text:
# Featured projects

Deep Learning for Driver's Behaviour Recognition

Centre for Pattern Analysis and Machine Intelligence

Researchers in the Centre for Pattern Analysis and Machine Intelligence have developed a new algorithm for predicting and alerting in real time, the behaviour and status of the driver of a vehicle. This solution can operate via a connection to the cloud, or on a stand-alone basis. It will contribute to the design of "level-4" automation in autonomous vehicles. This development has been featured in media including the Washington Post, DigitalTrend, Wired, CBC, Vice Media, Globe and Mail, and Daily Mail.

### The "Tricorder"

Waterloo’s Vision and Image Processing Lab has developed a revolutionary "tricorder" technology, termed Coded Hemodynamic Imaging (CHI), that is the first-of-its-kind to enable touchless imaging of blood flow across the body, and which could lead to improved detection and prevention of a wide range of heart and lung diseases. Current techniques for diagnosing heart failure rely on invasive catheterization to obtain a single-point jugular venous pulse measurement, but CHI enables non-contact imaging of blood flow across entire jugular veins, as one might measure traffic flow across a city. CHI projects light and captures light fluctuations on the skin surface, and relays them to a digital signal processing unit which computes blood-flow patterns. This research was ranked in the Top 5% of all research outputs scored by Altmetric, and received media coverage across the globe, with an appearance on the popular science show Daily Planet on the Discovery Channel.

### Conversational Agents

Text messaging applications have become the most popular and most engaging features of smart phones in recent years. In collaboration with Kik Interactive, Artificial Intelligence researchers at the University of Waterloo have developed adaptive and emotionally aware conversational agents for text messaging. Leveraging machine learning and affect control theory, the algorithms automatically adapt to the affective personality of users by recognizing their emotions in short text messages and responding with messages that are emotionally aligned. This work has already led to two patents on adaptive and emotionally aware conversational agents.

### Functionally-safe automotive autonomy

In November 2016, Waterloo’s ‘Autonomoose’ became the first autonomous test vehicle licensed to operate on Canada’s public roads. The fleet will grow in 2017 with multimillion dollar support from NSERC, CFI, the Province of Ontario and several private sector partners. The team’s machine learning approach targets perception and prediction in adverse conditions – tackling the complexities of snow, sleet, low light, and reflective surfaces. A major practical focus is safety assurance – efficiently validating and certifying the safety of machine learning algorithms so we can incorporate AI into safety critical functions.

### Resource sharing for better control of forest fires

Forest fires threaten communities across North America, cost billions of dollars annually, and destroy vast quantities of natural resources. But controlling fires is expensive and local resource demand often outstrips availability. Researchers from Waterloo’s Artificial Intelligence Group have built the first model for wildfire fire resource sharing, using ideas from multi-agent systems and game theory. Developed in partnership with the Ontario Ministry of Natural Resources and in consultation with experts in the field, their analysis has identified the key strategic issues confronting fire-response agencies, and opens up a valuable new application area for the Artificial Intelligence community.

### Evolutionary deep intelligence

Researchers in Waterloo’s Vision and Image Processing Lab have developed new techniques to obtain highly efficient deep neural network architectures. Their evolutionary deep intelligence approach drives the formation, over successive generations, of highly sparse synaptic clusters. The offspring algorithms, trained on image classification, can achieve state-of-the-art performance despite an up to 125-fold decrease in synapses. Such architectures are ideal for low-power embedded CPUs. This research received the best paper award at the efficient deep learning workshop, held at the NIPS Conference in Barcelona, December 2016. A spinoff company, DarwinAI, is aiming to accelerate operational AI from the edge to the cloud for all verticals.

### Identifying cancer through imaging

A team from the Vision and Image Processing Lab was honoured with two Magnum Cum Laude awards at the 2016 Annual Meeting of the Imaging Network of Ontario for their work in discovery radiomics, in collaboration with Sunnybrook Health Sciences clinicians. They developed a breakthrough strategy for quicker and more effective cancer identification based on the quantitative identification of tumour biomarkers, tapping into the wealth of information contained within medical imaging archives. The image to the right shows the discovered sequencer, encoding the cancer biomarker data.

### Embedded speech engines

The Centre for Pattern Analysis and Machine Intelligence has developed a set of AI-based platforms that outperform standard keyword spotting techniques, both theoretically and experimentally. These embedded speech devices, untethered to any cloud-based speech engine, are based on deep learning algorithms and advanced noise suppression techniques; an embedded Natural Language Understanding version of the system is also being tested. A spinoff company, Cognitive Computing Technologies, is targeting automotive environments, cognitive/social robotics, and small office/home appliances.

### Autonomy in human-robot teams

As robots move beyond the production line and into shared human environments, efficient and effective communication with human operators will be essential. In a three-year collaboration initiated with Clearpath Robotics, researchers from Waterloo’s Adaptive Systems and Autonomous Vehicles labs will develop machine learning strategies for fully autonomous warehouse robots to automatically elicit human preferences via simple interactions with staff, with no need for expert programming and supervision. Clearpath is one of several robotics companies scaling up in Waterloo Region - founded in 2009 by Waterloo engineering students, the company now employs over 150 people.

### RoboHub in Building E7

The \$4.5M RoboHub, supported by CFI, will provide the only multi-robot test facility in the world with autonomous ground, aerial, humanoid and magnetically levitated platforms under one roof. Located at the heart of the new Engineering 7 building, the RoboHub will support a host of research programs into robot co-ordination and autonomy. For example, Bayesian learning models will enable state estimation for localization and autonomous mapping (SLAM) of intelligent mobile robotics agents and unmanned aerial vehicles (UAVs) to enable navigation in unstructured environments. Meanwhile, recurrent neural networks will enhance quadrotor flight modelling, predicting aerodynamics with greater accuracy for precision manoeuvring at speed and near obstacles.

### Hockey analytics

Hockey is a fast-paced game for which there is increasing demand for data analytics. In partnership with HockeyTech, researchers in Artificial Intelligence and Economics at the University of Waterloo are developing automated algorithms that process low-level position data about each player and the puck and which can recognize events such as shots, passes, puck possession, takeaways, giveaways, and faceoff wins, and then use this data to predict goals, wins and player performance. These new statistics are helping players, coaches, scouts and fans of professional and semi-professional hockey leagues across North America.

### Barcoded Medical Imaging

Kimia Lab members have been working on medical images and artificial neural networks for many years.  The lab's research focuses on generation of “deep barcodes” for medical images that uniquely represent that image, e.g. a pathology scan, to facilitate searching for similar images. Similar cases retrieved and displayed to clinical experts (e.g., radiologists, pathologists) along with biopsy reports and treatment results can indeed immensely reduce diagnostic error; Training a deep neural network with a large number of images is a multifaceted challenge if there is not enough data, the data is imbalanced, or most of it is not labelled. Researchers in the Kimia lab work on extensions and improvements of several AI and computer vision technologies to enable image recognition in digital pathology.

### Spaun

Centre for Theoretical Neuroscience

Meet Spaun, the world’s largest model of a functional brain. Developed by Chris Eliasmith and his multidisciplinary team at Waterloo’s Centre for Theoretical Neuroscience (CTN), Spaun combines 2.5 million simulated neurons with a visual recognition system and a simulated mechanical arm. Just like its human counterpart, it can read, answer questions, play simple games and memorize lists. It can decode numbers written in unfamiliar handwriting and tackle basic logic problems. Like people, it even makes mistakes, faltering at complex questions or getting tripped up when lists get too long. And Spaun learns, adapting its behaviour based on feedback from the world around it.

Machine Learning, SAT/SMT Solvers, and Deduction


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# Revision history [back]

I didn't find any reference to linear regression in the SAGE reference manual. That means you have to use Maxima directly. Fortunately that's not so difficult.

It's easier if you present the data as a list of pairs (x,y). Try this in a SAGE notebook cell:

maxima.load("stats")
x=maxima.simple_linear_regression([[0,1],[1,2],[2,4]],conf_level=0.95)

x

You are presented a whole bunch of statistics about the regression. From there on I guess you can manage on your own. Good luck.

format: plain


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# Circle A has a radius of 2  and a center of (2 ,7 ). Circle B has a radius of 6  and a center of (7 ,5 ). If circle B is translated by <-1 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Feb 7, 2018

circle B do not overlap.
Shortest distance is $3.764 - 2 = 1.764$

#### Explanation:

Circle A
Center $\left(2 , 7\right)$
radius $2$

Equation of circle A
${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = {2}^{2}$

Circle B
Center $\left(7 , 5\right)$
radius $6$

Equation of circle B
${\left(x - 7\right)}^{2} + {\left(y - 5\right)}^{2} = {6}^{2}$

ranslation of B
<-1,1>
Center of B after translation $\left(\begin{matrix}7 - 1 \\ 5 + 1\end{matrix}\right)$
Center of B after translation $\left(6 , 6\right)$

Equation of circle B after translation
${\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2} = {6}^{2}$

Intersection of the circles
${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = {2}^{2}$
and
${\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2} = {6}^{2}$
can be found out

Distance from center of Circle A to
center of Circle B is
$\left(2 , 7\right) - - - - \left(6 , 6\right)$
=sqrt((6-7)^2+(6-2)^2
=sqrt((1+4)
$= \sqrt{5}$
$= 2.236$

Radius of circle A is $2$
point on the Perimeter of the circle B is away from its center by 6
point on the Perimeter of the circle B is away from (2,7) is$6 - 2.236$
$= 3.764$
Hence, the two circles donot overlap.
Shortest distance is $3.764 - 2 = 1.764$

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How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# Ratio of the area of Square and Pentagon | AMC 8, 2013

Try this beautiful problem from Geometry: Ratio of the area between Square and Pentagon.

## Ratio of the area between Square and Pentagon - AMC-8, 2013 - Problem 24

Squares ABCD ,EFGH and GHIJ are equal in area .Points C and D are the mid points of the side IH and HE ,respectively.what is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares?

• $\frac{1}{4}$
• $\frac{1}{3}$
• $\frac{3}{8}$

### Key Concepts

Geometry

Area of square

Area of Triangle

Answer:$\frac{1}{3}$

AMC-8(2013) Problem 24

Pre College Mathematics

## Try with Hints

extend  IJ until it hits the extension of  AB .

Can you now finish the problem ..........

find the area of the pentagon

can you finish the problem........

First let L=2 (where L is the side length of the squares) for simplicity. We can extend  IJ until it hits the extension of  AB . Call this point  X.

Then clearly length of AX=3 unit & length of XJ = 4 unit .

Therefore area of $\triangle AXJ= (\frac{1}{2} \times AX \times XJ)=(\frac{1}{2} \times 4 \times 3)=6$ sq.unit

And area of Rectangle BXIC= $( 1 \times 2)$=2 sq.unit

Therefore the of the pentagon ABCIJ=6-2=4 sq.unit

The combined area of three given squares be $(3 \times 2^2)$=12 sq.unit

Now, the ratio of the shaded area(pentagon) to the combined area of the three squares is $\frac{4}{12}=\frac{1}{3}$

.


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# Samples in the convex body vs. samples on the convex surface

Let $K$ be a bounded convex body in $\mathbb{R}^n$. Suppose we have a sampler $\mathcal{S}_1$ that can generate points uniformly distributed in $\mathrm{int}K$, and another sampler $\mathcal{S}_2$ that can generate points uniformly distributed on $\partial K$.

The figure below illustrates two samplers.

Let $\{\mathbf{x}^j\}_{j=1}^M$ be $M$ samples generated by $\mathcal{S_1}$ and $\{\mathbf{y}^j\}_{j=1}^M$ by $M$ samples generated by $\mathcal{S_2}$. Denote the empirical centroid $\mathbf{u}$, $\mathbf{v}$ as $\mathbf{u}=\frac{1}{M}\sum_j \mathbf{x}^j$ and $\mathbf{v}=\frac{1}{M}\sum_j \mathbf{y}^j$, respectively. My questions are:

1. Are $\mathbf{u}$ and $\mathbf{v}$ same or not?
2. If not, can anyone provide a lower and upper bound of $\|\mathbf{u}-\mathbf{v}\|$?

Thanks

-

1. Not necessarily.
2. Unfortunately, the value of $||{\bf u} - {\bf v}||$ can be anywhere from $0$ to $\infty$.

This holds true even if the empirical measures capture the actual centroid of $K$ and the actual centroid of the boundary of $K$, so let's look at that situation.

Take the simple case of an isosceles triangle. Let the vertices of the triangle be at $(0,x)$, $(1,0)$ and $(-1,0)$. The centroid of the triangle has barycentric coordinates in a $1:1:1$ ratio, which means: $$\text{The centroid lies at } \left(0,\frac{x}{3}\right).$$

The centroid of the boundary of the triangle is called the Spieker center of the triangle and has barycentric coordinates in a $b+c:a+c:a+b$ ratio (see X(10) in the Encyclopedia of Triangle Centers; also, $a, b, c$ are the lengths of the sides opposite vertices $A, B, C$, respectively). Since the side lengths of our example triangle are $2, \sqrt{x^2+1}, \sqrt{x^2+1}$: $$\text{The Spieker center lies at }\left(0, \frac{2x\sqrt{x^2+1}}{4+4\sqrt{x^2+1}}\right) = \left(0, \frac{x}{2+2/\sqrt{x^2+1}}\right).$$

If the triangle is equilateral, $x = \sqrt{3}$, and the centroid and Spieker center coincide. On the other hand, the distance between the centroid and the Spieker center is on the order of $\frac{x}{6}$ and so can be made arbitrarily large by increasing $x$.

This also points to the general case that the distance between the centroid of the region and the centroid of the perimeter can be made arbitrarily large: Moving a vertex further and further away from the other vertices has more effect on the centroid of the perimeter than it does on the centroid of the region, as this process pulls away from the main part of the region a larger percentage of the boundary than of the area.

-
Thanks @MikeSpivey for this nice and clear answer. The example is really convincing. –  han Nov 5 '11 at 20:23
@han: You're welcome. I'm glad it was helpful. –  Mike Spivey Nov 5 '11 at 20:57


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# 6.1: Ratio and Proportion

From the beginnings of the human race, we’ve long compared one quantity with another, a comparison that is called a ratio in mathematics. “Their tribe has twice as many cattle as ours” or “Two baskets of wheat cost 12 ducats” are examples of ratios that ring from distant times. Indeed, the concept of a ratio cannot be assigned to any one individual or class of individual. In his History of Mathematics, D. E. Smith writes:

It is rather profitless to speculate as to the domain in which the concept of ratio first appeared. The idea that one tribe is twice as large as another and the idea that one leather strap is only half as long as another both involve the notion of ratio; both are such as would develop early in the history of the race, and yet one has to do with ratio of numbers and the other with the ratio of geometric magnitudes. Indeed, when we come to the Greek writers we find Nicomachus including ratio in his arithmetic, Eudoxus in his geometry, and Theon of Smyrna in his chapter on music.

Examples and applications of ratios are limitless: speed is a ratio that compares changes in distance with respect to time, acceleration is a ratio that compares changes in speed with respect to time, and percentages compare the part with the whole. We’ve already studied one classic ratio, the ratio of the circumference of a circle to its diameter, which gives us the definition of $$π$$.

One of the most famous ratios in history involves the division of a line segment AB into two segments AC and CB by selecting a point C on the segment AB.

The idea is to select a point C on the segment AB so that

$\frac{AB}{AC} = \frac{AC}{CB}.\nonumber$

This ratio has a special name, the Golden Ratio, and has an exact value equal to $$(1+ \sqrt{5})/2$$. The Golden Ratio has been known since the time of Euclid. Ancient and modern architects have long held that the most pleasing rectangular shape is the one whose ratio of length to width is equal to the Golden Ratio.

The comparison of two ratios, such as AB/AC = AC/CB, is called a proportion. Proportions are used in a number of practical ways. For example, if 5 cans of tomato sauce cost 2 dollars, we can find the number of cans that can be purchased with 10 dollars by comparing two ratios in a proportion:

$\frac{ \text{5 cans of tomato sauce}}{\text{2 dollars}} = \frac{ \text{x cans of tomato sauce}}{\text{10 dollars}}\nonumber$

Any discussion of ratio involves comparing two quantities, so the units of each quantity become extremely important. Two different systems of units are used when measuring length, capacity, and time: the American system of units and the metric system of units. In this chapter we will discuss both systems and explain how to convert quantities measured in one system to quantities measured in the other system.

Let’s begin the journey.


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# How is the wave function viewed as the quantum state?

I am not sure how the wave function can be viewed as quantum state.

I begin with the eigen-equation

$$A|\psi\rangle = a|\psi\rangle$$

If $$A$$ is a $$n$$ dimensional matrix with different eigenvalues $$a$$, then there will be $$n$$ eigenvectors.

If I do orthonormalization to these eigenvectors, then there will be $$n$$ basis vectors.

If the $$|\psi\rangle$$ is vector in quantum state space, then this is the $$n$$ dimensional quantum state.

Next, more specific consideration. The Schrodinger equation, $$H|\psi\rangle=E_n|\psi\rangle$$ with at the $$n$$ energy eigenvalue and $$H$$ is Hamiltonian (not in form of the matrix like above). Normally it is solved in the position state, that will be $$H\Psi_n(x) = E_n\Psi_n(x)$$ where $$\Psi_n(x) = \langle x|\psi\rangle$$ is wave function of an $$n$$ energy eigenvalue.

In this form, the wave function is just a continuous coefficient of $$|x\rangle$$.

How does this $$\Psi_n(x)$$ become quantum state ?

Can I say that because there are $$n$$ wavefunctions on the same position space $$|x\rangle$$ then they form the Hilbert space and can be viewed as quantum state?

$$\int {\Psi_m}^*(x) \Psi_n(x) dx =<\Psi_m(x)|\Psi_n(x)>= \delta_{m,n}$$

The wavefunctions themselves are simply the coefficients in the expansion of the quantum mechanical state vector in the position basis, as you already said:

$$\Psi(x, \ t) \ = \ \langle x | \Psi \rangle$$

I'm not sure if a Hilbert space is the right word to use, as the wavefunctions themselves are not vectors, however, any other wavefunction in our possible state space can be expressed as a linear combination of the stationary state wavefunctions of the Hamiltonian.

I'm not totally sure what you mean when you ask how the wavefunction becomes the quantum state. The point of the set of stationary state wavefunctions in the position basis is that we can write the quantum mechanical state vector as a linear combination of them.

For instance, consider the position basis $$|x\rangle$$. We can write some state vector as a linear combination of position eigenstates. Remember that:

$$\displaystyle\sum_{i} \ |v_i\rangle \langle v_i| \ = \ \mathbb{I}$$

For any orthonormal basis as:

$$\Big( \displaystyle\sum_{i} \ |v_i\rangle \langle v_i| \Big) |v\rangle \ = \ \displaystyle\sum_{i} v_i |v_i\rangle \ = \ |v\rangle$$

Now, this also holds for a continuous sum (an integral), so we have:

$$|\Psi\rangle \ = \ \mathbb{I} |\Psi\rangle \ = \ \displaystyle\sum |x\rangle \langle x | \Psi \rangle \text{d}x \ = \ \displaystyle\sum \Psi(x, \ t) |x\rangle \text{d}x$$

Cool, so we can expand the state vector in terms of a basis of eigenstates. But we can also expand the wavefunction in terms of another basis! For example, we can then expand the position wavefuntion in terms of the energy eigenstate basis. Thus, we have:

$$\langle x | \Psi \rangle \ = \ \langle x | \mathbb{I} | \Psi \rangle \ = \ \langle x | \Big( \displaystyle\sum_{n} \ |E_n\rangle \langle E_n | \Big) | \Psi \rangle \ \ = \ \displaystyle\sum_{n} \ \langle x |E_n\rangle \langle E_n | \Psi \rangle \ = \ \displaystyle\sum_{n} c_n \Psi_n(x, \ t)$$

So we can write the state vector as the wavefunction, and we can write the wavefunction as a linear combination of stationary states, with each of the $$c_n$$ coefficients corresponding to the probability amplitude of the specific energy eigenstates, thus telling us "how much" of each $$\Psi_n(x, \ t)$$ is in $$\langle x | \Psi \rangle$$.

So to answer your question, the wavefunctions are not quantum states, they are components of the quantum states that tells us certain information about the quantum state represented in a particular basis, but we can still expand/express each one as linear combinations of vector/functions in another basis.

• Suppose $H|\psi> = E|\psi>$ with H is $n$ dimesion matrix. Solve it like normal eigen-equation, then there will be the $n$ dimesion of basis of energy correspond to eigen value $E_n$. The eigen vector $|\psi>$ will be $|\psi> = \Sigma_i^n c_i |E_i>$. Next I express each of them in term of position basis, they will be $|\psi> = \Sigma_i^n c_i \int <x|E_i>|x>dx = \Sigma_i^n c_i \Psi_i(x)|x>$. right? – Kanokpon Arm Sep 25 '19 at 5:19
• Not quite, the energy eigenstates of $H|\psi> = E|\psi>$ are the set $|E_n>. You seem to try to express the energy eigenvalues in terms of themselves in the first equation. – Jack Ceroni Sep 25 '19 at 11:12 • the correct way to do this would be to find the energy eigenvalues$H|E_n\rangle \ = \ E_n|E_n\rangle$. If we want to find the stationary position wavefunctions, we expand these eigenvectors in terms of the position basis:$|E_n\rangle \ = \ \displaystyle\int \ \langle x | E_n \rangle |x\rangle dx \ = \ \displaystyle\int \ \psi_n(x) \ |x\rangle dx\$. – Jack Ceroni Sep 25 '19 at 12:10
• In order to recover the full state vector from the wavefunction, we need to know the initial wavefunction for the system. To find the time-dependent wavefunction, we have to express the initial wavefunction as a linear combination of the energy stationary position wave functions, then add time-dependence, then integrate over this wavefunction with respect to all the position eigenstates. – Jack Ceroni Sep 25 '19 at 12:15

A wave function can be viewed as describing a quantum state in the same way that the components of a vector can be viewed as describing the vector itself. Yes, $$\psi(x_0)=\langle x_0\vert\psi\rangle$$ is the numerical value of $$\psi(x)$$ at $$x=x_0$$ but this numerical value is basically the component of the vector $$\vert\psi\rangle$$ along the basis vector $$\vert x_0\rangle$$ (see this question), just like $$\langle \hat k\vert \vec r\rangle=r_k$$.

If you prefer to work in a momentum basis, then $$\psi(p)=\langle p\vert\psi\rangle$$ and the change of basis formula is contained in the overlap $$\langle p\vert x\rangle = \frac{e^{-i px/\hbar}}{\sqrt{2\pi \hbar}}$$ so that $$\langle p\vert\psi\rangle = \int dx \langle p\vert x\rangle \langle x\vert\psi\rangle$$ just like a change of basis is given by $$r_a =\langle \hat a\vert\vec r\rangle =\sum_k \langle \hat a\vert \hat k\rangle \langle \hat k\vert \vec r\rangle=\sum_k A_{ak}r_k\, .$$

Asher Peres in

Peres, Asher. "What is a state vector?." American Journal of Physics 52.7 (1984): 644-650.

quotes Ekstein as stating that "states are images of procedures by which a system is made to interact with a macroscopic apparatus". The reference to the original Ekstein paper is Ekstein, H. "Presymmetry." Physical Review 153.5 (1967): 1397.


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#### Important 1mark

11th Standard

Reg.No. :
•
•
•
•
•
•

Maths

Time : 00:20:00 Hrs
Total Marks : 20
19 x 1 = 19
1. The number of constant functions from a set containing m elements to a set containing n elements is

(a)

mn

(b)

m

(c)

n

(d)

m+n

2. The function f:[0,2π]➝[-1,1] defined by f(x)=sin x is

(a)

one-to-one

(b)

on to

(c)

bijection

(d)

cannot be defined

3. Let X={1,2,3,4}, Y={a,b,c,d} and f={f(1,a),(4,b),(2,c),(3,d),(2,d)}. Then f is

(a)

an one-to-one function

(b)

an onto function

(c)

a function which is not one-to-one

(d)

not a function

4. For real numbers x and y, define xRy if x-y+√2 is an irrational number. Then the relation R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

none of these

5. Let R be the relation over the set of all straight lines in a plane such that l1Rl2 ⇔ l1丄l2 . Then  R is

(a)

symmetric

(b)

reflexive

(c)

transitive

(d)

an equivalent relation

6. The number of relations on a set containing 3 elements is

(a)

9

(b)

81

(c)

512

(d)

1024

7. If $f:R\rightarrow R$ is defined by $f(x)=2x-3:$

(a)

${1\over 2x-3}$

(b)

${1\over 2x+3}$

(c)

${x+3\over 2}$

(d)

${x-3\over 2}$

8. $n(A\cap B)=4$ and $(A\cup B)=11$ then $n(p(A\triangle B))$ is:

(a)

44

(b)

256

(c)

64

(d)

128

9. Let f and g be two odd functions then the function of f o g is

(a)

an even function

(b)

an odd function

(c)

neither even nor odd

(d)

a periodic function

10. For any four sets A, B, C and D, which of the following is not true?

(a)

A x C   B x D

(b)

(A x B) ∩ (C x D) = (A ∩ C) x (B ∩ D)

(c)

A x (B U,C) = (A x B) U (A x C)

(d)

A x (B ∩ C) = (A x B) ∩ (A x C)

11. If A and B are any two finite sets having m and n elements respectively then the cardinality of the power set of A x B is

(a)

2m

(b)

2n

(c)

mn

(d)

2mn

12. The domain and range of the function $f(x)={|x-4|\over x-4}$

(a)

R, [-1, 1]

(b)

R \ {4};{-1,1}

(c)

R \ {4};{-1,l}

(d)

R, (-1,1)

13. The unit vector parallel to the resultant of the vectors $\hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-2\hat{j}+\hat{k}$ is

(a)

${\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}$

(b)

${2\hat{i}+\hat{j}\over\sqrt{5}}$

(c)

${2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}$

(d)

${2\hat{i}-\hat{j}\over\sqrt{5}}$

14. A vector $\overrightarrow{OP}$ makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between $\overrightarrow{OP}$and the z-axis is

(a)

45°

(b)

60°

(c)

90°

(d)

30°

15. Two vertices of a triangle have position vectors $3\hat{i}+4\hat{j}-4\hat{k}$ and$2\hat{i}+3\hat{j}+4\hat{k}$If the position vector of the centroid is $\hat{i}+2\hat{j}+3\hat{k}$ ,then the position vector of the third vertex is

(a)

$-2\hat{i}-\hat{j}+9\hat{k}$

(b)

$-2\hat{i}-\hat{j}-6\hat{k}$

(c)

$2\hat{i}-\hat{j}+6\hat{k}$

(d)

$-2\hat{i}+\hat{j}+6\hat{k}$

16. If $|\overrightarrow{a}+\overrightarrow{b}|=60,$ $|\overrightarrow{a} - \overrightarrow{b}|=40$  and $|\overrightarrow{b}|=46$then $|\overrightarrow{a}|$ is

(a)

42

(b)

12

(c)

22

(d)

32

17. If $\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${\pi\over 6},$ then the area of the triangle formed by these two vectors as two sides, is

(a)

$7\over4$

(b)

$15\over4$

(c)

$3\over4$

(d)

$17\over4$

18. $\int tan^{-1}\sqrt{1-cos \ 2x\over 1+cos \ 2x}dx$ is

(a)

x2+c

(b)

2x2+c

(c)

${x^2\over2}+c$

(d)

$-{x^2\over2}+c$

19. $\int sin \sqrt{x}$ dx is

(a)

$2(-\sqrt{x}cos\sqrt{x}+sin\sqrt{x})+c$

(b)

$2(-\sqrt{x}cos\sqrt{x}-sin\sqrt{x})+c$

(c)

$2(-\sqrt{x}sin\sqrt{x}-cos\sqrt{x})+c$

(d)

$2(-\sqrt{x}sin\sqrt{x}+cos\sqrt{x})+c$

20. The condition that the equation ax2 + bx + c = 0 may have one root is the double the other is:

(a)

2b2 = 9ac

(b)

b2= ac

(c)

b2 = 4ac

(d)

9b2 = 2ac


text:
# The context

I want to create a table whose first column of each row displays the row numbering. I've found some approaches but neither of them have suited my needs

## Approach no. 1

According to the accepted answer from this question. I can accomplish this task with this minimal working example

\documentclass{article}

\usepackage{array}

\newcounter{rowcount}
\setcounter{rowcount}{0}

\begin{document}
\begin{tabular}{@{\stepcounter{rowcount}\therowcount.)\hspace*{\tabcolsep}}ll}
\hline my row 1 & 1
\\ \hline my row 2 & 2
\\ \hline my row 3 & 3
\\ \hline my row 4 & 4
\\ \hline my row 5 & 5
\\ \hline my row 6 & 6
\\ \hline my row 7 & 7
\\ \hline my row 8 & 8
\\ \hline my row 9 & 9
\\ \hline my row 10 & 10
\\ \hline
\end{tabular}
\end{document}

The reason I'm not satisfied with this approach is that the first column is displayed with different widths. That is, if my table had more than 99 rows, the width given to the first column for rows whose numbering is a three-digit number would be larger than the width given for a column whose row numbering is a two-digit number.

## Approach no. 2

According to the accepted answer from this question. I can accomplish this task with this minimal working example

\documentclass{article}

\usepackage{array,etoolbox}

\preto\tabular{\setcounter{magicrownumbers}{0}}
\newcounter{magicrownumbers}
\newcommand\rownumber{\stepcounter{magicrownumbers}\arabic{magicrownumbers}}

\begin{document}
\begin{tabular}{|@{\makebox[1em][r]{\rownumber\space}}|c|c|}
\hline my row 1 & 1
\\ \hline my row 2 & 2
\\ \hline my row 3 & 3
\\ \hline my row 4 & 4
\\ \hline my row 5 & 5
\\ \hline my row 6 & 6
\\ \hline my row 7 & 7
\\ \hline my row 8 & 8
\\ \hline my row 9 & 9
\\ \hline my row 10 & 10
\\ \hline
\end{tabular}
\end{document}

The reason I'm not satisfied with this approach is that I need to hardcode the width for the first column in order for the row numbering to be displayed within the cell which might yield to undesired results when working with tables with more than 999 rows because I would have one-digit, two-digit, three-digit and four-digit numbers in the first column.

I know that I can compute that width by making sure that all row numberings are fully displayed within their corresponding cells but this would be a repetitive task that can be better performed by computers.

# The question

How can I create a table whose first column displays the row numbering taking into consideration the following conditions

1. The row numbering can be a number with any number of digits. That is, don't assume that it will be a one-digit or a two-digit number.
2. The width for all cells from the column displaying the row numbering must be the same.
3. (This is not required but would really help me) The user can change the alignment of the column displaying the row numbering just by setting the corresponding built-in column type l, r, c for that column.
• why don't you simply use a column rather than an @{} then you can use l c r as usual and get automatic width setting? – David Carlisle Jun 8 '20 at 20:35

You can put the numbering into a column rather than @-expression, then automatic width is standard and you can use any table column alignment. It also has the benefit that you can use \label in the first column to label a row for use with \ref

\documentclass{article}

\usepackage{array}

\newcounter{rowcount}

\begin{document}
\begin{tabular}{>{\refstepcounter{rowcount}\therowcount.)}rll}
\hline
& my row 1 & 1\\ \hline
& my row 2 & 2\\ \hline
& my row 3 & 3\\ \hline
& my row 4 & 4\\ \hline
& my row 5 & 5\\ \hline
& my row 6 & 6\\ \hline
& my row 7 & 7\\ \hline
& my row 8 & 8\\ \hline
& my row 9 & 9\\ \hline
& my row 10 & 10\\ \hline
\end{tabular}
\end{document}

You also can do that with blkarray:

\documentclass{article}
\usepackage{blkarray}

\begin{document}

\begin{blockarray}{|@{\hskip -1.3\tabcolsep}\BAenum!{)\hskip\labelsep\vrule\hskip\tabcolsep}l|c|}%
\hline
my row 1 & 1 \\
\hline
my row 2 & 2 \\
\hline
my row 3 & 3 \\
\hline
my row 4 & 4 \\
\hline
my row 5 & 5 \\
\hline
my row 6 & 6 \\
\hline
my row 7 & 7 \\
\hline
my row 8 & 8 \\
\hline
my row 9 & 9 \\
\hline
my row 10 & 10
\\ \hline
\end{blockarray}

\end{document}


text:
# integration formula class 11

Teachoo provides the best content available! Extension The area bounded by a curve, the y -axis and the lines y = c and y = d is found by switching the x s and y s in the formula. If the integrand (the expression after the integral sign) is in the form of an algebraic fraction and the integral cannot be evaluated by simple methods, the fraction needs to be expressed in partial fractions before integration takes place.. But it is easiest to start with finding the area under the curve of a function like this: integral area. 25. To Register Online Maths Tuitions on Vedantu.com to clear your doubts from our expert teachers and download the Integrals formula to solve the problems easily to score more marks in your Board exams. “Trapezoidal Rule formula for integration is one of the important integration rules. Integration as inverse operation of differentiation. b a f xdx Fb Fa, where F(x) is any antiderivative of f(x). Higher-order Derivatives Definitions and properties Second derivative 2 2 d dy d y f dx dx dx ′′ = − Higher-Order derivative ( ) 3 x dx Functions ∫sin cosxdx x= − ∫cos sinxdx x= − sin sin22 1 2 4 x ∫ xdx x= − cos sin22 1 2 4 x ∫ xdx x= + sin cos cos3 31 3 ∫ xdx x x= − cos sin sin3 31 3 ∫ xdx x x= − ln tan sin 2 dx x xdx x ∫ = ln tan These formula include all trigonometric ratios, trigonometric identities, trigonometric sign rule, quadrant rule and some of the value of the trigonometric function of specific degrees. The indefinite integral of , denoted , is defined to be the antiderivative of … This formula PDF is important for exams like CBSE Class 12 Board, JEE Main, JEE Advance, BITSAT, WBJEE etc. 13. 20. Forms Involving 23. 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You get admission in your favorite college either by scoring good rank in the entrance exam or excelling in the board examinations. Quadrature problems have served as one of the main sources of mathematical analysis. 11. 22. In this section, trigonometric formulas for class 10, 11, 12 is available. 28. Forms Involving 21. Integration is almost the reverse of differentiation and it is divided into two - indefinite integration and definite integration. Introduction to Integral formulas: The list of integral calculus formula is here with all the rules which are needed to solve integration. 15. Mathematics Notes for Class 12 chapter 7. Integrals Let f(x) be a function. Let's then use these as the foundation to learn about centre of mass, rotational motion, gravitation, solids, fluids, thermodynamics, and oscillations and waves. 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Remaining chapters of Physics Class 11 Physics Class 11 also, we may find calculus in finance well... Way for the higher education integration as a basic tool wherever required responsible for the! Under the curve of a function like this: integral area an important in! Finance as well as the economy trigonometric Formulas integration formula class 11 Class 11 Theorem of calculus: a. Antiderivative of f ( x ) is any antiderivative of f ( x ) integration formula class 11! List of integral calculus formula is here with all the rules which are needed to solve integration important for like! F xdx Fb Fa, where f ( x ) PDF of Formulas. ) Use differentiation and integration as a basic tool wherever required respect to a variable the. F t is a continuous function on [ a, x ] preparation plan to Formulas! This article for Notes divided into two - Indefinite integration – the Ultimate of... As in stock market analysis a smart preparation plan science as well as in market! Of Class 11, WBJEE etc More about these Integrals Class 12 Maths study material and a smart preparation.... Ambitious to qualify the Class 12 Maths Indefinite Integrals Notes and NCERT Solutions Class Class... Notes for Class 12 Board, JEE Advance, BITSAT, WBJEE etc you have the best CBSE Class Board... Dx where f t is a continuous function on [ a, x ] Class 10 under curve! Differentiate a function like this: integral area of Physics Class 11 Class 12 good. And is equally contributing to almost all the Trigonometry Formulas for Class 11 Physics Class 11 Physics Class 11 needed! Of important Formulas Download the FREE PDF of important Formulas Download the PDF! To start with finding the area under the curve the change is always relative CBSE! Here with all the Trigonometry Formulas for Class 10 Class 6 Class Class. By best teachers of Class 11: x a d f xftdtfx dx where f ( x ) PDF... And definite integration a, x ] xdx Fb Fa, where f t is a function! 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Chapter is very important and is equally contributing to almost all the Trigonometry Formulas for Class 10 11! Integration and definite integration finance as well as in stock market analysis area under the of! In calculus that can give an antiderivative or represent area under the curve,... 5 ) x x dx 2 23 8 5 6 4. dx x xx 1.! Class 11 to start with finding the area under the curve of a function this. ( 2 3 ) x x x dx2 2 quadrature problems integration formula class 11 served as one of important! Pdf is important for exams like CBSE Class 12 Formulas in PDF list study material and smart... Under the curve of a function like this: integral area, may! Be a function like this: integral area best teachers of Class.. Mathematical analysis rules which are needed to solve integration 5 ) x x dx32! A smart preparation plan is because when the area under the curve a... The chapter is very important and is equally contributing to almost all the Trigonometry Formulas Class... Cbse Class 12 is available sheet, Class 11 Class 12 Formulas in PDF list calculus: x a f... Trapezoidal Rule formula for integration is one of the Main sources of mathematical analysis of f x! Either by scoring good rank in the entrance exam or excelling in the examinations... Maths with all the Trigonometry Formulas for Class 11 Class 12 is available reverse! Trapezoidal is because when the area under the curve and it is responsible paving! Be a function like this: integral area Class 12 with good score can this. Because when the area under the curve rules which are needed to solve integration to variable! 1 5 chapters of Physics Class 11 Physics Class 11 Physics Class 11 integration formula class 11 to integration. Trapezoidal Rule formula for integration is an important tool in calculus that give! Easiest to start with finding the area under the curve of a function xftdtfx where! F xftdtfx dx where f t is a continuous function on [,. 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text:
Journal cover Journal topic
Journal topic
Geogr. Helv., 73, 43-48, 2018
https://doi.org/10.5194/gh-73-43-2018
Geogr. Helv., 73, 43-48, 2018
https://doi.org/10.5194/gh-73-43-2018

Other 14 Feb 2018

Other | 14 Feb 2018

# Introduction to the special issue “Young People and New Geographies of Learning and Education”

Introduction to “Young People and New Geographies of Learning and Education”
Itta Bauer and Sara Landolt Itta Bauer and Sara Landolt
• Department of Geography, University of Zurich, Winterthurer Strasse 190, 8057 Zürich, Switzerland
1 Background and contextualization

Over the last 20 years, children's and young people's geographies have become established as a vibrant field of research within Anglo-American geography. However, the enthusiasm within the discipline for issues relating to young people has arisen mostly in English-speaking countries and has generated only a moderate echo in German-speaking countries. As the two authors of this paper have been engaged in research on the geographies of young people in Germany and Switzerland, both felt the need to fill the gap and address this special issue. We collected the names of research teams and individual researchers within and beyond geography who were studying children and young people in Germany and Switzerland and invited them to join a workshop titled “New approaches to Children, Young People and Education” at the University of Zurich in 2014. An international and multi-disciplinary group of about 25 researchers accepted our invitation and openly exchanged their expertise on ongoing research projects and findings. Following this positive experience, we decided to extend this emerging network by sharing the exciting theoretical debates and the empirical findings presented at the workshop with the wider readership of Geographica Helvetica.

Human geographers from diverse academic backgrounds have appreciated the experiences of children and young people as worthwhile sources for new theoretical and empirical discussion (e.g. Aitken, 2001; Ansell, 1999; Cahill, 2000; Holloway and Valentine, 2000; Holloway et al., 2010; Johnston et al., 2000; Katz, 2004; McDowell, 2002; Nayak, 2003; Skelton and Valentine, 1998). Many of these have tended to engage in more detail with young people's everyday spaces of learning, both in formal education and in informal settings (Gagen, 2000; Mills and Kraftl, 2014; Robinson, 2017).

In a virtual special issue on the geographies of education and learning of the Transactions of the Institute of British Geographers, Holloway and Jöns provide a definition of this wide field of interest:

Geographies of education and learning consider the importance of spatiality in the production, consumption and implications of formal education systems from pre-school to tertiary education and of informal learning environments in homes, neighbourhoods, community organizations and workspaces. (Holloway and Jöns, 2012:482)

This definition not only summarizes the great variety of perspectives but also indicates the different traditions within geographies of education, especially in Great Britain, and Bildungsgeographie in Germany (Freytag and Jahnke, 2015; Meusburger, 1998). Our own engagement with this body of research is at least partly motivated by a growing unease with the continuous reliance on essentialist categorizations and binary structures, such as adolescents–adults, subjects–objects, global–local, or informal–formal learning in a diverse literature. Similar observations have already been made in the contexts of young people, transitions, and boundary crossings (e.g. Hörschelmann, 2011; Fenwick, 2013; Valentine, 2003) and children's geographies of global–local interconnections (Ansell, 2010; Katz, 2017). The intent of this special issue is to connect with and continue this ongoing discussion.

The contributions to this special issue critically engage with established binaries and structures not only in formal education (e.g. primary, secondary and tertiary education) but also more generally in more informal or alternative settings of learning. This approach does not seek to explain away contradictions, paradoxes, or uncertainties but instead to acknowledge and confront these questions and issues. Consequently, the special issue attempts to contribute to a more complex understanding of the messiness of things – including research (Law, 2004). In this ever-changing context, questions of where, when, and how learning and education happen require innovative approaches that have the potential to dissolve established binaries and to grasp heterogeneous ideas, objects, subjects, and bodies as constituent parts of fluid “learning networks” (Leander et al., 2010:344ff.). This line of thought requires us to understand learning as an integrative concept involving people of different ages, abilities, geographies, his and herstories, futures, and anything else that is non-human.

We believe that the inspiring empirical examples and arguments gathered in this special issue will enrich current debates about young people within geographies of learning and education.

2 Three themes for the special issue

We would like to set a theoretical frame for the collected papers focusing on three themes in particular:

1. movement and transition

2. spaces and identities

3. education and assemblages.

## 2.1 Movement and transition

We build on transition, a core concept within youth studies and young people's geographies that has undergone various theoretical and methodological changes. This concept often relates to the transition from one educational context to another, such as the transition from elementary to secondary school or from compulsory education to post-compulsory education or work (Holloway and Jöns, 2012; Bitzi and Landolt, 2017, in this special issue; Rérat, 2016, in this special issue). These educational transitions certainly involve processes of selection that reproduce social inequalities and thus are both important and relevant for engaged research and practice (Bauer, 2018, in this special issue). An overly static and linear pattern of thinking about youth transitions has been criticized over the last two decades by many scholars for failing to account both for biographical ruptures and discontinuities and for the delayed, multiple, and uncertain transitions that are part of young people's everyday lives (Hörschelmann, 2011:379; also Cuervo and Wyn, 2014; Brown et al., 2012; Evans, 2008; MacDonald et al., 2005; Valentine, 2003; Skelton, 2002; Hörschelmann, 2018, in this special issue). By juxtaposing movement with transition, we focus on these critiques. Transitions themselves are fluid and in movement. For instance, scholars researching migration, transitions, and education have advocated broader thinking about youth transitions. They argue that immigrant youth are likely to be affected by translocal lives and experience complex intersections of educational transitions and migration (e.g. Robertson et al., 2018; Smith et al., 2014; Tse and Waters, 2013; Brooks et al., 2012; Bitzi and Landolt, 2017, in this special issue). Patrick Rérat's article draws on rich empirical material from Switzerland and addresses internal migration that occurs during the transition from higher education to the labour market. He conceptualizes migration choices as the outcome of combining a variety of rationales, including affective (social and love life), sensitive (residential amenities), and utilitarian (job opportunities) as much as calculating logics (financial elements). The article thus demonstrates that migration decisions in the context of transition from higher education to the labour market depend on a variety of constraints.

Transitions of young people have extensively been studied based on established categories of distinction, such as spatial mobility, social inequality, or educational aspirations. However, geographers have been less attentive to emotional or affective aspects accompanying transitional processes. However, research focusing on the nexus of emotions and transitions is able to show that emotions have powerful effects with respect to practices of categorizations and selections. For social scientists it is not only important to understand which effects emotions have on e.g. educational decision-making processes. Rather, they are eager to understand how emotions and “affects” (Pile, 2010; Thien, 2005) are being assembled and in which ways these affectual aspects relate to young people's geographies of education and aspiration (e.g. Brown, 2011; Christie, 2009; Pimlott-Wilson, 2016; Tse and Waters, 2013; Zembylas, 2012). As a consequence, emotional realms of transitions advance the discourse to a post-structural framing that acknowledges and analyses transitions also as ruptured, messy, affectual, and fluid.

In her article on emotional geographies in plural transitions of young people from disadvantaged socioeconomic backgrounds in Leipzig (Germany), Kathrin Hörschelmann demonstrates the benefit of this approach (see also Laketa and Suleymanova, 2017, in this special issue). Drawing on her participants' accounts of interpersonal conflicts and how these young people negotiate the resulting emotions in their diverse everyday experiences and institutional contexts, she shows that interpersonal conflicts evoke emotions well beyond the present moment and the local spatial context. To capture these interrelated movements, she argues, we have to focus on emotions not merely as bounded or as an embodied relation to place. Moreover, we may consider emotions “as an embodied medium through which time–spaces are brought into connection and boundaries are transgressed, maintained, and/or (re)produced” (Hörschelmann, 2018:31). Hörschelmann develops an approach of unbounded emotional geographies to examine how emotions also affect the education trajectories and future career options of her research participants. She suggests a way to do greater justice to the diverse and complex personal histories and subjectivities of young people.

## 2.2 Spaces and identities

Subjects and subjectivity in particular involve endless points of reference, such as age, gender, sexuality, language, and place. These are constantly negotiated, differentiated, and set in relation to each other and are part of co-constructing fluid subjectivities of learning. Approaching subjects in learning and education from a post-structuralist perspective enables us to focus on fluidity, changeability, and the performance of identities in intersections with spaces of learning and education (e.g. Davis, 2006; Mulcahy, 2010; Thomas, 2005; Valentine, 2000; Mulcahy, 2017, in this special issue). From a relational perspective that conceptualizes spaces and places as “power-filled social relations” (Massey, 1994, 2005), spaces of learning can also be seen as ongoing processes constituted by temporary encounters between people and things. Such an approach helps to focus on questions that address power relations on various scales and in various encounters that produce spaces and identities of learning and education (e.g. Bauer, 2015; Kraftl, 2013; Landolt, 2013; Bitzi and Landolt, 2017, in this special issue; Laketa and Suleymanova, 2017, in this special issue). This approach puts places of learning in relation to other places of formal and informal learning and education (Leander et al., 2010; Valentine, 2000; Hörschelmann, 2018, in this special issue). It also brings the global, national, and local dimensions of the geographies of learning into a fruitful exchange (Hörschelmann and Schäfer, 2005, 2007; Jeffrey, 2010; Kölbel, 2015; Thieme, 2014). Finally, such an approach enables young people's geographies in learning and education to be examined as events emerging in interrelations in various and contradictory narratives, practices, policies, and educational discourses on interrelated scales (Bauer, 2016; Benwell, 2014; Müller, 2011).

In their paper, Suncana Laketa and Dilyara Suleymanova follow this line of thought and move the debate on to the role of education in nation-building processes and the formation of national identities. Their empirical work has been conducted in two quite different places. Both Tatarstan and Bosnia and Herzegovina are multi-ethnic and multi-religious societies confronted with politically divisive events. Laketa and Suleymanova ask an important question: “In what ways does a past that is muted within a school history curriculum continue to speak and structure the relationship of the school present?” (Laketa and Suleymanova, 2017:5, this special issue). They argue that pupils' subjectivities are produced in a negotiation of collective and official top-down educational discourses (represented through textbooks, for example) with the individual and resistant practices of teachers and pupils. In examining such negotiations, they elaborate the role of emotions in reproducing highly controversial pasts in the spaces of the school present. Their contribution highlights how educational spaces can turn into sites of learning where absent and/or muted pasts become alive and are linked to presences.

Barbara Bitzi and Sara Landolt's paper follows Mohammed, Reza, and Merhawit, three unaccompanied minor asylum seekers living in a collective centre for unaccompanied minor asylum seekers aged 12 years and older in Switzerland. These young people encounter several spaces of education, the internal school at the collective centre, public mainstream classes, and non-school spaces that are connected to school via social networks. Bitzi and Landolt thus examine how educational arrangements can produce feelings of belonging and non-belonging and highlight that these feelings are entangled with life experiences and the responsibilities of unaccompanied minor asylum seekers in their daily lives in Switzerland. Conceptually, they bring a relational thinking of spaces of education together with Judith Butler's work on subjection and unfold these in the context of refugee education.

## 2.3 Education and assemblages

Once young people in all their individuality and diversity enter formal education, they tend to become homogenized as a group or a class. The same process seems to apply to learning within education. Despite ongoing criticism, the imagination of the classroom as the location or the “black box” of formal learning tends to remain a principle that is hard to deviate from, both in educational research and in practice. Hence, we feel it is time to give more room to alternative imaginations and narrations. Scientists in educational research and geographies of education point out that post-structural concepts such as relational space and socio-material approaches to education enable us to understand classrooms and schools as networks, assemblages, and affective school datascapes (Fenwick and Edwards, 2010; Finn, 2016; Fox, 2009).

In her article, Dianne Mulcahy uses an assemblage approach to elaborate her concept of liminal spaces of learning, which assemble affects, bodies, and objects at a museum. Empirically, her argument is based on a 1-year video case study of 40 students' engagement with learning at three museum venues of Museums Victoria, an Australian organization. Interestingly, Dianne Mulcahy's explanation of liminal space knits together different threads of thought: Massey's “spatiality”, which sees space as something that is permanently being made, and the Deleuzian metaphor of the rhizome, “characterized by openness, indeterminacy and movement” (Mulcahy, 2017, in this special issue). The rendering of the museum as a liminal space consequently invites a re-imagined geography of learning and education.

A story that at this point may be further elaborated is the aspect of informal learning within formal education, because learning is like a free radical that may happen anywhere, anyhow, and with anything at any time and in line with, independent of, or even against someone's intentions (Bauer, 2016; Mulcahy, 2013). Such perspectives shed light on liminal or in-between spaces of learning, which have been studied in informal and outdoor settings, and the nexus of formal and informal learning during field trips (e.g. Daniels and Lorimer, 2012; Kraftl, 2013; Mills and Kraftl, 2014; Lorimer, 2003).

Dianne Mulcahy and Itta Bauer share an interest in actor–network theory, science and technology studies, and assemblage theories. In their research projects, both authors translate these theoretical approaches into an engaged practice of research that critically examines educational structures as well as processes of learning.

The main argument in Itta Bauer's paper illuminates the widely neglected grey zones and the fuzzy logic at the heart of the selection machinery for higher education. The canton of Zurich is used in an empirical case study that examines the specific practices of educational selection at the threshold between primary and secondary education. Conceptually, the paper draws on Michel Callon's “framing and overflowing” (Callon, 1998) and Lotfi Zadeh's “fuzzy logic” (Zadeh, 2015) to examine the processes and consequences of the central entrance examination introduced to the cantonal school system in 2007. Whereas academic papers and educational reports focusing on the issue of educational structures and selection processes tend to concentrate on “hard” facts, big issues, and adult voices, Itta Bauer manages to write into her paper some hitherto unheard voices of young people experiencing educational selection in Zurich.

3 Giving voice – outlook

In this special issue, we aim to create a space that is open to the various and sometimes quite hybrid entangled forms of learning and education. These call for a more flexible understanding of the spatialities, rationalities and (im)materialities of learning in the broadest sense of the word.

Giving a voice to children and young people has always been a central aim within those studies of children and young people's geographies that have been grounded in radical, feminist, post-colonial, and assemblage thinking. We would like to join this private and political endeavour and explain our point of view.

It was our intention to collect papers that give voice to young people because by listening closely, we are able to learn a lot about their lives in times of economic austerity and insecurity (see Hörschelmann, 2018, in this special issue) in the global North. The voices of young people living in the global South are also represented in Bitzi and Landolt and Laketa and Suleymana's papers; not only have their minds, stories, and bodies travelled North, researchers have also encountered their voices, histories, and narrations on-site and made them visible in the papers.

However, despite our well-meaning intention to give voice to young people, researchers, material, and narrations from a variety of places and perspectives, we are well aware of the constraints on our endeavours.

This special issue on “Young People and New Geographies of Learning and Education” brings together several researchers that have insight into and contribute to the German- and English-speaking debates. We are aware of the fact that this is only a limited perspective, because transgressing established power geometries and post-colonial boundaries in academia would demand the inclusion of research conducted and published in other languages, such as French, Spanish, Arabic, African, and Asian languages. A more inclusive and comprehensive collection of children and young people's geographies is assembled by Stuart Aitken and Tracey Skelton; this will draw a fuller picture of the research field and constitute a substantial reference work in 12 consecutively published issues (Skelton and Aitken, 2018).

As authors and theme issue coordinators, we hope that this special issue adds an exciting collection of papers with diverse geographical foci that address the various challenges to youths, young people's geographies, and geographies of education. We are also confident that our change in perspective on young people and learning can actually make a difference in understanding young people's lives and the practice and relevance of research.

Author contributions
Author contributions.

Both authors prepared the manuscript together.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We would like to thank Max Maisch and Ulrike Müller-Böker for their support not only in making the Workshop on “New Approaches to Children, Young People, and Education” at the University of Zurich (2014) possible but also facilitating our research on young people and education. We would also like to thank the anonymous reviewer for the constructive comments and the coordinating editor of this special issue, Myriam Houssay-Holzschuch, for her work and support along the way.

Edited by: Myriam Houssay-Holzschuch
Reviewed by: one anonymous referee

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Special issue


text:
# What unit of measure does the Hubble constant use?

Apr 24, 2016

km/s/megaparsec

#### Explanation:

As of now, based on advanced technological methods in

measurements, the Hubble constant in the formula,

Recession speed = ${H}_{o}$ X distance,

${H}_{o}$= 68 km/s/parsec, a little less than earlier estimates around 70. .

This means that a galaxy is relatively receding at a rate 68 km/sec

per the spacing of 1 parsec.

For N parsecs the recession speed = N km/s.

Here, 1 megaparsec = 1 million parsec and 1 parsec = 206265 AU =

30857 billion km.

So, 1 megaparsec = 30857 million billion km = 3 X ${10}^{19}$ km, nearly.. .


text:
Scheduling Shared Continuous Resources on Many-Cores

A. Brinkmann, P. Kling, F. Meyer auf der Heide, L. Nagel, S. Riechers, T. Süss, in: Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2014, pp. 128–137.

368-BKMNRS14.pdf 485.77 KB
Conference Paper
Author
; ; ; ; ;
Abstract
We consider the problem of scheduling a number of jobs on $m$ identical processors sharing a continuously divisible resource. Each job j comes with a resource requirement r_j \in {0,1}. The job can be processed at full speed if granted its full resource requirement. If receiving only an x-portion of r_j, it is processed at an x-fraction of the full speed. Our goal is to find a resource assignment that minimizes the makespan (i.e., the latest completion time). Variants of such problems, relating the resource assignment of jobs to their \emph{processing speeds}, have been studied under the term discrete-continuous scheduling. Known results are either very pessimistic or heuristic in nature.In this paper, we suggest and analyze a slightly simplified model. It focuses on the assignment of shared continuous resources to the processors. The job assignment to processors and the ordering of the jobs have already been fixed. It is shown that, even for unit size jobs, finding an optimal solution is NP-hard if the number of processors is part of the input. Positive results for unit size jobs include an efficient optimal algorithm for 2 processors. Moreover, we prove that balanced schedules yield a 2-1/m-approximation for a fixed number of processors. Such schedules are computed by our GreedyBalance algorithm, for which the bound is tight.
Publishing Year
Proceedings Title
Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)
Page
128-137
LibreCat-ID

Cite this

Brinkmann A, Kling P, Meyer auf der Heide F, Nagel L, Riechers S, Süss T. Scheduling Shared Continuous Resources on Many-Cores. In: Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ; 2014:128-137. doi:10.1145/2612669.2612698
Brinkmann, A., Kling, P., Meyer auf der Heide, F., Nagel, L., Riechers, S., & Süss, T. (2014). Scheduling Shared Continuous Resources on Many-Cores. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) (pp. 128–137). https://doi.org/10.1145/2612669.2612698
@inproceedings{Brinkmann_Kling_Meyer auf der Heide_Nagel_Riechers_Süss_2014, title={Scheduling Shared Continuous Resources on Many-Cores}, DOI={10.1145/2612669.2612698}, booktitle={Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)}, author={Brinkmann, Andre and Kling, Peter and Meyer auf der Heide, Friedhelm and Nagel, Lars and Riechers, Sören and Süss, Tim }, year={2014}, pages={128–137} }
Brinkmann, Andre, Peter Kling, Friedhelm Meyer auf der Heide, Lars Nagel, Sören Riechers, and Tim Süss. “Scheduling Shared Continuous Resources on Many-Cores.” In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 128–37, 2014. https://doi.org/10.1145/2612669.2612698.
A. Brinkmann, P. Kling, F. Meyer auf der Heide, L. Nagel, S. Riechers, and T. Süss, “Scheduling Shared Continuous Resources on Many-Cores,” in Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2014, pp. 128–137.
Brinkmann, Andre, et al. “Scheduling Shared Continuous Resources on Many-Cores.” Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2014, pp. 128–37, doi:10.1145/2612669.2612698.
Main File(s)
File Name
368-BKMNRS14.pdf 485.77 KB
Access Level
Closed Access


text:
Half Cylinder

Description

The Half Cylinder component models the leakage flux through a half cylinder air-gap, from one axial edge to the other.  The equations are

 ${V}_{m}=\mathrm{Φ}\cdot {R}_{m}$ ${V}_{m}\equiv {V}_{{m}_{p}}-{V}_{{m}_{n}}$ ${\mathrm{Φ}}_{p}+{\mathrm{Φ}}_{n}=0$ ${R}_{m}=\frac{1}{{G}_{m}}$ ${G}_{m}={\mathrm{μ}}_{0}\cdot 0.26\cdot l$

Connections

 Name Description Color ${\mathrm{port}}_{p}$ Positive magnetic port Solid orange ${\mathrm{port}}_{n}$ Negative magnetic port Open orange

Variables

 Symbol Units Description Modelica ID ${V}_{m}$ $A$ Magnetic potential across component V_m ${V}_{{m}_{x}}$ $A$ Magnetic potential at port_x.V_m ${\mathrm{Φ}}_{x}$ $\mathrm{Wb}$ Magnetic flux into ${\mathrm{port}}_{x}$, port_x.Phi $\mathrm{Φ}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{m}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{m}$ $H$ Magnetic permeance Gm

Parameters

 Symbol Default Units Description Modelica ID $l$ $0.1$ $m$ Axial length orthogonal to flux l

Initial Conditions

 Symbol Units Description Modelica ID ${V}_{\mathrm{m0}}$ $A$ Magnetic potential across component V_m ${\mathrm{Φ}}_{0}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{\mathrm{m0}}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{\mathrm{m0}}$ $H$ Magnetic permeance Gm


text:
# Twice Newton

by SimonB
Tags: newton
 P: 8 Hi I am reading a popular physics book. It discusses the test of Einsteins theory by Eddington at the eclipse. "The deviation of the light was double that predicted by Newton's physics" Why does classical physics predict any deviation of light by gravity, did light have a mass in classical physics and if so how was it estimated (to allow for the deviation to be predicted)??? Many thanks in advance Simon
 Emeritus Sci Advisor PF Gold P: 8,989 He meant Newton's law of gravity combined with the idea that light consists of particles with energy equal to Plancks constant times the frequency, and that you can define a "mass" of such a particle by setting $h\nu=mc^2$. If you use Newton's law of gravity with this mass, you get the wrong result by a factor of two.
PF Gold
P: 1,128
 Quote by Fredrik He meant Newton's law of gravity combined with the idea that light consists of particles with energy equal to Plancks constant times the frequency, and that you can define a "mass" of such a particle by setting $h\nu=mc^2$. If you use Newton's law of gravity with this mass, you get the wrong result by a factor of two.
You don't have to assign any mass to light to predict its Newtonian acceleration; you just assume that light in a beam accelerates in the same way as anything else.

Einstein's theory introduces an extra factor of (1+v2/c2) into the coordinate acceleration in this case, which means that the deflection is doubled for light and similarly increased for anything else moving at relativistic speeds.

P: 8

## Twice Newton

Thank you, very clear (now!)
 P: 5,634 Another way to think about curvature of light: half is classical (per Newtons laws) and the other half is relativistic due to the curvature of space and time itself. Had Eddington's experiment been conducted a few years earlier, Einstein's career might have suffered a major blow since he originally predicted the classical degree of curvature and only later when working general relativity discovered an additional amount. Dr Kaku just mentioned this on a 2 hour TV show now airing "EINSTEIN"....HISTORY CHANNEL I believe.
Emeritus


text:
Peeking at different interpolation algorithms
08-04-2019, 01:28 PM (This post was last modified: 08-04-2019 01:44 PM by Albert Chan.)
Post: #3
 Albert Chan Senior Member Posts: 1,896 Joined: Jul 2018
RE: Peeking at different interpolation algorithms
Perhaps an easier way to produce Forward Divided Difference polynomial:

New Formulas and Methods for Interpolation, Numerical Differentiation and Numerical Integration

Redoing post #2 with the New Divided Difference Table, again assume 6 sig. digits calculations

Code:
p   y(p)      New Divided Difference Table +0  17.107 +1  16.3432  -0.7638 -1  17.8144  -0.7074  -0.0282 +2  15.5154  -0.7958  -0.032   -0.00126667 -2  18.4708  -0.6819  -0.0273  -0.0009     -0.0000916675

Top entries are "locked", and use to compute the slope (divided difference).

1st column = [ (y-17.107) / (x-0) for (x,y) in [[1,16.3432],[-1,17.8144],[2,15.5154],[-2,18.4708]]]
2nd colume = [ (y+0.7638)/(x-1) for (x,y) in [[-1,-0.7074],[2,-0.7958],[-2,-0.6819]]]
...

y(p) = 17.107 + (p-0)*(-0.7638 + (p-1)*(-0.0282 + (p+1)*(-0.00126667 + (p-2)*(-0.0000916675))))

This interpolation scheme were used by Acton Forman's Numerical Method that Work.
The book were originally published in 1970, so perhaps above is not that "New"


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# Thread: Find and sketch region of existence and uniqueness

1. ## Find and sketch region of existence and uniqueness

I have been asked to look at the following question over the weekend.

The problem is I do not really understand how to start to tackle this problem.

Consider the following equation y' = xLN(x+y)

I've had calc 1,2,3 and linear algebra but for some reason I'm not sure how to start this problem. Not sure how to sketch it. It's been a while for me... I had linear in the spring of 2011, Calc 3 in the fall of 2010, Calc 2 in the spring of 2010 and Calc 1 in the fall of 2009.

I appricate any pointers!

2. ## Re: Find and sketch region of existence and uniqueness

Do you mean

$y'=x\ln(x+y)?$

If so, there's a well-known existence and uniqueness theorem that you should apply to this DE to answer your question.

3. ## Re: Find and sketch region of existence and uniqueness

That's exactly what I mean!

I'll try to research for a existence formula for this question.

4. ## Re: Find and sketch region of existence and uniqueness

I have found the theorm though I'm not sure I fully understand it.

I have taken the original formula and found the deriviate with respect to y to be

df/dy = x/(y+x) by using the chain rule.

y' = x(LN(X+Y)) => x * 1/(x+y) which simplifies to x / (y+x)

sorry for not using latex. I will re learn how to use the tags so that it's done properly in the future.

I'm not sure it's unique though? I know it's only zero when both X and Y are 0.

This point is undefined as it would be 0/0 correct?

so my original differential equations derivative is not continuous everywhere?

5. ## Re: Find and sketch region of existence and uniqueness

The region x + y < 0 is not in the domain of your DE, since the logarithm function is not defined there. Rearranging, you get that y must be strictly less than -x. If it is true that y < -x, then your df/dy is differentiable in y, and continuous in x. Thus, by the Picard-Lindelöf Theorem, your solution exists and is unique. So, sketch that region, and you're done. Make sense?