the first four sections the authors review mainly descriptive poverties, distinguishing between distance and directional properties of spatial point patterns, starting from finite models, moving to the moment properties of line processes, and then revisiting space-time models. In the final section is given an introduction to modeling centered around the concept of Papangelou intensity. Some background and motivation from statistical and physical settings is provided and then an introduction to the more mathematical theory is given (modified Campbell measures, Papangelou kernels, Papangelou intensity, and exvisibility).
The comprehensive unified exposition of known and substantial new material given in this second volume will make this research monograph to a standard text book on advanced subjects of point processes. The many examples illustrate the mathematical results and the various applicability of point processes in various fields.

##### MSC:
 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures
Full Text:


text:
• UMA SHANKAR

Articles written in Journal of Earth System Science

• Estimation of gas hydrate saturation using isotropic and anisotropic modelling in the Mahanadi basin

A base of gas hydrate stability zone was established after coring and drilling under the National Gas Hydrate Program (NGHP) Expedition-01 in the Mahanadi basin. At two sites, logging-while-drilling log data, and, at one site, wireline log data, were acquired during the NGHP Expedition-01. Gas hydrate reservoirs modelling can be performed in two different ways. One way is isotropic (load bearing) and, on the other hand, anisotropic media (fracture filling with gas hydrate). Here, we have performed anisotropic modelling and estimated gas hydrate saturation using P-wave velocity, assuming an incidence angle of 75$^{\circ}$ represents the vertical fracture. The estimated gas hydrate saturation at sites NGHP-01-08 and NGHP-01-09, assuming anisotropic media, reduces the estimate by half compared to the saturation estimation by assuming isotropic media. The saturation at site NGHP-01-19 estimated from the isotropic and anisotropic P-wave velocity models are more or less similar except in the zone (175–210 m) just above the bottom simulating reflector depth, and this zone shows similar reduction in saturation as estimated at sites NGHP-01-08 and NGHP-01-09. Observations show that average gas hydrate saturations are relatively low (up to 5% of the pore space). The saturation of a gas hydrate estimated from an isotropic P-wave model varies from 5% to 20%. However, the saturation estimated from the anisotropic P-wave model shows a variation up to 10% of the pore spaces at three sites.

• Evaluation of site-speciBc characteristics using microtremor measurements in the Gorakhpur city of Uttar Pradesh, India

The microtremor measurements are carried out in and around the Gorakhpur city (Uttar Pradesh), India, overlain by alluvium at about 150 sites to understand the local site conditions. Horizontal-to-vertical spectral ratio (HVSR) conBrms that the majority of sites have a predominant frequency of ${\sim}$0.45 Hz, which may suggest the prevalence of thick soft sediments in the area. Conspicuously, a number of multiple peaks in HVSR curves at few sites may reflect the presence of different interfaces with significant impedance contrasts. Maximum amplification is observed of 4.0–5.3 to the NW–SE of the city, whilst few sites in the city are found to be associated with different values of peak amplification factor that varied between 2.0 and 4.0. It is also observed that the ground vulnerability index ($K_g$) in Gorakhpur city has values higher than 10.0 at most of the sites. Assimilation of 1-D velocity model for the city clearly shows that low shear wave velocity (${\sim}$300 m/s) down to the depth of ${\sim}$35 m, suggesting thick piles of sediments that may correspond to Cuvial river system in the area, whilst the peak frequency of about 0.45 Hz may correspond to the Quaternary–Tertiary sediment boundary that may exist at deeper layers (${\sim}$1000 m). The inference of this study may be used as inputs for earthquake risk management by reducing the severity of earthquake shaking through design of earthquake risk resilient structures.

• # Journal of Earth System Science

Volume 130, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Skip to main content

# Chapter11LaTeX Styles

Print and PDF output is created when PreTeXt outputs a file, which can subsequently be easily converted to PDF with engines like pdflatex and xelatex. There are many ways to influence the style (look, appearance) of this output, enhancing the content, but without changing it. This chapter is directed at publishers who desire to create an attractive and consistent design for their books or articles.

Please read this chapter thorougly once before embarking. Certain important points are made in certain contexts, yet are universally applicable. If you cherry-pick, you will miss them. Note also that some simple changes, like font size, are described in Chapter 5.


text:
PREPRINT
889834C5-39E7-4775-8432-042CA1D6FB10

OMC-2 FIR 4 under the microscope: Shocks, filaments, and a highly collimated jet at 100 au scales

L. Chahine, A. López-Sepulcre, L. Podio, C. Codella, R. Neri, S. Mercimek, M. De Simone, P. Caselli, C. Ceccarelli, M. Bouvier, N. Sakai, F. Fontani, S. Yamamoto, F. O. Alves, V. Lattanzi, L. Evans, C. Favre

Submitted on 8 September 2022

Abstract

Star-forming molecular clouds are characterised by the ubiquity of intertwined filaments. The filaments have been observed in both high- and low-mass star-forming regions, and are thought to split into collections of sonic fibres. The locations where filaments converge are termed hubs, and these are associated with the young stellar clusters. However, the observations of filamentary structures within hubs at distances require a high angular resolution that limits the number of such studies conducted so far. The integral shaped filament of the Orion A molecular cloud is noted for harbouring several hubs within which no filamentary structures have been observed so far. The goal of our study is to investigate the nature of the filamentary structures within one of these hubs, which is the chemically rich hub OMC-2 FIR 4, and to analyse their emission with high density and shock tracers. We observed the OMC-2 FIR 4 proto-cluster using Band 6 of the ALMA in Cycle 4 with an angular resolution of ~0.26"(100 au). We analysed the spatial distribution of dust, the shock tracer SiO, and dense gas tracers (i.e., CH${}_{3}$OH, CS, and H${}^{13}$CN). We also studied gas kinematics using SiO and CH3OH maps. Our observations for the first time reveal interwoven filamentary structures within OMC-2 FIR 4 that are probed by several tracers. Each filamentary structure is characterised by a distinct velocity as seen from the emission peak of CH${}_{3}$OH lines. They also show transonic and supersonic motions. SiO is associated with filaments and also with multiple bow-shock features. In addition, for the first time, we reveal a highly collimated SiO jet (~1${}^{\circ }$) with a projected length of ~5200 au from the embedded protostar VLA15. Our study shows that multi-scale observations of these regions are crucial for understanding the accretion processes and flow of material that shapes star formation.

Preprint

Subjects: Astrophysics - Astrophysics of Galaxies; Astrophysics - Solar and Stellar Astrophysics


text:
Distinguished Lectures in Natural Sciences Distinguished Lecture in Celebration of the 10th Anniversary of INS——Orthogonality Free Approaches for Optimization Problems on Stiefel Manifold
Yaxiang Yuan, Chinese Academy of Sciences, China
2020.11.17, 10:00-11:00

Distinguished Lectures in Natural Sciences Distinguished Lecture in Celebration of the 10th Anniversary of INS——On Inhibition of Thermal Convection Instability by a Magnetic Field under Zero Resistivity
Song Jiang, Institute of Applied Physics and Computational Mathematics, China
2020.11.15, 10:00-11:30

"AI + Math" Colloquia Statistical Efficiency and Optimization of Deep Learning from the View Point of Non-Convexity
Taiji Suzuki, The University of Tokyo, Japan
2020.11.12, 10:00-11:00

Gaussian Fluctuations from Nonlinear SPDEs
Yu Gu, Carnegie Mellon University
2020.11.12, 09:00-10:00

Distinguished Lectures in Natural Sciences Distinguished Lecture in Celebration of the 10th Anniversary of INS—— Optimal Transport: From Nature to Machine Learning
Alessio Figalli, ETH-Zürich, Switzerland
2020.11.11, 16:30-17:30

ZY-INS Colloquia 植物激素茉莉素和独脚金内酯受体感知机制

2020.11.11, 16:00-17:30

Distinguished Lectures in Natural Sciences Distinguished Lecture in Celebration of the 10th Anniversary of INS—— Dynamic Blocking Problems for a Model of Fire Confinement
Alberto Bressan, Penn State University, USA
2020.11.10, 10:00-11:00

Data Science Seminars Fourier Neural Operator for Parametric Partial Differential Equations
Zongyi Li, The California Institute of Technology, USA
2020.11.09, 10:00-11:00

"AI + Math" Colloquia An $O(S^R)$-Resolution ODE Framework for Discrete-Time Algorithms and Applications to Minimax Problems
Haihao Lu, The University of Chicago Booth School of Business
2020.11.05, 10:00-11:00

ZY-INS Colloquia 化学合成酵母基因组

2020.11.04, 16:00-17:30

The Forum on Scientific and Engineering Computation Normalized Gradient Flow for Computing Ground States of Bose-Einstein Condensates
Yongyong Cai, Beijing Normal University
2020.11.04, 14:00-15:00

Distinguished Lectures in Natural Sciences Distinguished Lecture in Celebration of the 10th Anniversary of INS—— Numerical Methods and Analysis for Phase Field Models
Tao Tang, Beijing Normal University - Hong Kong Baptist University United International College
2020.11.03, 15:30-16:30


text:
International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2010). Vol. E, ch. 5.2, pp. 413-414   | 1 | 2 |

Section 5.2.4.6.2. Inclined scanning, auxiliary tables

V. Kopskýa* and D. B. Litvinb

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:  kopsky@fzu.cz

5.2.4.6.2. Inclined scanning, auxiliary tables

| top | pdf |

Orientation orbit (mn0): Orientations of this orbit contain one of the three main cubic axes and are divided into three subsets corresponding to these axes for which the bases are separated by horizontal lines in the tables of orientation orbits and auxiliary bases. The orbit contains six orientations in groups of the Laue class () and 12 orientations in groups of the Laue class (). The orbit turns into a special orbit with fixed parameters for the special values , in groups of both the Laue classes () and (). The scanning changes from monoclinic/inclined to orthorhombic in the Laue class (), to tetragonal in the Laue class (). The symmetry of the orientation also increases to orthorhombic for special values , in groups of the Laue class ().

The choice of bases for the three subsets is the same as in orthorhombic groups, where the orientations of subsets are separated into three different orbits and the auxiliary bases are expressed in terms of vectors of the conventional cubic basis for the centring types P and I. For the centring type F, the Miller indices differ in the original and auxiliary basis. In this case, we express the Miller indices with reference to the original basis as and relate them to Miller indices with reference to the auxiliary bases. These relations are the same as in the case of F-centring in orthorhombic groups, see relations (5.2.4.2) and (5.2.4.3).

Orientation orbit (hhl): The orbit contains 12 orientations which divide into three subsets corresponding to the three main cubic axes. In each of the subsets, one of the vectors of the conventional cubic basis is chosen as the vector of the auxiliary basis. The orientations of the subsets are separated by horizontal lines across the table. The first subset corresponds to the vector c of the cubic basis and the orientations in this subset are the same as in the orbit for tetragonal groups of the Laue class (). The orientations within each subset are further divided into two pairs of orientations to which correspond two different unique axes of the monoclinic scanning group. These subsets are again separated by horizontal lines across the last two columns. For the centring types P and I and for the first subset of orientations, the description of orientations and bases coincides with the description of the orbit in tetragonal groups of the Laue class () and centring types P and I, including the choice of auxiliary and conventional bases of scanning groups and relations between Miller indices h, l and m, n. For the other subsets of orientations, the data in tables are obtained by the cyclic permutation of vectors a, b and c.

For the centring type F, orientation , we choose the auxiliary basis of the scanning group with the unique axis vectors , and , while for the orientation , we choose , and . The bases for the remaining orientations are again obtained by the cyclic permutation of vectors of the conventional cubic basis.

Transformation of Miller indices: The straight line in which a plane intersects with the plane has the direction of a vector or of the vector . As these two vectors can differ only by a numerical factor, the pair must be proportional to the pair and we obtain the relationsrecorded at the bottom row of the orientation-orbit table for the centring type F.

For the special values , , this orbit turns into an orbit with fixed parameters and an orthorhombic scanning group.


text:
In geometry, any polygon with seven sides and seven angles and seven vertices is called as Heptagon. The sum of angles in a Heptagon is 900°.

In real life, Heptagon shape can be seen in British fifty pence coin which is an equilaterally curved heptagon. The dome in the former Melbourne Magistrates Court in Melbourne, Australia, is a good architectural example for a regular heptagon.
A simple tablet organizer is also a good example for regular Heptagon.

## Regular Heptagon

A Heptagon having all sides of equal length and equal angles is called regular Heptagon.

The figure above shows a regular Heptagon. Each of its internal angles is equal to 128.571° (The formula to find an interior angle of an n-sided polygon is given by $\frac{(n-2)\pi}{n}$). In a polygon, the central angle is the angle made at the centre of the polygon by any two adjacent vertices and is given by the formula $\frac{360^o}{n}$. Thus the central angle measure of a regular Heptagon is 51.429°.
The number of distinct diagonals that are possible to draw from all vertices in a polygon is given in general by the formula $\frac{n(n–3)}{2}$. Thus the number of diagonals that can be drawn in a heptagon are 14.

The number of triangles that can be drawn by drawing the diagonals from a given vertex in general is given by n–2 and so we can draw 5 triangles in a regular Heptagon.

## Irregular Heptagon

An Irregular Heptagon has sides of different lengths.

Irregular polygons are not symmetric and so are not considered as having a centre. Thus have no central angle.

## Convex Heptagon

Geometrically, in any polygon if any line segment drawn between any two interior points lies entirely inside the polygon, then it is called a convex polygon. Similarly convex Heptagon is a 2D geometric figure in which any line segment drawn between any two interior points lies inside the Heptagon. Any regular Heptagon is a Convex Heptagon.

In a convex polygon with n sides, the sum of interior angles is given by the following equation: S = (n -2) × 180°.

So, in a convex Heptagon, the sum of angles is given by (7 -2) × 180° = 5 × 180° = 900°.

In a convex Heptagon, the sum of the degree measures of the exterior angles, one at each vertex, is 360°.

## Concave Heptagon

A concave polygon has one or more vertices which point towards the interior and looks like “pushed in”. A concave Heptagon is defined as a polygon with one or more interior angles greater than 180°. Also a Heptagon that is not convex is a concave Heptagon.

In this 3-pointed heptagon, we can see that the interior angles at B, D, E, G in figure 1 and B, C in the figure 2 are greater than 180°. On observing these figures we can see that in a concave polygon some diagonals will lie outside the polygon, like diagonals AC, CF and FA in figure 1 and AD, BD in figure 2.

## Area of a Heptagon Formula

The area of a Heptagon can be found in different ways. We can use various formulas depending on the data when the heptagon is regular.
Case (i) When the side length, s, is given the area of any n sided polygon is given by,
Area = $\frac{s^{2}n}{(4tan (\frac{180}{n}))}$, where the tangent function is always calculated in degrees.
Putting n = 7, we get the area of regular heptagon with side length s as, Area = 3.633 s2.

Case (ii) When an Apothem ‘a’(Apothem of a regular polygon is a perpendicular line segment drawn from the centre to the midpoint of
one of its sides) is given then the area of a polygon is given by :
Area = $a^{2}ntan(\frac{180}{n})$ where n is the number of sides of a polygon.
Putting n = 7, we get the area of regular heptagon with Apothem ‘a’, Area = 3.371 a2.

Case(iii) When a Radius ‘r’(Radius of a regular polygon is a distance from the centre to any of the vertex) is given, then the area of a polygon is given by
Area =$(\frac{1}{2}) r^{2}nSin(\frac{360}{n})$ where n is the number of sides of a polygon.
Putting n = 7, we get the area of regular heptagon with radius r, Area = 2.7364 r2.
If p is the Perimeter and A is the apothem, the area is also given by Area = ½ P A

In case of an irregular Heptagon, area can be found by breaking the heptagon into 7 triangles and finding the area of each triangle and adding them up.

## Interior Angles of a Heptagon

In a polygon, the interior angles always add up to a constant value and depends only on the number of sides no matter if it regular or irregular, convex or concave, or what size and shape it is. Each interior angle of any regular Heptagon is equal to 128.57° (An interior angle of any polygon is given by $\frac{(180n–360)}{n}$. Here n=7 for a heptagon).

## Exterior Angles of a Heptagon

An exterior angle of a Heptagon is formed by extending any one of its sides. The non-straight angle adjacent to an interior angle is the exterior angle. Each exterior angle of any regular Heptagon is 51.43° (In a polygon, an exterior angle forms a linear pair with its interior angle, so in general the exterior angle is given by 180-interior angle). In a convex polygon, the sum of the exterior angles is 360°.

## Heptagon Examples

Now let us learn how to find area of heptagon with following examples,

### Solved Examples

Question 1: Find the area of Heptagon to nearest tenth with a side length of 16 cms.
Solution:

The length of each side of a heptagon, s = 16 cms
Area of Heptagon = $\frac{s^{2}n}{4tan(\frac{180}{n})}$ = $\frac{s^{2}7}{4tan(\frac{180}{7})}$ =3.6339 s2
= 3.6339 X 16 X 16
= 930.28
= 930.28 sq.cms.
Thus the area of a heptagon with side length as 16 cms is 930.3 sq.cms.

Question 2: Find the area of a regular Heptagon with a radius of 10cms to the nearest hundredth.
Solution:

The radius of a heptagon,r = 10cms
Area of heptagon  =$(\frac{1}{2}) r^{2}nSin(\frac{360}{n})$ = $(\frac{1}{2}) r^{2}7Sin(\frac{360}{7})$  = 2.7364 r2
= 2.7364  x (10)2.
= 273.64 sq.cm
Thus the area of a heptagon with radius of 10 cms is 273.64 sq.cm.

### Octagon

 A Polygon A Concave Polygon is A Convex Polygon A Regular Polygon All Types of Polygons Area of any Polygon Congruent Polygon Frequency Polygon Pentagon Polygons Similar Polygon What is a Irregular Polygon The Octagon
 Area of Polygon Calculator Octagon Calculator Online Calculate Hexagon


text:
# zbMATH — the first resource for mathematics

Local error estimates for a finite element method for hyperbolic and convection-diffusion equations. (English) Zbl 0757.65104
Local error estimates are derived for a finite element method for two- dimensional hyperbolic and convection dominated equations. An approach for establishing local error estimates for the continuous method of W. H. Reed and T. R. Hill [Triangular mesh methods for the neutron transport equation. Los Alamos Sci. Laboratory Rep. LA-UR-73-479 (1973)] is used that is also applicable to the discontinuous Galerkin method, thus underscoring the close interrelationships between the two methods.
A continuous piecewise polynomial approximation of degree $$\geq 2$$ over a triangulation of $$\Omega\subset\mathbb{R}^ 2$$ is generated. It is shown that crosswind propagation of the numerical solution is limited to a distance $$O(\sqrt{h}\log h^{-1})$$, where $$h$$ is the mesh size. A rather simple test function depending only on the crosswind variable $$t$$ is applied for the analysis, what leads directly to stability results expressed in terms of this variable.
Reviewer: K.Zlateva (Russe)

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems
Full Text:


text:
kuCAu

2021-06-12

Find each unknown for problem. Check your Work. 46+n=8146+n=81

grbavit

Expert

"Determine the value of n in the given equation.
46+n=81
46+n-46=81-46
n=35
Check by substituting the value of n to the original equation.
46+n=81
46+35=81
81=81

Do you have a similar question?