text:
# latex one way gloves as seen on tv pdf download windows 10

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text:
Kattis

# Joint Attack

General Torstein has sent the $x$-coordinate for the next joint attack and is expecting you to promptly follow his orders in order to avoid impeding doom. Unfortunately Torstein hates numbers with more than 2 digits and loves horizontal line segments, and has therefore sent the coordinate as a continued fraction, i.e.

$x = x_0 + {1 \over {x_1 + {1 \over {\ldots }}}}$

Your rocket launcher only accepts coordinates as reduced fractions, so you need to quickly compute the correct numbers to feed it in order to commence the attack. Hurry! Failure may have dire consequences!

## Input

The first line of output is one integer $n$ ($1 \leq n < 40$), the number of coefficients in the continued fraction, followed by a line with $n$ integers ($1\leq x_ i < 100$) the coefficients of $x$.

## Output

The coordinate $x$ as a reduced fraction. It is guaranteed that the numerator and denominator are both less than $10^{18}$.

Sample Input 1 Sample Output 1
2
2 3

7/3

format: plain


text:
# student t pdf

Here's a code for student t pdf.

var("s n t")
assume(n, 'integer')
assume(n>0)
tdist=integral(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,infinity)
tdist

And it says, TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(n+1)/2>0)' before integral or limit evaluation, for example): Is (n+1)/2 an integer?

The funny thing is when I add n=5 or n=3, there's no error!

var("s n t")
assume(n, 'integer')
n=3
tdist=integral(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,infinity)
tdist

EDIT : Actually, I tried this on MATHLAB to fail.

   My last hope would be Mathematica? wolframalpha.com says it's timed out!

edit retag close merge delete

I think that once you say n=3, it's sort of substituted in. If you input sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)) you get 1/2*sqrt(1/3)*s*e^(-1/6*s*t^2 - 1/2*s)/pi, no n for Maxima to worry about.

( 2012-06-07 09:06:45 -0500 )edit

The problem is I want the integration for general n(positive integer), not for some specific n=3 or 5!

( 2012-06-07 20:06:19 -0500 )edit

I understand, I was just pointing out why there was no error.

( 2012-06-08 03:32:46 -0500 )edit

Sort by » oldest newest most voted

I don't think we're going to be able to fix this very easily at all. The problem is Maxima's framework, which there has been much discussion about and which is not going to change.

(%i5) assume(n>0);
(%o5)                               [n > 0]
(%i6) display2d:false;

(%o6) false
(%i7) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

no
;
(%o7) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
/(sqrt(n)*gamma(n/2)*sqrt(pi))
(%i8) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

yes;
Is n an integer?

yes;
(%o8) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
/(sqrt(n)*gamma(n/2)*sqrt(pi))

Don't even ask about why it asks if n is an integer when it knows (n+1)/2 is an integer; Maxima itself acknowledges that its assumption framework is weak, and of course every open source project has limited resources to fix big architectural issues like that. I don't think it would be trivial for them to fix that.

As for Sage, assuming n is an integer will help a little, but only so much.

(%i9) declare(n,integer);

(%o9) done
(%i10) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

yes;
(%o10) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
/(sqrt(n)*gamma(n/2)*sqrt(pi))

(%i11) declare((n+1)/2,integer);

declare: improper argument: (1+n)/2
-- an error. To debug this try: debugmode(true);

So we wouldn't even be able to try that, even if we wanted to; Maxima only lets "atoms" be declared to be something (i.e., variables), though we can assume positive or negative about them. And even though the answers are the same, many have pointed out that going through a decision tree of this kind and trying to check that can be arbitrarily computationally expensive, if it's even possible.

I'm sorry, but this is something at the core of how Maxima does business. On the plus side, the answer works nicely enough that one gets from Maxima, and you could use it in Sage.

more

What's about substituting n to make it an odd integer?

var("s k n t")
assume(k, 'integer')
assume(k>0)
term= sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2))
tdist=integral(term.substitute(n=2*k+1),s,0,infinity)
tdist

more

Yes, it works! But what about when n is an even integer? It also has solution when n is an even number, which is of course p.d.f of t-distribution!

( 2012-06-07 01:08:11 -0500 )edit

On the off chance you just need the Student t distribution:

See this Sage documentation and search for t-distribution.

sage: nu = 1
sage: T = RealDistribution('t', nu)
sage: T.get_random_element() # random
-0.994514581164
sage: T.distribution_function(0)
0.318309886184
sage: T.cum_distribution_function(1)
0.75
sage: T.cum_distribution_function_inv(.5)
0.0

The word "student's" was removed in the change associated with Trac 9080 which added F-distribution. It will be a lot more work to have the distributions well integrated with the rest of Sage, but we could put that word back in - I suppose it was my fault, since I reviewed the ticket.

Also, see this sagenb worksheet for someone plotting the actual pdf.

more

Does wrapping the expression with a python function fit your needs?

var("s n t")
term= sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2))

def tdist(k):
return integral(term.substitute(n=k),s,0,infinity)

tdist(4)

more

Thank you for the answer. But the full solution should be f(t,n). we could consider f(t,n) from f(t,1), f(t,2), f(t,3)... but it's not the exact solution!!!

( 2012-06-07 20:01:49 -0500 )edit
var("s t")
def f(t,n):
return integral(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,infinity)

print [f(t,k) for k in range(1,10)]

more

This is a modification of the solution by ndomes

( 2012-06-07 21:34:02 -0500 )edit


text:
## Sapling homework Question 9

$E=hv$

Jaclyn Dang 3B
Posts: 67
Joined: Wed Sep 30, 2020 10:02 pm

### Sapling homework Question 9

Hi, I'm not sure where to start on this problem, if anybody can give me the steps along with a quick reasoning/explanation, it'd be highly appreciated!

As you may well know, placing metal objects inside a microwave oven can generate sparks. Two of your friends are arguing over the cause of the sparking, with one stating that the microwaves "herd" electrons into "pointy" areas of the metal object, from which the electrons jump from one part of the object to another. The other friend says that the sparks are caused by the photoelectric effect. Prove or disprove the latter idea using basic physics.

Suppose the typical work function of the metal is roughly 4.260×10−19 J. Calculate the maximum wavelength in angstroms of the radiation that will eject electrons from the metal.

Wavelength=

Thank you!

darchen3G
Posts: 70
Joined: Wed Sep 30, 2020 9:39 pm

### Re: Sapling homework Question 9

E = hc/lambda

In your case, 4.26*10^-19 = hc/lambda then solve for lambda.

For the second part of the question, use E=hf to see if it has enough energy.

Jenny Lee 2L
Posts: 62
Joined: Thu Oct 08, 2020 12:15 am

### Re: Sapling homework Question 9

Hi Jaclyn!

Here is the formula that was used for the photoelectric effect: (E per photon) = (Work Function) + (KE)

The work function is the bare minimum you need to eject the electron from the metal surface with 0 kinetic energy. By maximum wavelength, it's asking us for the wavelength needed to match the energy of the work function (because E = Work Function if KE=0).

So we do:
1. Work function = E = 4.620x 10^-19 J
2. E= hv, find v (frequency)
3. Use c= (lambda)(v) to find the wavelength

Hope this helps!

Brianna Martilla 1C
Posts: 67
Joined: Wed Sep 30, 2020 9:58 pm

### Re: Sapling homework Question 9

Whenever I am given a problem I always see what is given and what I am being asked to find. We are given the value for energy and we are asked to find the wavelength. We can use E=hc/lambda, rewrite that as lambda=hc/E, then solve for lambda because h and c are constants. Just make sure to convert your answer to angstroms (1 angstrom=10^-10 m) because the answer is in meters.


text:
There are N gas stations along a circular route, where the amount of gas at station i isgas[i].

You have a car with an unlimited gas tank and it costscost[i]of gas to travel from station i to its next station (i+1). You begin the journey with an empty tank at one of the gas stations.

Return the starting gas station’s index if you can travel around the circuit once, otherwise return -1.

Note:
The solution is guaranteed to be unique.

total记录这个旅程的净油量变化，期间可以为负，当走完整个旅程后total是负的，就说明所有的油加起来走不完所有的路，即不可能走完。那么如果total是正的，则一定存在正确的方案。


text:
Browse Questions

# Which of the following can be removed by ion exchange method

$\begin{array}{1 1}(a)\;Chlorine\;ions\\(b)\;Sulphur\;ions\\(c)\;Flurine\;ions\\(d)\;Pottasium\;ions\end{array}$

Flurine ions.
Hence (c) is the correct answer.


text:
# Mathematical Sciences Research Institute

Home > Scientific > Colloquia & Seminars > Graduate Seminars

1. # Computing critical values of quadratic Dirichlet L-functions, with an eye toward their moments.

Location: MSRI: Simons Auditorium
Speakers: Matthew Alderson

Moments of L-functions has been a topic of intense research in recent years. Through the integration of random matrix theory and multiple Dirichlet series with traditional number theoretic arguments, methods for studying the moments of L-functions have been developed and, in turn, have lead to many well-posed conjectures for their behavior. In my talk, I will discuss the (integral) moments of quadratic DIrichlet L-functions evaluated at the critical point s=1/2. In particular, I will present formulas for computing the critical values for such L-functions and then compare the data for the corresponding moments to the (aforementioned) conjectured moments.
Created on Apr 15, 2011 09:13 AM PDT
2. # FBP-Informal Seminar

Location: MSRI: Baker Board Room
Speakers: TBA, Lihe Wang

Updated on Apr 01, 2011 03:03 AM PDT
3. # Moment Polynomials for the Riemann Zeta Function

Location: MSRI: Simons Auditorium
Speakers: Shuntaro Yamagishi

I will explain how we calculated the coefficients of moment
polynomials for the Riemann zeta function for k = 4,5.., 13
and numerically tested them against the moment polynomial conjecture.
Updated on Mar 31, 2011 04:37 AM PDT
4. # "Computing L-functions in SAGE"

Location: MSRI: Simons Auditorium
Speakers: Rishikesh

Created on Mar 23, 2011 08:26 AM PDT
5. # Elliptic curves of arbitrarily large rank (Over Function Fields)

Speakers: Kevin Wilson

Updated on Mar 14, 2011 08:03 AM PDT
6. # Empirical Evidence for an Arithmetic Analogue of Nevanlinna's Five Value Theorem

Location: MSRI: Baker Board Room
Speakers: James Weigandt

Updated on Feb 28, 2011 07:57 AM PST
7. # A problem related to the ABC conjecture

Speakers: Danial Kane (Harvard University)

Updated on Feb 13, 2011 03:01 AM PST
8. # Postdoctoral and Graduate Student Seminar TBA

Location: MSRI: Simons Auditorium

Pizza Lunch

Updated on May 13, 2013 11:01 PM PDT
9. # Gluing semiclassical resolvent estimates via propagation of singularities.

Location: MSRI: Baker Board Room
Speakers: Kiril Datchev

Pizza Lunch

Updated on Aug 14, 2014 02:45 PM PDT
10. # Lower bounds for the volume of the nodal sets

Location: MSRI: Simons Auditorium
Speakers: Hamid Hezari

Pizza Lunch

Updated on Dec 19, 2013 01:12 PM PST
11. # Non-intersecting Brownian Motions at a Tacnode: Soft and Hard Edge Case.

Location: MSRI: Simons Auditorium

Pizza Lunch

Updated on Nov 29, 2010 03:22 AM PST
12. # Harmonic maps into conic surfaces with cone angles less than $2\pi$

Updated on Nov 22, 2010 03:33 AM PST
13. # A tale of two tiling problems

Speakers: Benjamin Young

Updated on May 29, 2013 09:25 AM PDT
14. # Postdoctoral and Graduate Student Seminar TBA

Location: MSRI: Baker Board Room

Pizza Lunch

Updated on May 13, 2013 11:01 PM PDT
15. # Dihedral symmetry and the Razumov-Stroganov Ex-Conjecture

Location: MSRI: Baker Board Room

Pizza Lunch

Updated on Nov 05, 2010 07:12 AM PDT
16. # Geometric structures in the study of the geodesic ray transform

Location: MSRI: Simons Auditorium
Speakers: Juha-Matti Perkkio

Pizza Lunch

Updated on Oct 29, 2010 06:27 AM PDT
17. # "Edge scaling limits for non-Hermitian random matrices"

Location: MSRI: Simons Auditorium
Speakers: Martin Bender

Updated on Oct 29, 2010 07:57 AM PDT
18. # Postdoctoral and Graduate Student Seminar TBA

Location: MSRI: Simons Auditorium

Pizza Lunch

Updated on May 13, 2013 11:01 PM PDT
19. # From Oscillatory Integrals to a Cubic Random Matrix Model"

Speakers: Alfredo Deaño

Pizza Lunch

Updated on Oct 23, 2010 05:07 AM PDT
20. # Application of Riemann-Hilbert Problems in Modelling of Cavitating Flow

Location: MSRI: Simons Auditorium
Speakers: Anna Zemlyanova

Pizza Lunch

Updated on Oct 18, 2010 02:57 AM PDT
21. # Albrecht Durer, Magic Squares, and Unitary Matrix Integrals

Location: MSRI: Simons Auditorium
Speakers: Jonathan Novak

Pizza Lunch

Updated on Dec 04, 2013 12:45 PM PST
22. # Imaging Edges in Random Media

Location: MSRI: Simons Auditorium
Speakers: Fernando Guevara Vasquez

Pizza Lunch

Consider the problem of imaging a reflector (target) from recordings of the echoes resulting from probing the medium with waves emanating from an array of transducers (the array response matrix). We present an algorithm that selectively illuminates the edges or the interior of an extended target by choosing particular subspaces of the array response matrix. For a homogeneous background medium, we characterize these subspaces in terms of the singular functions of a space and wave number restricting operator, which are also called generalized prolate spheroidal wave functions. We discuss results indicating what can be expected from using this algorithm when the medium fluctuates around a constant background medium and the fluctuations can be modeled as a random field.
Updated on May 13, 2013 11:01 PM PDT
23. # Integrable Equations for Random Matrix Spectral Gap Probabilities

Location: MSRI: Simons Auditorium
Speakers: Igor Rumanov

Pizza Lunch

Connections are exposed between integrable equations for spectral gap probabilities of unitary invariant ensembles of random matrices (UE) derived by different --- Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) --- methods. Simple universal relations are obtained between these probabilities and their ratios on one side, and variables of the approach using resolvent kernels of Fredholm operators on the other side. A unified description of UE is developed in terms of universal, i.e. independent of the specific probability measure, PDEs for gap probabilities, using the correspondence of TW and ASvM variables. These considerations are based on the three-term recurrence for orthogonal polynomials (OP) and one-dimensional Toda lattice (or Toda-AKNS) integrable hierarchy whose flows are the continuous transformations between different OP bases. Similar connections exist for coupled UE. The gap probabilities for one-matrix Gaussian UE (GUE) or joint gap probabilities for coupled GUE satisfy various PDEs whose number grows with the number of spectral endpoints. With the above connections serving as a guide, minimal complete sets of independent lowest order PDEs for the GUE and for the largest eigenvalues of two-matrix coupled GUE are found.
Updated on May 13, 2013 11:01 PM PDT
24. # The Inverse Calderon Problem for Schrödinger Operator on Riemann Surfaces

Location: MSRI: Simons Auditorium
Speakers: Leo tzou

Pizza Lunch

We show that on a smooth compact Riemann surface with boundary (M0, g) the Dirichletto- Neumann map of the Schrödinger operator â g + V determines uniquely the potential V . This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them. This is joint work with Colin Guillarmou of CNRS Nice. The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.
Updated on May 13, 2013 11:01 PM PDT
25. # E. Nordenstam's Talk

Location: MSRI: Simons Auditorium
Speakers: Eric Nordenstam

Pizza Lunch

Updated on May 13, 2013 11:01 PM PDT
26. # Resistor Networks and Optimal Grids for Electrical Impedance Tomography with Partial Boundary Measurements

Location: MSRI: Baker Board Room
Speakers: Alexander Mamonov

Pizza Lunch

The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show two main advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct the conductivities of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods.
Updated on May 13, 2013 11:01 PM PDT


text:
# Maximum likelihood estimator - conditioning input on model parameters

Assume we have a dataset $$D=\{(x_i, y_i)\}$$ drawn from the joint distribution $$(x_i, y_i) \sim P (x=x_i, y=y_i)$$. We want to make predictions based on the dataset and for that we use a parametric model with parameter $$\theta$$ so that the prediction for a new $$x^*$$ is $$P(y^*|x^*, \theta)$$.

To estimate $$\theta$$, we maximize the likelihood $$P(D|\theta)$$ with respect to $$\theta$$, i.e. we find $$\theta$$ that maximizes the likelihood of observing the dataset $$D$$. To keep things simple, let us assume there is a single data point in the set, $$D=\{(x_1, y_1 \}$$. Then the likelihood is $$P(D|\theta) = P((x_1, y_1)|\theta)$$, i.e. the joint probability distribution of $$(x_1, y_1)$$ conditioned on $$\theta$$. We can expand the joint distribution as $$P((x_1, y_1)|\theta)=P(y_1|x_1,\theta)P(x_1|\theta)$$. However, in the usual definition of likelihood, only the first term appears. My question is, what happens to the second term $$P(x_1|\theta)$$? Is that somehow independent of $$\theta$$ and thus irrelevant to the estimator? If so, how can I see that?

• You wish to predict $y$ based on an observed $x$, so the only thing that matters is $p(y|x, \theta)$. $p(x|\cdot)$ doesn't matter because you will have observed $x$ at the time you are making your prediction, so it is fixed. Your likelihood function needs to reflect that. Jan 3 at 2:17


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# Could a human jump off Mimas without return?

A similar question has already been asked on dwarf planet Ceres: Could a Human reach escape velocity by jumping from the surface of Ceres (a dwarf planet)?

Ceres has 2.9% of Earth's gravity. Saturn's moon Mimas has 0.6% of Earth's gravity. If you jump strong enough, could you fly off into outer space from Mimas?

• Doesn't this also mistakenly assume that escaping a moon would let you "fly off into outer space"? You still have the gravity well of the planet to contend with. – R.. Oct 31 '18 at 18:32
• @R.. Not to mention the sun – Mad Physicist Oct 31 '18 at 20:40
• Definitely first read this as Minmus from KSP... – TemporalWolf Oct 31 '18 at 21:20
• @MadPhysicist: Yes, but I could see entering some wacky orbit around the sun as qualifying for "fly off into outer space". For a planet, less so, although of course it's just a matter of scale. – R.. Nov 1 '18 at 2:35

tl;dr: No chance, not even close!

The escape velocity from the surface of a round (spherically symmetric) body is given by

$$v_{esc} = \sqrt{\left(\frac{2 GM}{r_0} \right)},$$

showing that it is the $$\frac{mass}{radius}$$ ratio that's key here, not just the surface gravity given by

$$a_{g} = -\frac{GM}{r_0^2}.$$

So since

$$v_{esc} = \sqrt{a_g r_0},$$

a lower density but larger radius body with the same surface gravity would have a higher escape velocity. You can think of that as "the gravity extending outward farther" or better yet, just dropping off slower. Gravity drops by a factor of 4 at $$2r_0$$, so if $$r_0$$ is bigger, so is $$2r_0$$.

The problem is a little tougher because you have to look at the design of human legs. They are optimized to work in Earth gravity; they have mass an moments of inertia that work with muscle strength and the speed with which muscle fibers can contract. For that you can start with this excellent answer to bibliography for the question Any scholarly or serious work in Sports Science for the low surface gravity of Mars or the Moon? or other things tagged .

Let's look at what happens on Earth. Most people will find it a challenge to get to 1 meter in a standing high jump, and the world's record is 1.65 meters. Let's use 70 kg and 1 meter at $$g_0$$=9.8 m/s^2, some basic kinematics, and this page linking to the PDF Optimum Take-Off Range in Vertical Jumping to get a better picture

Source

Source

Published article Analysis of standing vertical jumps using a force platform Nicholas Linthorne, UNSW.

There's about a 1000 Newton force beyond the ~750 N supporting weight against gravity, or about 14 m/s^2, for about 0.25 meters. That's about 0.19 seconds and a take-off velocity of 2.6 m/s using $$v = \sqrt{2 g h}$$ and $$t = \sqrt{2x/g}$$.

If you could develop only the 1000 Newtons over 0.25 meters at those surface gravities, you would also achieve that ~2.6 m/s velocity.

However, the surface gravities of Mimas and Ceres are 0.064 and 0.28 m/s^2 respectively, and their escape velocities are 160 and 510 m/s, respectively!

So... no chance, not even close!

No.

Fermi estimate: Mimas' gravity is 0.064 m/s2, you need gravity to be about 1/20 of that to escape using a bike and a ramp (going by Deimos' surface gravity of 1/20 that of Mimas), lower still to escape by jumping :

• While more precisely, Mimas has 0.0636 m/s^2 or about 0.209 ft/s^2 which is about 0.006 g. – user27822 Oct 31 '18 at 9:45
• It's not an accepted answer (as of now), but you can always edit. – WBT Oct 31 '18 at 20:16


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Reference documentation for deal.II version GIT d2cc530c04 2023-03-22 15:10:02+00:00
NonMatching::ImmersedSurfaceQuadrature< dim, spacedim > Class Template Reference

#include <deal.II/non_matching/immersed_surface_quadrature.h>

Inheritance diagram for NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >:
[legend]

Public Member Functions

ImmersedSurfaceQuadrature (const std::vector< Point< dim >> &points, const std::vector< double > &weights, const std::vector< Tensor< 1, spacedim >> &normals)

void push_back (const Point< dim > &point, const double weight, const Tensor< 1, spacedim > &normal)

const Tensor< 1, spacedim > & normal_vector (const unsigned int i) const

const std::vector< Tensor< 1, spacedim > > & get_normal_vectors () const

bool operator== (const Quadrature< dim > &p) const

void initialize (const std::vector< Point< dim >> &points, const std::vector< double > &weights)

unsigned int size () const

const Point< dim > & point (const unsigned int i) const

const std::vector< Point< dim > > & get_points () const

double weight (const unsigned int i) const

const std::vector< double > & get_weights () const

std::size_t memory_consumption () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

bool is_tensor_product () const

const std::array< Quadrature< 1 >, dim > & get_tensor_basis () const

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

Static Public Member Functions

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

Protected Attributes

std::vector< Tensor< 1, spacedim > > normals

std::vector< Point< dim > > quadrature_points

std::vector< doubleweights

bool is_tensor_product_flag

std::unique_ptr< std::array< Quadrature< 1 >, dim > > tensor_basis

Private Types

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

Private Member Functions

void check_no_subscribers () const noexcept

Private Attributes

std::atomic< unsigned int > counter

std::map< std::string, unsigned int > counter_map

std::vector< std::atomic< bool > * > validity_pointers

const std::type_info * object_info

Static Private Attributes

static std::mutex mutex

Detailed Description

template<int dim, int spacedim = dim> class NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >

This class defines a quadrature formula to integrate over the intersection between an oriented surface, $$\hat{S}$$, and a cell or face. The word "immersed" in the class name reflects that the surface may intersect the cell/face in an arbitrary way.

The spacedim template parameter of this class is the dimension that the (spacedim-1)-dimensional surface is embedded in: $$\hat{S} \subset \mathbb{R}^{\text{spacedim}}$$. The dim parameter describes the dimension of the "object" that the surface intersects. That is, dim = spacedim corresponds to the surface intersecting a cell and dim = spacedim - 1 corresponds to the surface intersecting a face. The quadrature formula is described by a set of quadrature points, $$\hat{x}_q \in \mathbb{R}^{\text{dim}}$$, weights, $$w_q$$, and normalized surface normals, $$\hat{n}_q \in \mathbb{R}^{\text{spacedim}}$$.

Consider first the case dim = spacedim. We typically want to compute integrals in real space. A surface, $$S$$, intersecting a cell, $$K$$, in real space can be mapped onto a surface, $$\hat{S}$$, intersecting the unit cell, $$\hat{K}$$. Thus an integral over $$S\cap K$$ in real space can be transformed to an integral over $$\hat{S} \cap \hat{K}$$ according to

$\int_{S\cap K} f dS = \int_{S\cap K} f |d\bar{S}| = \int_{\hat{S}\cap\hat{K}} f \circ F_{K} \det(J) |\left( J^{-1} \right )^T d\hat{S}|,$

where $$F_K$$ is the mapping from reference to real space and $$J$$ is its Jacobian matrix. This transformation is possible since the continuous surface elements are vectors: $$d\bar{S}, d\hat{S} \in \mathbb{R}^{spacedim}$$, which are parallel to the normals of $$S$$ and $$\hat{S}$$. That is, the normal is needed to do the transformation. Thus, in addition to storing points and weights, this quadrature stores also the normalized normal for each quadrature point. This can be viewed as storing a discrete surface element,

$\Delta \hat{S}_q \dealcoloneq w_q \hat{n}_q \approx d\hat{S}(\hat{x}_q),$

for each quadrature point. The surface integral in real space would then be approximated as

$\int_{S\cap K} f dS \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) \det(J_q) |\left( J_q^{-1} \right)^T \hat{n}_q| w_q.$

When dim = spacedim - 1, this class represents a (spacedim-2)-dimensional integral. That is, if spacedim = 3 we have a line integral immersed in a face. Here, the transformation between the face, $$F$$, and reference face, $$\hat{F}$$, reads

$\int_{S\cap F} f dr = \int_{S\cap F} f |d\bar{r}| = \int_{\hat{S}\cap\hat{F}} f \circ F_{K} | J d\hat{r}| \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) |J_q \hat{t}_q| w_q,$

where $$\hat{t}_q = \hat{n}_q \times \hat{n}_F$$ is the tangent to the curve at $$\hat{x}_q$$ and $$\hat{n}_F$$ is the face normal. It would be possible to compute the tangent by only knowing the normal to the curve in the face plane (i.e. the dim-dimensional normal). However, when these quadratures are used, the weak form typically involves the so-called conormal, which can not be computed without knowing the surface normal in $$\mathbb{R}^{\text{spacedim}}$$. The conormal is the unit vector parallel to the projection of the face normal into the surface plane. This is essentially the same thing as the normalized boundary form.

Definition at line 101 of file immersed_surface_quadrature.h.

Member Typedef Documentation

template<int dim>
inherited

Define an alias for a quadrature that acts on an object of one dimension less. For cells, this would then be a face quadrature. A sub quadrature of a 0-dimensional quadrature is defined as still being 0-dimensional.

Definition at line 130 of file quadrature.h.

◆ map_value_type

 using Subscriptor::map_value_type = decltype(counter_map)::value_type
privateinherited

The data type used in counter_map.

Definition at line 230 of file subscriptor.h.

◆ map_iterator

 using Subscriptor::map_iterator = decltype(counter_map)::iterator
privateinherited

The iterator type used in counter_map.

Definition at line 235 of file subscriptor.h.

Constructor & Destructor Documentation

template<int dim, int spacedim = dim>
default

template<int dim, int spacedim>
 NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::ImmersedSurfaceQuadrature ( const std::vector< Point< dim >> & points, const std::vector< double > & weights, const std::vector< Tensor< 1, spacedim >> & normals )

Construct a quadrature formula from vectors of points, weights and surface normals. The points, weights and normals should be with respect to reference space, and the normals should be normalized.

Definition at line 22 of file immersed_surface_quadrature.cc.

◆ push_back()

template<int dim, int spacedim>
 void NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::push_back ( const Point< dim > & point, const double weight, const Tensor< 1, spacedim > & normal )

Extend the given formula by an additional quadrature point. The point, weight and normal should be with respect to reference space, and the normal should be normalized.

This function exists since immersed quadrature rules can be rather complicated to construct. Often the construction is done by partitioning the cell into regions and constructing points on each region separately. This can make it cumbersome to create the quadrature from the constructor since all quadrature points have to be known at time of creation of the object.

Note
This function should only be used during construction of the quadrature formula.

Definition at line 43 of file immersed_surface_quadrature.cc.

◆ normal_vector()

template<int dim, int spacedim>
 const Tensor< 1, spacedim > & NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::normal_vector ( const unsigned int i ) const

Return a reference to the ith surface normal.

Definition at line 59 of file immersed_surface_quadrature.cc.

◆ get_normal_vectors()

template<int dim, int spacedim>
 const std::vector< Tensor< 1, spacedim > > & NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::get_normal_vectors

Return a reference to the whole vector of normals.

Definition at line 70 of file immersed_surface_quadrature.cc.

◆ operator==()

template<int dim>
 bool Quadrature< dim >::operator== ( const Quadrature< dim > & p ) const
inherited

Test for equality of two quadratures.

Definition at line 304 of file quadrature.cc.

◆ initialize()

template<int dim>
 void Quadrature< dim >::initialize ( const std::vector< Point< dim >> & points, const std::vector< double > & weights )
inherited

Set the quadrature points and weights to the values provided in the arguments.

Definition at line 52 of file quadrature.cc.

◆ size()

template<int dim>
 unsigned int Quadrature< dim >::size ( ) const
inherited

◆ point()

template<int dim>
 const Point& Quadrature< dim >::point ( const unsigned int i ) const
inherited

Return the ith quadrature point.

◆ get_points()

template<int dim>
 const std::vector >& Quadrature< dim >::get_points ( ) const
inherited

Return a reference to the whole array of quadrature points.

◆ weight()

template<int dim>
 double Quadrature< dim >::weight ( const unsigned int i ) const
inherited

Return the weight of the ith quadrature point.

◆ get_weights()

template<int dim>
 const std::vector& Quadrature< dim >::get_weights ( ) const
inherited

Return a reference to the whole array of weights.

◆ memory_consumption()

template<int dim>
inherited

Determine an estimate for the memory consumption (in bytes) of this object.

Definition at line 313 of file quadrature.cc.

◆ serialize()

template<int dim>
template<class Archive >
 void Quadrature< dim >::serialize ( Archive & ar, const unsigned int version )
inherited

Write or read the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

◆ is_tensor_product()

template<int dim>
 bool Quadrature< dim >::is_tensor_product ( ) const
inherited

This function returns true if the quadrature object is a tensor product of one-dimensional formulas and the quadrature points are sorted lexicographically.

◆ get_tensor_basis()

template<int dim>
 std::conditional< dim==1, std::array< Quadrature< 1 >, dim >, const std::array< Quadrature< 1 >, dim > & >::type Quadrature< dim >::get_tensor_basis
inherited

In case the quadrature formula is a tensor product, this function returns the dim one-dimensional basis objects. Otherwise, calling this function is not allowed.

For dim equal to one, we can not return the std::array as a const reference and have to return it by value. In this case, the array will always contain a single element (this).

Note
The actual return type of this function is
std::conditional<dim == 1,

Definition at line 325 of file quadrature.cc.

◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 300 of file subscriptor.h.

◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

◆ check_no_subscribers()

 void Subscriptor::check_no_subscribers ( ) const
privatenoexceptinherited

Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.

Note
Since this function is just a consistency check it does nothing in release mode.
If this function is called when there is an uncaught exception then, rather than aborting, this function prints an error message to the standard error stream and returns.

Definition at line 53 of file subscriptor.cc.

◆ normals

template<int dim, int spacedim = dim>
 std::vector > NonMatching::ImmersedSurfaceQuadrature< dim, spacedim >::normals
protected

Vector of surface normals at each quadrature point.

Definition at line 155 of file immersed_surface_quadrature.h.

template<int dim>
protectedinherited

List of quadrature points. To be filled by the constructors of derived classes.

Definition at line 326 of file quadrature.h.

◆ weights

template<int dim>
protectedinherited

List of weights of the quadrature points. To be filled by the constructors of derived classes.

Definition at line 332 of file quadrature.h.

◆ is_tensor_product_flag

template<int dim>
protectedinherited

Indicates if this object represents quadrature formula that is a tensor product of one-dimensional formulas. This flag is set if dim==1 or the constructors taking a Quadrature<1> (and possibly a Quadrature<dim-1> object) is called. This implies that the quadrature points are sorted lexicographically.

Definition at line 341 of file quadrature.h.

◆ tensor_basis

template<int dim>
 std::unique_ptr, dim> > Quadrature< dim >::tensor_basis
protectedinherited

Stores the one-dimensional tensor basis objects in case this object can be represented by a tensor product.

Definition at line 347 of file quadrature.h.

◆ counter

 std::atomic Subscriptor::counter
mutableprivateinherited

Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).

The creator (and owner) of an object is counted in the map below if HE manages to supply identification.

We use the mutable keyword in order to allow subscription to constant objects also.

This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic class template.

Definition at line 219 of file subscriptor.h.

◆ counter_map

 std::map Subscriptor::counter_map
mutableprivateinherited

In this map, we count subscriptions for each different identification string supplied to subscribe().

Definition at line 225 of file subscriptor.h.

◆ validity_pointers

 std::vector *> Subscriptor::validity_pointers
mutableprivateinherited

In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.

Definition at line 241 of file subscriptor.h.

◆ object_info

 const std::type_info* Subscriptor::object_info
mutableprivateinherited

Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.

Definition at line 249 of file subscriptor.h.

◆ mutex

 std::mutex Subscriptor::mutex
staticprivateinherited

A mutex used to ensure data consistency when printing out the list of subscribers.

Definition at line 271 of file subscriptor.h.

The documentation for this class was generated from the following files:


text:
MetaDataset(dataset, labels_to_indices=None)

Descritpion

It wraps a torch dataset by creating a map of target to indices. This comes in handy when we want to sample elements randomly for a particular label.

Notes: For l2l to work its important that the dataset returns a (data, target) tuple. If your dataset doesn't return that, it should be trivial to wrap your dataset with another class to do that. TODO : Add example for wrapping a non standard l2l dataset

Arguments

• dataset (Dataset) - A torch dataset.
• labels_to_indices (Dict) - A dictionary mapping label to their indices. If not specified then we loop through all the datapoints to understand the mapping. (default: None)

Example

mnist = torchvision.datasets.MNIST(root="/tmp/mnist", train=True)

TaskGenerator(dataset, ways=2, shots=1, classes=None, tasks=None)

[Source]

Description

A wrapper to generate few-shot classification tasks.

tasks can both indicate predefined tasks, or just the number of tasks to sample. If specified as an int, a list of size task would be generated from which we'll sample. If specified as a list, then that list of tasks would be used to sample always.

The acceptable shape of list would be n * w, with n the number of tasks to sample and w the number of ways.

Each of the task should have w distinct elements all of which are required to be a subset of ways.

Arguments

• ways (int, optional, default=2) - Number of labels to sample from.
• shots (int, optional, default=1) - Number of data points per task to sample.
• dataset (MetaDataset or Dataset) - The (meta-) dataset to wrap.
• classes (list, optional, default=None) - List of classes to sample from, if none then sample from all available classes in dataset. (default: None)
• tasks (int or list, optional, default=None) - Tasks to be generated. If none, then all possible permutations are chosen from n classes and w ways.

## sample

TaskGenerator.sample(shots=None, task=None)

Description

Returns a dataset and the labels that we have sampled.

The dataset is of length shots * ways. The length of labels we have sampled is the same as shots.

Arguments

shots (int, optional, default=None) - Number of data points to return per class, if None gets self.shots. task (list, optional, default=None) - List of labels you want to sample from.

Returns

• Dataset - Containing the sampled task.


text:
# Thread: Gulliver and the King

1. ## Sigma

What does sigma mean I maths

2. ## Re: Gulliver and the King

Hello, baxteryay!

This is based on a classic problem.

He could cut one link like this: .$\displaystyle \subset\!\supset\!\!\!\subset\!\!\supset \;\;\subset\!\supset \;\;\subset\!\!\supset \!\!\!\subset\!\supset \!\!\! \subset\!\supset \!\!\!\subset\!\supset$

Then he can present from 1 to 7 links.

. . $\displaystyle \begin{array}{cc} \text{Number} & \text{Combination} \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & 1+2 \\ 4 & 4 \\ 5 & 1+4 \\ 6 & 2+4 \\ 7 & 1+2+4 \end{array}$

Since he needed two cuts, he had more than 7 links,
. . up to a maximum of 15 links.


text:
# Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)

Analytic and quasi-analytic vectorsGuided by researches in function theory, operator theorists gave the analogue to quasi-analytic classes. Let $A$ be an operator in a Banach space $X$. $A$ is not necessarily bounded hence the domain $D(A)$ is not necessarily to be the whole space. We say $x \in X$ is a $C^\infty$ vector if $x \in \bigcap_{n \geq 1}D(A^n)$. This is quite intuitive if we consider the differential operator. A vector is analytic if the series\sum_{n=0}^{\infty}\lVert{A^n x}\rVert\f...


text:
# 1 Pertemuan 06 Peluang Beberapa Sebaran Khusus Peubah Acak Diskrit Mata kuliah: A0392 - Statistik Ekonomi Tahun: 2010.

## Presentation on theme: "1 Pertemuan 06 Peluang Beberapa Sebaran Khusus Peubah Acak Diskrit Mata kuliah: A0392 - Statistik Ekonomi Tahun: 2010."— Presentation transcript:

1 Pertemuan 06 Peluang Beberapa Sebaran Khusus Peubah Acak Diskrit Mata kuliah: A0392 - Statistik Ekonomi Tahun: 2010

2 Outline Materi: Sebaran Binomial Sebaran Poisson Sebaran Hypergeometrik

3 Definitions A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories. 4. The probabilities must remain constant for each trial.

4 Notation for Binomial Probability Distributions S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p(p = probability of success) P(F) = 1 – p = q(q = probability of failure)

5 Notation (cont) n denotes the number of fixed trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials.

6 Important Hints  Be sure that x and p both refer to the same category being called a success.  When sampling without replacement, the events can be treated as if they were independent if the sample size is no more than 5% of the population size. (That is n is less than or equal to 0.05N.)

7 Method 1: Using the Binomial Probability Formula P(x) = p x q n-x ( n – x ) ! x ! n !n ! for x = 0, 1, 2,..., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p)

8 Method 2: Using Table A-1 in Appendix A Part of Table A-1 is shown below. With n = 4 and p = 0.2 in the binomial distribution, the probabilities of 0, 1, 2, 3, and 4 successes are 0.410, 0.410, 0.154, 0.026, and 0.002 respectively.

9 P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials Rationale for the Binomial Probability Formula

10 P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order Binomial Probability Formula

11 Any Discrete Probability Distribution: Formulas Std. Dev  = [  x 2 P(x) ] – µ 2 Mean µ =  [x P(x)] Variance  2  = [  x 2 P(x) ] – µ 2

12 Binomial Distribution: Formulas Std. Dev.  = n p q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials Mean µ = n p Variance  2  = n p q

13 This scenario is a binomial distribution where: n = 14 p = 0.5 q = 0.5 Using the binomial distribution formulas: Example Find the mean and standard deviation for the number of girls in groups of 14 births.

14 This scenario is a binomial distribution where n = 14 p = 0.5 q = 0.5 Using the binomial distribution formulas: µ = (14)(0.5) = 7 girls  = (14)(0.5)(0.5) = 1.9 girls (rounded) Example (cont)

15 For this binomial distribution, µ = 50 girls   = 5 girls µ + 2  = 50 + 2(5) = 60 µ - 2  = 50 - 2(5) = 40 The usual number girls among 100 births would be from 40 to 60. So 68 girls in 100 births is an unusual result. Example Determine whether 68 girls among 100 babies could easily occur by chance.

16 Sebaran Hipergeometrik Besarnya peluang bagi peubah acak hipergeometrik yang menyatakan banyak keberhasilan dalam contoh acak berukuran n adalah :

17 Sebaran Hipergeometrik Nilai tengah peubah acak X adalah: Ragam peubah acak X adalah: Simpangan baku peubah acak X adalah:

18 Sebaran Poisson Sebaran peluang Poisson dapat digunakan untuk mendekati peluang banyak sukses dalam sejumlah percobaan, jika banyak percobaan n  20 dan peluang sukses p  0,05 atau p  0,95 atau Jika n makin besar dan p makin mendekati 0 atau 1 penggunaan rumus Poisson akan semakin baik.

19 Definition The Poisson distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit. P(x) = where e  2.71828 µ x e -µ x!x! Formula

20 Poisson Distribution Requirements  The standard deviation is  = µ.  The random variable x is the number of occurrences of an event over some interval.  The occurrences must be random.  The occurrences must be independent of each other.  The occurrences must be uniformly distributed over the interval being used. Parameters  The mean is µ.

21 Difference from a Binomial Distribution The Poisson distribution differs from the binomial distribution in these fundamental ways:  The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean μ.  In a binomial distribution the possible values of the random variable are x are 0, 1,... n, but a Poisson distribution has possible x values of 0, 1,..., with no upper limit.

22 Example World War II Bombs In analyzing hits by V-1 buzz bombs in World War II, South London was subdivided into 576 regions, each with an area of 0.25 km 2. A total of 535 bombs hit the combined area of 576 regions If a region is randomly selected, find the probability that it was hit exactly twice. The Poisson distribution applies because we are dealing with occurrences of an event (bomb hits) over some interval (a region with area of 0.25 km 2 ).

23 Example The mean number of hits per region is number of bomb hits number of regions The probability of a particular region being hit exactly twice is P(2) = 0.170.

24 Poisson as Approximation to Binomial Rule of Thumb  n  100  np  10 The Poisson distribution is sometimes used to approximate the binomial distribution when n is large and p is small.

25 Poisson as Approximation to Binomial - μ Value for μ  = n p  n  100  np  10

26 SELAMAT BELAJAR SEMOGA SUKSES SELALU

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# Le Monde puzzle [#754]

December 25, 2011
By

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

The pre-X’mas puzzle in Le Monde weekend edition is about “magical numbers” having as digits all digits between 0 and n (at least once) and being multiple of all digits between 1 and (n+1). Easy, isn’t it?! I thought so while driving down to the Alps on Saturday and (on Monday early morning) I tried a brute force solution

magi=function(n){
prdct=prod(1:(n+1))

for (t in 1:10^6){
num=sum(sample(0:n)*10^(0:n))
if (num==prdct*trunc(num/prdct)){
print(num)
break()}
}
}

which worked for n=2,3, but failed for n=4. Maybe too brute-force? So I imposed the basic divisibility by 2×5=10 from the start, namely the last digit had to be 0:

magi=function(n){
prdct=prod(1:(n+1))
if (n>3){
for (t in 1:10^6){
num=sum(sample(0:n)*10^(0:n))
if (num==prdct*trunc(num/prdct)){
print(num)
break()}
}
}else{
for (t in 1:10^6){
num=10*sum(sample(1:n)*10^(0:(n-1)))
if (num==prdct*trunc(num/prdct)){
print(num)
break()}
}
}
}

Still did not work for n=4… Actually, there was a mistake in prdct in both of the above, in that the solution number only needs to be divided by the largest powers to the prime numbers between 2 and n+1, as implemented below.

I thought a wee more on the conditions and realised that any random permutation of the digits {1,…,4} could not be divided by 3, since their sum is 10. Therefore, the solution for n=4 must have at least 6 digits. This led me to a more general R function which works for the cases n=4,5,6 and has a second parameter, k, namely the number of digits in the proposed solution. It also includes imposing a correct second digit based on the divisibility by 4:

magi456=function(n,k){

proo=3*4*5*(1+6*(n==6)) #only required divisors
digi=10^(0:(k+1))
sol=rep(9,k+2)*digi
for (t in 1:10^6){

a0=0 #multiple of 2*5
a1=sample(2*(1:trunc(n/2)),1) #multiple of 4
b=sample((1:n)[-a1]) #all other integers
b=sample(c(b,sample(0:n,(k-n+1),rep=TRUE)))
a=sum(c(a0,a1,b)*digi)
if (a%%proo==0){
sol=a
print(a)
break()
}

}

return(sol)
}

A similar solution can be proposed for the cases n=7,8,9, with a different constraint on the three first digits due to the divisibility by 8. There again is a special case when n=7, since the sum of all integers from 1 to 7 is 28, not divisible by 3. On the other hand, the sum of all integers from 1 to 8 is 36 and the sum of all integers from 1 to 9 is 45, both divisible by 9. Here is the corresponding R code:

magi789=function(n,k){

proo=(1+2*(n==7))*5*7*8*(1+8*(n>7)) #only required divisors
digi=10^(0:(k+1))
sol=rep(9,k+2)*digi

for (t in 1:10^6){
a0=0 #multiple of 2*5
#multiple of 8 (4, 2)
a1=sample(2*(1:trunc(n/2)),1)
a2=sample((1:n)[-a1],1)
while ((a1+2*a2)%%4!=0){
a1=sample(2*(1:trunc(n/2)),1)
a2=sample((1:n)[-a1],1)
}
b=sample((1:n)[-c(a1,a2)]) #all other integers
b=sample(c(b,sample(0:n,(k-n+1),rep=TRUE)))
a=sum(c(a0,a1,a2,b)*digi)
if (a%%proo==0){
sol=a
print(a)
break()
}

}

return(sol)
}

Le Monde was furthermore asking for the smallest solution for each n. I ran the R code a few thousand times for every n and obtained

 > magi456(4,4)
122340
> magi456(5,4)
123540
> magi456(6,5)
1235640
> magi789(7,7)
122437560
> magi789(8,7)
123487560
> magi789(9,8)
1234759680

Of course, these are upper bounds on the smallest solutions (and there are more clever ways of R coding the above as well as of solving the puzzle in a mathematical way). Being away till Tuesday, I will not check the solution till then…

Filed under: Mountains, R, Statistics, University life Tagged: mathematical puzzle

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 Another Derivative Method of Mode Equation of Wave Guide Coupled with Prisme 棱镜耦合波导模式方程的另一导出法 短句来源 ANOTHER EXPRESSION METHOD OF SPACIAL CONCENTR- ATEDNESS OF EARTHQUAKES AND ITS APPLICATION TO EARTHQUAKE PREDICTION 地震空间集中度的另一种表示方法及其在地震预报中的应用 短句来源 ANOTHER MATHEMATICAL PROOF OF LIEBOWITZ FORMULA Liebowitz公式的另一数学证明 短句来源 ANOTHER METHOD FOR FINDING THE SEQUENCES{ g2i + 1 (x)}&{ f2i+1 (x) } , AND SOME OTHER PROPERTIES OF THESE SEQUENCES 函数序列{g_(2j+1)(x)}及{f_(2j+1)(x)}的另一种求法及另一些性质 短句来源 ANOTHER POSSIBLE EXPLANATION FOR THE θ—τ PUZZLE θ—τ疑难的另一种可能解释 短句来源 更多
 the other
 The Other Form of Regression Coefficient Covariance Matrix of a Normal Equation 正规方程另一形式的回归系数协方差矩阵 短句来源 Bridging problems concretely In one case preventing Great Salt Lake flooding, in the other promoting traffic safety: Major concrete bridge projects 具体解决疑难问题——混凝土大桥工程:一处防止大盐湖湖水泛滥,另一处改善交通安全 短句来源 on the other, how to unify the identity consciousness of Northeast China to form a new nation-state-nationalism. 一种是如何对待苏联的中国民族主义,另一种是如何把东北地区的认同意识凝聚成为新的国民国家民族主义。 短句来源 The effects of the studied anions on Raman spectrum of water could be summed up into two kinds. F-, OH-, S2-, CO2-3 belonged to one kind, and Cl-, Br-, I-, NO-3, and SO2-4 belonged to the other. 通过比较阴离子对水的羟基伸缩振动拉曼光谱的影响,可将所研究的阴离子分为两类,一类阴离子有F-、OH-、S2-、CO33-,另一类阴离子有Cl-、Br-、I-、NO3-和SO24-. 短句来源 The other recognizing unit with TCR-CD4 model can specifically recognize the MHC-Ⅱmolecule but promiscuously recognize the peptide from high affinity to lower affinity. 另一种为复合受体模式(TCR-CD4 model),对MHC-Ⅱ分子特异性要求很高(NHC制约),但有可以不同亲合度结合抗原肽的混杂性; 短句来源 更多
 the other one
 The other one is the same size at two ends,and 0.13±0.24 cmin length. 另一种两端大小基本相同,长1.13±0.24cm。 短句来源 The sedimentation rates are the lowest in the middle region of the lake, in which two samples could not be determined and the other one is 0.14 cm/a. 全湖以湖心区的沉积速率最低:三个测点中有二个因沉积率太低未能检出,另一测点的沉积率为0.14cm/a。 短句来源 (Ⅱ) the intermediate(INT1b) then isomerizes to the other one intermediate(INT2), with an energy barrier of 22.31 kJ/mol; ( )中间体 ( INT1 b)异构化成另一中间体 ( INT2 ) ,其势垒为 2 2 .3 1 k J/mol; 短句来源 Based on [106],we have researched the limits and inverse limits in the category STML,given the other one kind of product of -smooth topological molecular lattices. 以文[106]的相关概念为基础，研究了范畴STML(L)中的极限与逆极限问题，从而给出了范畴STML(L)的另一种乘积，且这两种乘积在范畴意义下是同构的。 短句来源 SDS-PAGE appeared two ribbons, one is α- and β-subunit with molecular weight about 16～18kD, while the other one is γ-subunit with molecular weight about 31kD. SDS-PAGE电泳出现2条染色带,一条着色较深带为α和β亚基,分子量为16～18kD左右,另一条较浅为γ亚基,分子量大约为31kD。 短句来源 更多
 other one
 The other one is the same size at two ends,and 0.13±0.24 cmin length. 另一种两端大小基本相同,长1.13±0.24cm。 短句来源 The sedimentation rates are the lowest in the middle region of the lake, in which two samples could not be determined and the other one is 0.14 cm/a. 全湖以湖心区的沉积速率最低:三个测点中有二个因沉积率太低未能检出,另一测点的沉积率为0.14cm/a。 短句来源 (Ⅱ) the intermediate(INT1b) then isomerizes to the other one intermediate(INT2), with an energy barrier of 22.31 kJ/mol; ( )中间体 ( INT1 b)异构化成另一中间体 ( INT2 ) ,其势垒为 2 2 .3 1 k J/mol; 短句来源 Based on [106],we have researched the limits and inverse limits in the category STML,given the other one kind of product of -smooth topological molecular lattices. 以文[106]的相关概念为基础，研究了范畴STML(L)中的极限与逆极限问题，从而给出了范畴STML(L)的另一种乘积，且这两种乘积在范畴意义下是同构的。 短句来源 The bi-parabolic equation is decomposed into two second order parabolic equation, the one is nonhomogeneous and other one is homogeneous, the two equations are all solved by stable difference schence of truucation error O(τ~2+h~4). 双抛物型方程分解为两个二阶抛物型方程,其一为非齐次,另一为齐次,每一个均用局部截断误差为O(τ2+h4)的稳定差分格式来解. 短句来源 更多

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 another
 We study the modificationA→A' of an affine domainA which produces another affine domainA'=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. Another proof of Joseph and Letzter's separation of variables theorem for quantum groups Another interesting subset of Vλ ? Vμ is the set of decomposable tensors. A similar result is proved for another class of regular 2-dimensional real algebras (the principal isotopes of the algebra ${\mbox{\bf R}}\oplus{\mbox{\bf R}}$). Another generalization is obtained in the context of representations of Jordan algebras, in the spirit of Herz's previous work on matrix spaces. 更多
 the other
 On the other hand, there is a locally trivialGa-action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2). We describe two opposite direction functors between Kronecker webs and integrable bihamiltonian structures: one is left inverse to the other. Extending this theory, we show how to use correlations between two processes to predict one from the other. Extending this theory, we show how to use correlations between two processes to predict one from the other. Specifically, for1>amp;lt;p>amp;lt;∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding Lp weighted inequality holds for T*. 更多
 the other one
 While for the other one, 2,7,12-tertriazine-5,5',10,10',15,15'-hexatetradecyltruxene (tertriazine-truxene-C14) form the intermolecular multi-hydrogen bonds. One form is constituted by formate dehydrogenase H and hydrogenase 3, and the other one is the same formate dehydrogenase and hydrogenase 4; the presence of small protein subunits, carriers of electrons, is also probable. One of them uses glass containing sodium ions as electrolyte, and the other one uses beta-ceramics. The equatorial plane in one Np polyhedron is formed by the oxygen atoms of four chromate ions and water molecule and by the oxygen atoms of five chromate ions in the other one. One is the well known electrochemical charge transfer process, while the other one is a direct chemical reaction of the metal, i.e. 更多
 other one
 While for the other one, 2,7,12-tertriazine-5,5',10,10',15,15'-hexatetradecyltruxene (tertriazine-truxene-C14) form the intermolecular multi-hydrogen bonds. One form is constituted by formate dehydrogenase H and hydrogenase 3, and the other one is the same formate dehydrogenase and hydrogenase 4; the presence of small protein subunits, carriers of electrons, is also probable. One of them uses glass containing sodium ions as electrolyte, and the other one uses beta-ceramics. The equatorial plane in one Np polyhedron is formed by the oxygen atoms of four chromate ions and water molecule and by the oxygen atoms of five chromate ions in the other one. One is the well known electrochemical charge transfer process, while the other one is a direct chemical reaction of the metal, i.e. 更多
 Emission and absorption band spectra of sulphur dioxide were studied in the region between 2600 to 2000 A.U. A new system of emission bands (about 150 bands) was found. When these bands were compared with the absorption bands and Mr. Lotmar's fluorescent bands, coincidences were found which amount to 40% and 50% of the total number of absorption and fluorescent bands respectively. These agreements and the very different structure as compared with the SO bands recorded on the same plate made it very probable... Emission and absorption band spectra of sulphur dioxide were studied in the region between 2600 to 2000 A.U. A new system of emission bands (about 150 bands) was found. When these bands were compared with the absorption bands and Mr. Lotmar's fluorescent bands, coincidences were found which amount to 40% and 50% of the total number of absorption and fluorescent bands respectively. These agreements and the very different structure as compared with the SO bands recorded on the same plate made it very probable that SO2, is the emitter of these emission bands. Evidences were also found that these emission bands arc not the known bands of O2, Oa+ and S2 in the same region. As a further support a vibra-tional level scheme was worked out, using the three fundamental frequencies of the normal SO2, molecule (1150, 525, and 1360 cm-1) in the lower state and 750 and 350 cm-1 (possible also 1110 cm-1) as the frequencies in the upper state. This scheme accounts for 1Q% of the absorption bands, 70% of the emission bands and a small fraction of the fluorescent bands. While the scheme may not be the final due to the complexity of the vibrational formula, one feels fair, as far as the present evidence goes, to conclude that the emission bands observed are actually emitted by the SO2, molecules and that they can be fitted by a vibrational level scheme based on the three known fundamental frequencies of the normal state. 此文目的在报告二氧化硫气体分子之吸收光谱及发射光谱(emission speetrun)之研究光谱区域约在2000A与2600A间有一新组发射光带(band)发现。数约百五十在吸收光带中有百分之四十之光带与此发射光带附合。在Lotmar先生之弗光光谱(fluorescent speetrum)中,有百分之五十之光带与此发射光带附合并此所得之发射光带之结构与SO之光带显有区别,令人设信此发射光带为SO_2分子所发者。在另一方面,找到证据,所得之发射光带亦非0_2,O_2~+及S_2诸分子之光谱为求更进一步之证实,试以SO_2之三基本振动数(fundamental frequency)1150,525及1360cm~(-1)为能力阶梯之差而作一常态(normal state)下之能力阶梯。再用750,350cm~(-1)(或1110亦用)为分子在激态(excited state)时之振动数,而作一激态下之能力阶梯所得之能力图(energydiagram)可说明吸收光带之70%,发射光带之70%及弗光光带之一小部份,此能力图并不完全,故不能望其为最终者。以三原子分子情形复杂,完全正确之分析,尚有待焉。之...此文目的在报告二氧化硫气体分子之吸收光谱及发射光谱(emission speetrun)之研究光谱区域约在2000A与2600A间有一新组发射光带(band)发现。数约百五十在吸收光带中有百分之四十之光带与此发射光带附合。在Lotmar先生之弗光光谱(fluorescent speetrum)中,有百分之五十之光带与此发射光带附合并此所得之发射光带之结构与SO之光带显有区别,令人设信此发射光带为SO_2分子所发者。在另一方面,找到证据,所得之发射光带亦非0_2,O_2~+及S_2诸分子之光谱为求更进一步之证实,试以SO_2之三基本振动数(fundamental frequency)1150,525及1360cm~(-1)为能力阶梯之差而作一常态(normal state)下之能力阶梯。再用750,350cm~(-1)(或1110亦用)为分子在激态(excited state)时之振动数,而作一激态下之能力阶梯所得之能力图(energydiagram)可说明吸收光带之70%,发射光带之70%及弗光光带之一小部份,此能力图并不完全,故不能望其为最终者。以三原子分子情形复杂,完全正确之分析,尚有待焉。之 能 力唯协所得之能 力 圆 山。。。,。w山。…。。。。\可。Z5t 叫 吸收 几 卅之70九传 时 光排 上 7,％及 比 儿 儿 卅上 一 个 部 份 此 能 力 闲 伙 不 正 全;故个能 望IC iX侦相 许 nt原一户 外 广似形 应 输,完全 正雕之分 析。尚有 Measurement of the intensity of total scattering of x-rays by a number of polyatomic gases was made for scattering angles between 15° and 130 ° using an ionization method of recording the scattered intensity. Balanced filters of ZrO2, and SrO were used to separate the MoKa rays and Soller slits were placed in front of the ionization chamber to obtain a definite scattering angle. The gases studied are CL2, CO2, N2O, H2S, CC14 and CHCl3. In each case the absolute values of the scattered intensity were determined... Measurement of the intensity of total scattering of x-rays by a number of polyatomic gases was made for scattering angles between 15° and 130 ° using an ionization method of recording the scattered intensity. Balanced filters of ZrO2, and SrO were used to separate the MoKa rays and Soller slits were placed in front of the ionization chamber to obtain a definite scattering angle. The gases studied are CL2, CO2, N2O, H2S, CC14 and CHCl3. In each case the absolute values of the scattered intensity were determined by comparison with the scattering from oxygen, the results of Wollan for the latter gas being taken as correct. The experimental results are actually compared with Woo's theory of the scattering of x-rays by polyatomic gases and the agreement seems to be satisfactory. 吴有训氏最近对于多原子气体散射线之理论,曾作详尽的探讨。吴氏得到一个公式,表示由多原子气体所散射之强度,其中一部为相干的散射,另一部为不相干的散射。 以前关于多原子气体散射X-线之实验,为数甚少,且为定性的结果。最近美人Wollan,对于由O_2及N_2(双原子气体)所散射钼的K_3α线之强度,曾作绝对的度量。Wollan的结果,与吴有训氏的理论,甚属相符。本篇目的,在测定由 Cl_2,CO_2,N_2O,H_2S,CCl_4及 CHCl六种气体所散射X-线之强度,每一实验,均与由0_2者互相比较,根据Wollan的结果,每种气体所散射之绝对强度,皆一一量得。所用之入射X-线为钼之Kα线,系藉Ross的平衡过滤法分出。强度之测量,系用一游离方法。散射角度的范围,自15度至130度。每种气体的实验结果,均与吴氏的理论,互相比较,证明理论与实验,甚属相符。在计算时,原子的“构造因数”,系由Hartree的方法算得,一分子中两原子的相隔距离,则由带光谱的结果推得。 The absorption band at 2240 A discovered by Stucklen in an underwater spark does not agree with the band observed by Hulthen and Zumstein in a high temperature oven and later studied by Grundstrom. The important difference lies in the complete lacking of the R-branch lines in Stucklen's band. New experimental investigation led to the discovery of the unfound R-branch lines. This is a strong evidence that the bands are actually the same. Stcklen氏发现用铜极在水中生火花有一吸收光带位於2240A区域此光带与Hulthen氏及Zumstein氏及Grund-strom氏在高温度炉中用铜之蒸气加以氢气所得之吸收光带(亦在2240A附近)不能一致最大之分别在Stucklen之光带全无R支谱线新实验工作发现R支谱线之存在得R(0)至R(13)之十四根线确实证明Stucklen之光带并非另一新带。 << 更多相关文摘
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You are here: Home » Content » Solid State Physics and Devices-the Harbinger of Third Wave of Civilization » SSPD_Chapter 7_Part 7_Scaling of MOS circuits_concluded.

# SSPD_Chapter 7_Part 7_Scaling of MOS circuits_concluded.

Module by: Bijay_Kumar Sharma. E-mail the author

Summary: SSPD_Chapter 7_Part 7_Scaling of MOS circuits_concluded. gives the definition of half-pitch and node. It gives limitation posed by Interconnect delays and the ways to overcome them. It gives importance of thermal management because of exponentially growing dissipation per die with scaling. It also gives the new challenges being posed due to scaling especially the sub-threshold leakages and the use of High K-metal to overcome it.

SSPD_Chapter 7_Part 7_Scaling of MOS circuits_concluded.

7.7.8. Definition of half pitch and node.

International Technology Roadmap for Semiconductors uses ‘node’ to refer to the smallest feature size on Logic Chips. This is the length of the Gate of MOSFET. Whereas in Memory Chips ‘half pitch’ is used to define the smallest feature size.

Half-pitch is the half distance between two adjacent aluminium pathways. Nodes and half-pitch are illustrated in Figure 7.7.8.1

As is evident from the figure, Half-Pitch = Node. But with the evolution of Technology the MOS structure has been modified and today Half-Pitch > Node. In Figure 7.7.8.2 we see the evolution of MOS structure from 1970 to 2005.

As we see that with scaling LDD(Lightly Doped Drain) had to be introduced under the edges of the Gate to mitigate the short channel effect. With the scaling of the dimensions, VDD was not proportionately scaled down under General Scaling Scheme as a result sharp electric field is created which results in “hot electrons effect”. Hot Electron Effects has deleterious effects on the Device. These are the following:

1. It injects electrons into the oxide layer leading to instability of the device.
2. It limits the velocity to scatter limited velocity which in effect saturates the electron velocity and we forfeit the advantage of reduction in transit delay with scaling. In long channel devices transit delay scales down by 1/α in constant field scaling and by 1/α2 in constant voltage scaling.
3. In short channel when electron velocity saturates , we can obtain the scale down by 1/α in constant field scaling but we cannot obtain 1/α2 scaling in constant voltage scaling.
4. Also under short channel effect channel length modulation parameter gets aggravated.

To mitigate the short channel effects we introduce Side-Wall Spacer at the edges of the Poly-Si Gate and LDD(Lightly Doped Drain) on source and drain side as shown in the lower half of Figure 7.7.8.2.. LDD extends below the the edge of the Poly-Si Gate. This has been achieved by tilted angle implant.

By grading the doping of Drain and Source from N+ to N- (LDD), we allow a more gradual drop of voltage leading to the reduction in the peak of the electric field along the channel. Even modest reduction in field leads to significant improvement in the performance of the device. It suppresses the hot electron generation, it prevents the saturation of the drift velocity of the electrons in the channel and the shallow junction of LDD helps minimize the chanel length modulation parameter.

The net result of this modification has been that in recent years Half-Pitch > Node as seen in the following tables.

In Table 7.7.8.1 we give the scaled down dimensions of MOSFET over the last 4 decades.

Table 7.7.8.1.Dimension Scaling in MOSFET over the last 4 decades.

 Year 1967 1997 1999 2001 2003 2006 L(μm) 10 0.25 0.18 0.13 0.1 0.07 DRAM(Gbit/cm2) 64M 0.18 0.38 0.42 0.91 1.85 Jn.Depth(xj nm) 1000 100 70 60 52 40 Interconnection Pitch(nm) 2000 600 500 350 245 130

In Table 7.7.8.2. we give the scaling in terms of Half-Pitch and Node.

Table 7.7.8.2. Evolution of Nodes and Half-Pitch with advancing Generations.

 Year Node(nm) Half-Pitch(nm) Gate Length*(nm) 2009 a 32 52 29 2007 a 45 68 ? 2005 b 65 90 32 2004 b 90 90 37 2003 b 100 100 45 2001 c 130 150 65 1999 c 180 230 140 1997 d 250 250 200 1995 d 350 350 350 1992 d 500 500 500

a-ITRS data 2008 update.

a-ITRS data 2006

a-ITRS data 2001

a-ITRS data 1997

*The physical length has become smaller than the printed length.

7.7.9.Limits due to Interconnection.

In Figure 7.7.9.1 we have defined the width(W), thickness(t) and spacing (S) of the metal pathway/metal interconnection at the same level and height (h)of the oxide layer separating two adjacent level metal pathways.

We have two scaling models. One model is fixed thiockness and the other is scaled thickness.

Wire length will be short for ‘local innterconects’ and will be chip long for ‘global interconnects’.

Die size will be scaled by DC = 1.1. As we scale down, the die size is scaled up so as to build more and more complex ‘System-on-Chip’.

In Table 7.7.9.1 we give the performance of local and global interconnects.

Table 7.7.9.1. Influence of Scaling on Interconnect Characteristics.

 Parameter ___________ Var.’t’ Const.’t’ Width(W) 1/S 1/S Spacing(S) 1/S 1/S Thickness(t) 1/S 1 Interlayer Oxide height(h) 1/S 1/S Characteristics per unit length Wire Resistance per unit length RW=1/(Wt) S2 S Fringing Cap.per unit length CWf=t/S 1 S Parallel Plate Cap.per unit length CWp=W/h 1 1 Total Wire Cap per unit length CW= CWf+ CWp 1 1 to S Unrepeated RC per unit length tWU= RW CW S2 S to S2 Repeated RC per unit length* tWR=√(RC RW CW) √S 1 to √S Cross Talk (Noise) t/S 1 S Local/scaled Interconnect Delay Length(L) L 1/S 1/S Unrepeated Wire RC delay L2tWU 1 1/S to 1 Repeated Wire RC delay LtWR 1/√S 1/S to 1/√S Global Interconnect Delay Length(Chip length) L DC DC Unrepeated Wire RC delay L2tWU S2DC2 SDC2to S2DC2 Repeated Wire RC delay LtWR DC√S DC to DC√S

*Asuming constant field scaling of gates.

7.7.9.1.Interconnect Delays are limiting the performance with advanced Generations of Technology

Inspection of Table 7.7.9.1 leads us to the following conclusions

1. Wire Capacitances per micron is remaining constant at 0.2femto Farads/μm. This is roughly 1/10 of gate capacitance;
2. Local wires are getting faster but not quite as fast as the Gates but that is not a major problem.
3. Global wires are getting slower.

In Figure 7.7.9.1 we make a comparative study of the Gate Delays, Interconnect delays and the overall delay. We clearly see a minimum dealy of 12 psec at Node 250nm for Al and SiO2 and a minimum delay of 8 psec at Node 180nm for Cu and low-K interconnect. This is the reason why IBM has completely switched over to Copper and low-K interconnects.

7.7.9.2.Ways and means for overcoming the Interconnect Bottleneck of cost and performance.

Some of the logical solutions for overcoming the Interconnect bottleneck are:

1. More levels of interconnect. We had to resort to 7to 8 levels of metallization at 100nm Node.
2. Hierarchy of design rules for local, semi-local and global routing.
3. Once the practical limits have been achieved we can further improve by going for Copper interconnects and low refractory index dielectric materials for higher propogation velocity. These dielectrics could be organic or aerogols.

7.7.10. Thermal Management due to excessive heating consequent to scaling.

Scaling reduces power dissipation per Gate but scaling increases the number of gates per chip. This coupled with higher speed scales up Chip Power Dissipation. Therefore Thermal Management becomes the central theme of Advanced Electronic Systyems.

In 1960, we had SSI chips which dissipated 0.1W to 0.3W per chip. These were cooled by Air and Liquid Cooling Techniques.

In 1980 we had BJT LSI and CMOS VLSI chips which dissipated 1 to 5W.

In 1990 CMOS VLSI were dissipating 15 to 30 W of heat.

CMOS μP chips at 1 to 2 W level were managed by heat sinks and heart spreaders.

When the dissipation level reached 2 to 3 W/cm2 totalling to 300W at substrate level in multi-chip modules there we went for Liquid cooling.

Water cooled multi-chip modules evolved to refined indirect liquid cooling designs.

In 1990, third generation cooling modules were introduced which could handle 2 to 5kW dissipation in 225 to 1000cm2 footprint.

7.7.10.1.Methods of Cooliing .

Some of the popular methods of cooling which were adopted were:

1. Compact heat exchangers;
2. Miniature refrigerators in Si – cooling system was imntegrated with the chip.
3. Direct liquid cooling of bare chips.
4. Chip packages in dielectric liquid;
5. In late 1980s, low cost , air-cooled multi-chip modules became popular. These had spatial and volumetric cooling facilities.
6. Water cooled modules were also developed.

The method of cooling was categorized according to the product categories.

• Low cost electronic systems costing less than $300 had bouyance, induced natural circulation of air method of cooling. • Handheld equipments costing less than$1000 had heat spreaders. This would suffice for 1 to 2 W ICs.
• Desktop and note-books computers costing less than $3000 had several options. • One option was heat sink with air-cooling. • Second option was cooling by remotely located fans. • Third option was Fan cooled heat sinks. • Fourth was natural convecting air, circulating past low-fin heat sinks, heat pipes and metal cases. This sufficed for 3 to 5W per chip dissipation. • High performance Systems costing more than$3000 had forced cooling systems.
• The harsh environment had still more elaborate methods of cooling.

By 1997, IC power dissipation reached 100 to 150 W heatdissipation. Chip back side is allocated to heat removal. The heat removal may be done by any one of the following methods:

• Conducting away the heat through solids with high thermal conductivity;
• Through convection the heat is removed;
• Through pumped liquid transport.
• Throuugh Vapour diffusion.

Thermal and Thermofluid modeling software is being accepted for simulation based development and implementation of advanced packages. High performance, air-cooled heat sinks will dominate future applications.

7.7.11.The Challenges being posed by Scaling.

7.7.11.1. High-K dielectric for Gate Insulation.

The biggest challenge is the unacceptably thin gate oxide with advanced scaling.

Table 7.7.11.1 gives the thicknesses of gate oxide with advanced Generation of MOS devices.

Table 7.7.11.1. The Gate Oxide Thickness (D nm) in advanced Generations of Devices.

 Generation 5μm 2μm 1.2μm 90nm 45nm 30nm D(nm) 88nm 40nm 22nm 1.3nm ? ?

We see that as we reach 90nm Generattion of Devices, the oxide layer becomes:

At 5 atomic layer thick Gate Oxide direct Quantum Mechanical tunneling is enabled leading to high Gate leakage and increased static power dissipation. The only solution was to make it physically thick but keep it electrically thin so that threshold voltage is unaffected., So we go for high K dielectric material for gate insulation.

Since Gate Capacitance has to remain constant therefore:

Therefore:

So if a high K material 5× the dielectric constant of SiO2 is used we will have 5× thick oxide layer that is 6.5nm thick dielectric material at the Gate which will completely cutoff the leakage and hence we normalize the statric power dissipation to nW in CMOS logic..

High K dielectrics are HfO2 with K=15 to 30 or HfSiO2 with K= 12 to 16..

7.7.11.2. Poly-Silicon Gate should be replaced by Metal as Silicon Dioxide replaced by High-K insulator.

The scaling forces SiO2 to be replaced by high-K insulator but it simultaneously demands Poly-Si to be replaced by Metal. As we progress with scaling Poly-Si becomes less effective as Gate. There are three reasons for this:

1. Because of Fermi-level pinning we get higher threshold voltage;
2. High-K material has high elasticity hence results in higher phonon or lattrice scattering. This results in the deterioration of channel conductivity;
3. Poly-Si has poor bonding with high-K insulator.

Therefore Poly-Si has to be replaced by Metal again as it was in the start of IC

Technology.

7.7.11.3.Strained Silicon for enhanced channel mobility.

To further solve the problem of reduced channel mobility, Uniaxial Process induced Stress could be used for enhanced mobility and hence for higher channel conductivity.

7.7.12. Concluding Remarks on Scaling.

Table 7.7.12.1. gives the typical scaling factors which have been used for the advanced Generations of the Devices.(Reference:”Device Scaling:The Treadmill that fuelled Semiconductor Industry Growth” by Pallab Chatterjee, i2 Technologies, Inc.)

Table. 7.7.12.1.Semiconductor Association Association 1992 Overall Road Map Technology Characteristics.

 1992 1995 1998 2001 2004 2007 Node(2λ) nm 500 350 250 180 120 100 Scaling factor(α) - 1.43 2 2.8 4.16 5 Desktop,VDD(V) 5 3.3 2.2 2.2 1.5 1.5 Portable,VDD(V) 3.3 2.2 2.2 1.5 1.5 1.5 Desktop, (β) - 1.515 2.27 2.27 3.33 3.33 Portable, (β) - 1.5 1.5 2.2 2.2 2.2 High Per.-PD(W per die) 10 15 30 40 40-120 40-200 Portable.-PD(W per die) 3 4 4 4 4 4 Clock(MHz)-off chip 60 100 175 250 350 500 Clock(MHz)-on chip 120 200 350 500 700 1000 f0 is scaled by α2/β-offchip 60 81 105 207 312 450 f0 according to SIA 60 100 175 250 350 500 f0 is scaled by α2/β-on-chip 120 162 211 414 623 901 f0 according to SIA 120 200 350 500 700 1000 Power per die(W) 10 15 30 40 80 120 No. of Gates per die(SIA) 300k 500k 2M 5M 10M 20M Pg(power per gate)×10-5W 3.33 3 1.5 0.8 0.8 0.6

In addition to speed and packing density, scaled CMOS provides new opportunities in low power management. Speed can be traded off fiir reduction in power consumption.

40N. AA NiMH Battery Cells are required for FULL MOTION VIDEO PLAYER in 0.5μm technology.

1No. AA NiMH Battery Cells are required for FULL MOTION VIDEO PLAYER in 0.1μm technology.

This gives the kind of accessibility which can be developed for the electreonic goods with Scaling.

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text:
Hello everybody, can someone help me? This riddle really gave me the gilles so far.
I need to solve this for school but I have no idea! joking ;)
Not sure if it will help - check this out. So funny!

You're very close
but can't be fed

You can have
a look inside
there will be
no place to hide

However,
You try to enter me
and more room arises
suddenly

Can you explain this trick to me?

This is so easy!

• Seems like a book. But some lines don't make sense. – QuyNguyen2013 Nov 21 '14 at 15:41
• Sounds like the "future" – dmg Nov 21 '14 at 16:25
• +1 for "gave me the gilles" :-p Could it be the Tardis? (Multiple dose of cyanide coming up from me soon!) – Rand al'Thor Nov 21 '14 at 18:23
• I looked up "Tardis". No, to all of you:) – Avigrail Nov 21 '14 at 18:28
• There is another Gilles. This is going to be confusing! – Rand al'Thor Nov 22 '14 at 11:50

Hat

You're very close But I'm always ahead

Ahead -> A-head -> A-het -> A-hat (just keep saying it over and over again (see title))

but can't be fed

Indicates inanimate - but since "can be beheaded" seems a contradiction - thus is the art of riddles.

You can have a look inside there will be no place to hide

This inanimate object is hollow but not large enough to hide inside.

However, You try to enter me

"U" try to enter. "U" displaces "A".

and more room arises suddenly

Hat -> Hut. A hut is larger than a hat. By putting a "U" in the hat (displacing "A") more room has been created :)

• Congratulations. You found out about the trick. Now you can do a magic show :) – Avigrail Nov 28 '14 at 19:51
• Good one 'cop! :-) @avigrail - that third verse was really evil :-[ :-) – Rand al'Thor Nov 28 '14 at 23:21
• @randal'thor I already told cop that I didn't expect it to be this hard :/ – Avigrail Nov 29 '14 at 9:28

I know this isn't the right answer, but this question posted on Puzzling Stack Exchange almost applies...

1. "You're very close but I'm always ahead"

This could mean our answers are close to correct, that we are physically located close to the right answer (right on our screens), or could be a reference to the fact that the question will always be located just ahead (above) the correct answer on our screens.

1. "You can have a look inside, there will be no place to hide"

We can attempt to edit your question, where you will not be able to conceal any additional hints from us that are hidden when viewed normally. Also there is no place to fit additional text when editing the question...

1. "However, try to enter me, more room arises suddenly" until we hit "enter" on the keyboard and more room arises for us to type in.

Just thought that was interesting... Since you said you need this for school, it sadly can't be the correct answer. :(

• Thank you very much for sharing! I dont need this for school actually. I was just hinting to the latest rumors and annoying some people. I wanted this puzzle to fit the Q&A-Style :D – Avigrail Nov 22 '14 at 4:30
• "We can attempt to edit your question, where you will not be able to conceal any additional hints from us that are hidden when viewed normally" - that's exactly what he did with his last riddle! Nice idea though. – Rand al'Thor Nov 22 '14 at 10:07

Could it be a

conjuror's hat?

You're very close

When wearing a hat, you are very close to it, but it's always above ('ahead of'?) your head (so the word 'ahead' is sort of wordplay too).

but can't be fed

'Beheaded' is here used to mean 'removed from the head'. (The OP is not a native English speaker!) And certainly a hat can't be fed.

You can have
a look inside
there will be
no place to hide
However,
try to enter me
more room arises
suddenly

These verses are what makes it a

conjuror's hat

rather than just a

hat.

You can look inside this object, and indeed there appears to be no place to hide. But

when the conjuror is performing 'magic' with it on stage, he can produce all sorts of objects, most famously rabbits, from the apparently empty hat, and also make them disappear into it. So for a rabbit, trying to enter the hat may lead to 'more room arising suddenly'.

Credit to @d'alar'cop for the idea of a

hat,

but he couldn't make this fit with the 3rd verse. The only remaining mystery is the relevance of the multi-'had' sentence linked to by the OP - maybe it's a red herring, or just a hint to a word which sounds like 'had'?

A pair of eye glasses

For obvious reasons; the first two verses, respectively: location of application and translucency. Taking poetic license for the third verse, that while in use, they have been 'entered' and provide more 'room' that you can:

-see. "It's just a trick of the light."

• Good guess but no :) – Avigrail Nov 28 '14 at 14:24

Could it be a

I guess explanation won't be required after you revisit your's:-)

You're very close But I'm always ahead Can be beheaded but can't be fed

A shadow is always very close to you and ahead of you. Sometimes, because of the light's angle or broad daylight, you can behead it ( remove the head or kill its existence).

You can have a look inside there will be no place to hide

Wherever you go, it follows.

However, You try to enter me and more room arises suddenly

Since the shodow moves with you, when you try to catch it ( as we must have tried as a child :) ), it moves with us and sometimes growing bigger.