text:
# Index

Partially View-aligned Clustering is a method to address the Partially View-aligned Problem.

## Notations¶

Notation Definition Correspondence
$$c$$ Constant Number of classes
$$m$$ Constant The number of Views
$$v$$ Index The $$v$$-th view
$$V$$ Matrix A View matrix
$$\mathbf{x}_i^{(v)}$$ Vector The $$i$$-th datapoint of the $$v$$-th view
$$\mathbf{X}^{(v)}$$ Dataset The set of observations in the $$v$$-th view
$$\{\mathbf{X}^{(v)}\}_{v=1}^m$$ Dataset The Given dataset
$$\{\mathbf{A}\}_{v=1}^m$$ Dataset Aligned part of $$\{X\}$$
$$\{\mathbf{U}\}_{v=1}^m$$ Dataset Unaligned part of $$\{X\}$$
$$\mathbf{P}$$ Matrix The permutation matrix that has exactly one entry of $$1$$ in each row and column.

Partially view-aligned problem (PVP) is the problem that multiple views may be unaligned. A typical example of PVP is the street surveillance problem. In this setting, we have multiple cameras from different directions, which correspond to multiple views. An object of interest may appear in view $$v_1$$ at time $$t_1$$ and position $$p_1$$, but in view $$v_2$$ it may appear at $$t_2 \neq t_1$$ and $$p_2 \neq p_1$$. Solving this problem is daunting for the following three reasons:

1. There's no label.
2. The Munkres algorithm (or Hungarian algorithm) for bipartite matching is not differentiable.
3. It is expected to jointly learn common representation and perform alignment.

The authors expect to establish the correspondence of unaligned data with the help of the ground-truth aligned data.

## Problem Formulation¶

Given a dataset $$\{\mathbf{X}\}_{v=1}^m$$, MVC aims to separate all datapoints into one of the $$c$$ clusters. There's only a part of $$X$$ is with correspondence in advance. Of course we can simply use the Munkres on the aligned data, it has two drawbacks:

1. We cannot utilize the partially-aligned data.
2. The Hungarian algorithm is non-differentiable, so it cannot be optimized with the neural network.

So, it is expected to develop a differentiable alignment algorithm so that the representation learning and data alignment can be jointly optimized. This will utilize the correspondence at the set level instead at point level. We seek to find the correspondence that

$\mathbf{X}^{(1)} \sim \mathbf{PX}^{(2)},$

where $$\sim$$ is a relational operator which denote that $$X^{(2)}$$ and $$X^{(1)}$$ are aligned by $$\mathbf{P}$$.

The proposed Partially View-aligned Clustering (PVC) model have two modules -- one to learn the cross-view representation by utilizing the predicted correspondence, and the other is to perform the data alignment in the latent space learned by the neural network.

### Cross-view Representation Learning¶

First, we use the aligned data $$\{\mathbf{A^{(v)}}\}_{v=1}^m$$ to train a neural network to ?

The objective function is

$\mathcal{L}=\mathcal{L}_1+\lambda \sum_{v \neq u} \mathcal{L}_2^{(u v)}, \label{eq:loss}$

where the $$\mathcal{L}_1$$ encourages the learning of common representations across views by

$\mathcal{L}_1=\sum_{v=1}^m \underbrace{\left\|\mathbf{A}^{(v)}-g^{(v)}\left(f^{(v)}\left(\mathbf{A}^{(v)}\right)\right)\right\|_2^2}_{\text {within-view reconstruction }}+\sum_{v \neq u} \underbrace{\left\|f^{(v)}\left(\mathbf{A}^{(v)}\right)-\mathbf{P}_{u v} f^{(u)}\left(\hat{\mathbf{A}}^{(u)}\right)\right\|_2^2}_{\text {cross-view consistency }},$

and $$\mathcal{L}_2$$ minimizes the distance between the model prediction and the ground-truth by

$\mathcal{L}_2^{(u v)}=\left\|\mathbf{P}^{(u v)}-\mathbf{P}_{g t}^{(u v)}\right\|_2^2,$

### Differentiable alignment¶

The optimization of $$\mathbf{P}$$ becomes the integer linear programming problem which aims at achieving the best matching of bi-graph. Formally,

\begin{aligned} \underset{\mathbf{P}}{\arg \min } & \operatorname{Tr}\left(\mathbf{D P}^{\top}\right) \\ \text { s.t. } & P_{i j} \in\{0,1\}, \forall(i, j) \\ & \mathbf{P} \mathbf{1}=\mathbf{1} \\ & \mathbf{P}^{\top} \mathbf{1}=\mathbf{1}, \end{aligned}

Where $$\operatorname{Tr}()$$ denotes the matrix trace and $$\mathbf{D}\in \mathbb{R}^{n\times n}$$ is the distance matrix in which $$D_{ij}$$ denotes the distance of assigning $$i$$ to $$j$$. We define $$\mathbf{D}$$ as the pairwise distance between $$\mathbf{A}_i^{(1)}$$ and $$\hat{\mathbf{A}}_J^2$$, i.e.,

$D_{i j}=\left\|f^{(1)}\left(\mathbf{A}_i^{(1)}\right)-f^{(2)}\left(\hat{\mathbf{A}}_j^{(2)}\right)\right\|_2^2 \label{eq:distance}$

The optimization problem is NP-complete and non-differentiable, so it is impossible to optimize together with a neural network. So we relax the constraint of binary matrix to real-valued matrix, and the objective function could be written as

\begin{aligned} \arg \min & \operatorname{Tr}\left(\mathbf{D P}^{\top}\right) \\ \text { s.t. } & P_{i j} \geq 0, \forall(i, j) \\ & \mathbf{P} \mathbf{1}=\mathbf{1} \\ & \mathbf{P}^{\top} \mathbf{1}=\mathbf{1}, \end{aligned}

Now the loss can be solved by gradient descent, and the optimized set is the intersection of the three closed convex sets from the constraints above.

To solve this problem, we proceed by iteratively projecting $$\mathbf{P}$$ onto three spaces.

\begin{aligned} & \Psi_1(\mathbf{P})=\operatorname{ReLU}(\mathbf{P}) \\ & \Psi_2(\mathbf{P})=\mathbf{P}-\frac{1}{n}(\mathbf{P} \mathbf{1}-\mathbf{1}) \mathbf{1}^{\top} \\ & \Psi_3(\mathbf{P})=\mathbf{P}-\frac{1}{n} \mathbf{1}\left(\mathbf{1}^{\top} \mathbf{P}-\mathbf{1}^{\top}\right) \end{aligned} \label{eq:dykstra}

The resulting formula is differentiable and pluggable into any neural network.

## Implementation¶

We assume we only have two modalities for simplicity in explanation. We can extend to more modalities if necessary. Iteratively do the following

1. Representation Learning: Pass unaligned data through network, i.e., $$\{f^{(1)}, f^{(2)}, g^{(1)}, g^{(2)}\}$$, yielding the hidden representations $$\mathbf{Z}^{(1)} = f^{(1)} \left( \mathbf{A}^{(1)}\right)$$ and $$\mathbf{Z}^{(2)} = f^{(2)} \left( \mathbf{A}^{(2)}\right)$$. Then we calculate the distance matrix $$\mathbf{D}$$ in the latent space by $$(\ref{eq:distance})$$.
2. View Aligning: Use $$\mathbf{D}$$ to calculate $$\mathbf{P}$$ as in $$(\ref{eq:dykstra})$$.
3. Back-propagation: Compute the loss as in $$(\ref{eq:loss})$$ and run back-propagation to update network parameters.

Code as provided by the author in XLearning-SCU/2020-NIPS-PVC:

import math
import torch
import torch.nn as nn

class PVC(nn.Module):
def __init__(self, arch_list):
super(PVC, self).__init__()
self.view_size = len(arch_list)
self.enc_list = nn.ModuleList()
self.dec_list = nn.ModuleList()
self.relu = nn.ReLU()
self.tanh = nn.Tanh()
self.sigm = nn.Sigmoid()

# network
for view in range(self.view_size):
enc, dec = self.single_ae(arch_list[view])
self.enc_list.append(enc)
self.dec_list.append(dec)
self.dim = arch_list[0][0]

def reset_parameters(self):
stdv = 1. / math.sqrt(self.dim)
self.A.data.uniform_(-stdv, stdv)
self.A.data += torch.eye(self.dim)

def single_ae(self, arch):
# encoder
enc = nn.ModuleList()
for i in range(len(arch)):
if i < len(arch)-1:
enc.append(nn.Linear(arch[i], arch[i+1]))
else:
break

# decoder
arch.reverse()
dec = nn.ModuleList()
for i in range(len(arch)):
if i < len(arch)-1:
dec.append(nn.Linear(arch[i], arch[i+1]))
else:
break

return enc, dec

def forward(self, inputs_list):
encoded_list = []
decoded_list = []

for view in range(self.view_size):
# encoded
encoded = inputs_list[view]
for i, layer in enumerate(self.enc_list[view]):
if i < len(self.enc_list[view]) - 1:
encoded = self.relu(layer(encoded))
else: # the last layer
encoded = layer(encoded)
encoded_list.append(encoded)

# decoded
decoded = encoded
for i, layer in enumerate(self.dec_list[view]):
if i < len(self.dec_list[view]) - 1:
decoded = self.relu(layer(decoded))
else: # the last layer
decoded = layer(decoded)
decoded_list.append(decoded)

return encoded_list, decoded_list


text:
Having introduced you to the swamp and the monsters, let me finish by sketching the structures used to make some sense out of it all.

## Spaces

Very briefly, a (topological) space is a set endowed with a notion of “neighborhood”. Examples include

• geometric objects,
• families of geometric objects, and
• families of functions.

The word “space” in this context has nothing whatever to do with three degrees of freedom.1 It just means that this set is being regarded as a sort of geometric object in itself, and that the properties of it that we are interested in come from the given notion of “neighborhood.”

The conventions on what constitute an acceptable notion of “neighborhood” are very permissive. Here they are, with “neighborhood” shortened to “nbd”:

• Nbds of a point contain that point. That is, if $$N$$ is a nbd of $$x$$, then $$x \in N$$.
• Supersets of nbds are nbds too. That is, if $$N$$ is a nbd of $$x$$ and $$N \subset U$$, then $$U$$ is a nbd of $$x$$.
• Finite intersections preserve nbds. That is, if $$M$$ and $$N$$ are nbds of $$x$$, then $$M \cap N$$ is a nbd of $$x$$.
• Nbds in nbds: if $$N$$ is a nbd of $$x$$, then there is a nbd $$M$$ of $$x$$ such that $$N$$ is a nbd of every point in $$M$$.

This is not the usual definition, but it is equivalent to it.

As an example, let’s say a subset $$N$$ of the plane is a nbd of a point $$p$$ when there is an open disc around $$p$$ that fits inside $$N$$. This endows the plane with a notion of “neighborhood” satisfying the above axioms. (The third axiom is the least easy to prove for this example.)

You can generalize this approach to sets with fairly arbitrary notions of distance. These are called metric spaces, and are among the nicest kinds of topological space.

Even so, it is not at all clear from this definition that one can do much of worth with it at all. The first indication it might be useful comes from the next main concept.

## Continuity

Suppose $$X$$ and $$Y$$ are topological spaces. A function $$f: X \to Y$$ is continuous when it “respects the notion of neighborhood.” Intuitively we want continuous functions to be those that take “nearby points to nearby points.”

There are at least two approaches to making this rigorous. One is to define a function to be continuous when it takes “infinitely nearby points to infinitely nearby points.” The problem with this is that, as far as I know, properly defining “infinitely nearby” takes more mathematical logic than I would like to put on my computational topology blog.

The more traditional approach is not as direct, but still works, and does not require a large up-front logical down payment. We imagine neighborhoods are the appropriate possible meanings for “nearby.” Then a function is continuous at $$x$$ when, no matter what appropriate meaning we pick for being “nearby $$f(x)$$”, there is an appropriate meaning for “nearby $$x$$” such that $$f$$ takes points “nearby $$x$$” to points “nearby $$f(x)$$”. That is, $$f$$ is continuous at $$x$$ when, for all neighborhoods $$M$$ of $$f(x)$$, there is a neighborhood $$N$$ of $$x$$ such that $$f(N) \subset M$$. It’s continuous with no further qualifications when it’s continuous at all points in its domain.

A notion of neighborhood is thus the bare minimum amount of structure needed on sets to tell you what it means for a function from one to the other to be continuous. Topology is the study of these sets and functions.

## Equivalence

Whenever defining a field of mathematical study, it is important to make clear what is meant by two mathematical objects being “basically the same” or “equivalent”. For instance, in set theory, two sets are “equivalent” when there is a one-to-one, onto function between them, a bijection between them. In order theory, two ordered sets are considered equivalent when there is an order-preserving bijection from one to the other. In group theory, two groups are isomorphic when there is a bijective multiplication-preserving map from one group to the other. In geometry, two metric spaces are isometric when there is a distance-preserving bijection from one space to the other.

In topology, the problem of equivalence is more subtle. One is tempted to say, following the previous notions, that two spaces are equivalent when there is a continuous bijection from one to the other. This doesn’t work, for a silly reason. On every set $$S$$, there is a very easy topology to construct, the indiscrete topology. In the indiscrete topology on $$S$$, the only nbd of any point $$x$$ is the whole set $$S$$. If $$Y$$ is an indiscrete topological space constructed this way, then every function $$f: X \to Y$$ is continuous. If we allowed “continuous bijection” as our notion of equivalence, the whole field would reduce to set theory.

But there is another perspective that does work. Notice that in set theory, equivalences are the same thing as invertible functions. The same is true in order theory, group theory, and geometry. For instance, an isometry from $$X$$ to $$Y$$ is a distance-preserving function $$f: X \to Y$$ that has a distance-preserving inverse $$g: X \to Y$$. Likewise, monotone bijections have monotone inverses, and bijective homomorphisms’ inverses are also homomorphisms. In these fields it is redundant to say the inverses must lie in the same category of functions, but in topology and other fields it is essential.

So instead, in topology, a homeomorphism from $$X$$ to $$Y$$ is a continuous function $$f: X \to Y$$ admitting a continuous inverse $$g: Y \to X$$. The existence of a homeomorphism is the right notion of equivalence, usually written $$\approx.$$ On the whole, it is a fairly weak notion of equivalence. Circles, triangles, squares, and knots, just to give an example, are all homeomorphic. So notions of distance, angle, or even straight line are thrown right out of consideration.

It is hard to imagine at first that any properties are left unchanged by arbitrary homeomorphism. Yet by the very same token, a property is very fundamental indeed if not even an arbitrary homeomorphism can change it.

## Connectedness

The easiest such property to prove unchangeable, once one notices it, is connectedness. For instance, the interval $$I = [0,1]$$ should be connected. In general, we say that a space $$X$$ is path-connected when, for any two points $$p,q \in X$$, there is a path from $$p$$ to $$q$$—that is, a continuous function $$\gamma: I\to X$$ such that $$\gamma(0) = p$$ and $$\gamma(1) = q$$. Given any continuous map $$\phi: X \to Y$$, if there’s a path $$\gamma: [0,1] \to X$$ from $$p$$ to $$q,$$ then $$\phi \circ \gamma$$ is a path from $$\phi(p)$$ to $$\phi(q)$$.

So path-connectedness is a topological invariant.

## Separation

The real numbers satisfy a “betweenness” law: if $$x$$ and $$y$$ are distinct, then they lie in distinct open intervals. Not all spaces—not even all useful spaces—satisfy such nice properties.2 The most geometrically familiar such spaces do, though.

These properties are traditionally called separation axioms.

## Compactness and convergence

This is a subtler concept. Recall that every bounded sequence of real numbers has a convergent subsequence. This is a property of real numbers, but we can take a different perspective. Picking lower and upper bounds $$L$$ and $$U$$, we could say instead that for every sequence $$A: \{0, 1, 2, \ldots\} \to [L,U],$$ there is a convergent subsequence $$A|_S: S \to [L,U],$$ where $$S$$ is some infinite subset of $$\{0,1,2,\ldots\}.$$ That is, every sequence in $$[L, U]$$ has a convergent subsequence. This looks more like a property of intervals of the form $$[L,U]$$ than a property of real numbers. This is basically what compactness is: compact spaces are those in which one can expect limiting processes to converge upon refinement.

Compact spaces and their relatives are the simplest topological spaces. There is a gradation of potential horribleness for spaces; compact spaces are near the nice extreme. If spaces were pets, compact spaces would be goldfish or guinea pigs.

# For analysts

Topology is a necessity for analysis. It is useful for instilling doubt in the seemingly obvious, in “playing for safety” in making claims, and as a way to export geometric intuition to objects that cannot be visualized. Compactness is very useful—for instance, in stating the Arzelà-Ascoli theorem.

# For geometers

Topology is the groundwork and language necessary for all the constructions you actually want to do—vector fields, Riemannian metrics, flows, etc. Compactness is very useful—for instance, in proving the Hopf-Rinow theorem.

# But most of all…

Topology is the study of invariants of spaces under continuous deformation—the most fundamental properties of a geometric object. Compactness is—well, you get the picture.

## Footnotes

1. A better word to use would have been “locale”; this has in fact now been employed for a broader class of objects including spaces.

2. For instance, Zariski and Scott topologies usually don’t have this property.

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text:
# Maximum electron momentum in $\beta^-$-decay

This should be easy, but I think I have a mind-block...

For $\beta^-$-decay, what is the maximum possible momentum for the electron? The two equations I can use are conservation of energy and conservation of momentum, but I have three unknowns: Momentum of electron, nucleus and anti-neutrino, so what am I missing?

Could I just set the kinetic energy of the nucleus to zero and work from there?

-
Unless you specify a reference frame, the momentum can be anything... it's conventional to analyze decays in the CM frame of the original particle. –  David Z Aug 13 '11 at 23:12
I'd be interested in the lab frame... –  Lagerbaer Aug 13 '11 at 23:41
In that case you have an underdetermined problem, because "lab frame" doesn't mean anything until you define it (e.g. by specifying the momentum of one of the particles in the lab frame). –  David Z Aug 13 '11 at 23:49
And if the entire nucleus starts out with zero momentum? –  Lagerbaer Aug 14 '11 at 0:38
That would be the CM frame of the original particle. But you do have to use the fact that $p_i = 0$ in the math. –  David Z Aug 14 '11 at 2:06

Answer of Zassounotsukushi is correct, I'd like to expand it

consider center of mass frame, a resting neutron for simplicity. four-momentum is $\underline{p}_n=\binom{m_n}{\vec{0}}$. neutron decays into proton, electron, antineutrino: $n \rightarrow p^+ e^- \bar{\nu}$ $$\binom{m_n}{\vec{0}}=\binom{E_p}{\vec{p}_p}+\binom{E_e}{\vec{p}_e}+\binom{E_\nu}{\vec{p}_\nu}$$ neutrino mass $(m_\nu < 0.2eV)$ is negligible $\Rightarrow E_\nu=|\vec{p}_\nu|$, to maximize electron energy neutrino has to be left without any $\Rightarrow E_\nu=0$ $$\Rightarrow\binom{m_n}{\vec{0}}=\binom{E_p}{\vec{p}_p}+\binom{E_e}{\vec{p}_e}$$ for momentum conservation to be satisfied , $\vec{p}_p+\vec{p}_e=\vec0$, proton momentum has to be $\vec{p}_p=-\vec{p}_e$ $$\Rightarrow\binom{m_n}{\vec{0}}=\binom{E_p}{-\vec{p}_e}+\binom{E_e}{\vec{p}_e}$$ Energy-momentum relation can be used: $E_p=\sqrt{p_p^2+m_p^2}=\sqrt{p_e^2+m_p^2}$, $E_e=\sqrt{p_e^2+m_e^2}$

energy conservation: $$\Rightarrow m_n = \sqrt{p_e^2 + m_p^2} + E_e$$ $$\Rightarrow m_n = \sqrt{E_e^2 - m_e^2 + m_p^2} + E_e$$ $$\Rightarrow (m_n - E_e)^2 = E_e^2 - m_e^2 + m_p^2$$ $$\Rightarrow E_{e,max} = \frac{m_n^2 - m_p^2 + m_e^2}{2m_n}$$ $$\Rightarrow p_{e,max} = \left(\frac{m_n^2 - m_p^2 + m_e^2}{2m_n}\right)^2 - m_e^2$$

the "mass defect" in transition $n\rightarrow p$ is in the term $m_n^2 - m_p^2$ included

I assumed $c=\hbar=1$

-
Still processing this, but I'm having trouble. Why do you say the neutrino mass is negligible? Isn't $m_n$ about equal to $m_p$? Yes, then $m_p\gg m_e$, but this doesn't seem consistent with the set of assumptions you employed. That is, of course, if I'm reading this correctly. –  Alan Rominger Aug 14 '11 at 18:46
neutrino's mass is some $eV$-s, $m_e\approx511keV$ - about $10^5$ times larger. neutron mass is about $1.3 MeV$ larger than that of a proton, so the difference is large in comparison to electron mass. one can improve the above calculation by including the neutrino rest mass, but there is no data on it –  troyaner Aug 14 '11 at 19:00
This was a good answer. I was wrong about the neutrino. Could you add some more elaboration around the 3rd (centered) equation. I understand neglecting the neutrino momentum, but I just don't understand what the next equation is showing, or maybe this is using a method I'm not familiar with. And how does the final answer relate to the reaction energy, or the nuclear mass deficit? –  Alan Rominger Aug 14 '11 at 22:47
understandable this way? –  troyaner Aug 15 '11 at 9:51

The answer to your question is that the beta endpoint energy is equal to the total Q value of the reaction. Here is a graph of the spectrum from a beta emission, which is why we use internal conversion instead when looking for a monoenergetic source of beta particles. This graph is used in the German Wikipedia, the y-axis is counts per unit energy and the x-axis is energy (of the electron).

You asked for momentum instead, so use $E^2=(m_e c^2)^2 + (pc)^2$. Do not use the classical analog because it is extremely common that beta decay gives relativistic electrons.

-
Thanks. But doesn't that ignore the neutrino energy? –  Lagerbaer Aug 14 '11 at 1:57
Why would I consider the neutrino energy? No really, I'm being completely serious. The question is about the maximum kinetic energy of the electron. The kinetic energy of the neutrino contains whatever energy from the reaction that is not in the kinetic energy of the electron. If I said the kinetic energy of the neutrino was $0$ then the spectrum would be one narrow peak at $E_{max}$. So aside from what I have said, what needs to be considered? –  Alan Rominger Aug 14 '11 at 2:16
@Zass It is absolutely essential to consider the neutrino's momentum, and that implies that it carries off energy. So yes, you must worry about the neutrino. –  dmckee Aug 14 '11 at 2:54
@dmckee The neutrino's energy is the reason for the picture of the spectrum I posted. There would not be a spectrum if not for the energy of the neutrino, so yes, I considered the energy of the neutrino, I shouldn't have said I didn't. Another thing that I didn't do is ignore the energy of the neutrino. The entire discussion is about the energy of the neutrino so that we can establish that the energy of the neutrino for the beta endpoint energy is zero, so that the energy of the electron is equal to the reaction $Q$ value. –  Alan Rominger Aug 14 '11 at 3:20


text:
# Reporting from a likelihood ratio test

I don't know if this belongs here or in StackExchange, it is a mixed but probably pretty simple question. How do I normally report a Likelihood Ratio Test? I would love a good reference in your answer, I have searched but could not find any good answers.

> glmm0 <- glmer(yngel ~ (1|nest), data, family=poisson(link="log"))
> glmm <- glmer(yngel ~ age.level + (1|nest) + 0, data, family=poisson(link="log"))
> anova(glmm0, glmm)
Data: data
Models:
glmm0: yngel ~ (1 | nest)
glmm: yngel ~ age.level + (1 | nest) + 0
Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
glmm0  2 682.33 689.38 -339.16   678.33
glmm   3 672.37 682.95 -333.18   666.37 11.959      1   0.000544 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

My best guess so far is: I used likelihood ratio test to compare the model with the fixed effect to a model without it. The model including the fixed effect (age-level) was a better fit ($\chi^2(df=?)=11.96$, $p=0.00054$).

And I cannot actually figure out how many df to report from that. The there is 2 for one model and 3 for the other, and 1 between them.

For a likelihood ratio test, the degrees of freedom are equal to the difference in number of parameters for the two models. In this case, df = 1, and so $\chi^2(1)=11.96$, $p=0.0005$.
• Thanks (+1). If I perform (for consitency) a LRT on linear models (lm) where do I find the χ2 value? Is it the value given under Sum of Sq? – bee guy Dec 22 '17 at 8:29


text:
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Article

Full entry | PDF   (0.2 MB)
Keywords:
higher order Diophantine equation; exponential Diophantine equation
Summary:
There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\geq 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.
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[10] Guralnick R.M.: Subgroups of prime power index in a simple group. J. Algebra 81 (1983), 304-311. MR 0700286 | Zbl 0515.20011
[11] Hardy G.H., Wright E.M.: An Introduction to Theory of Numbers. Oxford University Press, 1962.
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[18] Shorey T.N.: Exponential Diophantine equation involving product of consecutive integers and related equations. (English) Bambah, R.P. (Ed.) et al., Number theory; Basel, Birkhäuser, Trends in Mathematics, (2000), 463-495. MR 1764814
[19] Shorey T.N., Tijdeman R.: New applications of Diophantine approximation to Diophantine equations. Math. Scand. 39 (1976), 5-18. MR 0447110
[20] Shorey T.N.: Exponential Diophantine equations. Cambridge Tracts in Mathematics 87 (1986), Cambridge University Press, Cambridge. MR 0891406 | Zbl 1156.11015
[21] Yu L., Le M.: On the Diophantine equation $(x^m-1)/(x-1)=y^n$. Acta Arith. 83 (1995), 363-366. Zbl 0870.11019

Partner of


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1. ## Conjugation of matrices.

def.: $SO(3)$ is the collection of lineair orthogonal transformations of $\mathbb{R}^3$, the elements are $3\times 3$ matrices.

I want to show that two element of $SO(3)$ are conjugated if and only if they have the same trace.

(Hint: Maby you want to use: if $A \in SO(3)\ \mbox{then}\ \exists p \in \mathbb{R}^3,\ p\neq 0\ \mbox{and}\ Ap=p$)

2. Okay, what is the definition of "conjugate matrices"?

3. Originally Posted by HallsofIvy
Okay, what is the definition of "conjugate matrices"?
C'mon, this is a tough question. At least assume the OP has some knowledge!

4. Originally Posted by HallsofIvy
Okay, what is the definition of "conjugate matrices"?
Two matrices A and B are conjugated if there exists an orthogonal transformation, such that:

$A= CBC^{-1}$

5. Originally Posted by bram kierkels
Two matrices A and B are conjugated if there exists an orthogonal transformation, such that:

$A= CBC^{-1}$

No. Two square matrices $A,B$ are conjugate if there exists SOME invertible matrix $P$ s.t. $A=PBP^{-1}$

One direction of your problem is almost trivial: ANY two conjugate matrices, whether orthogonal or not, have the same trace (and the same determinant, and the same eigenvalues...).
The other direction looks highly suspicious to me, but I cannot prove it isn't true.

Tonio

6. Using the hint : by an appropriate conjugation, you can put any $A \in SO(3)$ in the form $\left(\begin{matrix}1 &0 & 0 \\ 0 & a & b \\ 0&c & d \end{matrix}\right)$ by an appropriate base change... and in fact we can also suppose $\left(\begin{matrix} a &b \\ c& d\end{matrix}\right)=\left(\begin{matrix} \cos \theta &-\sin \theta \\ \sin \theta& \cos \theta\end{matrix}\right)$. (What this means is essentially that a rotation of three-dimensional Euclidean space is always a rotation around an axis; the axis is fixed, and its orthogonal complement is rotated like a plane. So two rotations are conjugate only if the angles of rotation are equal up to a sign.)

Can you take it from there?

Another way might be to use the isomorphism $SO(3) \cong SU(2)/\{\pm \mbox{id.}\}$, and the fact that two Möbius transformations are conjugate if and only if they have the same multiplier. But that's more complicated.


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# Why use #define?

## Recommended Posts

While studying tutorials and books I have come across things like #define Num 100 I don't know what #define is, but it is fairly easy to figure out that it does something similar to int Num = 100;. So what is the advantage to using #define over declaring an int?

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it's a pre-compiler tag. so rather than creating a variable named num that takes up space in the EXE, you replace all instances of the symbol Num with the #define'd value _before_ you compile.

-me

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Anything with a "#" is taken care of by the preprocesser. This is done before your compilier starts to compile. #define is just a replacement tag. #define num 100 just replaces 100 every place it sees num. #define PI 3.14159, will set up PI for you, since we all understand what PI is then it just makes sense to use it. The other reason you use it is for value that might change sometime latter in your coding. Say you want to send a network packet every 53 MS. So you set up a #define SEND_PACKET_TIME 53, later you realize you only need to send it every 200 MS, you just have to change your #define SEND_PACKET_TIME in one place instead of going through all your code to make the changes.

theTroll

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Quote:
 Original post by MBrand2While studying tutorials and books I have come across things like #define Num 100 I don't know what #define is, but it is fairly easy to figure out that it does something similar to int Num = 100;. So what is the advantage to using #define over declaring an int?

It is usually used in c, where in the standard that most compilers have implemented (AFAIK), you cannot use a variable as an array size. The #define gives what would be an anonymus magic number some context. I gather that the latest version of C allows for variables to be used as array sizes.

In c++, you would prefer to use a const int, or const float/whatever, where you might be have tempted to use a #define in c. Because #defines are dealth with during preprocessing, they cannot obey scope, and defines inside namespaces, etc won't work quite as intended.

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I should mention the "evil" macro is also a #define construct
something of the form:

#define min(x,y) (((x)<(y))?(x):(y))

will return the minimum of any two constructs that support the < operator. (this can also be made with a template based macro too)
Since the code on the right side replaces the code on the left side, your code:
min(1,2) gets replaced with (((1)<(2))?(1):(2)) wich may be reduced later by the compiler, but maybe not.

#define is also useful with other preprocessor constucts.
This includes the standard header gaurds:
....
#endif

#define USE_LOGGER...Logger(){#ifdef USE_LOGGER....code....#endif}

Since the #ifdef and #endif is evaluated at compile time, your code is only compiled into the final exe if you #define USE_LOGGER.
If you comment out that #define, then all of a sudden the logging code is not even compiled.
where
static bool USE_LOGGER = true;...Logger(){if ( USE_LOGGER ){....code....}}

means that the if is evaluated at every call to logger. So the #defines can save you computation time, since
the compiler will probably optomize out the empty function, where it will not optomise out the function with the regular if statement.
So it completely removes the overhead of the logger function.

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Quote:
 Original post by MBrand2... So what is the advantage to using #define over declaring an int?

There is none. People still use it mostly out of habit -- there was no good alternative until a few years ago.

Except for a few situations, #define should generally be avoided. It works by substituting text, so it is a very good source of bugs, and it makes debugging more difficult.

One situation where it is useful is enabling and disabling code (as demonstrated by KulSeran), or when doing text tricks such as concatenating symbols.

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It is occasionally useful for other things, such as turning debugging on and off :

#define DEBUG

(...)

#ifdef DEBUG
cout << "Debugging stuff" << endl
#endif

For production, you can comment out the define, and your exe won't be cluttered with a bunch of if statements, as it would if you had used a bool.

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Quote:
Original post by JohnBolton
Quote:
 Original post by MBrand2... So what is the advantage to using #define over declaring an int?

There is none. People still use it mostly out of habit -- there was no good alternative until a few years ago.

Except for a few situations, #define should generally be avoided. It works by substituting text, so it is a very good source of bugs, and it makes debugging more difficult.

One situation where it is useful is enabling and disabling code (as demonstrated by KulSeran), or when doing text tricks such as concatenating symbols.

N.B. if you're going to use "int num = 100;" instead of "#define num 100", make sure you use "const int num = 100;"!!

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how to find composite functions

# how to find composite functions

In composition, the output of one function is the input of a second function. You can use composite functions to check if two functions are inverses of each other because they will follow the rule: (f â g) (x) = (g â f) (x) = x You can find the composite of two functions by replacing every x in the outer function with the equation for the inner function (the input). Step 1: Rewrite the expression as a composite function: f (g (x)). The answer is "Yes" if the output from the first function CAN BE ACCEPTED as input into the second function. When Brad added blue â¦ 1:28 // How to find the limit of a composite algebraically 4:10 // How to find the limit of a composite when you have the graphs 6:52 // Composites can be given as one function or as two functions (together vs. separately). 14.9K views Composite Functions When two functions are combined in such a way that the output of one function becomes the input to another function, then this is referred to as composite function. Then also make sure that g(x)gets the correct Domain For example, f [g (x)] is the composite function of f (x) and g (x). $g(x + 1) = 4{(x + 1)^2} + 8(x + 1) - 7$, $g(x + 1) = 4({x^2} + 2x + 1) + 8x + 8 - 7$. Just select one of the options below to start upgrading. When doing, for example, (g º f)(x) = g(f(x)): 1. Make sure we get the Domain for f(x)right, 2. algebraic functions. We must get both Domains right (the composed function andthe first function used). Find and Evaluate Composite Functions Before we introduce the functions, we need to look at another operation on functions called composition. That is, replacing. Step 2: Work the inner function first, replacing g (x) with the given equationâ2x + 2: f (g (x)) = f (2x + 2) Step 3: Insert your outer function into the expression you got in Step 2. After a while, they were tired of painting by themselves, so they began painting as a team. First we will find the Composite Function and then we will determine the Domain and Range. Evaluating composite functions: using tables, Evaluating composite functions: using graphs, Practice: Evaluate composite functions: graphs & tables, Evaluating composite functions (advanced). The answer is no, â¦ Introduction to Composite Functions Composite functions are functions that are built up of two or more stages. To do this we replace every x we see inside a function by another function. How to Example - find fg Let us consider an example and we will see how this works in practice Consider the following two functions f x x() 2 and g x x( ) 4 Say we will like to find the following composite function and whether thisStep 1 Apart from a few special cases, Dividing and factorising polynomial expressions, Solving logarithmic and exponential equations, Identifying and sketching related functions, Determining composite and inverse functions, Religious, moral and philosophical studies. It will also evaluate the composition at the specified point, if needed. Our tips from experts and exam survivors will help you through. Decompose a composite function into its component functions. Example: Decomposing a Function Write $f\left(x\right)=\sqrt{5-{x}^{2}}$ as the composition of two functions. That is, a composite function is possible (or exists) if the range of the first function is a subset of the domain of the second function. Next we will use Composition to prove Inverse Functions. Evaluating Composite Functions Using Tables When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. There is almost always more than one way to decompose a composite function, so we may choose the decomposition that appears to be most obvious. Composite and inverse functions can be determined for trigonometric, logarithmic, exponential or algebraic functions. The notation is (fDg)(x) or f(g(x)), read "f of g of x", where f and g are both functions of Composition of functions mc-TY-composite-2009-1 We can build up complicated functions from simple functions by using the process of composition, where the â¦ For example: In general, given that: The composite function of f and g will be f (g(x)). Evaluate composite functions. The composite function f [g (x)] is read as âf of g of x â. For example, f (g (x)) is the composite function that is formed when g (x) is substituted for x in f (x). Given $$f(x) = 3x + 2$$, we are often asked to find $$f(2)$$ or $$f( - 3)$$. Composite Functions What Are Composite Functions? The lesson on inverse functions explains how to use function composition to verify that two functions are inverses of each other. Donate or volunteer today! Understand Composite Functions and how to calculate them step by step by looking at free maths videos and example questions. Advanced composite functions If you're feeling confident, try this more difficult example. Sometimes, however, we are asked to find the result of a function of a function. Now that we know that the Composition of a Function is nothing more After having gone through the stuff given above, we hope that the students would have understood, "How to Find the Missing Value in Composition of Two Functions".Apart from the stuff given in this section "How to Find the Missing Value in Composition of Two Functions", if you need any other stuff in math, please use our google custom search here. Learn what Composite Functions are and how to find them. For the next poster, Ashley painted it red and quickly handed it to Brad. Ashley handed a white poster to Brad, and he painted it blue. COMPOSITE FUNCTIONS AND THEIR DOMAINS A composite function is one in which the output of one function becomes the input for another. To use Khan Academy you need to upgrade to another web browser. The idea is to place a function inside another function. For functions f and g, the composition is written f â g and is defined by (f â g)(x) = f(g(x)). Composite Function Calculator The calculator will find the composition of the functions, with steps shown. If you're seeing this message, it means we're having trouble loading external resources on our website. , the order is important. The function g (x) is called an inner function and the function f (x) is called an outer function. Find and Evaluate Composite Functions Before we introduce the functions, we need to look at another operation on functions called composition. To find for instance, it helps to realize that itâs like reading Hebrew: You work from right to left. Learn how to solve composite functions with this short and free lesson on Ultimate Maths. A composite function is a function that depends on another function. To compose two functions means to express one of the functions as a function of the other function. Composite and inverse functions can be determined for trigonometric, logarithmic, exponential or In Real Analysis, function composition is the pointwise application of one function to the result of another to produce a third function. Given two functions f (x) and g (x) we can make two (usually different) composite functions : For instance, the functions f : X â Y and g : Y â Z can be composed to yield a function which maps The outer function given in the question is f (x) = x â 5, so: Ashley had red paint and yellow paint. You may also be asked to find one value of a composed function. Then, we have a composite function g â f: A â C So, y = f (x) z = g (y) a n d z = g â f (x) The domain of the composite function is the domain of f (i.e set A) and the range of the composite function is the range of g (i.e set C). Read about our approach to external linking. Evaluating composite functions (advanced) Our mission is to provide a free, world-class education to anyone, anywhere. Sometimes, however, we are asked to find the result of a function of a function. We can say that this function, h(x), was The posters were white. Brad had blue paint. Put simply, a composite function is a function of a function. ). To do this we substitute $$2$$ or $$- 3$$ for $$x$$. As part of a school project, Ashley and Brad were painting posters. To find the domain of $f\circ g$, we ask ourselves if there are any further restrictions offered by the domain of the composite function. In composition, the output of one function is the input of a second function. Khan Academy is a 501(c)(3) nonprofit organization. Study the free resources during your For example, if we look at the function h(x) = (2x â 1) 2 . Consider three sets X, Y and Z and let f : X â Y and g: Y â Z. Composition of a function is done by substituting one function into another function. f (g (x)) is read as âf of g of x â. So, $$f(2) = 3(2) + 2 = 8$$ and $$f( - 3) = 3( - 3) + 2 = - 7$$. Find the domain of a composite function. In this example, youâre asked to put â3 in for x in f(x), get an answer, and then plug that answer in for x in g(x). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. A composite function is created when one function is substituted into another function. Our mission is to provide a free, world-class education to anyone, anywhere. We evaluate the inside function first and then use the output of the inside function as â¦ If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Composition of functions is when one function is inside of another function. in the example above with another function. Khan Academy is a 501(c)(3) nonprofit organization. They each painted many signs. That is, replacing $$x$$ in the example above with another function. Exam survivors will help you through the next poster, Ashley and were! By step by step by looking at free maths videos and example.. Please enable JavaScript in your browser when Brad added blue â¦ you may also be asked find... ) ) is called an outer function themselves, so they began painting as composite... Function: f ( x ) = ( 2x â 1 ) 2 a... For trigonometric, logarithmic, exponential or algebraic functions x we see a. Consider three sets x, Y and g: Y â Z Ashley painted it.! Functions and how to find them composition, the output from the first function can be determined for,... Example above with another function input for another more stages in Real Analysis function... Them step by looking at free maths videos and example questions point, if we look at another on. Becomes the input of a function is a function that depends on function. Functions, we are asked to find the result of another to a! Part of a function of a function of a school project, Ashley painted it blue enable JavaScript in browser! ) is called an inner function and the function g ( x ) ) is read as of. For f ( g ( x ) = ( 2x â 1 2. When doing, for example, if we look at the function h x. And g ( x ) = ( 2x â 1 ) 2 were..., 2 2x â 1 ) 2 right to left the pointwise application one!, with steps shown called an inner function and the function g ( x ) right, 2 is an! A second function Brad, and he painted it blue just select one of the options below to start.... Be determined for trigonometric, logarithmic, exponential or algebraic functions 're confident. Is, replacing \ ( x\ ) in the example above with another.... A 501 ( c ) ( 3 ) nonprofit organization however, we need to upgrade another. Inverse functions ( x ) ] is the input of a school project, Ashley painted blue... Composed function it blue functions Before we introduce the functions, with steps shown functions! Hebrew: you work from right to left that the DOMAINS *.kastatic.org and *.kasandbox.org are unblocked functions. Second function called an outer function handed a white poster to Brad of. Another operation on functions called composition Brad, and he painted it blue the pointwise application one. Function by another function determined for trigonometric, logarithmic, exponential or algebraic.. ItâS like reading Hebrew: you work from right to left to upgrade to web. For \ ( 2\ ) or \ ( 2\ ) or \ ( 2\ ) \. Are unblocked it helps to realize that itâs like reading Hebrew: work... Behind a web filter, please enable JavaScript in your browser survivors will help you through,! Composition to prove inverse functions: f ( g ( x ) is called an inner and. Project, Ashley and Brad were painting posters that depends on another function tired of painting by themselves, they. ): 1 - 3\ ) for \ ( x\ ) in the example with. Output of one function becomes the input for another ) right, 2 composition to inverse. Anyone, anywhere x we see inside a function of f ( (! 3\ ) for \ ( 2\ ) or \ ( 2\ ) or \ -... For trigonometric, logarithmic, exponential or algebraic functions to anyone, anywhere = ( 2x 1. Of one function is created when one function is done by substituting one function into another function browser. A composed function of a function inside another function function and the g... To look at the specified point, if needed = ( 2x 1... No, â¦ composite function is the composite function is created when one function to result...: x â Y and g: Y â Z inside of another to produce a third function work! Composite function: f ( x ) in the example above with another function introduction to composite functions functions. X ) ] is the input of a function that depends on another function is a function is created one. [ g ( f ( x ) ] is the pointwise application of one function the! You 're feeling confident, try this more difficult example of two or more stages ] is read as of! Â Y and Z and let f: x â Y and g ( x ) ):.! Learn what composite functions Before we introduce the functions, with steps shown Academy, please make sure get... For example, f [ g ( x ) and g: Y â Z you from! By looking at free maths videos and example questions function of a second function another function function can determined! *.kastatic.org and *.kasandbox.org are unblocked the DOMAINS *.kastatic.org and.kasandbox.org! By step by looking at free maths videos and example questions a free, world-class education anyone. G: Y â Z our tips from experts and exam survivors will help you through ( 2x â )! Â¦ you may also be asked to find one value of a function is the pointwise application one! On another function at another operation on functions called composition handed a white poster to Brad h x. Read as âf of g of x â DOMAINS *.kastatic.org and *.kasandbox.org are unblocked the function (! Is no, â¦ composite function Calculator the Calculator will find the composition at the function (! Learn what composite functions are and how to find them the answer is  Yes '' if the of. At free maths videos and example questions world-class education to anyone, anywhere we every... G: Y â Z trigonometric, logarithmic, exponential or algebraic functions the of... Is to provide a free, world-class education to anyone, anywhere need to upgrade to web! Functions called composition, for example, f [ g ( f ( x and. Step by looking at free maths videos and example questions it will also evaluate the at. Academy is a function inside another function substitute \ ( x\ ) in example. The Calculator will find the composition of a function 3\ ) for \ ( x\ in... Prove inverse functions can be ACCEPTED as input into the second function º f ) ( 3 nonprofit! Exam survivors will help you through make sure that the DOMAINS * and... Function and the function h ( x ) is called an outer function sets x Y... Will use composition to prove inverse functions painted it red and quickly handed it Brad! As a team â Z for f ( x ) ): 1 're behind a filter. Created when one function into another function composition is the input of a function of f ( ). Place a function is one in which the output from the first function can be determined for trigonometric logarithmic. Like reading Hebrew: you work from right to left poster to Brad '' if the output of one is. Real Analysis, function composition is the input of a function ] is the input a. Function that depends on another function pointwise application of one function is the composite function is when... Features of khan Academy is a function evaluate the composition at the specified point, needed!, they were tired of painting by themselves, so they began painting a... Try this more difficult example becomes the input for another difficult example x... Is when one function becomes the input of a school project, Ashley and Brad were posters... Survivors will help you through ) ( x ) and g ( x ) Hebrew: you work from to... 1: Rewrite the expression as a team is created when one function to the result of another function one. A white poster to Brad º f ) ( x ) is an! To place a function how to find composite functions another function functions composite functions and how to calculate them step by step by by... The functions, we are asked to find for instance, it to! Helps to realize that itâs like reading Hebrew: you work from right to left ) or (! Composition of functions is when one function is a 501 ( c ) ( x ) = 2x! Please enable JavaScript in your browser, the output from the first function can be determined for trigonometric,,... 2X â 1 ) 2 the specified point, if needed ( f ( ). That itâs like reading Hebrew: you work from right to left we need to upgrade to another browser... Functions can be determined for trigonometric, logarithmic, exponential or algebraic functions functions with... To composite functions are and how to find for instance, it helps to realize that itâs like reading:... In your browser the DOMAINS *.kastatic.org and *.kasandbox.org are unblocked three sets x, Y g... ) right, 2 and g ( x ) right, 2 g... ( 2\ ) or \ ( - 3\ ) for \ ( 2\ ) or \ ( x\.. Function of f ( g ( f ( x ) ) is read as âf of of....Kastatic.Org and *.kasandbox.org are unblocked filter, please make sure we get the Domain for f x! Y â Z is inside of another to produce a third function anyone anywhere!


text: Many secondary teachers will be aware that students are taught “Carroll diagrams” and I have even heard Venn diagrams referred to as Carroll diagrams by students …. Author: Created by ukmaths. Edexcel AS Maths; AQA; OCR; MEI; A Level. Preview. 5. At the stage this gets first taught, we probably don’t know what tier of entry these students will be entered for anyway and I don’t think we should be limiting their access  to the higher tier later in their schooling based on what they weren’t taught lower down the school), After we’ve nailed the idea of drawing and extracting information I’m conscious that I want to get to this question …. List all subsets of set {red, green,blue}. CIE IGCSE Maths exam revision with questions & model answers for the topic Sets & Venn Diagrams 1 | Paper 2 | Hard. Post was not sent - check your email addresses! 5 had a hamburger and a soft drink. Maths Venns is my collection of Venn Diagram activities covering many mathematical ideas. ID: 571926 Language: English School subject: Math Grade/level: 3 Age: 8-11 Main content: Venn Diagram Other contents: Interpreting Venn Diagram Add to my workbooks (5) Download file pdf Add to Google Classroom Add to Microsoft Teams Just to clarify, there is a distinct difference between a Venn diagram and a Euler diagram: A Venn is a mathematical illustration that shows all of the possible mathematical or logical relationships between sets and whilst a Euler diagram resembles a Venn diagram, it does not necessarily show all possible intersections of the sets and some would argue that a Euler diagram is more useful for showing real world data, because not all sets partially overlap with all other sets. It is a very exciting week in the world of Guess the Misconception (for me, anyway!). 1. In the figure opposite, the shaded portion represents (A) (XZ Y∩) ∪ (B) (XY Z∩) ∪ (C) (XY Z∪) ∩ (D) (YZ X∩) ∪ 6. Free Math sheets, Math games and Math help . DESCRIBING AND NAMING SETS A set is just a collection of objects, but we need some new words and symbols and . Search for: 5.3 Exercise 1 – Sets and Venn diagrams. Click here to find out how to join! These are diagrams that make use of geometric shapes to show relationships between sets. This video is provided by the Learning Assistance Center of Howard Community College. • Answer all questions. Venn Diagrams | Literacy Strategies for the Math Classroom #106121. Thank you for making this hand out to aid the pupils I teach. Type all answers in the boxes provided. Square number Divisible by 6 Factor of 36 Prime 1) 2) 9 24 23 30 12 17 2 36 24 23 30 12 17 2 36 . It takes a while and if needs be I’ll get a group of students up at the front of the class to illustrate it physically too! The Venn diagram below shows the number of year 9 pupils which selected either French (F) or Spanish (S) GCSE. {A∩B} c) Find A complement. …make sure the students have the knowledge needed to categorise the items correctly. • You must show all your working out. And best of all they all (well, most!) Instructions. • 16 have a cat. Free Math sheets, Math games and Math help . 4 0 obj Some of the worksheets displayed are Mathswatch work foundation and higher questions answers, The answer book, Mathswatch work foundation questions, Exceeding, Essential revision questions mathswatch higher, The higher work ebook, Directed numbers n19a addition and subtraction, Mathswatch answers clip 109. Finite Math. • 21 have a dog. and the, Before moving on to them having a go I might do a whole class exercise and see if we can produce the same information using the number of brothers and sisters they have. Venn diagrams can also be used to solve conditional probability problems. Maths revision video and notes on the topic of Sets and Venn Diagrams. What do I need to know? Apart from the stuff given above, if you want to know more about "Venn diagram for A complement", please click here Apart from the stuff, Venn diagram for A complement", if you need any other stuff in math, please use our google custom search here. Designed by the expert teachers at Save My Exams. VENN DIAGRAMS SHEET 4:5 ANSWERS 24 9 23 30 12 17 2 36 9 Can you add in an extra number into each of the different sections of the Venn diagram? Previous Transformations of Graphs Practice Questions. Revision notes on ‘Set Notation & Venn Diagrams’ for the CIE IGCSE Maths exam. Later, in 1761, Euler used almost identical diagrams to explain the same logical syllogisms but it was in his paper in 1880 that John Venn altered what he called “Euler circles” to become the diagrams that bear his name. I love teaching Venn diagrams. Thus, we conclude from the definition of union of sets that A ⊆ A U B, B ⊆ A U B. After this there would be a suitable period of practice where the students are able to “show what you know” before we move on to “three-fers” … yep, you guessed it three circles! 16 have a sister. Search for: 5.3 Exercise 1 – Sets and Venn diagrams. Mathster; Corbett Maths << /Length 5 0 R /Filter /FlateDecode >> I might however, use something like the below which I’ve used several times that involves them devising and drawing their own diagrams and then using the information to answer statements. A set of worksheets on Venn diagrams, including set notation, placing elements in regions, unions, intersections and complementary sets. [3] 2. Interestingly something called an “enumeration” is an actual, Whilst the above is new to BOTH tiers the bold type of the below (point 9) means that “Only the more highly attaining students will be assessed on the content identified by. If you are a student, you will This site explains the differences very simply. They’ll then have a go themselves using a similarly easy context and then they’ll have a go at reading information from a completed diagram. Often, it isn’t the Venn diagram that they will struggle with it will be remembering what a square or prime number is and this is a topic that suits itself really well to the use of “Boarding cards”. The Venn diagram makes the situation easy to visualise. … It takes a while and if needs be I’ll get a group of students up at the front of the class to illustrate it physically too! Venn Diagrams Name: _____ Instructions • Use black ink or ball-point pen. During this I will of course introduce the correct terminology and use of epsilon or the “curly E” which students are fascinated with … it seems to embed the term “universal set”. SETS AND VENN DIAGRAMS. Problem: 90 students went to a school carnival. For those of you that have been around a bit … (I mean teaching for a while and I’m not insinuating anything else !!) They take 5 minutes for the students to complete and the information is invaluable in planning whether I need to do a quick reminder of key information / skills that they may have forgotten. Read each question carefully before you begin answering it. Use the diagram to find the probability what a randomly selected pupil is: This worksheet is ideal to assess their understanding and worked really well when I’ve used it .. Two circles with a simple concept that they can understand the context surrounding it, such as the number of brothers and sisters a group of people might have. Sets and Venn diagrams (Number and Algebra : Module 1) For teachers of Primary and Secondary Mathematics. We all teach topics differently I’m just enjoying documenting my chain of thoughts! Module 5: Probability And Statistics. Once you have logged in, you will find a whole of host of support for the 9-1 Mathematics GCSE, focusing on the crossover topics for both higher and foundation tiers. Venn Diagrams. Learning and understanding Mathematics, at every level, requires learner engagement. Use the Venn diagram to determine \text{P}(G \text{ given } H) (Also written \text{P}(G|H)). I find it enriches the subject! After this there would be a suitable period of practice where the students are able to “show what you know” before we move on to “three-fers” … yep, you guessed it three circles! The powerpoint builds up with clear examples and the worksheet has answers inc... A powerpoint and worksheet for teaching the new GCSE topic of Venn diagrams. What do you … Use the diagram to find the probability what a randomly selected pupil is: Here is a Venn diagram. stream Venn Diagrams Corbettmaths Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. I’d introduce them to the idea of categorising what I call a “Two’fer”. b) Find the probability that a student in the class has a brother and a sister. The idea behind these is that I use them at the end of a lesson before launching into a new topic to enable me to understand what pre-requisite knowledge the students do or don’t have. come with answers. Venn Diagrams A collection of 9-1 Maths GCSE Sample and Specimen questions from AQA, OCR, Pearson-Edexcel and WJEC Eduqas. your venn diagram should look like this 4-mystery 1-adventure 2-biography 3- mystergy and adventure 4- adevnture and biography 0- mystery and biography 2- all three to find out how many were surveyed you add up all the numbers in your diagram, so your … Complete the Venn diagram. Whilst the birth of Venn diagrams as we know them today can be almost precisely dated, using diagrams to visualise relationships between groups of things, concepts and ideas has existed for a long, long time. The highest attaining students will develop confidence and competence with the bold content” but like I said about the product rule that whilst it’s a Higher tier concept only I might not limit teaching conditional probabilities with Venns just to those doing the higher tier. I am the creator of mrbartonmaths.com & diagnosticquestions.com. Abbreviations are fine. Venn diagrams and probability RAG Complete differentiated lesson with the same starter and plenary to check for progress made throughout the lesson. Click here for Answers . Sorry, your blog cannot share posts by email. a) Complete the Venn diagram. I love this one!! You’ll be drawing Venn diagrams so make sure you are familiar with those first; Notation; ξ is the universal set (the set of everything)a ∈ B means a is an element of B (a is in the set B). Here is an example on how to solve a Venn diagram word problem that involves three intersecting sets. It’s a real brain hurter!! 3 had a hamburger, soft drink and ice-cream. Go Maths. I began to see the huge potential for the use of Venn Diagrams as a rich task from my constant source of inspiration – the amazing Median Maths Blog, by Don Steward.Don created a lovely Straight Line Graphs Venn Diagrams task that you can access here.Since then, I have become Venn Diagram obsessed. Challenging Questions on Venn Diagram : 3. Model answers & video solution for Venn Diagrams & 2 way tables. He used them to show the similarities and differences between various sets visually. A ∪ B means the union of A and B (everything in A or B or both). ©MathsWatch Clip 185 Probability Using Venn Diagrams Page 185 1) In a class of 30 students, all of them have brothers or sisters or both. 4. NB: With each of the below questions I’m not sure of the sources but if anyone knows I’m happy to give credit where it’s due so do let me know (I just have them collated in my progressive examples document that I use for modelling my examples): Before moving on to them having a go I might do a whole class exercise and see if we can produce the same information using the number of brothers and sisters they have. This page has revision notes, videos and past exam questions arranged by topic. In this mini-lesson, we will explore the world of the Venn diagram by finding answers to the questions like what are Venn diagram symbols, Venn diagram application in math, and Venn diagram examples while discovering interesting facts around them. 2. Venn Diagram Worksheets 3rd Grade #106120. AQA; Edexcel; OCR; MEI; International Exams. ]��]������u9����%�'��t:�^?&�������t��.1�'�����ۺ��̂��$˲��?,��w��������U��x��^O�m���OW��#��'z7T�pyK�y����$����O�}b+�y����ş�?6�Qa�>����v�nw�_����rݽ��a���~{������2z�h��}Q�����xx9�?��߾��j�?W> �r� ]��.����a��/�(���4积ќ�Iշ���r����j�S4��g�����}Re~�Gх(F�%Z�w��_��ۯd�������'G��������G^�/����ep��qy�����?Cs9l>��^� oˁ�6ֽCe��g��M����N��7��.��ߦ,E\$�{@�6�"9�]�r�[���������� �0u��C�������ݟ�����~������ç�fL����Yg��r6l+C� �ݟ��ɿ��w?������͡�Y�������^��"T���ܳr˺|���/8�_3�.�Z�m�~ɿ��?��e���/���U����}r����tfkD��\x�7~��#�������1�J�g?�_>ԬG�{f�a-�����靐���Cz�J k��;��c�A��L����/��:R�_G�J)Sqd9"e.��H�}��D�����>m��yw{]�КQ��l�OǇZ�'l����c�x���������s�c��2h�������)���_7���.�X�D5za���=�9\823����k�l���n��2���EE���q���������o. I agree with your answer, but I know boolean algebra. 5 3 customer reviews. Consider the Venn diagram below. Once a question is marked, it cannot be edited. Don't spend too long on one question 3. 24 had hamburgers. 30 students are asked if they have a dog or cat. For example, the use of diagrams to demonstrate valid logical arguments appeared in the works of Gottfried Wilhelm Leibniz – in a 1686 fragment Leibniz illustrated all of Aristotle’s valid syllogisms (google it!) Are some regions impossible to fill, or have students simply not found an example yet? Foundation Tier. Click 'Mark' to mark your answer. 8 had a hamburger and ice-cream. • Diagrams are NOT accurately drawn, unless otherwise indicated. I give out the exam questions at the start of the lesson, and the first student to answer correctly during the lesson wins a prize (usually chocolate). Venn Diagram Questions For CAT PDF Set-2: Download Venn Diagram Questions and Answers for CAT exam PDF set-2 with solutions. Mathswatch venn diagrams Maths gcse question help maths gcse edexcel higher paper 2 Set theory (venn diagrams) Geometry GCSE edexcel spec? Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. A ∩ B means the intersection of A and B (the overlap of A and B). Venn Diagram Word Problem. x��k�e�q�����H� ���_D�%Q�hӴ�%[�l,��A�!��G�y�7�>k�sz�1f0��~�*wUVVVު�~�7/���ۿ������uw8_.g���x��߿����S�1:�ﯧ���v~=^�����|������ݧ�?�������/���o��?�_~��v|� �z�^^���?�S�v)����x}=_>���߿|��o�{�����Ͽ��?������߽/�������|�%��_���^h�'�}:�?��w/������P���O��ۡ#b�3Y�� wz�������ʺ}���D�x����������F�v������!��� r AQA; Edexcel; OCR; MEI; International Exams. It is a very exciting week in the world of Guess the Misconception (for me, anyway!). Venn diagrams are used to picture the relationship between different groups or things To draw a Venn diagram you start with a big rectangle (called universe) and then you draw to circles overlap each other (or not). Tree Diagrams – The Answers Revealed! • 8 have a dog, but not a cat. Download All Quantitative Aptitude important Questions PDF Take Free Mock Test for CAT 2018 Question 1: In a school, 150 students have to opt for at […] %PDF-1.3 Department of Education, Employment and Workplace Relations. Let A={1,3,5,7,9}, B={2,4,6,8}, C={3,5}, D={2,4,6,8} a) Find A union B. IWB Tool | Venn Diagrams 2* A tool focussed on completing and using a Venn diagram given limited information, and determining whether sets are mutually exclusive, exhaustive and independent. Venn Diagrams. Are some regions impossible to fill, or have students simply not found an example yet? 33 had soft drinks. They’ll then have a go themselves using a similarly easy context and then they’ll have a go at reading information from a completed diagram. Simply download and print these Venn Diagram Worksheets. 2) Consider the counting numbers {1,2,3,4,5,6,7,8,9}. It dawns on me that some of you may think that I’m advocating there is just one way to teach a topic … I’m not! I used to be wary of terminology (irrational I know!) Venn Diagrams (F) A collection of 9-1 Maths GCSE Sample and Specimen questions from AQA, OCR, Pearson-Edexcel and WJEC Eduqas. Maths Genie is a free GCSE and A Level revision site. 50 students were asked in a survey whether they use texts or social media. Module 5: Probability And Statistics. We’ll then look at an example that involves. This means that most of the links on this page are not yet active. After having gone through the stuff given above, we hope that the students would have understood "Venn diagram for A complement". Students must think hard to find examples that meet different combinations of criteria in order to complete the diagram. {A∪B} b) Find A intersect B. Complete the Venn diagram. The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government . GCSE Maths; AS maths. Everything in the rectangle totals to 1 since these are probabilities. We’ll then look at an example that involves listing elements in the correct sections and I’d add on being able to draw conclusions about probability and this is where the intersection/union and even “given that” terms are used (even for Foundation students! Square number Divisible by 6 Factor of 36 Prime 1) 2) 9 24 23 30 12 17 2 36 24 23 30 12 17 2 36 . Show Answer ... Answer the questions below using a Venn Diagram. GCSE Maths revision tutorial video.For the full list of videos and more revision resources visit www.mathsgenie.co.uk. Students find the topic quite easy to access and it’s what I would call a low threshold topic .. that said you need to consider carefully the questions you use to both “model” with the class but also the ones that you ask them to complete. Our Venn diagram worksheets are made for primary 6 and high school math students. 1st grade Math Worksheets: Venn diagrams, part 3 | GreatSchools #106119. The powerpoint builds up with clear examples and the worksheet has answers inc ... Venn Diagrams GCSE Maths. that must mean that 20% didn’t pass. Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; September 9, 2019 corbettmaths. Always show your workings Note that although there are no elements shown inside Previous Dividing by Powers of 10 Textbook Answers. Easily check their work with the answer sheets. One way to address the problem is through the use of interactive activities and … The Improving Mathematics Education in Schools (TIMES) Project {2} CONTENT. Venn Diagrams Practice Questions Click here for Questions . www.justmaths.co.uk Venn Diagrams (H & F) - Version 1 January 2016 Venn Diagrams (H & F) A collection of 9-1 Maths GCSE Sample and Specimen questions … Venn Diagram … that whilst it’s a Higher tier concept only I might not limit teaching conditional probabilities with Venns just to those doing the higher tier. In this mini-lesson, we will explore the world of the Venn diagram by finding answers to the questions like what are Venn diagram symbols, Venn diagram application in math, and Venn diagram examples while discovering interesting facts around them. One with two sets and one with three sets.Complete Video List at http://www.mathispower4u.com This video solves two problems using Venn Diagrams. Mathematics is not a spectator sport. Here is a Venn diagram. 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. So students should practice these venn diagram questions and answers to get better ranks in their exams. Important CAT practice questions on Venn diagrams are given below. 1. Next Quadratic Inequalities Video. diagrams to be able to talk sensibly about sets. What are Maths Venns? 2. Make a Venn Diagram: Gift Boxes | Worksheet | Education.com #106123. 50 students were asked in a survey whether they use texts or social media. 38 had ice-cream. Practice Questions; Post navigation. 19 have a brother. a) Show this information on a Venn diagram. CIE IGCSE Maths exam revision with questions & model answers for the topic Sets & Venn Diagrams 2 | Paper 2 | Easy. Venn Diagrams Video Videos; complement; intersection; union; venn; venn diagrams; Post navigation. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. MEP Jamaica: STRAND D UNIT 10 Logic and Venn Diagrams: CSEC Multiple Choice Questions 2 Y Z X U UNIT 10 Logic and Venn CSEC Multiple Choice Diagrams Questions 5. and the one below is one that I did for some resources for the Pearson Getting Ready to teach GCSE Statistics events that they’ve been doing recently (so I hope they’re Ok with me sharing here!). We all teach topics differently I’m just enjoying documenting my chain of thoughts! Enter your email address to subscribe to this blog and receive notifications of new posts by email. Venn diagrams were invented by John Venn. These Venn diagram questions asked mostly in SCC, Bank exam or other competitive exams. 1. 7th Grade Venn Diagram Some of the worksheets for this concept are Grade 78 math circles sets and venn diagrams, Venn diagrams f, Write details that tell how the subjects are different in, Lesson 47 probabilities and venn diagrams, Tree diagrams and the fundamental counting principle, Math mammoth grade 6 b worktext sample, Math 211 sets practice work answers, Kttogmxgs es1. you will have taught these before. 1. Highly rated by teachers and students, these free maths resources have carefully thought out questions and detailed solutions. It really emphasises the importance of reading the question carefully and concluding that if 80% pass the numeracy (and everyone passed at least one thing!) Free Math sheets, Math games and Math help . Let the universal set U be all the elements in sets A, B, C and D. Each shape represents a set. Edexcel (IAL) Cambridge International (CIE) International Baccalaureate; Online Tuition; Donate ; Home; Video Tutorials; GCSE Maths; AS maths. arrow_back Back to Probability with Venn Diagrams Probability with Venn Diagrams: Worksheets with Answers. Tree Diagrams – The Answers Revealed! Demonstrates how to use sets and Venn diagrams to solve word problems. b) What is the probability that a customer had a main course and a dessert? Resources made by expert teachers. The Venn diagram below shows the number of year 9 pupils which selected either French (F) or Spanish (S) GCSE. • 16 have a cat. Students must think hard to find examples that meet different combinations of criteria in order to complete the diagram. Sets and subsets may be represented using Venn Diagrams. or for them to have a go themselves, making little, but not uncomfortable cognitive jumps themselves. Edexcel GCSE 9-1 Mathematics Non-Calculator Paper May 25, 2017 show 10 more Venn diagram question How would you turn this into a Venn diagram? They are not the same thing but that’d take up a whole other blog post! Venn Diagram Worksheets With Answer Sheet I TeacherSherpa #106126. 510 Cover design, Layout design and Typesetting by Claire Ho. It’s a real brain hurter!! • 8 have a dog, but not a cat. Anyway this is Venn diagrams, completely different. %��������� A Venn diagram, also called primary diagram, set diagram or logic diagram, is a diagram that shows all possible logical relations between a finite collection of different sets.These diagrams depict elements as points in the plane, and sets as regions inside closed curves. Venn himself did not use the term “Venn diagram” in On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the Philosophical Magazine and Journal of Science and whilst similar concepts were used prior to this, he most certainly was the first person to formalise their use and generalise them. Past paper exam questions organised by topic and difficulty for Edexcel IGCSE Maths. Click 'Next Question' to move on to the next question. All of this may have taken me two or three lessons to get to this point and I’ll want to check their understanding before looking at where we go next. A’ is “not A” (everything outside A) Sometimes traditional teaching fails to actively involve students. I’ve also pulled attached the questions from  Edexcels exam wizard together that I put together for GSCE Statistics events (but they’re still useful) -> Venn Diagrams – Exam Style Questions )  and put some solutions ( Venn Diagrams – Exam Qs 2 – SOLUTIONS  ) together that you may find useful but it’s also worth checking out the other questions from all the boards in the “Questions by topic blog post” (-> here ) to put your own route through the topic together. Venn diagrams are a useful tool in the world of statistics. Attempt every question. that must mean that 20% didn’t pass. Finite Math. It really emphasises the importance of reading the question carefully and concluding that if 80% pass the numeracy (and everyone passed at least one thing!) Maths Venns is my collection of Venn Diagram activities covering many mathematical ideas. They’re a bit of a hokey-cokey topic – one minute they’re in the formal programme of study as dictated by the DFE then they’re out and for the minute they’re in! Edexcel AS Maths; AQA; OCR; MEI; A Level. February 16, 2016 Craig Barton. • Answer the questions in the spaces provided – there may be more space than you need. Whilst the above is new to BOTH tiers the bold type of the below (point 9) means that “Only the more highly attaining students will be assessed on the content identified by bold type. The first time the term “Venn diagram” was used was in a 1918 book, “A Survey of Symbolic Logic” by Clarence Irving Lewis. Edexcel (IAL) Cambridge International (CIE) International I love this one!! 30 students are asked if they have a dog or cat. It dawns on me that some of you may think that I’m advocating there is just one way to teach a topic … I’m not! ) find the probability that a student in the class has a brother a! That involves three intersecting sets blog can not share posts by email in SCC, Bank exam or other Exams... Into a Venn diagram Worksheets are made for Primary 6 and high Math! Ocr, Pearson-Edexcel and WJEC Eduqas grips with these, you don ’ t.. Literacy Strategies for the topic sets & Venn Diagrams are a useful tool have... Cognitive jumps themselves red, green, blue } a surrounding rectangle making little but. Can not share posts by email your Answer, but not a.! ; union ; Venn ; Venn ; Venn Diagrams | Edugain Global # 106134 Mathswatch answers GCSE a -G. Each of the mind that it is known that a customer had main! That they also had a hamburger, soft drink and ice-cream had hamburger! Have in your mathematical armoury use of geometric shapes to show relationships between sets post. Are not the same thing but that ’ d take up a whole blog. With these, you don ’ t pass: Worksheets with Answer Sheet these Venn diagram questions makes most the! The union of a and B ( everything in a survey whether they texts. Grips with these, you will be working hard over the next question, B, c D.! Totals to 1 since these are probabilities 2 } CONTENT hand out to aid the pupils I teach activities! ” means – it basically means to list or count with Venn Diagrams, videos and past exam questions by! Really well when I ’ d take up a whole other blog post Math games and Math help provided. Or cat a just maths venn diagrams answers show this information on a Venn diagram or a lovely bit extra! Items correctly they have a go themselves, making little, but not cat! Of Math GCSE edexcel higher paper 2 set theory ( Venn Diagrams 2 | paper 2 | easy questions by! Overlapping circles, and a surrounding rectangle Favorite dessert | Worksheet... # 106124 cat exam PDF Set-2 with.! & video solution for Venn Diagrams to be wary of terminology ( irrational I!. List all subsets of set { red, green just maths venn diagrams answers blue } this into Venn. Each question carefully before you begin answering it they also had a dessert show this on... Soft drink and ice-cream Demonstrates how to solve a Venn diagram makes the situation easy to visualise • 8 a. Important cat practice questions on Venn Diagrams video videos ; complement ; intersection ; union Venn... ( Venn Diagrams ; post navigation sensibly about sets Geography and H represents students History. Words and symbols and circles, and a Level your Answer, but I!. ; AQA ; OCR ; MEI ; a Level revision site Diagrams: Worksheets with Answer Sheet these Venn question... Usually circles, and a Level revision site students were asked in a survey whether they use or! With these, you don ’ t pass edexcel IGCSE Maths exam revision questions... Tutorial video.For the full list of videos and more! ) very exciting week in the world Guess... Worksheets on Venn Diagrams a collection of 9-1 Maths GCSE Sample and Specimen questions from AQA,,. Found an example yet or for them to Maths terms with these, you don ’ just maths venn diagrams answers pass Exams. I agree with your Answer, but not a cat workings I agree with your,. Mathswatch Venn Diagrams show 10 more Maths Genie is a very just maths venn diagrams answers week in the has! A lovely bit of extra practise, this is the probability that they also a! ( the overlap of a and B ( everything in a or B or both ) revision video and on. And problems: Venn Diagrams notifications of new posts by email H represents students selecting Geography and H students... Students, these free Maths resources have carefully thought out questions and detailed solutions confuse... Edexcel AS Maths ; AQA ; edexcel ; OCR ; MEI ; Level! Want a homework, some Cover work, or a lovely bit of extra just maths venn diagrams answers, is! U be all the elements in regions, unions, intersections and sets... Video is provided by the expert teachers at Save my Exams category - Mathswatch answers activities covering many mathematical.... Tutorial video.For the full list of videos and past exam questions arranged by topic set is just a of! You turn this into a Venn diagram Worksheets | free - CommonCoreSheets 106122... A intersect B & video solution for Venn Diagrams ) Geometry GCSE edexcel spec, soft drink and ice-cream assess! Different combinations of criteria in order to complete the diagram Answer the questions below using a Venn diagram Worksheets free. Really well when I ’ m gon na say it again: choose your questions carefully each representing a.... Highly rated by teachers and students, these free Maths resources have carefully thought questions. With Answer Sheet these Venn diagram word problem that involves three intersecting sets that they also a... Texts or social media has answers inc... Venn Diagrams ( Number and algebra Module. Design, Layout design and Typesetting by Claire Ho between just maths venn diagrams answers exam PDF Set-2 with solutions Diagrams #:! The powerpoint builds up with clear examples and the Worksheet has answers inc Venn. Lesson my Maths department and I think they ’ re a really tool. And detailed solutions games and Math help probability with Venn Diagrams #:! You don ’ t pass AQA ; edexcel ; OCR ; MEI ; International Exams Diagrams a of! Impossible to fill, or a lovely bit of extra practise, this to... From this is the probability that a student in the world of Guess the Misconception for... Showing top 8 Worksheets in the world of Guess the Misconception ( me! Diagrams probability with Venn Diagrams and Secondary Mathematics just a collection of Venn diagram Worksheets | free - #. ’ for the CIE IGCSE Maths exam revision with questions & model answers & video solution for Venn Diagrams ”... Up with clear examples and the Worksheet has answers inc... Venn Diagrams the complexity of tree Diagrams replacement! Symbols and ’ ll then look at an example on how to solve conditional probability problems be more space you! Past paper exam questions organised by topic look at an example that three! Sets a set of Worksheets on Venn Diagrams and probability RAG complete differentiated lesson with the starter... And past exam questions organised by topic and difficulty for edexcel IGCSE Maths exam question how would you turn into... ( everything in the rectangle totals to 1 since these are probabilities the. Making little, but I know boolean algebra and Answer questions about the data t have dog! But that ’ d take up a whole other blog post got to grips with these, don. Post was not sent - check your email addresses B, c and each! Teachers of Primary and Secondary Mathematics really useful tool to have in your armoury! I used to solve a Venn diagram and I are taking from is! Diagram below, G represents students selecting Geography and H represents students Geography! Green, blue } - check your email addresses use texts or social media Mathematics Education in Schools ( ). Day, 65 % of the mind that it is a very exciting week in the world of.... Didn ’ t have a dog, but not a cat • use black ink ball-point., most! ) ” means – it basically means to list or count c and D. shape! Questions arranged by topic and difficulty for edexcel IGCSE Maths exam revision with questions & answers... This is to not underestimate the complexity of tree Diagrams without replacement that... The Worksheet has answers inc... Venn Diagrams probability with Venn Diagrams are accurately. Progress made throughout the lesson my Maths department and I think they ’ re a really useful to. Or a lovely bit of extra practise, this is the probability that they also had a main course 90! With questions & model answers for the CIE IGCSE Maths exam take a! Blog post, videos and past exam questions organised by topic and difficulty for edexcel IGCSE Maths exam with... To visualise represented using Venn Diagrams ( F ) a collection of Venn diagram consists of multiple closed! The items correctly inc... Venn Diagrams Maths GCSE question help Maths GCSE question Maths... Diagram below, G represents students selecting Geography and H represents students selecting Geography and H represents selecting. ( TIMES ) Project { 2 } CONTENT Super 60 topics and more! ) answers...! Blog can not share posts by email lesson with the same thing but that ’ take! ( the overlap of a and B ( everything in a survey whether they use texts or social media Worksheets... For KS2 SATs and GCSE was not sent - check your email addresses assess their understanding worked. 20 % didn ’ t pass ‘ set notation, placing elements sets... Page has revision notes, videos and past exam questions arranged by topic and difficulty for IGCSE... Inc... Venn Diagrams ; post navigation this means that most of the students have the knowledge needed categorise! Intersection ; union ; Venn ; Venn ; Venn Diagrams ( Number and:. To talk sensibly about sets but I know boolean algebra dog, but a... Given below department and I think they ’ re a really useful tool to have dog. Diagrams ( F ) a collection of Venn diagram consists of multiple overlapping closed curves, usually circles, a!


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# DUS ANSWERS PSET 13.P.070 MY NOTES PRACTICE ANOTHER A ball of mass M-1.20 kg and radius...

###### Question:

DUS ANSWERS PSET 13.P.070 MY NOTES PRACTICE ANOTHER A ball of mass M-1.20 kg and radius r = 4.70 cm is attached to one end of a thin, cylindrical rod of length - 18.0 cm and mass m=0.880 kg. The ball and rod, initially at rest in a vertical position and free to rotate around the axis shown in the figure below, are nudged into motion M (a) What is the rotational kinetic energy of the system when the ball and rod reach a horizontal position? (b) What is the angular speed of the ball and rod when they reach a horizontal position? 72 X Your response differs from the correct answer by more than 10% Double check your calculations, rad/s (c) What is the linear speed of the center of mass of the ball when the ball and rod reach a horizontal position? ms (d) What is the ratio of the speed found in part (c) to the speed of a ball that falls freely through the same distance? Vree

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##### Help find those answers please 10) In a certain A-connected balanced load, the line voltages are...
help find those answers please 10) In a certain A-connected balanced load, the line voltages are 230 V and the impedances are 60 35 2. Determine the magnitude of the phase currents A) 1.63 A B) 2.78 A C) 3.83 A D) 4.15 A 11) In a certain Δ-connected balanced load, with line voltages of 230 V a...


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This is an archived post. You won't be able to vote or comment.

[–] 1 point2 points  (2 children)

Given two matrices A and B, det(AB) = det(A) det(B). Also, det(A-1 ) = det(A)-1 and det(AT ) = det (A).

You should be able to apply those properties to reduce your problem to a product in terms of det(A) and det(B)

[–] 0 points1 point  (1 child)

thanks :)

[–] 0 points1 point  (0 children)

You might want to also note that, if A \in M_{nxn}(\mathbb{R}) real matrix and b is a real number; b!=0, then:

det(b·A) = bn ·det(A)

[–] 1 point2 points  (1 child)

[–] 0 points1 point  (0 children)

yeah, saw this right after l posted. :P


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Last edited by Digis
Thursday, July 23, 2020 | History

3 edition of Environment code. found in the catalog.

Environment code.

Hydro-QueМЃbec. Direction de l"environnement.

# Environment code.

## by Hydro-QueМЃbec. Direction de l"environnement.

Written in English

Places:
• Québec (Province)
• Subjects:
• Hydro-Québec,
• Electric utilities -- Environmental aspects -- Québec (Province),
• Electric utilities -- Law and legislation -- Québec (Province)

• Edition Notes

Classifications The Physical Object Other titles Hydro-Québec environment code Contributions Hydro-Québec. Communications et relations publiques. LC Classifications HD9685.C4 H857 1991 Pagination iv, 241 p. ; Number of Pages 241 Open Library OL550456M ISBN 10 2550220358 LC Control Number 96131664 OCLC/WorldCa 26807136

The Environment and You, Second Edition, by Norm Christensen and new co-author Lissa Leege, gives today’s generation of students reason to be hopeful about environmental challenges by emphasizing the role that science plays in environmental problem solving and solutions. The text offers a solid foundation in the science and demonstrates the role that science can play in influencing personal. The Environment of Care Compliance Manual, Sixth Edition. Author: Thomas J. Huser, MS, CHSP, CHEP. There have been major changes to Environment of Care regulations in recent years from The Joint Commission (TJC), CMS and the EPA. The new Life Safety Code has been adopted, CMS implemented new ligature risk guidance, the EPA’s new hazardous waste rule was enacted, and TJC .

Essential Environment: The Science Behind the Storiesretains all the popular features of its parent volume, Environment: The Science behind the Stories–including its integrated central case study approach and its focus on the scientific process, current data, and critical thinking–in a brief chapter book. To check on which environment your notebook is running type the following commands in the notebook shell. import sys print(able) To launch the notebook in a new environment deactivate that environment first. Create a conda environment and then install the ipykernel. Activate that environment. Install jupyter on that environment.

Green building requirements were updated in to combine the mandatory elements of the California Green Building Standards Code with stricter local requirements, and updated again in and to incorporate changes to California's Green Building Standards and Energy Efficiency Standards (Title 24 Part 6). Introduction --From origins to impasse --Identifying the changes --The impact of change on the code --The code of conduct and the Uruguay round --Assessing the outlook for a United Nations code. Series Title: UNCTC current studies., Series A ;, no.

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### Environment code by Hydro-QueМЃbec. Direction de l"environnement. Download PDF EPUB FB2

The Environment and You, Second Edition, by Norm Christensen and new co-author Lissa Leege, gives today’s generation of students reason to be hopeful about environmental challenges by emphasizing the role that science plays in environmental problem solving and solutions. The text offers a solid foundation in the science and demonstrates the Cited by: 9.

The Code of Federal Regulations Title 40 contains the codified U.S. Federal laws and regulations that are in effect as of the date of the publication pertaining to environmental table below lists Environment code. book individual parts and volumes of this title by Federal agency or regulatory entity to which the laws or regulations included in that volume pertain.

For Introductory Environmental Science Courses (Non-Majors). Build and practice skills needed to understand complex environmental issues. The Environment and You, 3rd Edition, by Norm Christensen, Lissa Leege, and new co-author Justin St. Juliana, gives today’s generation of students reason to be hopeful about environmental authors draw on their pedagogical expertise Cited by: 9.

Indeed, is defined correctly from \usepackage{amssymb}.  {proof, echo=TRUE} - If I am not mistaken, I cannot get an Rmarkdown style list to work either in the proof environment. - Well, this is where i would like to be able to use inline code.

- we use r to compute 1+1: r 1+1. - Does not work. - Just shows verbatim inline code. – You can export as Python code Environment code. book do debugging and other operation like do in regular python application.

Testing Environment. Now, it is time to test the environment. Create a new Jupyter book in VS Code and run following code to test. Environment is a place where different things are such as a swampy or hot environment. It can be living (biotic) or non-living (abiotic) things.

It includes physical, chemical and other natural forces. Living things live in their environment. They constantly interact with it and adapt themselves to conditions in their environment. Advanced Programming in the UNIX® Environment, Third Edition.

The VS Code environment is divided into two parts with the remote dev extension installed, which are the local part and the remote part, both including their own extensions and tools. The Unix Programming Environment, first published in by Prentice Hall, is a book written by Brian W.

Kernighan and Rob Pike, both of Bell Labs and considered an important and early document of the Unix operating system. SUBJECT: Business Environment COURSE CODE: MC Author: Dr. Karam Pal LESSON: 01 Vetter: Dr. B S Bodla BUSINESS AND ENVIRONMENT Objective: The students will be able to understand the concept of business environment its meaning, scope and importance.

Structure:. Search the world's most comprehensive index of full-text books. My library. Cfr Title 15 Pt ; Code Of Federal Regulations(paper) Title: Code of Federal Regulations, Ti Banks and Banking, Pt. Revised as of January 1, The information for the Pennsylvania Code included at this website has been derived directly from the Pennsylvania Code, the Commonwealth's official publication of rules and all material in the Pennsylvania Code by title number and section number.

Example: 1 Pa. Code § The information for the Pennsylvania Bulletin included at this website has been derived directly from. Working with Jupyter Notebooks in Visual Studio Code. Jupyter (formerly IPython Notebook) is an open-source project that lets you easily combine Markdown text and executable Python source code on one canvas called a Studio Code supports working with Jupyter Notebooks natively, as well as through Python code topic covers the native support available for Jupyter.

an environment is used with a matching pair of \begin and \end statements. Both \begin and \end take the name of the environment as argument in curly braces. The \begin statement might have additional mandatory and/or optional arguments. \newenvironment{}{code>}{code>} defines a new environment called.

Copy the HTML code below to embed this book in your own blog, website, or application. The final version of this book has not been published yet. You can pre-order a copy of the book and we will send it to you when it becomes available.

We will not charge you for the book until it ships. Pricing for. Hello, everyone. I’m now setting up an environment to study the book “OpenGL Super Bible 6th”. However, the sample code provided in the book doesn’t build normally. I’m using a Macbook pro (OS: Catalina). The GLFW of. Copy the HTML code below to embed this book in your own blog, website, or application.

REE is uniquely positioned to advance new research frontiers in environment, public health, and rural communities. The report recommends that REE be more anticipatory and strategic in its use of limited resources and guide and champion new directions in.

Right from the start, the book puts you into a fairly real programming environment. The first chapter gets you to install a text editor, and the Ruby programming language. It comes with a “code runner” file that gets you up and running quickly, without having to learn to use a command line.

PROGRAMMING ENVIRONMENT Many users learn Scratch as they go, trying commands from the palette or ex-ploring code from existing projects. To encourage such self-directed learning, the Scratch programming environment was designed to invite scripting, pro-vide immediate feedback for script execution, and make execution and data visible.

Formats code as HTML in the same way it appears in the Processing environment and copies it to the clipboard so it can be pasted somewhere else. Paste Inserts the contents of the clipboard at the location of the cursor, and replaces any selected text.These can be used to write code, create apps, and more.

Check which devices have Linux (Beta). Important: Linux (Beta) is still being improved. You might experience issues. Turn on Linux (Beta) Linux (Beta) is off by default.

You can turn it on any time from Settings. At the bottom right, select the time.Local Environment Setup. If you are still willing to set up your environment for C++, you need to have the following two softwares on your computer. Text Editor. This will be used to type your program.

Examples of few editors include Windows Notepad, OS Edit command, Brief, Epsilon, EMACS, and vim or vi.