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How To Solve Algebraically Systems Of Equations - Tessshebaylo
How To Solve Algebraically Systems Of Equations
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How to extract patterns from traffic data for better insights into mobility
Mobility patterns are a cornerstone of mobility demand modelling. The idea is to estimate patterns from mobility observations, such as vehicle counts, and use them to build different demand models
that explain or emulate people’s mobility needs. In other words, to understand when, why and how people want to go from A to B, we need to start gathering mobility observations and find patterns.
Back in the days when road traffic data was scarce, mobility patterns were inferred from “field knowledge”, that is, the intuition of the traffic engineer plus the variables that influenced traffic
according to the engineer’s experience. In other words, there were a set of variables that described the type of day. For example, the day of the week, the weather, or if it was a holiday. Then, a
typical day for each type of day was sampled (usually by manually counting vehicles) and there they had the road traffic patterns for the corresponding city.
The advent of loop detectors and other sensors made it possible to automate road data gathering and provide data for each type of day in order to calculate a pattern that better described all days of
the same type. So, if the explanatory variables were weekdays (Monday-Friday) and weekends/holidays, one could take all data for all weekends/holidays, calculate the average or the median or any
other reduction metric, and have the pattern for weekends/holidays. The same could be done for all weekdays that were not holidays to get the pattern for workdays.
Basically, this procedure implicitly performs clustering by grouping days according to a set of features, such as day or the week, that according to the engineer’s experience, minimizes both the
number of patterns and the traffic flow variance in each group. Then the pattern, or the exemplar, of each group is its centroid according to Euclidean distance if it is calculated using the mean of
all of them. This can be generalized to any other method for calculating or choosing an exemplar for each group of days, so we could say that each type of day is a cluster of days, and each pattern
is the exemplar that represents each cluster.
Obviously, this methodology can be systemized and automated with a top-down or bottom-up splitting/merging of the types of days. For example, we could start by taking all 14 different types of days
(7 days of the week being a holiday or not a holiday, makes 14 types of day) calculate the pattern of each group, merge the most similar pair, and iterate these two steps until we meet a stopping
criterion. Alternatively, we could start with a single group with all the days, calculate each pattern and select the sub-group whose pattern differs most from the parent pattern. The first
methodology is more efficient, but both should reach similar results with limited explanatory variables, as in this example.
However, is this the right way to define mobility patterns? Clearly not: there is no point in using a set of predefined explanatory variables for grouping days if want to maximize the homogeneity of
a set of variables, such as flow, that we already have. In such a case, the right way to extract daily patterns is to cluster days according to the metrics whose homogeneity we want to maximize. For
example, if we want to minimize the distance between days according to their daily flow profile, we can use the Euclidean distance of the flow between days. Up for discussion are points like
the right distance metric, or which variables to use to calculate the distance, or even how we define patterns or clusters, because these depend on the purpose the patterns.
To illustrate, let’s use the dataset described used in the previous article and sample data from 2018 and 2019, which happen to be the most homogenous sets. We could take the whole dataset, but this
would mean that we would have to add year-explanatory variables if we do not use clustering. I regret to say that I cannot share the dataset and I must keep the name of the city anonymous, but you
can replicate the following exercise with any dataset that includes loop detector data.
We’ll start by taking the whole dataset and try to group days according to [day-of-the-week and not holidays] and [holidays]. The 2-year dataset has a total of 22 holidays, so there should be enough
to have a good pattern if holidays are similar. With this procedure we get 8 groups. For each group we can take the average, or if we think that there may be outliers, we can take the median. Once we
have the 8 patterns, we can calculate the distance between them. Since we are doing patterns according to vehicle flow, I will use the GEH<5 as the similarity metric since it is widely used, and it
is a relative metric that give values from 0 (lowest similarity) to 100 (highest similarity) so it’s easy for everyone to understand.
The image below shows the GEH<5 for each pair of the 8 patterns. On the left, there are the patterns calculated using the average, and on the right, patterns calculated using the median. According to
that, weekdays (Monday-Friday) are very similar, Saturdays and Sundays are different, and holidays are like Sundays. Thus, we could merge weekdays and Sunday-Holidays and have 3 patterns that explain
the mobility in the city’s dataset if we prefer to have fewer than 8 patterns.
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Select and Format Information on the Vote & Report Screen
The Vote & Report screen is where you will start your voting process and review the results in a table form. This flexible screen enables you to view average scores, vote totals, and measures of
agreement; highlight high, medium and low scores with a custom set of colors, add mathematical formulas to your voting results and decide what items, what criteria and what formulae results should be
included and excluded from reporting.
Select the Values to Display
The standard setting for the Voting and Results screen for multiple choice and rating scales voting types is to display the average of the votes received, displayed to one decimal place. Several
other options are available.
1. To change the information displayed, navigate to Vote & Report > Data.
2. Click the required option (from the first five menu items as described below).
• Averages: The default setting. This is the arithmetic mean of all votes received.
• Response Totals: The sum of all the votes. For example, out of eight voters, three voted 1, three voted 2 and two voted 3, the response total would be 15, calculated as follows:
• Agreement – Spreads: This calculates the standard deviation of the values, which is the deviation from the mean for the population, with the result shown to two decimal places.
• Agreement – Standard Error: Measure of deviation from the mean for a sample. This is similar to standard deviation but takes into account the number in the sample (voters) in order to quantify
the accuracy of the deviation measure. (For example, a sample of 100 voters would provide a more accurate assessment of the variation than a group of 20.)
• Modes - Multiple Choice: For multiple choice votes, the mode may displayed instead of the mean, as the mean is typically not a meaningful measure. The mode is the most commonly observed value in
a sample (i.e. the item voted for most often). For example, for a question with three possible answers, option 1 receives 6 votes, option 2 receives 3 votes and option 3 receives 1 vote, the mode
is 6, whereas the mean is 4.33.
Select the Values to be Displayed for Groups and Formulae Using Group Values
If you are using the Groups feature, you can select which values appear in the table for each of the groups, using the following two options:
Group Values: Select the average of the underlying item values or the sum of the values.
Formula Values for Groups: Choose either Apply the Formula which calculates the formula based on the group sum or group total; or Average of Item Formula Values, which shows the mean of the
calculated value for each item in that group. Depending on the formula used, these options may produce different results.
Have more questions? Submit a request
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This lecture covers:
• Chemical reaction networks (CRNs) to describe species and reactions among them
• Determinstic ODE kinetics of CRNs
• Stochastic Markov chain kinetics of CRNs
• Their link via the Langevin equation
Chemical Reactions Networks¶
A chemical reaction network is described by a set $\mathcal{S}$ of species and a set $\mathcal{R}$ of reactions. A reaction is a triple $(\mathbf{r}, \mathbf{p}, \alpha)$ where $\mathbf{r}, \mathbf
{p}\in \mathbb{N}_0^{\mathcal{S}}$ and $\alpha\in\mathbb{R}_{\geq0}$. The species with positive count in $\mathbf{r}$ are called the reaction's reactants and those with positive count in $\mathbf{p}$
are called its products. The parameter $\alpha$ is called the reaction's rate constant.
The dynamical equations¶
For simple protein production, we have the following species
• DNA. We assume there is a single promoter followed by a protein-coding gene in the cell
• mRNA, where $m$ is the current number of mRNA corresponding to the above gene
• protein, where $p$ is the current number of proteins corresponding to the above gene
as well as the following reactions among them: \text{DNA} &\rightarrow \text{mRNA} + \text{DNA} &\text{(transcription)}\\ \text{mRNA} &\rightarrow \emptyset &\text{(mRNA degradation and dilution)}\\
\text{mRNA} &\rightarrow \text{protein} + \text{mRNA} &\text{(translation)}\\ \text{protein} &\rightarrow \emptyset &\text{(protein degradation and dilution)}
Macroscale equations (deterministic, ODE semantics)¶
As we've seen before, the deterministic dynamics, which describe mean concentrations over a large population of cells, are described by the ODEs
\frac{\mathrm{d}m}{\mathrm{d}t} &= \beta_m - \gamma_m m, \\[1em] \frac{\mathrm{d}p}{\mathrm{d}t} &= \beta_p m - \gamma_p p.
In general, the rate of a reaction $(\mathbf{r}, \mathbf{p}, \alpha)$ at time $t$ is $\alpha \prod_{s\in \mathcal{S}} s(t)^{\mathbf{r}(s)}$ where $s(t)$ is the concentration of species $s$ at time
$t$. The differential equation for a species $s \in \mathcal{S}$ is: $$ \frac{ds}{dt} = \sum_{(\mathbf{r}, \mathbf{p}, \alpha) \in \mathcal{R}} (\mathbf{p}(s) - \mathbf{r}(s)) \cdot \alpha \prod_{\
tilde{s}\in \mathcal{S}} \tilde{s}(t)^{\mathbf{r}(\tilde{s})} $$
The Chemical Master equation (stochastic, Markov chain semantics)¶
We can write a master equation for these dynamics. In this case, each state is defined by an mRNA copy number $m$ and a protein copy number $p$. States can transition to other states with rates. We
assume that the state fully described the system state. That is the probability to transition from a state $(m,p)$ to a state $(m',p')$ with in an infinitesimal time $\Delta t$ is independent if how
long our system already is in state $(m,p)$. It is approximately $\Delta t \cdot \gamma_i(m,p)$, where $\gamma_i(m,p)$ is the rate at which reaction $i$ happens if in state $(m,p)$.
The following image shows state transitions and their corresponding reactions for large enough $m$ and $p$. Care has to be taken at the boundaries, e.g., if $m = 1$ or $m = 0$.
Denote by $P(m, p, t)$ the probability that the system is in state $(m,p)$ at time $t$. Then, by letting $\Delta t \to 0$, it is
\frac{\mathrm{d}P(m,p,t)}{\mathrm{d}t} &= \beta_m P(m-1,p,t) & \text{(from left)}\\ &+ (m+1)P(m+1,p,t) & \text{(from right)}\\ &+\beta_p mP(m,p-1,t) & \text{(from bottom)}\\ &+ \gamma (p+1)P(m,p+1,t)
&\text{(from top)}\\ &- mP(m,p,t) & \text{(to left)}\\ &- \beta_m P(m,p,t) & \text{(to right)}\\ &- \gamma p P(m,p,t) &\text{(to bottom)}\\ &- \beta_p mP(m,p,t)\enspace. & \text{(to top)}
We implicitly define $P(m, p, t) = 0$ if $m < 0$ or $p < 0$. This is the master equation we will sample from using the stochastic simulation algorithm (SSA) also called Gillespie algorithm.
The Gillespie algorithm¶
The rhs terms in the above equation are familiar to us: they almost look like reaction rates with a single difference of not being functions of concentrations, but of species counts. For example, a
reaction $$ A + B \rightarrow C $$ with mass-action kinetics and rate constant $\gamma$ in units of $\text{L} s^{-1}$ has rate $$ \gamma \cdot [A] \cdot [B] $$ in units of $\text{L}^{-2} \cdot \text
{L} \text{s}^{-1} = \text{L}^{-1} \text{s}^{-1}$. Concentrations may as well be given in molar units.
By contrast, the propensity of the reaction is $$ \gamma' \cdot A \cdot B $$ in units of $\text{s}^{-1}$, where $\gamma' = \gamma / \text{vol}$ is in units of $\text{s}^{-1}$ and vol is the volume of
the compartment in which the reactions happen.
The propensity for a given transition/reaction, say indexed $i$, is denoted as $a_i$. The equivalence to notation we introduced for master equations is that if transition $i$ results in the change of
state from $n'$ to $n$, then $a_i = W(n\mid n')$.
In general, the propensity of a reaction $(\mathbf{r}, \mathbf{p}, \alpha)$ at time $t$ is $$ \frac{\alpha}{v^{o-1}} \prod_{s\in \mathcal{S}} \binom{s(t)}{\mathbf{r}(s)} $$ where $s(t)$ is the count
of species $s$ at time $t$, $v$ is the volume, and $o = \sum_{s\in\mathcal{S}} \mathbf{r}(s)$ is the order of the reaction. Its effect is subtracting $\mathbf{r}(s)$ from the count of species $s$ and
adding $\mathbf{p}(s)$ to the count of species $s$.
Switching states: transition probabilities and transition times¶
To cast this problem for a Gillespie simulation, we can write each change of state (moving either the copy number of mRNA or protein up or down by 1 in this case) and their respective propensities.
\begin{array}{ll} \text{reaction, }r_i & \text{propensity, } a_i \\ m \rightarrow m+1,\;\;\;\; & \beta_m \\[0.3em] m \rightarrow m-1, \;\;\;\; & m\\[0.3em] p \rightarrow p+1, \;\;\;\; & \beta_p m \\
[0.3em] p \rightarrow p-1, \;\;\;\; & \gamma p\enspace. \end{array}
We will not carefully prove that the Gillespie algorithm samples from the probability distribution governed by the master equation, but will state the principles behind it. The basic idea is that
events (such as those outlined above) are rare, discrete, separate events. I.e., each event is an arrival of a Poisson process. The Gillespie algorithm starts with some state, $(m_0,p_0)$. Then a
state change, any state change, will happen in some time $\Delta t$ that has a certain probability distribution (which we will show is exponential momentarily).
transition probabilities¶
The probability that the state change that happens is because of reaction $j$ is proportional to $a_j$. That is to say, state changes with high propensities are more likely to occur. Thus, choosing
which of the $k$ state changes happens in $\Delta t$ is a matter of drawing an integer $j \in [1,k]$ where the probability of drawing $j$ is
\frac{a_j}{\sum_i a_i}\enspace.
transition times¶
Now, how do we determine how long the state change took? Let $T_i(m,p)$ be the stochastic variable that is the time that reaction $i$ occurs in state $(m,p)$, given that it is reaction $i$ that
results in the next state. The probability density function $p_i$ for the stochastic variable $T_i$, is p_i(t) = a_i\, \mathrm{e}^{-a_i t}\enspace, for $t \geq 0$, and $0$ otherwise. This is known as
the exponential distribution with rate parameter $a_i$ (related, but not equal to the rate of the reaction).
The probability that it has not occurred by time $\Delta t$, is thus P(T_i(m,p) > \Delta t \mid \text{reaction } r_i \text{ occurs}) = \int_{\Delta t}^\infty p_i(t) \mathrm{d}t = \mathrm{e}^{-a_i \
Delta t}\enspace.
However, in state $(m,p)$ there are several reactions that may make the system transition to the next state. Say we have $k$ reactions that arrive at times $t_1, t_2, \ldots$. When does the first one
of them arrive?
The probability that none of them arrive before $\Delta t$ is P(t_1 > \Delta t \wedge t_2 > \Delta t \wedge \ldots) &= P(t_1 > \Delta t) P(t_2 > \Delta t) \cdots = \prod_i \mathrm{e}^{-a_i \Delta t}
= \mathrm{exp}\left(-\Delta t \sum_i a_i\right)\enspace. This is the equal to $P(T(m,p) > \Delta t \mid \text{reaction } R \text{ occurs})$ for a reaction $R$ with propensity $\sum_i a_i$. For such a
reaction the occurrence times are exponentially distributed with rate parameter $\sum_i a_i$.
The algorithm¶
So, we know how to choose a state change and we also know how long it takes. The Gillespie algorithm then proceeds as follows.
1. Choose an initial condition, e.g., $m = p = 0$.
2. Calculate the propensity for each of the enumerated state changes. The propensities may be functions of $m$ and $p$, so they need to be recalculated for every $m$ and $p$ we encounter.
3. Choose how much time the reaction will take by drawing out of an exponential distribution with a mean equal to $\left(\sum_i a_i\right.)^{-1}$. This means that a change arises from a Poisson
4. Choose what state change will happen by drawing a sample out of the discrete distribution where $P_i = \left.a_i\middle/\left(\sum_i a_i\right)\right.$. In other words, the probability that a
state change will be chosen is proportional to its propensity.
5. Increment time by the time step you chose in step 3.
6. Update the states according to the state change you choose in step 4.
7. If $t$ is less than your pre-determined stopping time, go to step 2. Else stop.
Gillespie proved that this algorithm samples the probability distribution described by the master equation in his seminal papers in 197690041-3) and 1977. (We recommend reading the latter.) You can
also read a concise discussion of how the algorithm samples the master equation in Section 4.2 of Del Vecchio and Murray.
The Chemical Langevin Equation¶
The Chemical Master Equation can be difficult to solve, which is why several approxmations were developed. One of these approximations is the Chemical Langevin Equation, which we will now derive. It
is perfectly useful on its own, in particular from a computational perspective. We present it here for another reason as well: to show that the ODE kinetics of a CRN are an approximation to the
expected stochastic kinetics, at least for linear CRNs.
It starts
$$ X_i(t + \tau) = X_i(t) + \sum_{j=1}^M \nu_{j,i} K_j(X(t), \tau) $$ where $\nu_{j,i}$ is the net change in the count of species $i$ in reaction $j$ and $K_j(X(t), \tau)$ is the random variable that
specifies how many times reaction $j$ occurs in the time interval $[t, t+\tau]$.
Assumption 1 of the chemical Langevin equation is that the propensity functions do not change significantly during the time interval $[t, t+\tau]$, i.e., $a_j(X(t')) \approx a_j(X(t))$ for all $t' \
in [t,t+\tau]$.
Then, we can rewrite the above equation as $$ X_i(t + \tau) = X_i(t) + \sum_{j=1}^M \nu_{j,i} \mathcal{P}_j(a_j(X(t)) \cdot \tau) $$ where the $\mathcal{P}_j(a_j(X(t)) \cdot \tau)$ are independent
Poisson variables with parameter $a_j(X(t)) \cdot \tau$.
Assumption 2 of the chemical Langevin equation requires the expected number of occurrences of each reaction to be big, i.e., $a_j(X(t))\cdot \tau \gg 1$.
Note that the two assumption require a tradeoff: assumption 1 wants $\tau$ to be small whereas assumption 2 wants $\tau$ to be big. It is very well possible that no choice of $\tau$ satisifies both
assumptions for a given system.
It is well-known that the Poisson distribution with parameter $\lambda$ is well approximated by the normal distribution $\mathcal{N}(\lambda,\lambda)$ with expected value $\mu = \lambda$ and variance
$\sigma^2 = \lambda$ for large values of $\lambda$.
Assumption 2 thus suggests using this approximation to get $$ \mathcal{P}_j(a_j(X(t)) \tau) \approx a_j(X(t))\tau + \sqrt{a_j(X(t))\tau} \cdot \mathcal{N}(0,1) $$ where $\mathcal{N}(0,1)$ is a
standard normally distributed random variable.
Now, switching to $X(t)$ being real-valued and setting $\tau = dt$, in the limit $dt \to 0$, we finally get the chemical Langevin equation: $$ \frac{dX_i(t)}{dt} = \sum_{j=1}^M \nu_{j,i} a_j(X(t)) +
\sum_{j=1}^M \nu_{j,i} \sqrt{a_j(X(t))} \Gamma_j(t) $$ where the $\Gamma_j$ are independent standard Gaussian white noise processes.
From Stochastic to Deterministic Kinetics¶
If the assumptions of the chemical Langevin equation hold sufficiently well, it is easy to make the connection between the stochastic and the deterministic kinetics of CRNs. In this case, taking the
expected value (over an ensemble of sample paths) on both sides of the equation, we get: $$ \frac{d \mathbb{E} X_i(t)}{dt} = \sum_{j=1}^M \nu_{j,i} \mathbb{E} a_j(X(t)) $$
If the propensities follow a mass-action law, the expected value $\mathbb{E} a_j(X(t))$ is sometimes an approximation to $a_j(\mathbb{E} X(t))$. If there are only unary reactions, they are equal for
The detour via the chemical Langevin equation is not the only way to show the above formula. For a large-volume (or large-species-count) limit, it can be derived directly from the Chemical Master
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Review using column subtraction with regrouping from hundreds to tens | Oak National Academy
Hello, my name's Mrs. Hopper, and I'm really looking forward to working with you on this lesson from our unit on reviewing column edition and subtraction.
Are you ready to work hard? Are you ready to remember perhaps some things that you've learned before? Well, if so, let's make a start.
So in this lesson, we're going to be using column subtraction, including regrouping from the hundreds to the tens.
We've got two key words, column subtraction and regroup.
So I'll take my turn, then it'll be your turn.
My turn, column subtraction.
Your turn.
My turn, regroup.
Your turn.
Well done.
They may well be words you're very familiar with.
Let's just remind ourselves what they mean.
They are important in the lesson.
Column subtraction is a way of subtracting numbers by writing a number below another, helping us to keep track of our subtractions and to work effectively and efficiently.
The process of unitizing and exchanging between place values is known as regrouping.
For example, 10 tens can be regrouped for 100.
100 can be regrouped for 10 tens.
And that's gonna be really important to us today we've got two parts to our lesson.
In the first part, we're going to be regrouping from hundreds to tens.
And in the second part, we're going to be regrouping from hundreds to tens and from tens to ones.
So let's make a start.
And we've got Alex and Lucas helping us in our lesson.
Alex is unsure how to calculate 215 subtract 133.
He says, "I really can't remember how to do this.
I wonder if you can." Could you help him out? Lucas says, "What's the problem Alex?" And Alex says, "The tens digit of the subtrahend is greater than the tens digit of the minuend." Let's remind
ourselves what those words mean.
The subtrahend is the number we are subtracting.
So 133.
And the minuend is our whole, the number we're starting with, 215.
And the tens digit in 215 is smaller than the tens digit in 133.
So 110 subtract 310 would be difficult to do in our column subtraction.
Lucas says, "It's not a problem, you just need to regroup one of the hundreds as 10 tens." Ah, we know that 100 is equal to 10 tens.
So we can take one of those hundreds and regroup it as 10 tens and add them into the tens column.
Lucas says, "Let's start by representing 215 using base 10 blocks." So there they are.
2 hundreds, 1 ten, and 5 ones.
And from that we're going to subtract 133.
Let's start with the ones.
Five ones subtract three ones is equal to two ones.
So there we go.
We can see our three ones have faded out.
3 tens is greater than 1 ten.
We have to regroup one of the hundreds as 10 tens.
We haven't got enough tens on their own, but we've got lots of tens locked up in those hundreds.
So we've regrouped one of the hundreds into 10 tens.
There are now 11 tens and 11 tens subtract 3 tens is equal to 8 tens.
So we've got plenty of tens now.
We can subtract our 3 tens and we've got 8 tens remaining.
100 subtract 100 is equal to zero 'cause we've only got one whole hundred left and we need to take away one of the hundreds for the 133.
So our whole hundred has disappeared.
As Alex said, "215 subtract 133 is equal to 82.
That really helped, Lucas, thank you." He's going to use column subtraction to calculate 215, subtract 133 now.
Can you picture what was happening with the base 10 blocks maybe to help you? He says, "I need to start with the smallest place value first." 5 ones subtract 3 ones is equal to 2 ones.
The tens digit of the subtrahend is greater than the tens digit of the minuend.
Three is greater than one.
I'll regroup 100 as 10 tens.
So now there's 1 hundred and 11 tens.
11 tens subtract 3 tens is equal to 8 tens and 100 subtract 100 is equal to zero hundreds.
Well done, Alex.
215 subtract 133 is equal to 82.
And I hope you were picturing what was happening with the base 10 blocks as we were doing the crossings out and rewriting the numbers in the minuend of our calculation.
Over to you now.
You're going to look at each column subtraction.
Do they need regrouping in the tens? So you're going to focus in on those tens digits.
Can you spot whether they need regrouping? Is the tens digit of the subtrahend greater than the tens digit of the minuend? So pause the video, have a look, and we'll come back and talk through them
How did you get on? Yes, in the first one, regrouping is needed.
We've got 1 ten in our minuend and 4 tens in our subtrahend.
And regrouping is needed again for the middle one.
There are 6 tens in the minuend and 8 tens in the subtrahend that we're subtracting.
What about the last one? Nope, regrouping is not needed in the tens here.
We've got 3 tens and we're subtracting 3 tens.
We just won't have any tens.
Well done if you've got those right.
It's really useful to look carefully at the numbers in a column subtraction so that you can identify whether regrouping is going to be needed or not.
So Alex is using column subtraction to calculate 536 subtract 84.
He says, "I need to start with the numbers with the smallest place value." So he's gonna start with the ones.
6 ones subtract four ones is equal to two ones.
The tens digit of the subtrahend is greater than the tens digit of the minuend.
Eight is greater than three.
I'll regroup 100 into 10 tens.
So now there are 4 hundreds and 13 tens.
13 tens subtract 8 tens is equal to 5 tens and 4 hundreds subtract zero hundreds is equal to 4 hundreds.
536 subtract 84 is equal to 452.
You're getting really good at this, Lucas.
Over to you.
Can you use column subtraction to calculate 628 subtract 277? Alex says, "Start with the numbers with the smallest place value." So pause the video, have a go, and we'll look at the answer together.
How did you get on? So we started with the ones.
8 ones subtract 7 ones is equal to 1 one, but the tens digit of the subtrahend is greater than the tens digit of the minuend.
Seven is greater than two.
So we're going to regroup 100 as 10 tens.
So now there are 5 hundreds and there are 12 tens.
12 tens subtract 7 tens is equal to 5 tens and 5 hundreds subtract 2 hundreds is equal to 3 hundreds.
So 628 subtract 277 is equal to 351.
Time for you to do some more practise.
Complete each calculation using column subtraction and you will need to regroup from the hundreds to the tens.
So pause the video, have a go, and we'll look at the answers together.
How did you get on? So here are the answers to the first three.
So in A, we only had 3 tens in our minuend and we were subtracting 4 tens in the subtrahend.
So we regrouped one of the hundreds.
So we now have 2 hundreds and 13 tens.
And in B, we had to subtract 9 tens from seven.
So we had to regroup one of the hundreds.
So we had 3 hundreds and 17 tens.
And then in C, we were only subtracting a two digit number, but we still needed to regroup.
So we regrouped one of the hundreds.
So we had 4 hundreds and 15 tens so that we could do the subtraction.
And Alex is reminding us the tens digit of the subtrahend is greater than the tens digit of the minuend in all of these cases.
And for the second set, similar thinking.
But E was quite interesting.
The tens digit of the minuend is zero.
100 is regrouped into 10 tens, so that there are 10 tens in total.
So in E we regrouped to hundreds.
So there were six hundreds and 10 tens.
I hope you were successful with those and you've got your regrouping sorted out.
Let's move on to the second part of our lesson.
This time we're going to regroup from the hundreds to the tens and the tens to the ones.
So Alex is looking at this calculation.
What can you see? He says, "I'll need to regroup in the tens.
The tens digit of the subtrahend is greater than the tens digit of the minuend." Yup, that's right, Alex.
4 tens is greater than 2 tens.
There we are.
Have you spotted something else? Ah, Lucas has.
He says, "You'll also need to regroup in the ones." The ones digit of the subtrahend is greater than the ones digit of the minuend.
4 ones is greater than 2 ones.
So we've got a lot of regrouping to do here.
Alex uses column subtraction to calculate it.
He says, "First, I'll regroup 1 ten as 10 ones." So now there is 1 ten and there are 12 ones.
12 ones subtract 4 ones is equal to 8 ones.
I'll regroup 100 as 10 tens.
So now there are 4 hundreds and we had 1 ten left from our regrouping before and we've put 10 more in.
So now we've got 11 tens.
11 tens subtract 4 tens is equal to 7 tens and 4 hundreds subtract 2 hundreds is equal to 2 hundreds.
So 522 subtract 244 is equal to 278.
Alex is looking at this calculation.
He says, "I'll need to regroup the ones.
The ones digit of the subtrahend is greater than the ones digit of the minuend." That's right, nine is greater than six.
Lucas says, "Do you need to regroup in the tens as well?" Hmm, we've got zero tens.
Take away zero tens, it doesn't look like it.
He says, "I don't think so.
Zero tens subtract zero tens is equal to zero tens." So he is going to use his column subtraction to calculate.
606 subtract 209.
First, I need to regroup 1 ten as 10 ones.
But the tens digit of the minuend is zero.
Where's he going to get a 10 from? "Ah," he says, "I'll need to group 100 as 10 tens first." We know we've got tens, but they're locked away in our hundreds.
So he's going to regroup one of the hundreds.
So now there are 5 hundreds and there are 10 tens.
"Now," he says, "I can regroup 1 ten as 10 ones." So there are now 9 tens and there are 16 ones.
So we've regrouped a hundred for 10 tens, and then we can use those 10 tens to regroup one for 10 ones.
So now we can complete our column subtraction.
16 ones subtract 9 ones is equal to 7 ones.
9 tens subtract zero tens is equal to 9 tens and 5 hundreds subtract 2 hundreds is equal to 3 hundreds.
So 606 subtract 209 is equal to 397.
"You had to regroup in the ones and the tens," says Lucas.
You might want to have another look at that, have a go at it yourself, and make sure you can see why all that regrouping was needed so that we could have an extra 10 ones in our ones column.
So time for you to have a look.
Look at each calculation and explain how you will need to regroup in each case.
Will you need to regroup a 10 into 10 ones? And will you need to regroup a hundred into 10 tens? Or maybe you'll need to do both.
Look carefully at each calculation.
And Lucas is asking a question, do you need to regroup in the tens because you had to regroup a 10 as 10 ones? That might be something to look for.
Pause the video, have a look at the calculations, and we'll discuss your thinking afterwards.
How did you get on? So in this calculation, regrouping is only needed in the ones.
If we regroup one of those tens, we'll still have 5 tens and we can subtract three.
In this calculation, regrouping is needed in the ones and the tens.
In both cases, the digit in the subtrahend is bigger than the digit in the minuend.
What about the final one? Well, regrouping is needed in the tens here because regrouping is also needed in the ones.
So if we regroup one of the tens to give us 12 ones, we'll then only have 6 tens.
So we will need to regroup a hundred into 10 tens so that we can complete the subtraction of the tens.
Really useful to look carefully at the numbers before you start doing your calculation just to make sure that you've spotted when regrouping is going to be needed.
Alex has these number cards.
He says, "Choose three of the cards and make a three-digit number." So Alex says, "I'll make the number 954." Lucas says, "Now reverse the digits to get another three-digit number." He says, "That
gives me the number 459." So 954, reverse the order of the digits, 459.
Lucas says, "Now subtract the smaller number from the larger number." So Alex says, "Let's calculate 954, subtract 459." Can you see if you'll need to do any regrouping? You're going to have a go.
Can you complete the calculation for Alex? He says, "Start with the numbers with the smallest place value." Do you think you're gonna need to do any regrouping? Pause the video and have a go and then
we'll talk through your answer together.
How did you get on? Alex says, "We need to regroup 1 ten as 10 ones." So now there are 4 tens and 14 ones.
14 ones subtract 9 ones is equal to 5 ones.
Ah, and this is one of those cases where we now need to regroup in the hundreds to the tens as well.
We need to regroup 100 as 10 tens.
So now there are 8 hundreds and 14 tens.
14 tens subtract 5 tens is equal to 9 tens and then 8 hundreds subtract 4 hundreds is equal to 4 hundreds.
So 954 subtract 459 is equal to 495.
Time for you to do some practise.
You're going to complete each column subtraction and think carefully about whether you need to do any regrouping.
And for part two, and you're going to play Lucas's game.
You're going to use these number cards.
Choose three of the number cards and make a three digit number.
So we've got 327.
Now, reverse the digits and get another three-digit number.
Next, subtract the smaller number from the larger number.
So here we do 723 subtract 327, and we'll use a column subtraction to do it.
Lucas says, "Do this a few times.
Do you notice anything?" Pause the video, have a go at your tasks, and we'll get together for some feedback.
How did you get on? Here are the answers for one.
So we had a lot of regrouping to do, didn't we? In the first two, we knew we were going to have to regroup from the tens to the ones and from the hundreds to the tens.
And in C, we had to regroup from the hundreds to the tens because we had to regroup from the tens to the ones.
So Lucas says, "Each column subtraction needed regrouping in the ones and the tens." How did you get on with two? Here are some possible answers you might have got.
So we had 413 subtract 314.
So lots of regrouping needed there to get an answer of 99.
And again, lots of regrouping needed here to get an answer of 396.
Did you do this a few times? Did you notice anything? So you may have noticed Alex says that you kept getting the same differences.
99, 198, 297, 396 perhaps.
Ah, Lucas spotted something.
When you add the digits of the difference, you always get 18.
So nine add nine is equal to 18, and nine add three add six is equal to 18 here.
So you always get a digit sum, we call it, of 18.
I wonder why that happens.
I hope you enjoyed having a go.
Maybe you could try this task out on somebody else, a friend, or maybe somebody at home.
And we've come to the end of our lesson.
So I hope you've enjoyed reviewing, using column subtraction with regrouping from hundreds to tens.
It may be something you've done before and you've reminded yourself about it, or it may be something new.
So what have we learned about? We've learned that we need to regroup when the ones or tens digit of the subtrahend is greater than the ones or the tens digit of the minuend.
So we are subtracting a larger value digit than the value of the digit in our whole.
We can regroup a 10 into 10 ones and we can regroup a hundred into 10 tens.
And we've also learned that if the tens digit of the minuend is a zero, you need to regroup the hundreds before you can regroup the tens.
And that's a really useful strategy to have ready.
Thank you for all your hard work today.
I hope you've enjoyed it, and I hope I get to work with you again soon.
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Mean Field Methods for Classification with Gaussian Processes
Part of Advances in Neural Information Processing Systems 11 (NIPS 1998)
Manfred Opper, Ole Winther
We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi(cid:173) cation models with Gaussian processes. In contrast to
previous ap(cid:173) proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given.
1 Modeling with Gaussian Processes
Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation
community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written
in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order
correlations C(s, S') == E[h(s)h(S')], where E denotes expectations
M Opper and 0. Winther
with respect to the prior. Interesting examples are
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class lsst.daf.butler.Timespan(begin: TimespanBound, end: TimespanBound, padInstantaneous: bool = True, _nsec: tuple[int, int] | None = None)¶
Bases: BaseModel
A half-open time interval with nanosecond precision.
Raised if begin or end has a type other than astropy.time.Time, and is not None or Timespan.EMPTY.
Raised if begin or end exceeds the minimum or maximum times supported by this class.
Timespans are half-open intervals, i.e. [begin, end).
Any timespan with begin > end after nanosecond discretization (begin >= end if padInstantaneous is False), or with either bound set to Timespan.EMPTY, is transformed into the empty timespan, with
both bounds set to Timespan.EMPTY. The empty timespan is equal to itself, and contained by all other timespans (including itself). It is also disjoint with all timespans (including itself), and
hence does not overlap any timespan - this is the only case where contains does not imply overlaps.
Finite timespan bounds are represented internally as integer nanoseconds, and hence construction from astropy.time.Time (which has picosecond accuracy) can involve a loss of precision. This is of
course deterministic, so any astropy.time.Time value is always mapped to the exact same timespan bound, but if padInstantaneous is True, timespans that are empty at full precision (begin > end,
begin - end < 1ns) may be finite after discretization. In all other cases, the relationships between full-precision timespans should be preserved even if the values are not.
The astropy.time.Time bounds that can be obtained after construction from Timespan.begin and Timespan.end are also guaranteed to round-trip exactly when used to construct other Timespan
Attributes Summary
begin Minimum timestamp in the interval, inclusive.
end Maximum timestamp in the interval, exclusive.
model_computed_fields A dictionary of computed field names and their corresponding ComputedFieldInfo objects.
model_config Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].
model_fields Metadata about the fields defined on the model, mapping of field names to [FieldInfo][pydantic.fields.FieldInfo] objects.
Methods Summary
contains(other) Test if the supplied timespan is within this one.
difference(other) Return the one or two timespans that cover the interval(s).
fromInstant(time) Construct a timespan that approximates an instant in time.
from_day_obs(day_obs[, offset]) Construct a timespan for a 24-hour period based on the day of observation.
from_yaml(loader, node) Convert YAML node into _SpecialTimespanBound.
intersection(*args) Return a new Timespan that is contained by all of the given ones.
isEmpty() Test whether self is the empty timespan (bool).
makeEmpty() Construct an empty timespan.
overlaps(other) Test if the intersection of this Timespan with another is empty.
to_yaml(dumper, timespan) Convert Timespan into YAML format.
Attributes Documentation
EMPTY: ClassVar[_SpecialTimespanBound] = 1¶
Minimum timestamp in the interval, inclusive.
If this bound is finite, this is an astropy.time.Time instance. If the timespan is unbounded from below, this is None. If the timespan is empty, this is the special value Timespan.EMPTY.
Maximum timestamp in the interval, exclusive.
If this bound is finite, this is an astropy.time.Time instance. If the timespan is unbounded from above, this is None. If the timespan is empty, this is the special value Timespan.EMPTY.
model_computed_fields: ClassVar[Dict[str, ComputedFieldInfo]] = {}¶
A dictionary of computed field names and their corresponding ComputedFieldInfo objects.
model_config: ClassVar[ConfigDict] = {'json_schema_extra': {'description': 'A [begin, end) TAI timespan with bounds as integer nanoseconds since 1970-01-01 00:00:00.'}}¶
Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].
model_fields: ClassVar[Dict[str, FieldInfo]] = {'nsec': FieldInfo(annotation=tuple[int, int], required=True, frozen=True)}¶
Metadata about the fields defined on the model, mapping of field names to [FieldInfo][pydantic.fields.FieldInfo] objects.
This replaces Model.__fields__ from Pydantic V1.
yaml_tag: ClassVar[str] = '!lsst.daf.butler.Timespan'¶
Methods Documentation
contains(other: Time | Timespan) bool¶
Test if the supplied timespan is within this one.
Tests whether the intersection of this timespan with another timespan or point is equal to the other one.
otherTimespan or astropy.time.Time
Timespan or instant in time to relate to self.
The result of the contains test.
If other is empty, True is always returned. In all other cases, self.contains(other) being True implies that self.overlaps(other) is also True.
Testing whether an instantaneous astropy.time.Time value is contained in a timespan is not equivalent to testing a timespan constructed via Timespan.fromInstant, because Timespan cannot
exactly represent zero-duration intervals. In particular, [a, b) contains the time b, but not the timespan [b, b + 1ns) that would be returned by Timespan.fromInstant(b)`.
difference(other: Timespan) Generator[Timespan, None, None]¶
Return the one or two timespans that cover the interval(s).
The interval is defined as one that is in self but not other.
This is implemented as a generator because the result may be zero, one, or two Timespan objects, depending on the relationship between the operands.
Timespan to subtract.
A Timespan that is contained by self but does not overlap other. Guaranteed not to be empty.
classmethod fromInstant(time: Time) Timespan¶
Construct a timespan that approximates an instant in time.
This is done by constructing a minimum-possible (1 ns) duration timespan.
This is equivalent to Timespan(time, time, padInstantaneous=True), but may be slightly more efficient.
Time to use for the lower bound.
A [time, time + 1ns) timespan.
classmethod from_day_obs(day_obs: int, offset: int = 0) Timespan¶
Construct a timespan for a 24-hour period based on the day of observation.
The day of observation as an integer of the form YYYYMMDD. The year must be at least 1970 since these are converted to TAI.
offsetint, optional
Offset in seconds from TAI midnight to be applied.
A timespan corresponding to a full day of observing.
If the observing day is 20240229 and the offset is 12 hours the resulting time span will be 2024-02-29T12:00 to 2024-03-01T12:00.
classmethod from_yaml(loader: SafeLoader, node: MappingNode) Timespan | None¶
Convert YAML node into _SpecialTimespanBound.
Instance of YAML loader class.
YAML node.
Timespan instance, can be None.
intersection(*args: Timespan) Timespan¶
Return a new Timespan that is contained by all of the given ones.
All positional arguments are Timespan instances.
The intersection timespan.
isEmpty() bool¶
Test whether self is the empty timespan (bool).
classmethod makeEmpty() Timespan¶
Construct an empty timespan.
A timespan that is contained by all timespans (including itself) and overlaps no other timespans (including itself).
overlaps(other: Timespan | Time) bool¶
Test if the intersection of this Timespan with another is empty.
otherTimespan or astropy.time.Time
Timespan or time to relate to self. If a single time, this is a synonym for contains.
The result of the overlap test.
If either self or other is empty, the result is always False. In all other cases, self.contains(other) being True implies that self.overlaps(other) is also True.
classmethod to_yaml(dumper: Dumper, timespan: Timespan) Any¶
Convert Timespan into YAML format.
This produces a scalar node with a tag “!_SpecialTimespanBound” and value being a name of _SpecialTimespanBound enum.
YAML dumper instance.
Data to be converted.
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The Stacks project
Lemma 42.54.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme with virtual normal sheaf $\mathcal{N}$. Let
$Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $c$ and $c(Z \to X, \mathcal{N})$ commute (Remark 42.33.6).
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Paint by Symbols (P.I.HUNT 3) - Puzzle WikiP.I.HUNT 3/Paint by Symbols
Paint by Symbols is one of the fifteen investigation puzzles from P.I.HUNT 3. It is a nonogram with numbers.
For answer checking purposes, the final enumeration is this length(1 7).
Each symbol substitutes with a number for a nonogram. Bounding arguments are useful for deducing what each symbol can represent.
The final grid depicts a mathematical expression with the symbols. Evaluate the expression to get a number.
Nonogram - Explicitly described as "No No. Nonograms".
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What are two types of scientific models? | Socratic
What are two types of scientific models?
1 Answer
physical models
mathematical models
conceptual models
** -replica of the original but in suitable for learning size
(larger atom\smaller solar system)
** -using various mathematical structures to represent real world situations.
(graph of climate change)
** -diagram shows of a set of relationships between factors that impact or lead to a target condition
(diagram of food web)
Impact of this question
8598 views around the world
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Variational Discrete Action Theory
2021 Theses Doctoral
Variational Discrete Action Theory
This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances
efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the
advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians.
VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter
decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the
each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation
function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the
SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation.
We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate
that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue
of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2
provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics
with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single
framework. VDAT will have broad applications in condensed matter and materials physics.
In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground
state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the
equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density
matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between
kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible
computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability
in lattice model Hamiltonians, in addition to real materials systems.
The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron
scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should
find broad applications in both model Hamiltonians and real materials.
• Cheng_columbia_0054D_16268.pdf application/pdf 1.48 MB Download File
More About This Work
Academic Units
Thesis Advisors
Marianetti, Chris A.
Ph.D., Columbia University
Published Here
November 16, 2020
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What is an Equilateral Triangle? Definition and Examples
What is an equilateral triangle? Definition and examples
An equilateral triangle is a type of triangle that has all sides of equal length and all angles of equal measure.
Some interesting properties of an equilateral triangle
• An equilateral triangle has three lines of symmetry. Each line of symmetry starts from a vertex and is perpendicular to the side opposite the vertex
• Since all sides of an equilateral triangle are equal, the perimeter is just three times the length of one of its sides
• Since the sum of the angles in a triangle is equal to 180 degrees, each angle in an equilateral triangle measures 60 degrees
• Each line of symmetry divides an equilateral triangle into two congruent triangles
Real-life examples of equilateral triangles
The Star of David made by two interlocking equilateral triangles is a good real-life example.
The traffic sign below warning someone of a danger is an equilateral triangle
The rack used to place billiard balls in their starting positions is shaped like an equilateral triangle
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SAT Syllabus: What is the Section Wise Syllabus & Key Topics?
What subjects are covered in the SAT syllabus?
The SAT syllabus includes three main sections: Reading, Writing and Language, and Math. The Reading and Writing sections assess comprehension, grammar, and vocabulary, while the Math section covers
algebra, problem-solving, data analysis, and some advanced math topics.
What math topics are included in the SAT Math section?
The SAT Math section focuses on algebra, geometry, statistics, problem-solving, data analysis, and some advanced math topics like trigonometry and complex numbers.
Are there any science or social science topics on the SAT?
While there is no dedicated science or social science section, SAT Reading passages may come from science, history, or social studies, and may require interpreting charts, graphs, and data.
Does the SAT Writing and Language section test vocabulary?
Yes, but instead of testing obscure vocabulary, the SAT focuses on words in context, requiring students to understand the meaning of words based on how they are used in a passage.
How important is the essay in the SAT?
The SAT Essay is now optional and no longer a mandatory part of the exam. Some colleges may still recommend or require it, so it's important to check the admissions requirements of the schools you're
applying to.
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there is a syntax for query boxes, we don’t use it much these days, with discussion being held here instead, but you could still use it. That would help separate your questions/comments from what
David C wrote.
@#69 Peter, regarding editing pages, the general rule is that anything on nLab can be edited, with announcement here if substantial. For others’ private webs, I just correct typos.
As for my own, where that Brandom note is, I just collect together some sketchy thoughts there. I’d be happy to read your thoughts there, if you could designate them as yours.
@#65: Peter, concerning the passage you criticize though I would admit the sin of ’rhetoric’ I would deny the charge of ’dark age rhetoric’. The intention there is to provoke the reader with the idea
that the ’logical lightweight’ Hegel outdoes Kant when it comes to having a critical attitude to traditional logic (note the implied suggestion to view Hegel as an expansion of the Kantian project to
logic) and more generally that the postKantian philosophers of the 1790 were quick to dismiss practically all preceding ’dogmatic’ metaphysics but often took the traditional laws of formal reasoning
for granted. I am probably willing now to exempt at least some of the postKantians from this charge since some of them felt indeed that the critical philosophy demanded a revision of traditional
logic e.g. Salomon Maimon published a ’Neue Theorie des Denkens’ in 1794, Jacob Siegismund Beck, a mathematician from Kant’s inner circle published a ’Lehrbuch der Logik’ in 1820 introducing
transcendental concepts into traditional logic, and Fichte in 1808 lectured on ’transcendental logic’ producing a large posthumously published text. The point is that Kant did not feel this need, the
Jaesche-Logik contains the famous quote that general and pure logic is dull and short and basically a closed chapter since antiquity (or something like this), a quote that made it into the 1928
textbook of Hilbert and Ackermann who obviously did not think that chapter quite as closed neither did Leibniz before them.
This does not mean that the Jaesche-Logik is unimportant for the philosophy of logic nor that Kant’s transcendental logic cannot not fruitfully confronted with geometric logic, though calling the
later ’Kant’s logic’ runs into the problem that Kant admitted traditional logic as a valid form of reasoning regardless of the objective content of the concepts employed i.e. to the extent that Kant
’had’ a logic traditional logic is a better candidate for it, in my view.
That Kant’s reasoning is inherently constructive is due to his attempt to model philosophy on the reasoning with constructions in Euclid’s geometry and the later is also an albeit remote source of
geometric logic. Anyway, I am the last person to belittle Kant who is in fact one of the brightest stars on my philosophical firmament. In the later passages of the nLab article the continuity
between Kant and the postkantian systems and Hegel is stressed. I generally find it useful to view thinkers like Kant, Fichte, Schelling and Hegel to be involved in a common project of transcendental
philosophy which in my view is highly relevant to contemporary philosophy or cognitive science and deserves to be formalized by methods of modern mathematics.
Concerning Ploucquet, there is a German-Latin edition of his Logic by Michael Franz available as well as an article by Redding exploring the connection between Hegel and Ploucquet called THE ROLE OF
LOGIC “COMMONLY SO CALLED” IN HEGEL’S SCIENCE OF LOGIC presumably available from his homepage as a preprint. In the context of cognitive underpinning for sheaf theory the link to the Petitot paper at
Aufhebung might be interesting as well.
In any case, feel free to edit or expand Aufhebung when you cannot stomach certain passages. Additional insights or views are always appreciated and generally encouraged by the nLab!
I'll pass on to you anything that gets written up, or perhaps recordings of the talk if they appear first.
This is probably a silly question, but would the right etiquette be to edit your Brandom note if I wanted to respond to your thoughts on the topic?
Well you’d have been very welcome to the workshop anyway even if just to attend.
Interesting you mention Brandom. I jotted down a note which sounds like it may be in the same direction as your criticism. (By the way, I overlapped here with Ken Westphal for a couple of years.)
When you have something to read on what you describe in the 3rd paragraph, I’d be very interested.
Unfortunately, the reading group is just a bunch of us in a room thrashing things out.
Hi David, thanks for the welcome. I still feel somewhat out of my depth here, even if I've been browsing nLab for at least a year at this point. I considered submitting something for your HoTT and
philosophy workshop, but nervousness and the need to move to South Africa on short notice mitigated against it.
I haven't written anything up on the Kant-Vickers connection yet. It's part of a cache of insights I've been slowly tripping over since discovering the A&L paper on transcendental logic and getting
serious about understanding sheaves and Grothendieck topoi. Strangely, I got into all this by trying to figure out what was so unsatisfactory about Brandom's formal incompatibility semantics, which
has some Hegelian philosophical inspiration, but ends up being horribly gerrymandered into classical rubbish. One of the inconsistencies between Brandom's philosophical inspirations and his formalism
(as pointed out to me by Ken Westphal) is precisely that it ignores Kant's distinction between negative and infinite judgments, collapsing everything back into (classical) propositional negation, and
thereby being completely unable to account for the sorts of concrete incompatibilities between predicates he starts from (e.g., between blue and the various predicates - green, red, yellow, etc. -
that fall under non-blue). There are more complaints I could make on this front, as learning what's wrong in Brandom's project has been quite enlightening, but I'll stop there.
The overarching project that these ideas belong to is the development of what I'm calling 'computational Kantianism', reading Kant's transcendental psychology as essentially already the project of
AGI, using contemporary work in logic/maths/compsci to make sense of Kant and using Kant to provide some overarching structure connecting this same work. I gave a talk in Dublin recently that went
over some of the overarching methodological ideas of this approach, but it's not written up. I'm due to give a seminar at a Summer school in NYC directly on computational Kantianism later this month,
but I'm still working from notes, trying to condense things down into something tractable. I think one can draw a useful line between Kant's insistence on the primacy of judgment as a starting point
for transcendental psychology and Harper's idea of computational trinitarianism, and that this (along with certain stories one can tell about subterranean connections between Kant and constructivism
in the history of mathematics: i.e., Brouwer-Heyting, topos theory, Curry-Howard, HoTT) opens up the possibilities for reflecting back and forth between Kant and contemporary work I'm proposing. The
really novel thing I think can be imported back from Kant is his conception of the relationship between mathematical and empirical judgment/cognition, which I think can be roughly understood through
the duality between intuitionistic and co-intuitionistic logic (judgment) and computational data and co-data (cognition). I think this line of thinking inevitably leads you from Kant to Hegel, as the
relationship between imagination and understanding needs to be supplemented with that between understanding and reason, but it's nice to start with Kant's emphasis on our (computational) finitude and
build up to Hegel from there. It also has the advantage of suggesting how to extend computational trinitarianism beyond HoTT, insofar as one can project something like a co-intuitionistic dual of
HoTT for empirical cognition. There's a few more things I could say about this, but I'll stop myself before I get further out of my depth!
Is this reading group online? If so, I'd very be interested in tagging along. One of the things reading the work being done here has taught me is how much was lost in the transition from the
Aristotelian to the Fregean logical paradigm.
Welcome Peter, from another philosopher.
I’m very interested in your “potted example”. The reading group I belong to will soon be reading Paul Redding on the lost subtleties of negation possible in term logic in his ’Analytic Philosophy and
the Return of Hegelian Thought’.
Have you written on the Kant-Vickers connection?
Disclaimer: This is my first post here. I'm a philosopher by training, but I'm trying to teach myself enough category theory in order to eventually participate in the project in a meaningful way.
I wanted to offer a reason to reconsider the following quote from the entry:
"However critical these idealist systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It
is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself."
I can't speak for Fichte, but there's some good evidence that this is not the case for Kant. Dorothea Achourioti and Michiel van Lambalgen have an excellent paper titled 'A Formalization of Kant's
Transcendental Logic' (http://philpapers.org/rec/ACHAFO) that makes the case that Kant's logic is actually geometric logic. The paper is aimed at philosophers, and so goes out of its way to not use
category theory (it uses inverse limits of a system of inverse sets), but I have confirmed with van Lambalgen that this was a very deliberate de-categorification for philosophical consumption. I
actually think that the semantics they provide is probably way too simple, but the exegetical case for interpreting Kant's logic as geometric is very good in my view, and the connection to
Grothendieck topoi provides various interesting connections between Kant's transcendental psychology and mathematics/compsci (e.g., the idea that object synthesis is about tracking local invariants
in varying heterogeneous data (intuition) attached to/organised by some base topology (forms thereof)).
If you want a potted example, Kant's distinction between negative judgments and infinite judgements makes perfect sense from the perspective of Steve Vickers's ideas about geometric type theory. For
Kant, the crucial difference between the two (corresponding to propositional vs. predicate negation) is that the latter has existential import (relation to an object) and the former does not.
Vickers's idea is that the infinitary disjunctions of geometric logic can be re-interpreted using typed existential quantification, and this is essentially what provides the existential import/
objective validity of the infinite judgment from the Kantian perspective. Using a standard and overly simplistic example, judging that an extended object is non-blue excludes a determinate range of
possibilities for that object, because the type of extended objects includes a colour attribute with a strictly delimited but potentially infinite range of possible variations. There's more that
could be said here, but I think this indicates that Kant is more interesting for the nLab project than the above quote suggests.
I'll close with one more observation about the history of logic. I'm no expert, but the history of logic between Leibniz and Boole is really not very well articulated, and it seems that there are a
lot of lines of influence that really aren't properly understood. It's all too easy to represent it as a sort of logical dark age where nothing took place. I honestly don't know how well Hegel would
have understood Kant's perspective on logic, given that much of what we now understand comes from the lectures he gave on the topic, which wouldn't have been widely available at the time. However, it
seems that Hegel and Schelling were significantly influenced by Ploucquet, from whom they seem to have contracted the idea of the reversibility of subject and predicate. Ploucquet is one of the
significant figures recognised in the tradition post-Leibniz, but his work doesn't seem to be widely studied (it's all in latin, with no English translation from what I can gather) either on its own
terms or as an influence on Hegel's logical views. It's worth avoiding 'logical dark age' rhetoric, whether it is between Aristotle and Frege or Leibniz and Hegel, as there are hidden lines of
research and influence that were still in the process of uncovering and reconstructing.
From a quick glance at the section you link, to the disjointness property is meant only for the particular example. In the general case, the two subcategories are usually far from disjoint, in fact,
one can think of the process of Aufhebung as a gradual level-to-level augmentation of the objects in the intersection, that contains the ’true thoughts’ where content (left inclusion) coincides with
notion (right inclusion), starting from $0\cap 1=\emptyset$ up to $id_\mathcal{E}\cap id_\mathcal{E}=\mathcal{E}$.
under this section, we read:
the functors L and R must actually correspond to inclusions of disjoint subcategories
I take it this is not necessary in general, and is only true here since the composites $T \circ L, T \circ R$ are equal to the identity. I think that in general, when the composites are merely
isomorphic to the identity, the subcategories have intersection given by the equalizer of the subcategory inclusions. Is this reasoning sound?
The quote from Hegel is 113 from here:
Cancelling, superseding, brings out and lays bare its true twofold meaning which we found contained in the negative: to supersede (aufheben) is at once to negate and to preserve.
At MPI Bonn this Aufhebungs-announcement is flying around (full pdf by In Situ Art Society).
I have take the liberty to add it to the entry Aufhebung.
added to the discussion here of Aufhebung $\sharp \empty \simeq \empty$ over cohesive sites pointer to lemma 4.1 in
• William Lawvere, Matías Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015,
No. 26, pp 909-932. (TAC)
which obverseves this more generally when pieces-have-points.
I wrote out a more detailed proof of the statement here that the bosonic modality $\rightsquigarrow$ preserves local diffeomorphisms.
It seems the proof needs not just Aufhebung in that
$\rightsquigarrow \Im \simeq \Im$
but needs also that this is compatible with the $\Im$-unit in that $\rightsquigarrow$ sends the $\Im$-unit of an object $\stackrel{\rightsquigarrow}{X}$ to itself, up to equivalent
$\rightsquigarrow( \stackrel{\rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} ) \;\;\; \simeq \;\;\; ( \stackrel{\
rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} )$
This is true in the model of super formal smooth $\infty$-stacks, so I am just adding this condition now to the axioms. But it makes me wonder if one should add this generally to the concept of
Aufhebung, or, better, if I am missing something and this condition actually follows from the weaker one.
That makes me want to experiment with re-thinking about a possibly neater way of defining differentially cohesive toposes.
Something like this:
A differential cohesive topos is (…of course…) a topos $\mathbf{H}$ equipped with two idempotent monads $\sharp,\Re : \mathbf{H} \to\mathbf{H}$ such that there are adjoints $\int \dashv \flat \dashv
\sharp$ and $\Re \dashv ʃ_{inf} \dashv \flat_{inf}$ (…but now:) and such that
1. (clear:) $ʃ {}_{\ast}\simeq {}_{\ast}$ and $\sharp \emptyset \simeq \emptyset$
2. (maybe:) $\flat_{inf} \Pi \simeq \Pi$ and $\Re \flat \simeq \flat$.
I need to go through what I have to see what the minimum needed here is. I certainly need $\flat_{inf} \flat \simeq \flat$ for the relative infinitesimal cohesion to come out right. Also $\Re \ast \
simeq \ast$, which would follow from the above.
In any case, I feel now one should think of these axioms as describing a picture of the following form (the Proceß)
$\array{ &\stackrel{}{}&& id &\stackrel{}{\dashv}& id \\ &\stackrel{}{}&& \vee && \vee \\ && & \Re &\dashv & ʃ_{inf} & \\ &&& \bot && \bot \\ &&& ʃ_{inf} &\dashv& \flat_{inf} \\ &&& \vee &&
\vee \\ &&& ʃ &\dashv& \flat & \\ &&& \bot && \bot \\ &&& \flat &\dashv& \sharp & \\ &&& \vee && \vee \\ &&& \emptyset &\dashv& \ast & \\ }$
and the question is what an elegant minimal condition is to encode the $\vee$-s.
coming back to #16:
I used to think and say that in the axioms of cohesion the extra exactness condtions on the shape modality seem to break a little the ultra-elegant nicety of the rest of the axioms. There is an
adjoint triple of (co-)monads, fine… and in addtition the leftmost preserves the terminal object – what kind of axiomatics is that?!
But Aufhebung now shows the pattern: that extra condition on the shape modality
$ʃ {}_\ast \simeq {}_\ast$
is just a dual to the “Aufhebung of becoming”
$\sharp \emptyset \simeq \emptyset\,.$
Maybe a co-Aufhebung, or something.
coming back to #50:
on the other hand, of course $\flat_{inf}$does provide Aufhebung of cohesion in the sense that $\flat_{inf} \int \simeq \int$.
Of course this follows trivially here, since we are one step to the left and both of $\int$ and $\flat$ correspond to the same subcategory.
Ah, I should be saying this more properly (and this maybe highlights a subtlety in language that we may have not properly taken account of somewhere else in the discussion):
in the topos over the site of formal smooth manifolds, the sub-topos of $\flat^{rel}$-modal types is “infinitesimally cohesive” in that restricted to it the map $\flat \to \int$ is an equivalence.
Yes, right, in the given model the $(\flat^{rel} \dashv \sharp^{rel})$-level exhibits what “we” here had decided to call “infinitesimal cohesion”, which is essentially another word for what Lawvere
had called a “quality type”.
And yes, I’d agree that it would make much sense to regard $(\flat^{rel} \dashv \sharp^{rel})$ as being the “next” level after $(\flat \dashv \sharp)$. After all, the sequence of inclusions of levels
$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv & \sharp \\ \vee && \vee \\ \emptyset && \ast }$
reads in words “a) the single point, b) collections of points, c) collections of points with infinitesimal thickening”.
And it seems clear in the model (though I’d have to think about how to prove it) that $\flat^{rel} \dashv \sharp^{rel}$ (when given by first-order infinitesimals) should be the smallest nontrivial
level above flat $\flat \dashv \sharp$.
Unfortunately, I am not sufficiently familiar with your example to fully understand the details but I asked because I have the understanding that $\flat^{rel}\dashv\sharp^{rel}$ is actually a quality
type hence its own Aufhebung !? So my idea is that in some sense it comes reasonably closest to provide Aufhebung for a level which otherwise lacks Aufhebung.
In order for this to make sense, one would probably like to demand that $\flat^{rel}\dashv\sharp^{rel}$ is the smallest quality type that subsumes $\flat\dashv\sharp$.
I haven’t really thought about under which conditions $(\flat \dashv \sharp)$ has Aufhebung, all I meant to say here is that there is naturally this level $(\flat^{rel} \dashv \sharp^{rel})$ sitting
above it, but that in the standard model this level, at least, is not even a resolution of $(\flat \dashv \sharp)$. There might be others that are, though.
And regarding this being canonical: the claim is that if differential cohesion is given, then $(\int^{rel} \dashv \flat^{rel})$ is canonically given. Think of it this way: differential cohesion is
not really a level above cohesion, because of the “carrying” of the adjoints to the left. But it canonically induces $\flat^{rel}$ and so as a soon as that happens to have an adjoint $\sharp^{rel}$,
then thereby it induces something that is a level over cohesion.
So maybe it’s good to think of this as some kind of “carrying back to the right”-operation, if you wish.
So, do you mean that $\flat\dashv\sharp$ lacks Aufhebung ? If this is the case, couldn’t $\flat^{rel}\dashv\sharp^{rel}$ be regarded as a sort of homotopy approximation to the Aufhebung i.e. is
construction of $\flat^{rel}\dashv\sharp^{rel}$ from $\flat\dashv\sharp$ sufficiently canonical !?
This is long overdue: I have started at differential cohesion – relation to infinitesimal cohesion to add some first notes on how differential cohesion
$\array{ \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp }$
induces “relative” shape and flat $ʃ^{rel} \dashv \flat^{rel}$ – such that when this extends to a level
$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast }$
then this level exhibits infinitesimal cohesion.
This is the case in particular for the model of formal smooth ∞-groupoids and all its variants (formal complex-analytic $\infty$-groupoids, etc.).
But I think in all these cases $(\flat^{rel} \dashv \sharp^{rel})$ does not provide Aufhebung for $(\flat \dashv \sharp)$.
This is because: for $X$ being $\flat$-modal hence being a discrete object, then maps $U \to \sharp^{rel} X$ out of any object $U$ in the site, which are equivalently maps $\flat^{rel}U \to X$, are
maps out of the disjoint union of all formal disks in $U$ into $X$. These are again representable (we are over the site of formal smooth manifolds) and so these maps are equivalent to $X(\flat^{rel}
U)$. But $X$ is discrete and hence constant as a sheaf on the Cahiers-site, and so these are equivalent to $X(\flat U)$ which in turns is equivalent to maps $\flat U \to X$ and hence to maps $U \to \
sharp X$. So by Yoneda we conclude that $\sharp^{rel} X \simeq \sharp X$ in this case, but this is in general not equivalent to $X$.
It seems we’d have the freedom to make a definition either way, after all this is to formalize something that Hegel said, and depending on how we feel about that and depending on which mathematics we
would like to see developed, we may feel that different definitions are appropriate. I am just thinking that in any case there should be some kind of justification. The order relation via
subcollections of modal types seems well motivated, but then switching to a different relation along the way seems to call for a reason.
I’ll be happy with whatever definition leads to something interesting. In any case I am presently short of examples of subtoposes which would be in relation in one of the senses under consideration,
but not in the other. Maybe we should try to get hold of some examples for such a phenomenon to get a feeling for which kind of subtlety the definition should try to take care of.
I start to see your point here, the way I stated the definition is cooked up out of Lawvere 89 who defines this for graphic toposes and states everything in terms of the pretty-well behaved ideals in
the underlying category, so I kept his terminology but looked for the definitions in terms of subtoposes to KRRZ11 which I’ve read in the way that $\prec$ and $\leq$ are two different things, but
actually life becomes much easier if they are the same. I was already looking for a proposition showing their compatibility in order to prove that quintessential localizations are their proper
Aufhebung. I guess you are right.
Thomas, if it is as intended, then my question would be why this is intended. Why use in the last line of the definition not the same order relation as introduced in the first line? What’s the
rationale for this?
ad 44#: to me the definition appears to be as intended unless I forgot to switch notation at some place. The problem was previously it used $\ll$ for the minimality where it should have used the
essential subtopos order which I then thought best to denote with the least marked $\leq$. I think I’ll try later to use $\sqsubset$ and $\lhd$ instead of $\prec$ and $\ll$ as they have rotated
versions $\sqcup$ and $\bigtriangledown$.
Urs, re #40, I was trying to do something like that back here.
I had the feeling Neel was onto something too a few comments later.
Let me see if I understand – you changed the notation in the first clause in the definition, but should you not also change it in the last line then? Maybe I am missing something.
By the way, regarding your email: the reason that I oriented these diagrams as I did as opposed to with levels going upwards is just because I wouldn’t know how to typeset the order relation going
Sorry for the interruption, but I had to fix the definition of Aufhebung as the minimality is defined relative to the usual order of the levels in the literature. For this I reused $\leq$ as the
order of levels and tried to use predecessor-equality for the resolution relation but the local teX doesn’t know my dialect.
coming back to #34-#35 and also to our discussion in person on the justification of calling a $\Box$-modality the “necessity” modality:
if we turn what Hermida does around, then everything makes sense to me.
Namely: if we
• read “necessarily” as the name specifically for the “for all”-operation turned into a comonad;
• read “possibly” as the name specifically for the “there exists”-operation turned into a monad;
then first of all $\Box$ is a comonad as desired and moreover then its interpretation as formalizing “necessity” is indeed justified:
for let $X$ be a “context” which you may want to think of as the “type of all possible worlds”, and if $P$ is a proposition about terms of type $X$, then
• the statement that “for $x \colon X$ it is necessarily true that $P(x)$” is just another way to say in Enlish that “for all $x \colon X$ it is true that $P(X)$”;
• the statement that “for $x \colon X$ it is possibly true that $P(x)$” is just another way to say in Enlish that “there exists $x \colon X$ such that it is true that $P(X)$”;
So, more formally, given the adjoint triple of dependent sum $\dashv$ context extension $\dashv$ dependent product
$\mathbf{H}_{/X} \stackrel{\stackrel{\sum_X}{\longrightarrow}}{\stackrel{\stackrel{X^\ast}{\longleftarrow}}{\underset{\prod_X}{\longrightarrow}}} \mathbf{H}$
then it makes justified sense to call the induced (co-)-monads
$(\lozenge_X \dashv \Box_X) \coloneqq ( X^\ast \underset{X}{\sum} \dashv X^\ast \underset{X}{\prod}) \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$
“possibility” and “necessity”, respectively, because
• by the established interpretation of $\sum_X$ as “there exists $x \colon X$” we have that $\lozenge_X P$ holds for a given $x\in X$ precisely if it holds for at least one $x \in X$, hence if it
is possible in the context $X$ for it to hold at all;
• by the established interpretation of $\prod_X$ as “for all $x \colon X$” we have that $\Box_X P$ holds for a given $x\in X$ precisely if it holds for all $x \in X$ hence if it is inevitable,
hence necessary, in the context $X$ that if holds.
It seems to me that this here is proof that over a cohesive site $(\flat \dashv \sharp)$ is Aufhebung of $(\emptyset \dashv \ast)$. But check that I am not being stupid here:
+– {: .num_prop #OverCohesiveSiteBecomingIsAufgehoben}
Let $\mathcal{S}$ be a cohesive site (or ∞-cohesive site) and $\mathbf{H} = Sh(\mathcal{S})$ its cohesive sheaf topos with values in Set (or $\mathbf{H} = Sh_\infty(S)$ its cohesive (∞,1)-topos ).
Then in $\mathbf{H}$ we have Aufhebung, def. \ref{Aufhebung}, of the duality of opposites of becoming $\emptyset \dashv \ast$.
+– {: .proof}
By prop. \ref{OverCohesiveSiteBecomingIsResolved} we have that $(\flat\dashv \sharp)$ resolves $(\emptyset \dashv \ast)$ and so it remains to see that it is the minimal level with this property. But
the subtopos of sharp-modal types is $\simeq$Set which is clearly a 2-valued Boolean topos. By this proposition these are the atoms in the subtopos lattice hence are minimal as subtoposes and hence
also as levels.
I have edited the formatting of the central definition a bit in order to make the central ideas spring to the eye more vividly. Now it reads as follows:
Let $i,j$ be levels, def. \ref{Level}, of a topos $\mathcal{A}$ we say that the level $i$ is lower than level $j$, written
$\array{ \Box_i &\leq& \Box_j \\ \bot && \bot \\ \bigcirc_i &\leq& \bigcirc_j }$
(or $i\leq j$ for short) when every i-sheaf ($\bigcirc_i$-modal type) is also a j-sheaf and every i-skeleton ($\Box_i$-modal type) is a j-skeleton.
Let $i\leq j$, we say that the level $j$resolves the opposite of level $i$, written
$\array{ \Box_i &\ll& \Box_j \\ \bot && \bot \\ \bigcirc_i &\ll& \bigcirc_j }$
(or just $i\ll j$ for short) if $\bigcirc _j\Box_i=\Box _i$.
Finally a level $\bar{i}$ is called the Aufhebung of level $i$
$\array{ \Box_i &\ll& \Box_{\bar i} \\ \bot &\searrow& \bot \\ \bigcirc_i &\ll& \bigcirc_{\bar i} }$
iff it is a minimal level which resolves the oppose of level $i$, i.e. iff $i\ll\bar{i}$ and for any $k$ with $i\ll k$ then it holds that $\bar{i}\ll k$.
Thanks for the feedback. For the argument in prop. 2 the homotopy theory is irrelevant (I should have pointed to cohesive site!), you may just as well consider just plain presheaves. It’s meant to be
a trivial argument: the site by assumption has a terminal object and there is a morphism from that terminal object to every other object. That implies that if a presheaf (of sets) assigns the empty
set to that terminal object, it has to assign the empty set to every other object of the site, too.
Assuming that you use corollary II of Yoneda lemma in the last step of the proof of prop.1 at Aufhebung I think that prop.1 is ok.
As far as I can tell the proof of prop.1 uses only the strictness of $\empty$ and the rest is purely general using just the adjointness and the specific equivalences you assume, so this seems to be
valid more generally for extensive cats with an adjunction that relates to $\empty$ in the appropriate way without being necessarily $\flat\dashv\sharp$, no !?
To see through prop.2 I’d need time to acquire some knowledge on infinity-cohesive-sites. This should be easier to see for someone with better grounding in the higher categorical point of view
though, I guess.
In any case, it would be nice to have this somewhat more general result for the Aufhebung of $\empty\dashv {\ast}$.
Yes, that’s exactly what we seem to have agreed on over at modality – Notation.
I’ll change it at Aufhebung, too.
BTW, any comment on #17? We talked about it by email. It now seems to me that it does work for the “standard examples” of cohesion. But let me know if I am missing something.
Sorry for bringing this up again, as the entry now has $\Box$ on the left, I would suggest wispering to use $\bigcirc$ on the right in the context of Aufhebung as this nicely suggests ’being’ unless
this interferes negatively with the convention you just set up. As this is just a tiny detail I would very much like to avoid a discussion on this and propose to drop the subject when you don’t like
the idea.
In that remark 3.3 he says that his $\langle R\rangle$ is the composition of pullback followed by dependent sum, and that his $[R]$ is the composition of pullback followed by dependent product. That
makes $[R]$ a monad.
Where? I see him say
Monadic interpretation of $\langle - \rangle$,
but that’s expected.
Hm, on the other hand, Hermida’s article has $\Box$ be a monad, not a comonad.
Thanks. I see it now in remark 3.3. Good, so I’ll add that to the entry modality, too. So then maybe “tendency to use” is not that bad after all. Do we known an author who explicitly does not use
that convention?
It seems that $[R]$ is always on the right.
Ah, right. So under this translation, does he have $\Box$ as the right adjoint?
Hermida uses $\Box$ and $\lozenge$ in the abstract, but then angled and square parentheses around a relation symbol.
Okay, I have changed “tendency to use” to “some authors use”. Then I added one more example, namely
• Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)
(where the left part of cohesion appears in terms of adjoint modalities on p. 367).
Maybe “some authors” is just “Gonzalo Reyes”, though? Hermida does not seem to use the symbols from modal logic, does he?
Regarding notation for differential cohesion: I may not know what candidates there are from traditional theory. Maybe none? I don’t know. The situation of differential cohesion seems to have been
missed, by and large.
When adjunctions between modalities matter, there is a tendency …
My knowledge of these situations is so small I couldn’t speak of a tendency. I don’t know of any philosophers who have raised the issue of adjunctions. Do people know about computer science, etc.?
Even Hermida as mentioned here seems to have the dichotomies lined up.
But what about my final point in #21? Should one use just $\Box$ for left/right adjoint comodalities $Red$ and $\flat_{inf}$? Sometimes people have used L and M for necessity/possibility.
Okay, I have tried to edit accordingly here. Please feel invited to further edit if you see further need.
If anything I’d say $\lozenge$ is the more commonly used of the two. The SEP modal logic entry chooses it. So, all things equal, I’d have it as the left adjoint monadic modality.
Thanks. I’ll rephrase the statement in the entry then. Just a moment…
As far as I’m aware it’s just an arbitrary choice of notation, like $\wedge$, $\cdot$, or & is a choice for conjunction.
How do modal logicians choose between $\lozenge$ and $\bigcirc$? What does the choice indicate for them, if anything?
Cancel what I just wrote having actually read what was written.
If that’s suggesting in modality that $\lozenge$ is less used by modal logicians for possibility, then that’s wrong. As far as I’m concerned it’s used as much as $\bigcirc$:
Also $\lozenge$ is used for a modality, in particular if it is left adjoint to a $\Box$.
On the other hand we could choose to make such a convention of distinguishing left and right adjoint monads. But let’s be explicit.
What about the differential cohesive comonads? Shouldn’t we have notation to distinguish left and right adjoints there too?
I did this just by exclusion principle: according to #12 the diamond has been used for the left adjoint, so the circle remains for the right adjoint.
But I am happy with any other convention/tendency.
Now I’ve looked at your notation remark at modality; should we use $\bigcirc$ rather than $\lozenge$ at Aufhebung since it is right adjoint to $\Box$? (What’s the origin of $\lozenge$ being left
adjoint to $\Box$ and $\bigcirc$ being right adjoint to it? I didn’t think that necessity and possibility were adjoint in either direction; are they?)
Ok, I’ve switched $\Box$ and $\lozenge$ in the entry.
What I am talking about is now cleaned up here. Let me know if I am hallucinating (it’s late here).
Back to the Aufhebung via global points:
so over every infinity-cohesive site it is true that $\sharp\emptyset \simeq \emptyset$, I suppose.
Since here indeed only the initial sheaf has no global points. (E.g. if a sheaf on $CartSp$ has no global points, that means it assigns the empty set to $\mathbb{R}^0$, but that means it must assigns
the empty set to each $\mathbb{R}^n$ since there do exist maps $\mathbb{R}^0\to \mathbb{R}^n$).
I added a remark at modality in a section Notation. Everyone is kindly invited to expand and/or edit.
I agree, my choice was hasty as I looked only for an adjunction between the modal operators without realizing that they were defined dually. Though I must admit that I found the choice esthetically
pleasing - it would be good to have some suggestive symbols.
I think Mike’s point is that for the notation here its not the side of the adjunction but whether we have a monad or comonad that decides what the box is. The box should be a comonad as for necessity
. That is consistent with what you cite from Reyes, but not with using box for $\sharp$.
(We are lacking an $n$Lab page that says this comprehensively. Something should go at modal operator )
My decision came from a short look at p.116 in the ’ generic figures’ book of Reyes et.al. where they define $\lozenge:=i^*\circ i_!$ and $\Box :=i^*\circ i_*$ in context of an inclusion of posets;
in which case then $\lozenge\dashv \Box$ whereas the nlab entry defines the composition dually in reverse order. so this should probably be changed, sorry for the confusion!
Thomas, can you explain further your reasons for assigning $\lozenge$ and $\Box$ in that order? I don’t understand why necessity would be on the right side.
@#8: the notation was chosen by me probably after having looked in which way the modal operators are adjoint to replace $L\dashv R$ of Lawvere (1989) or the $sk\dashv cosk$ used in KRRZ11. In
complies broadly with the intuition to have necessity on the ’right’ side of being/sheaf but feel free to choose something more suitable.
re #6:
this refers to the Idea section here. I just meant two levels where one includes the previous one, I have changed it now to read as follows:
If for two levels $\mathbf{H}_{1} \hookrightarrow \mathbf{H}_2$ the second one includes the modal types of the idempotent comonad of the first one, and if it is minimal with this property, then
Lawvere speaks of “Aufhebung” (see there for details) of the unity of opposites exhibited by the first one.
Are you sure that $\lozenge _i$ and $\Box _i$ are named correctly? If the are intended to suggest the “possibly” and “necessarily” modalities of classical modal logic, then $\lozenge$ should be the
monad and $\Box$ the comonad.
re #3:
to say that in a more pronounced way, the condition $\sharp \emptyset \simeq \emptyset$ implies that the ambient $\infty$-topos has homotopy dimension $\leq 0$ relative to the sub-$\infty$-topos of $
\flat$-modal objects. This is a property enjoyed by all models for which the $\flat$-subtopos is $\infty Grpd$ (by this proposition). So is this “Aufhebung” a strong classicality condition on the $\
flat$-modal subtopos?
re #5: I’d say “yes, of course”, but that means I am probably missing some subtlety that you have in mind :-)
What does it mean for two levels to be “consecutive”?
Let me connect this with something more familiar to me. To say that $(i^*,i_*):E\to F$ is an essential subtopos, with essentiality $i_!$, is almost the same as to say that $(i_!,i^*):F\to E$ is a
connected local geometric morphism, except that we don’t ask $i_!$ to be left exact. Right?
I guess my wording in the entry is a bit careless, as I brought in the general context of categories of being in order to motivate the metaphysical lingo for $\empty\dashv\ast$ and its Aufhebung,
though Lawvere proves this only in special cases e.g. in the 1991 ’Hegelian taco’ paper. so unless, a proof for the general case can be cooked up, I am afraid I’ve to row back a bit in the entry.
It maybe worth mentioning that a negative result of a (sufficiently) cohesive topos where this Aufhebungs relation fails to hold would also show the limits of the particular categorical
interpretation of Hegel’s logic.
Let me look at the most basic case. That $(\flat \dashv \sharp)$ is Aufhebung for $(\emptyset \dashv \ast)$ means that
$\sharp \emptyset \simeq \emptyset$
and that $\sharp$ is minimal with this property. What is the statement regarding conditions under which this is the case?
By adjunction and using that the initial object in a topos is stict, it follows that $\sharp \emptyset \simeq \emptyset$ is equivalent to the statement that the only object with no global points is $
\emptyset$ itself, because
$(X \to \sharp \emptyset) \simeq (\flat X \to \emptyset) \simeq (\flat X \simeq \emptyset) \,.$
Is that automatic? (This is probably elementary, please bear with me.) And how to see or check that $\sharp$ is minimal with this property?
For completeness I have added an brief entry level of a topos and cross-linked a bit.
Thomas Holder has been working on Aufhebung. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references,
cross-linked with duality of opposites).
Thanks guys. I'll try to figure out the neatest way of commenting that separates out my comments from David's thoughts.
Thomas, thanks for the reference to the Redding paper, it looks great!
As for Kant/Hegel, I appreciate that what you're doing is provocation. It is of course incredibly important to get people to consider Hegel as an actual logician, and indeed, thereby to broaden their
understanding of what logic is. I simply think it is worth pushing the envelope further back and giving Kant the same treatment, and indeed, thereby allowing us to draw more interesting *logical*
lines from Kant to Hegel. Here are a couple further reasons to reconsider Kant's status as a logician, responding to your points:
1. One of van Lambalgen's students, Riccardo Pinosio, wrote an excellent MSc thesis reconstructing Kant's philosophy of mathematics based on the idea that transcendental logic is geometric logic
(https://www.illc.uva.nl/Research/Publications/Reports/MoL-2012-13.text.pdf). It's really good, and it does an excellent job of dismissing Friedman's claim that Kant's constructivism was motivated by
the inability of his logic to formulate ∀∃ judgements. There's more work to be done along these lines, as the role of temporality in Kant's account of construction is still not completely explicated,
but I think it's a great start.
2. I think focusing on Kant's comments on general logic is to sell short the *genuinely logical* character of his transcendental logic. By analogy, one might say that 'Well, classical first-order
predicate calculus and set theory is perfectly adequate, as far as it goes...' and then go on to talk about what it *isn't* adequate for, such as reasoning about things that aren't either
unstructured points or set-theoretic constructions upon them, but it would be a terrible mistake for someone to focus on the first comment at the expense of the rest of the explanation. For a long
time people have simply dismissed the idea that Kant's transcendental logic is logic at all, much as they have dismissed Hegel's logic, and it is important to see that there are good reasons for
counteracting the former dismissal as well as the latter. Furthermore, I would argue that combining these argumentative strategies bolsters them, both rhetorically (it is often easier to get people
to take Hegel seriously by exhibiting his advances on Kant), and conceptually (it is often easier to actually understand Hegel by exhibiting his advances on Kant). However, to make good on this
promise I have to justify Kant's idea that there is something *inherently logical* about relation to an object (objective validity), and show that this is encapsulated in a significant way by
geometric logic. That might take some time, and is probably better to do on its own page/thread. For now, you might want to think about the issue in the following way: for Kant, general logic is fine
for reasoning about both mathematical and empirical matters only insofar as it is entirely indifferent to the difference between them, and in this indifference it is also perfectly good for reasoning
about Harry Potter, unicorns, or even colourless green ideas, and engaging in a panoply of other language games that might be perfectly internally consistent yet have nothing resembling purchase on
the world. It is not too far a jump to interpret him as circling around the same issues that Hegel (per Lawvere) is talking about in terms of the difference between subjective and objective logic.
Added a reference to the recent article by Marmolejo-Menni on “level $\epsilon$”.
diff, v102, current
Added a reference.
diff, v103, current
Added reference to
• {#Menni19}M. Menni, Monic skeleta, Boundaries, Aufhebung, and the meaning of ‘one-dimensionality’, TAC 34 no.25 (2019) pp.714-735. (tac)
• {#Menni22}M. Menni, Maps with Discrete Fibers and the Origin of Basepoints, Appl. Categor. Struct. 30 (2022) pp. 991–1015.
diff, v107, current
Added a reference to
• Michael Roy, The topos of ball complexes, TAC Reprints no.28 (2021) pp.1-62. (abstract)
diff, v108, current
Added reference to
• {Menni23} M. Menni, The successive dimension, without elegance, arXiv:2308.04584 (2023). (pdf
diff, v111, current
best practice not to link directly to a pdf, but always to its HTML landing page, if available (as in the present case: [arXiv:2308.04584])
(on Twitter the arXiv even has a bot running to remind people of this netiquette — here I am that bot)
diff, v112, current
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Fibonacci Series Program in Java
Last Updated on September 22, 2023 by Mayank Dham
Within this article, we will delve into the intricacies of the Fibonacci series Program in Java. Furthermore, we shall acquire the knowledge to construct a Java program for generating the Fibonacci
series, employing diverse techniques including iterative, recursive, and recursive with memoization. Additionally, we will analyze the time and space complexities associated with each of these
What is the Fibonacci series?
In the Fibonacci series, the next term is an addition of the previous two terms. For example, if the previous two terms are 3 and 5 then the next term will be 8 (3+5). The first two terms of the
Fibonacci series are 0 and 1 respectively. Now, if we look at the third term it will be the summation of the first term(0) and the second term(1) is 1 (0+1). If we look for the fourth term it will be
the addition of the second term(1) and the third term (1) is 2(1+1). Using the same approach we can find the remaining terms easily.
In the above image, we can see the first ten terms of the Fibonacci series. We can clearly observe that apart from the first two terms every term is the sum of its previous two terms. Now, we have an
idea about how the Fibonacci series is constructed. So let’s see various approaches to writing the Fibonacci series program in java.
Approach 1: Iterative method for the Fibonacci series program in java
In this iterative method, we will use the first two terms of the Fibonacci series 0 and 1, and then we will find further terms using these two terms. Let’s see how to write the Fibonacci series
program in java using this method.
// iterative fibonacci program in java
class PrepBytes {
public static void main(String[] args) {
int first_term=0;
int second_term=1;
int second_last_term=0;
int last_term=1;
System.out.println("The first 10 terms of the Fibonacci series are:");
for(int i=0;i<10;i++){
System.out.print(first_term+" ");
else if(i==1){
System.out.print(second_term+" ");
int current_term=second_last_term+last_term;
System.out.print(current_term+" ");
The first 10 terms of the Fibonacci series are:
In the above Fibonacci program in java, first, we have defined the initial two terms 0 and 1. After that, we have also created two more variables last_term and second_last_term to keep track of the
last and second last term so far. Now, we will run a for loop 10 times to print the first 10 terms of the Fibonacci series. In the loop, if i is less than 2 then we will directly print the first two
terms, and if i is greater than or equal to 2 then first we will compute the current term which equals the sum of the last and second last term. Now, we will change the second last term as the last
term and the last term as the current term to calculate further terms. In the output, we can see that the first ten terms of the Fibonacci series are printed.
Time Complexity: O(n)
Auxiliary Space Complexity: O(1)
Approach 2: Recursive method for the Fibonacci series program in java
In this approach, we will use base case as the first two terms and our recursive formula for the nth term will be (n-1)th term +(n-2)th term. Let’s see how to write the Fibonacci series program in
java using the recursive method.
// recursive fibonacci program in java
class PrepBytes {
static int fibonacci(int n)
if (n < 2)
return n;
return fibonacci(n - 1)
+ fibonacci(n - 2);
public static void main(String args[])
System.out.println("The first 10 terms of the Fibbonaci series are:");
for (int i = 0; i < 10; i++) {
System.out.print(fibonacci(i) + " ");
The first 10 terms of the Fibonacci series are:
In the above Fibonacci program in java, we have written a recursive method Fibonacci. In that method, we have given base case (recursion termination condition) if the given number is less than 2 then
we will return that number. Otherwise, we will find the n-1 term [fibonacci(n-1)] and the n-2 term [fibonacci(n-2)] and we will return the sum of both terms. In the output, we can see that the first
ten terms of the Fibonacci series are printed.
Time Complexity: O(2^n)
Auxiliary Space Complexity: O(1)
Approach 3: Recursion + Memoization method for the Fibonacci series program in java
In this method, we will use the above recursion method the only difference is that we will use the memoization method with this approach. Memoization means we will store the answer to all the
calculated terms so that we can use them again and we do not need to find that term again and again.
// recursive + memoization fibonacci program in java
class PrepBytes {
static int fibonacci(int n,int[] memo)
if (memo[n]!=-1){
return memo[n];
if (n < 2)
return n;
int current_term= fibonacci(n - 1,memo) + fibonacci(n - 2,memo);
return current_term;
public static void main(String args[])
int[] memo= new int[10];
for(int i=0;i<10;i++){
System.out.println("The first 10 terms of the Fibbonaci series are:");
for (int i = 0; i < 10; i++) {
System.out.print(fibonacci(i,memo) + " ");
The first 10 terms of the Fibonacci series are:
In the above Fibonacci program in java, we have created an array of size 10 to store calculated terms and we will initialize the first two terms as 0 and 1 in that array. After that, we will use the
same recursive method as above to find the nth term of the Fibonacci series. Now, when we call a method Fibonacci to find the nth term we have two options. One is, if the nth term is already
calculated then we will directly return the answer. Otherwise, we will calculate the answer and store the answer in an array while returning from the recursive call. In the output, we can see that
the first ten terms of the Fibonacci series are printed.
In conclusion, the Fibonacci series is a fundamental mathematical sequence that holds significant relevance in various fields, including computer science and mathematics. Implementing the Fibonacci
series program in Java offers insights into different programming techniques, such as iterative, recursive, and recursive with memoization. Understanding the trade-offs between these methods in terms
of time and space complexity can aid in choosing the most suitable approach for specific use cases. Mastering the Fibonacci series program not only showcases programming skills but also provides a
deeper understanding of algorithmic thinking and problem-solving.
Frequently Asked Questions (FAQs) about Fibonacci Series Program in Java:
Some of the FAQs related to Fibonacci Series Program in Java are discussed below:
1. What are the different methods to generate the Fibonacci series in Java?
There are several methods to generate the Fibonacci series in Java, including iterative, recursive, and recursive with memoization approaches.
2. What is an iterative approach to generating the Fibonacci series?
In the iterative approach, each term is calculated based on the two previous terms using a loop.
3. What is a recursive approach to generating the Fibonacci series?
In the recursive approach, a function calls itself to calculate the Fibonacci terms. However, this method can lead to exponential time complexity.
4. What is the recursive approach with memoization?
The recursive approach with memoization involves storing the results of expensive function calls and reusing them to avoid redundant calculations. This significantly improves the time complexity of
the algorithm.
5. What is the time complexity of the iterative Fibonacci program?
The time complexity of the iterative Fibonacci program is O(n), where n is the term number in the series.
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Section: New Results
Performance analysis of cellular networks with opportunistic scheduling using queueing theory and stochastic geometry
In [38] submitted this year, combining stochastic geometric approach with some classical results from queuing theory,we propose a comprehensive framework for the performance study of large cellular
networks featuring opportunistic scheduling. Rapid and verifiable with respect to real data, our approach is particularly useful for network dimensioning and long term economic planning. It is based
on a detailed network model combining an information-theoretic representation of the link layer, a queuing-theoretic representation of the users' scheduler, and a stochastic-geometric representation
of the signal propagation and the network cells. It allows one to evaluate principal characteristics of the individual cells, such as loads (defined as the fraction of time the cell is not empty),
the mean number of served users in the steady state, and the user throughput. A simplified Gaussian approximate model is also proposed to facilitate study of the spatial distribution of these metrics
across the network. The analysis of both models requires only simulations of the point process of base stations and the shadowing field to estimate the expectations of some stochastic-geometric
functionals not admitting explicit expressions. A key observation of our approach , bridging spatial and temporal analysis, relates the SINR distribution of the typical user to the load of the
typical cell of the network. The former is a static characteristic of the network related to its spectral efficiency while the latter characterizes the performance of the (generalized) processor
sharing queue serving the dynamic population of users of this cell.
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Surface Roughness - The Teeth in Copper Jaw (Details of Surface Roughness Effect) The Review Log
You might be wondering how a shark could be related to the surface roughness topic? As a child, I always thought animals have teeth similar to human beings until I watched the deep blue sea movie.
Similarly, I always thought that copper foils and metallic surfaces are quite smooth in nature until I learned the nuances of the copper foil manufacturing process. We will discuss in this topic of
how the shark-like teeth of copper (surface roughness) impacts the signal integrity performance.
Surface roughness is a concept that we see at least one DesignCon paper every year. I am expecting the coming DesignCon conferences no different. Upon deeper review of these topics, we will find that
they are loaded with mathematical models as well as complicated jargons such as cannon ball model, profilometer measurements and SEM scanning, which makes all of these topics sound like a PHD
research topic.
In the end, everything boils down to the following discussion: Is the copper foil a smooth surface or a rough surface. If it is a rough surface, how rough it is and what is the deviation of that
roughness from what is considered being smooth ? Why do we need to understand this topic?
The total loss is characterized in the following way:
Total Loss = Insertion Loss of Dielectric + (Surface roughness coefficient * Insertion Loss of conductor)
Let us understand why a copper will introduce loss into the signal. To understand this, let us first discuss what is skin effect and why the currents will flow outside the conductor.
Skin Effect and Skin Depth:
The skin effect is a phenomenon whereby alternating electric current does not flow uniformly with respect to the cross-section of a conductive element, such as a wire [1].
The current density is highest near the surface of the conductor and decreases exponentially as distance from the surface increases. Skin depth is a measure of how far electrical conduction takes
place in a conductor, and is a function of frequency. At DC (0 Hz) the entire conductor is used, no matter how thick it is. At microwave frequencies, the conduction takes place at the top surface
only. As we go down the surface, the current density decrease. This can be seen even in the animation presented below.
Eddy Currents and Magnetic Fields:
When a good electrical conductor (like copper or aluminum) is exposed to a changing magnetic field, a current is induced in the metal, commonly called an eddy current. Eddy currents flow in closed
loops within conductors, in planes perpendicular to the magnetic field.
Why is Copper Foil roughened?
Copper foil is roughened to promote adhesion of the dielectric resin to the conductor in printed circuit boards. Adhesion at the interface between conductor and insulator must be very robust due to
conditions during manufacturing, assembly, and standard usage to which a printed circuit board is subjected. This interface is exposed to corrosive chemicals during processing and to high
temperature, high humidity, cold, shock, vibration, and shear stresses during use. Technologies that optimize surface and resin chemistries, as well as surface area, are utilized by foil
manufacturers and laminators to promote and retain adhesion.
As shown in the Figure3 ( Reference [1]), the 90° peel strength is a measure of how well a conductor adheres to the dielectric material and is directly proportional to the fourth root of the
thickness of deformed resin (yo) and other parameters like foil thickness, etc. The adhesive interlayer thickness is correlated to the treatment height. Higher roughness increases the interlayer
thickness, the surface area, and the chemical contributions to peel strength. Though roughness contributes positively to peel, it has a negative impact on signal integrity.
Copper Foil Manufacturing Process:
There are various ways of copper foil manufacturing process available in PCB industry. Electro-deposited (ED) copper is widely used in the PCB industry. A finished sheet of ED copper foil has a matte
side and drum side. The drum side is always smoother than the matte side.
The matte side is usually attached to the core laminate. For high frequency boards, sometimes the drum side of the foil is laminated to the core. In this case it is referred to as reversed treated
(RT) foil. Various foil manufacturers offer ED copper foils with varying degrees of roughness. Each supplier tends to market their product with their own brand name. Presently, there seems to be
three distinct classes of copper foil roughness:
Some other common names referring to ULP class are HVLP or eVLP. Profilometers are often used to quantify the roughness tooth profile of electro-deposited copper. Tooth profiles are typically
reported in terms of 10-point mean roughness (Rz ) for both sides, but sometimes the drum side reports average roughness (Ra ) in manufacturers’ data sheets. Some manufacturers also report RMS
roughness (Rq ). Now we know why we have roughness in the copper profile.
Simulation vs Measurement (Chicken and Egg Story)
It is quite tough to interpret the manufacturer’s data sheet. Some authors argue to use design feedback methodology.
Design a test coupon –>fabricate the board –> Measure the frequency domain parameters performance –>Extract the Dk,Df and surface roughness of the board –> Cross Verify and compare against the
simulated parameters –> 100% it will vary –> Now add back into the simulation channel performance –>Re simulate the analysis ! This is called Design feedback methodology. In a competitive cut throat
market this is time consuming procedure. Money and resources go for a kick !Consider the analysis demonstrated in Fig 6 and 7, we can observe two different set of designs. Figure 6 consists of
measured calibration traces of the same design and same manufacturer and So is with Figure 7 analysis
Which sample should we consider for measuring the material parameters? To solve this issue to certain extent, empirical models will allow us to reach our goal sooner. I always used to listen this
statement from my manager: Sometimes an OK answer NOW is better than a good answer. This statement is made by Eric Bogatin.
Empirical Models: Surface Roughness Coefficient
Calculating the surface roughness coefficient is done via an empirical model fit. Finding the right empirical fit to the measured data is the ball game in the research papers and the tussle to be
better than previous used models.
Empirical Models that are available in market for calculating the surface roughness of the conductors:
1) Morgan – Hammerstad 2) Groisse Model 3) Huray “Snowball” 4) Hemispherical model (Hall, Pytel) 5) Stochastic models (Sanderson, Tsang). May be there could be more than mentioned but I restricted
the analysis for the first three models.
There is one more advanced model developed by Bert Simonvich based on Huray Snowball. Polar instruments has implemented this model in their software. If you want to learn more, all his application
notes are sufficient to understand the analysis [4].
Also, I want to mention about Yuriy Shlepnev, Simberian Inc application notes regarding this topic. The following paper : Conductor surface roughness modeling: From “snowballs” to “cannonballs”
discusses in detail about the myth surrounding the models [5]
Morgan -Hammerstad:
Power absorption factor Ksr is defined as ratio of power dissipated by the eddy currents in a conductor per unit area when the surface is rough, Pa,rough, to the power dissipated when the surface is
smooth, Pa,smooth. Consider the formulation no 4 and 5 which are interpreted as per above discussion.
Ksr is used as a correction factor to account for the conductor surface roughness effect when calculating attenuation. The attenuation constant, αc,rough, for wave propagation in waveguides involving
conductors with rough surfaces can be estimated by multiplying the attenuation constant calculated for a smooth conductor, αc,smooth, with Ksr.
Samuel Morgan: Reference Paper -[2]
Highlights of research paper: Morgan determined that, at 10 GHz, current flow transverse to periodic structures could increase loss by up to 100%! If the current flow was parallel, the losses
increased by up to 33%. Morgan hypothesized the cause for this additional loss assumed that the loss was a function of the RMS distortion of a rough surface relative to the electromagnetic skin depth
of a perfectly smooth metal surface with conductivity equal to that of the bulk metal.
In the paper, Morgan experimented the analysis on equilateral, triangular and square shapes and considered the cases to be treated as two dimensional; the surface roughness is assumed to consist of
infinitely long parallel grooves or scratches either normal to or parallel to the direction of induced current flow. From the analysis, Morgan figured out that transverse grooves have a considerably
greater adverse effect than grooves parallel to the current.
Morgan -Hammerstad Empirical Fit:
In 1975, Hammerstad proposed an empirical fit for the Morgan’s published data and it was is quite ok at that time, however it saturates for the rough conductor at low frequencies to a value of 2.
As observed in the results, the empirical fit is very limited and hence advised not to use this method for validating the copper profile roughness.
Groisse Model (Available in the ANSYS Software):
I would give a very brief analysis about this model. This model is slightly better than Hammerstad model. However even this model saturates to a value of 2. The symbols used in the equation holds the
same as explained earlier. This model is seldom used at our end.
Huray Model (Snow Ball Model):
Before we move into this model, there are few insights Paul.G.Huray provided along with the other authors in this paper. I wish all SI engineers read this reference paper without fail. It is quite
lengthy and may take time but worth a read.
Does surface roughness increase the length and resistance ? Does the current take longer time to travel in case of rough surface ? Actually the answer is NO. In case you felt yes, then the
propagation delay should have been increased in case of rough copper surface.
Let us understand what does this author means: The answer is that local surface charge density (Quantity of charge per unit area) is formed by the displacement of conduction electrons transverse to
the surface profile so that a wave of charge density propagates at whatever speed is needed to support the external electric field intensity in the transmission line. No charged particles actually
move at relativistic speeds ( a speed comparable to speed of light).
A current flowing on the rough surface of a conductor cannot be regarded as a current flowing on a flat surface of the conductor with a longer equivalent path. When current flows in a conductor,
assuming that the current distribution decays exponentially within the depth of the conductor, the model is not correct either.
As discussed earlier, high profile samples resemble copper nodule pyramid structures arranged in a nominal hexagonal pattern on a matte finish surface. By comparison, the low profile samples appear
to be made up of similar size copper nodules randomly scattered on a flat plane; the nodules vary in radius but the average size is about 1 μm in diameter.
Now let us come back to the empirical model developed by Paul Huray, we shall neglect the non uniform model architecture developed by the team as it does not contribute much but gave a pathway to
develop EDA based empirical model.
Uniform Snowball Architecture:
So now that the Huray Model equation is understood, the next task is how to fit these complicated equations into the EDA toolkit. Below is the simulation analysis for various empirical model fits.
What do I do in my daily life simulation and measurement analysis:
Get from the vendor (TTM/Sanmina/Founder) the data sheet of the dielectric materials being used in the stack up design. It could be either MEG 6 or MEG 7 or Tachyon 100 G and obtain the surface
roughness measurement value (eg Rz) which is roughly divided into low/medium/high roughness, understand the roughness difference between the top surface and the bottom surface, and apply all of them
into the simulation model. For Dk and Df fit, I use Djordjevic-Sarkar model (Wide Debye Model). Then use the design feedback methodology.
That’s all Amigos ! This took really a lot of time as I went back and forth to understand. Next time, when you see the canon ball or surface roughness as topic at DesignCon, you will be more educated
on this topic 😉
Thanks for reading, I hope you enjoyed reading this discussion.
1) Non-Classical Conductor Losses due to Copper Foil Roughness and Treatment by Gary Brist and Stephen Hall Intel Corporation Hillsboro, Sidney Clouser, Tao Liang
2)S.P. Morgan Jr., “Effect of Surface Roughness on Eddy Current Losses at Microwave Frequencies,” Journal of Applied Physics, Vol. 20, p. 352-362, April 1949.
3) Practical methodology for analyzing the effect of conductor roughness on signal losses and dispersion in interconnects.
4)Simonovich, Bert, “Practical Method for Modeling Conductor Surface Roughness Using The Cannonball Stack Principle”, White Paper.
5)Conductor surface roughness modelling: From “snowballs” to “cannonballs” Yuriy Shlepnev, Simberian Inc.
Written by: Kalyan Vaddagiri
Senior SI Engineer at Molex
Original Article on LinkedIn
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Requirements for a Major or Minor in Physics - Department of Physics
Requirements for a Major or Minor in Physics
The Physics Department offers programs leading to a B.S., an A.B. or a minor in Physics. Students planning to attend graduate school in physics or engineering will typically pursue a B.S. degree.
Students interested in careers in medicine, law, finance, or other areas will benefit from any of the programs, but they may find that the A.B. or the minor allows more time to pursue other degrees
or interests. The first 5 courses taken for the B.S. and the A.B. are the same, so it is not necessary to decide between the two degrees until at least the end of the 2nd year.
Any student contemplating becoming a physics major or minor is strongly encouraged to consult with the Director of Undergraduate Studies in Physics as early as possible.
Note: Effective Fall 2023, all main campus courses have been renumbered using a new 4-digit numbering system.
Requirements for a B.S. in Physics
• 2101 (Mechanics) or 2051
□ Co- or Pre-requisite: MATH 1350 (Calculus I)
• 2102 (Electromagnetic Phenomena) or 2052
□ Co- or Pre-requisite: MATH 1360 (Calculus II)
• 2103 (Relativity and Quantum Physics)
□ Co- or Pre-requisite: MATH 2370 (Multivariable Calculus)
• 2104 (Modern Experimental Physics)
• 2105 (Mathematical and Computational Methods)
• 3101 (Intermediate Mechanics)
• 3102 (Intermediate Electricity and Magnetism)
• 3103 (Quantum Mechanics)
• 3104 (Statistical Physics)
• 4998-4999 (Independent Research; 3 credits)
• Physics elective: 2002-3701 or >= 4202
• Any approved physics or math/science course
Students considering the B.S. major are strongly encouraged to begin the major during their freshman year; however, with careful planning, it is possible to complete the B.S. starting in the first
semester of sophomore year. Typical schedules for various situations and printable requirement checklists can be found on the Planning Your Physics or Biological Physics Major page.
Note: Students earning the B.S. in Physics are exempt from the College Social Science General Education Requirement.
Requirements for an A.B. in Physics
• 2101 (Mechanics) or 2051
□ Co- or Pre-requisite: MATH 1350 (Calculus I)
• 2102 (Electromagnetic Phenomena) or 2052
□ Co- or Pre-requisite: MATH 1360 (Calculus II)
• 2103 (Relativity and Quantum Physics)
□ Co- or Pre-requisite: MATH 2370 (Multivariable Calculus)
• 2104 (Modern Experimental Physics)
• 155 (Mathematical and Computational Methods)
• 3101 or 3102 or 3103 or 3104
• Physics elective: 2002-3701 or >= 4202 or >= 4202
• Physics elective: >= 2002
• Any approved physics or math/science course
• Any approved physics or math/science course
Students considering the A.B. major are encouraged to begin the major during their freshman year to allow the greatest flexibility; however, it is straightforward to complete an A.B. starting in the
first semester of sophomore year. Typical schedules for various situations and requirement checklists can be found on the Planning Your Physics or Biological Physics Major page.
Departmental Honors
The faculty may award Honors in Physics to Physics or Biological Physics majors who have performed exceptionally well both in coursework and in independent research. Students who are awarded Honors
in Physics typically have a GPA in physics lecture courses of 3.7 or better. Students must also have exhibited excellence in independent research (including at least six credits of research
coursework) and must have presented their work in written and oral forms to the faculty. To be eligible for consideration, a physics or biological physics major must have completed at least 4
upper-level physics lecture courses (PHYS 2002-3701 or >= 4202), including at least two courses from PHYS 3101, 3102, 3103, and 3104.
Requirements for a Minor in Physics
A minor in physics consists of the first four courses of the core sequence plus one additional physics course, as follows:
• 2101 (Mechanics)
□ Co- or Pre-requisite: MATH 1350 (Calculus I)
• 2102 (Electromagnetic Phenomena)
□ Co- or Pre-requisite: MATH 1360 (Calculus II)
• 2103 (Relativity and Quantum Physics)
□ Co- or Pre-requisite: MATH 2370 (Multivariable Calculus)
• 2104 (Modern Physics and its Experimental Methods)
• One additional physics course (preapproved PHYS-2105 or higher)
Alternates for these courses may be approved on a case-by-case basis.
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Economic Risk Theory (SPbSTU Press, 1999)
Books, Completed, Published, Russian
The modern approaches to forecasting critical conditions and catastrophic events in complex economic systems are introduced. Examples of application of quantitative methods of the economic risk
theory to analysis of stability and diagnostics in economic systems based on statistical and/or nonlinear processing of time series are represented. The main attention is devoted to ways of holistic
modeling of stochastic economic system behavior and problems of quantitative estimation of economic risks.
The textbook is designed for students of economic, marketing, management departments and for everyone who is interested in the problem of risk analysis in complex economic systems.
O. Y. Uritskaya. Introduction to Economic Risk Theory: Lecture Textbook // St. Petersburg: SPbSTU Press, 1999.
Fig. 1. An example of bifurcation in model time series.
Fig. 2. Mont
hly fluctuations in world prices for gold (top) and the corresponding simulated cycle. (Cycles, 1993).
Fig. 3. Dependence of the form of the Pareto distribution tail on the Pareto index value.
1. The subject of economic risk theory
1. Origin of risk in economic system
2. Characteristics of economic systems
3. Modeling of economic processes
4. Time series analysis
2. Basis of economic risk theory
1. Self-organized criticality theory
2. Self-organized criticality models
3. Fractals and their properties
4. Fractal objects
5. Fractal properties of economic systems
3. Methodology of economic risk theory
1. Economic risk estimation
2. Data preparation (measurements, aggregation, calendar problems)
3. Fractal dimension of time series (ruler method, box counting statistics, correlation function method, spectral method, Hurst rescaled range analysis)
4. Forecasting long-term tendencies on the base of time series fractal dimension
5. Analysis of system stability and catastrophe forecasting
6. Risk estimation by statistical methods
7. Risk estimation by fractal methods
4. Problems
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Model Selection
Model selection is an important step in the machine learning workflow. It is about choosing the best model from a set of potential models to get the highest predictive performance on a particular
dataset. The process Model selection can highly impact the effectiveness of the model and, ultimately, the success of the machine learning project
Model Selection- why it matters?
A primary motive of machine learning is to develop a model that generalizes well to new, unseen data (usually called test data). Selecting an inappropriate model can lead to overfitting, which means
it will work extremely well on the train data where target values are already known and poorly on the test data where raw data is used as an input to predict target values, it is equally probable
that an inappropriate model selection process can lead to underfitting where it fails to track the patterns and isn't accurate.
Steps in Model Selection
• Define the Problem and Targets: Understand the problem domain and the goals of the project. Is it a classification or regression problem? What metrics will be used to evaluate the model's
performance? Defining the target variable (dependent variable) and input features (independent variables) is the first step.
• Data Preparation: Clean and preprocess the data. This includes handling missing values, encoding categorical variables, scaling features, and splitting the data into training, validation, and
test sets. A proper preprocessing procedure makes sure of the input characteristics are suitable for modeling.
• Choose Candidate Models: Select a set of potential models to evaluate. This can include linear models, decision trees, support vector machines, ensemble methods, and neural networks. The variety
of model selection allows us to compare both simple models and complicated models.
• Evaluate Models: Use techniques like cross-validation to evaluate the performance of each candidate model on the training data. Cross-validation techniques, such as k-fold cross-validation, help
in estimating the generalization error by averaging the performance across different validation folds.
• Hyperparameter Tuning: Optimize the hyperparameters of the models to improve their performance. This step is very important for complicated models where the number of additional parameters can
significantly affect the outcome.
• Model Comparison: Compare the models based on their performance metrics and select the best one. This involves looking at relevant measures like mean squared error, absolute error, and other
error metrics.
• Validate the Model: Test the selected model on the test set to ensure it generalizes well to new data. This final step helps in verifying the model’s performance on unseen data.
Model Selection for classification problem
For example imagine we are working on a classification problem to predict whether a customer will buy a product based on their browsing history. The steps of approach to model selection would
• Define the Problem and Targets: our target here is to maximize the accuracy and F1-score of the predictions (these are Evaluation metrics typically used across classification problems).
• Data Preparation:
□ Data Collection: Gather browsing history data for customers, including features like page views, time spent on pages, and previous purchases.
□ Data Cleaning: Handle missing values, remove duplicates, and ensure data consistency across the dataset.
□ Feature Engineering: Create new features based on the existing data, such as the average time spent on a page or the number of pages viewed in a session, time of day or night purchased at
most i.e morning,afternoon,evening or night. etc.
□ Encoding Categorical Variables: Convert categorical variables to numerical format using techniques like one-hot encoding.
□ Scaling: Normalize the numerical features to ensure they are on the same scale.
□ Data Splitting: Split the data into training (70%), validation (15%), and test (15%) sets.
• Choose Candidate Models:
□ Logistic Regression:
☆ A simple yet effective linear model for binary classification.
□ Decision Tree:
☆ A non-linear model that splits the data based on feature values.
□ Random Forest:
☆ An ensemble method that combines multiple decision trees to improve performance.
□ Support Vector Machine (SVM):
☆ A model that finds the hyperplane that best separates the classes.
□ Neural Network:
☆ A complex model capable of capturing intricate patterns in the data.
• Evaluate Models: Use 5-fold cross-validation to evaluate the the aforementioned models. Calculate the average accuracy and F1-score for all 5 models.
Example code for evaluating models
from sklearn.model_selection import cross_val_score, GridSearchCV
from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import RandomForestClassifier
from sklearn.svm import SVC
from sklearn.neural_network import MLPClassifier
from sklearn.metrics import accuracy_score, f1_score
# Sample data
X_train, X_test, y_train, y_test = ... # Load and preprocess your data
# Define models
models = { 'Logistic Regression': LogisticRegression(),
'Decision Tree': DecisionTreeClassifier(), 'Random Forest': RandomForestClassifier(),
'SVM': SVC(), 'Neural Network': MLPClassifier() }
# Evaluate models
for name, model in models.items():
scores = cross_val_score(model, X_train, y_train, cv=5, scoring='accuracy')
print(f'{name} Accuracy: {scores.mean()}')
# Example of output:
# Logistic Regression Accuracy: 0.78
# Decision Tree Accuracy: 0.75
# Random Forest Accuracy: 0.80
# SVM Accuracy: 0.77
# Neural Network Accuracy: 0.79
• Hyperparameter Tuning: Use grid search to find the best hyperparameters for the SVM model, as shown in our hypothetical example it showed promising performance.
# Hyperparameter tuning for the best model
param_grid = {
'C': [0.1, 1, 10],
'kernel': ['linear', 'rbf']
grid_search = GridSearchCV(SVC(), param_grid, cv=5, scoring='accuracy')
grid_search.fit(X_train, y_train)
print(f'Best parameters: {grid_search.best_params_}')
# Output: Best parameters: {'C': 1, 'kernel': 'rbf'}
• Model Comparison: Compare the models based on their cross-validation performance and select the correct model (e.g., Random Forest).
• Validate the Model: Test the best model on the test set to confirm its performance.
# Validate the model
best_model = RandomForestClassifier(n_estimators=100, max_depth=None)
best_model.fit(X_train, y_train)
y_pred = best_model.predict(X_test)
print(f'Test Accuracy: {accuracy_score(y_test, y_pred)}')
print(f'Test F1-Score: {f1_score(y_test, y_pred)}')
# Output: Test Accuracy: 0.82
# Output: Test F1-Score: 0.81
Model Selection for Regression problem
For example lets take a regression problem where our target is to predict house prices based on various features such as location, size, and number of rooms. The steps would include:
• Define the Problem and Targets: our target is to minimize the mean squared error (MSE) (This is an evaluation metric typically used for Regression problems)
• Data Preparation:
□ Data Collection: Gather data on house prices and features such as location, size, number of rooms, age of the house, etc.
□ Data Cleaning: Handle missing values, remove duplicates, and ensure data consistency.
□ Feature Engineering: Create new features, like the price per square foot. categorical features like premium location etc
□ Encoding Categorical Variables: Convert categorical variables like neighborhood to numerical format using techniques like one-hot encoding.
□ Scaling: Normalize the numerical features to ensure they are on the same scale.
□ Data Splitting: Split the data into training (70%), validation (15%), and test (15%) sets.
• Choose Candidate Models:
□ Linear Regression: A simple linear model (Linear reg ) for predicting continuous values.
□ Decision Tree Regressor: A non-linear model that splits the data based on feature values.
□ Random Forest Regressor: An ensemble method that combines multiple decision trees to improve performance.
□ Support Vector Regressor (SVR): A model that finds the hyperplane that best fits the data.
□ Neural Network: A complex model capable of capturing intricate patterns in the data.
• Evaluate Models: Use 5-fold cross-validation to evaluate the models. Calculate the average MSE for all 5 models.
# Example code for evaluating models
from sklearn.model_selection import cross_val_score, GridSearchCV
from sklearn.linear_model import LinearRegression
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import RandomForestRegressor
from sklearn.svm import SVR
from sklearn.neural_network import MLPRegressor
from sklearn.metrics import mean_squared_error
# Sample data
X_train, X_test, y_train, y_test = ... # Load and preprocess your data
# Define models
models = {
'Linear Regression': LinearRegression(),
'Decision Tree': DecisionTreeRegressor(),
'Random Forest': RandomForestRegressor(),
'SVR': SVR(),
'Neural Network': MLPRegressor()
# Evaluate models
for name, model in models.items():
scores = cross_val_score(model, X_train, y_train, cv=5,
print(f'{name} MSE: {-scores.mean()}')
# Example of output:
# Linear Regression MSE: 120000.0
# Decision Tree MSE: 135000.0
# Random Forest MSE: 110000.0
# SVR MSE: 115000.0
# Neural Network MSE: 125000.0
• Hyperparameter Tuning: Use grid search to find the best hyperparameters for the Random Forest model, as it showed promising performance.
# Hyperparameter tuning for the best model
param_grid = {
'n_estimators': [50, 100, 200],
'max_depth': [None, 10, 20] }
grid_search = GridSearchCV(RandomForestRegressor(), param_grid, cv=5, scoring='neg_mean_squared_error')
grid_search.fit(X_train, y_train)
print(f'Best parameters: {grid_search.best_params_}')
# Output: Best parameters: {'max_depth': None, 'n_estimators': 100}
• Model Comparison: Compare the models based on their cross-validation performance and select the best one (e.g., Random Forest).
• Validate the Model: Test the best model on the test set to confirm its performance.
# Validate the model
best_model = RandomForestRegressor(n_estimators=100, max_depth=None)
best_model.fit(X_train, y_train)
y_pred = best_model.predict(X_test)
print(f'Test MSE: {mean_squared_error(y_test, y_pred)}')
# Output: Test MSE: 105000.0
Model Selection Techniques
Cross - Validation
Cross-validation is a very unique technique for model evaluation. It involves splitting the data into multiple subsets and training the model on some subsets while validating it on others. Common
techniques include k-fold cross-validation, leave-one-out cross-validation, and nested cross-validation.
For example In a 5-fold cross-validation, the dataset is first divided into 5 equal parts. The model is trained on 4 parts and tested on the remaining part(this means 4 parts of training data and 1
part of test data). This process is repeated 5 times, with each part being used as the test set once. And the average of performance metric is considered for all 5 iterations in this process.
Feature Selection
Feature selection is the process of selecting the most relevant features for the model. It helps in reducing the dimensionality of the data and improving the model's performance. Techniques include
forward selection, backward elimination, and recursive feature elimination. There are important libraries in python like scikit-learn offers SelectKBest, RFE, and feature_selection module for various
feature selection methods.
Model Averaging
Model averaging involves combining multiple models to improve performance. This can be done by averaging their predictions or using more sophisticated techniques like stacking or boosting.
For example In an ensemble model, predictions from a decision tree, a logistic regression model, and a support vector machine (SVM) are averaged to make a final prediction. This technique is chosen
to reduce chances of overfitting and improve generalization of prediction model.
Penalty Terms
Penalty terms, such as L1 (Lasso) and L2 (Ridge) regularization, are used in linear models to prevent overfitting by adding a penalty for large coefficients.
For example In a linear regression model predicting house prices, L1 regularization (Lasso) can be used to enforce sparsity, setting some coefficients to zero, while L2 regularization (Ridge) shrinks
all coefficients towards zero (both of which is considered potential techniques to reduce overfitting).
Nested Cross-Validation
Nested cross-validation is very powerful method used for hyperparameter tuning and model selection at the same time. It involves two layers of cross-validation: the inner loop for hyperparameter
tuning and the outer loop for model evaluation.
For example In a 5x5 nested cross-validation, the outer 5-fold cross-validation loop checks model performance, while the inner 5-fold loop optimizes hyperparameters. This process ultimately gives a
true estimate of the model's performance.
Model Assumptions and Selection Criteria
Each model has its own assumptions and selection criteria. For instance, linear regression assumes a linear relationship between the input features and the target variable, while random forests do
not have such assumptions.so In case of predicting sales based on advertising spend, linear regression might be chosen only if the relationship is linear, and a decision tree would be a better fit
iif the relationship is more complex.
Model Uncertainty
Model uncertainty can be addressed by using probabilistic models that provide confidence intervals or probabilities for their predictions. Bayesian models and ensemble methods are examples of
approaches that incorporate model uncertainty.
For example Bayesian linear regression provides a distribution of possible outcomes rather than a single prediction, offering insights into the uncertainty of predictions. and there are python
libraries like PyMC3, Stan, and scikit-learn’s BayesianRidge helpful in implementing this method.
Model Evaluation and Assessment
Model evaluation and assessment involve using relevant measures like mean squared error, accuracy, F1-score, and others to compare the performance of different models. This step ensures that the
selected model meets the project goals and generalizes well to new data (unseen data or test data).
For example In a classification problem predicting customer churn, the accuracy, precision, recall, and F1-score are calculated for different models to select the best performing one. And in python
important libraries from scikit-learn offers metrics like accuracy_score, f1_score, and mean_squared_error.
Computational Models and Criteria
Computational models and criteria for model selection include various algorithms and techniques used to compare and select models. These may involve mathematical models, model-based inference, and
model identification criteria.
For example Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to compare models on the basis of their likelihood and complexity, selecting the model that best
balances fit and simplicity. And python library from statsmodels provides functions to calculate AIC and BIC.
When to choose Classical ML Models vs. Deep Learning Models
Deciding between whether to choose Classical machine learning models vs deep learning models primarily depends upon the amount of data available, this decision would impact the model's complexity and
computational resources required to get the expected performance.
Data Size: Deep learning models can leverage high amounts of data to learn complex patterns. hence Lrger datasets typically favor deep learning models to learn complex data patterns.
Model Interpretability: Classical ML models are typically more interpretable, making it preferable when understanding the model's decisions, which is very important for the problem.
Computational Resources: for Deep learning models we require more computational power and longer training times in comparison to classical ML models.
Task Complexity: Tasks that involve complex patterns and high-dimensional data, such as image and speech recognition, benefit more from deep learning models.
Classical ML Models
Classical ML models, like linear regression, support vector machines (SVM), and random forests, are usually more effective when the datasets are smaller. These models are generally easier to
interpret and need less computational power compared to deep learning.
For example a dataset with 10,000 rows, predicting house prices based on features like square footage, number of bedrooms, and location, a random forest model or linear regression might be
sufficient and efficient. These models can handle the dataset size well and provide interpretable results.
Deep Learning Models
Deep learning models, like neural networks (NN), convolutional neural networks (CNN), and recurrent neural networks (RNN), are better option for large datasets. They can automatically learn complex
patterns and features from the data, their performance is commendable in tasks like image and speech recognition.
For example a dataset with 200,000 rows of image data, where the goal is to classify objects inside the images, a CNN would be more appropriate. CNNs are better at processing image data due to their
ability to learn hierarchical feature representations through convolutional layers.
Which models to select from Deep Learning models
The choice of deep learning models depends on the data modality, the specific task (classification, regression, segmentation, retrieval, etc.), and other considerations like model size and inference
Data Modality and Model Selection
Different types of data (modality) require different deep learning architectures. The following table outlines common modalities and corresponding suitable models:
• Images, Videos (CNN, ViT)
□ For example, CNNs use convolutional layers to extract features from images, which makes them efficient for image classification tasks. Vision Transformers (ViT) have also shown great
performance in image classification by using transformer architectures.
□ The use case would be Classifying images of handwritten digits (MNIST dataset).
• Representation Learning (Autoencoder)
□ For example, Autoencoders are used for unsupervised learning to learn efficient codings of input data. They can be used for tasks like anomaly detection or data compression.
□ The use case would be Reducing the dimensionality of high-dimensional data for visualization.
• Generative Models (VAE, GANs, Transformers)
□ For example, Variational Autoencoders (VAE) and Generative Adversarial Networks (GANs) generate new data samples similar to the training data. Transformers are used for generating text or
translating languages.
□ The use case would be Generating realistic images or synthesizing new text.
• Language (Text) (Transformer)
□ For example, Transformers, such as BERT or GPT, are used for natural language processing tasks like text classification, translation, and summarization.
□ The use case would be a Sentiment analysis of customer reviews.
• Speech (LSTM, Transformer)
□ For example, Long Short-Term Memory (LSTM) networks and Transformers handle sequential data, making them suitable for speech recognition and language modeling.
□ The use case would be Transcribing spoken language into text.
• Graphical Data (Graph Neural Nets)
□ For example, Graph Neural Networks (GNN) work with data structured as graphs, capturing relationships and dependencies between nodes.
□ The use case would be Predicting molecular properties in chemistry.
• 3D Data (GNN, PointNets, CNNs)
□ For example, Models like PointNets process 3D point cloud data, while CNNs can be adapted for 3D data by using 3D convolutions.
□ The use case would be 3D object recognition in autonomous driving.
"Pattern Recognition and Machine Learning" by Christopher M. Bishop : This link directs you to a FREE PDF of this book published on Microsoft website!
"Machine Learning: A Probabilistic Perspective" by Kevin P. Murphy : A detailed book focusing on the probabilistic approach to machine learning.
YouTube Channels
StatQuest with Josh Starmer : Simplified explanations of various statistical and machine learning concepts, including model selection.
Sentdex : Practical tutorials on machine learning with Python, including model selection.
Data School : A clear introduction to cross-validation and its importance in model selection.
This article was written by Kartikey Vyas, and edited by our writers team.
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OpenStax College Physics for AP® Courses, Chapter 14, Problem 68 (Problems & Exercises)
Frozen waste from airplane toilets has sometimes been accidentally ejected at high altitude. Ordinarily it breaks up and disperses over a large area, but sometimes it holds together and strikes the
ground. Calculate the mass of $0^\circ\textrm{C}$ ice that can be melted by the conversion of kinetic and gravitational potential energy when a 20.0 kg piece of frozen waste is released at 12.0 km
altitude while moving at 250 m/s and strikes the ground at 100 m/s (since less than 20.0 kg melts, a significant mess results).
Question by
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Solution video
OpenStax College Physics for AP® Courses, Chapter 14, Problem 68 (Problems & Exercises)
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Video Transcript
This is College Physics Answers with Shaun Dychko. 20.0 kilograms of frozen toilet waste from an airplane is falling from an initial height of 12.0 kilometers which is 12.0 times 10 to the 3 meters.
It starts with a speed of 250 meters per second and just before it hits the ground, it has its speed reduced to 100 meters per second and this is presumably due to friction and there's gonna be some
heat provided by friction and that's gonna be melting some of the frozen waste. It has a temperature of 0 degrees Celsius initially and this detail means that all the energy that it loses due to
gravitational potential energy being lost and due to kinetic energy being lost all of that goes directly into the phase change from ice to liquid— there's no need for the energy to be absorbed to
change temperature. Okay! So the total heat that's going to be absorbed is going to be the change in potential energy plus the change in kinetic energy and I put absolute value bars here just to
avoid negative sign issues; we know that this potential energy is going to be reduced and that's going to be turned into heat so it's going to be a positive amount of heat and we know the kinetic
energy is also going to be reduced so this ΔkE is technically going to be a negative number but we just wanted magnitude... we wanna make it just positive because that's going to be the amount added
to this heat as well. So we have the heat then is gonna be the total mass times g times h—that's the change in potential energy— plus one-half total mass times initial speed squared minus one-half
total mass times final speed squared. So this is how much kinetic energy is lost but written as a positive number by putting the initial speed before the final. So we can factor out the total mass
over 2 from all the terms and we have to introduce the number 2 here if we wanna do that and so we have total mass over 2 equals or times 2gh plus v i squared minus v f squared; all of this equals
the heat absorbed. So heat absorbed goes directly into phase change and so that's gonna equal the mass of water that's melted multiplied by the latent heat of fusion and we can replace Q with this
and we do that here and now we can figure out the mass that's melting by dividing both sides by latent heat of fusion. So the melted mass then is the total mass that we start with divided by 2 times
the latent heat of fusion times 2gh plus v i squared minus v f squared. So that's 20.0 kilograms divided by 2 times 334 times 10 to the 3 joules per kilogram times 2 times 9.8 newtons per kilogram
times 12.0 times 10 to the 3 meters plus 250 meters per second— initial speed squared— minus 100 meters per second—final speed squared just before it hits the ground— and that is 8.61 kilograms. So
there's still gonna be a chunk of ice that's, you know, 11.39 kilograms of mass that's going to be hitting a single spot and creating quite a mess.
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The Pythagorean Theorem Lesson Plan | Congruent Math
Ever wondered how to teach the Pythagorean Theorem in an engaging way to your 8th grade students?
In this lesson plan, students will learn about the Pythagorean Theorem proofs, legs and hypotenuse, right triangles, and their real-life applications. Through artistic, interactive guided notes,
check for understanding, a color by code activity, and a real-life application example, students will gain a comprehensive understanding of the Pythagorean Theorem.
The lesson culminates with an exploration of baseball diamond highlighting the importance of understanding the Pythagorean Theorem in real-world scenarios.
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You Are On Multi Choice Question Bank SET 617
30852. The saturated and dry densities of a soil are respectively 2000 kg/m3 and 1500 kg/m3. The water content (in percentage) of the soil in the saturated state would be
30853. The beam ABC shown in the given figure is horizontal. The distance to the point of contraflexure from the fixed end 'A' is
30855. A timber beam of rectangular section 100 mm x 50 mm is simply supported at the ends, has a 30 mm x 10 mm steel strip securely fixed to the top surface as shown in the given figure. The
centroid of the "Equivalent timber beam" in this case from the tope surface
30861. Which among the following assumptions are made in the design roof trusses ? 1. Roof truss is restrained by the reactions 2. Axes of the members meeting at a joint intersect at a common point
3. Riveted joints act as frictionless hinges 4. Loads act normal to roof surface Select the correct answer using the codes given below:
30862. A ladder AB of weight W and length l is held in equilibrium by a horizontal force P as shown in the figure given above. Assuming the ladder to be idealized as a homogeneous rigid bar and the
surfaces to be smooth, which one of the following is correct ?
30864. Wearing locations of rails and their reasons are listed below : 1. Wear at end of rails : Loose fish bolts 2. Wear at sides of rail head : Constant lerake application 3. Wear on the top of
rail head on tangent truck : Rigidity of wheel base 4. Wear on top of rail head on curves : Lesser area of contact between wheel and rail Which of the pairs given above are correctly matched ?
30867. In a laminar boundary layer, the velocity distribution can be assumed to be given, in usual notations, as Which one of the following is the correct expression for the displacement thickness δ
for this boundary layer ?
30868. In a plate load test, how is the ultimate load estimated from the load settlement curve on a log-log graph ?
30869. The standard Proctor compaction curve of a clay is depicted in the above figure. Points A, B and C correspond to three compaction states of the soil, which fall on this curve. For which point
(s) is the coefficient of permeability minimum ?
30871. Ratio of the width of the car parking area required at kerb for 30° parking relative to 60° parking is approximately
30872. An equipment is purchased for Rs. 40, 000.00 and is fully depreciated by straight line method over 8 years. Considering interest on average annual cost at 15% p.a., the charge on the Company
due to use of this equipment, if made uniformly over the 8 years, is
30875. Gross flange area for a riveted plate girder is to be designed considering net area as 80% of its gross area. Consider width of the flange as 500 mm while web plate as 1000 mm x 12 mm. The
girder is to resist a maxi-mum BM of 4500 kNm. Maximum allowable bending stress in tension is 150 MPa. Gross flange area is
30877. For the compound bar shown in the below figure, the ratio of stresses in the portions AB : BC : CD will be
30880. What is the volume of a 6 m deep tank having rectangular shaped top 6 m x 4 m and botton 4 m x 2 m (computed through the use of prismoidal formula) ?
30882. Under load, the void ratio of a submerged saturated clay decreases from 1.00 to 0.92. What will be the ultimate settlement of the 2 m thick clay due to consolidation ?
30890. In the figure shown aside, which one of the following correctly represents effective stress at C ? (Symbols have the usual meanings).
30893. A rod shown in the given figure is such that the lower part is made of steel and the upper part is made of copper. Both the ends of the rod are rigidly clamped. If an axial force P is applied
at the junction of the two metals, given that the ratio Esteel/ECopper is equal to 2, then the force in the copper rod would be
30895. For a vertical concentrated load acting on the surface of a semi-infinite elastic soil mass, the vertical normal stress at depth z is proportional to
30898. Two reservoirs are connected by two pipes A and B of same and length in series. If the diamerer of A is 30% larger than that of B, what is the ratio of head loss in A to that of B ?
30899. A smooth sphere of weight W is supported in contact with a smooth vertical wall by a string fastened to a point on its surface, the other end being attached to a point in the wall as shown in
the given figure. If the length of the string is equal to the diameter of the sphere, then the tension in the string and the reaction of the wall respectively will be
30900. In an external focussing tacheometer, the fixed interval between the stadia hairs is 5 mm; the focal length of the objective is 25 cm ; and the distance of the vertical axis of the instrument
from the optical centre of the objective is 15 cm. The constants of the tacheometer are
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Question ANSWER With Solution
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Books written and edited by Prof. Hitoshi KIYA
1. Hitoshi Kiya, Digital Signal Processing, Ohmsha, 2014 August (195 pages)
2. Hitoshi Kiya, Essence of Digital Signal Processing, Ohmsha, 2014 August (185 pages)
3. Hitoshi Kiya, Introduction to Fourier Analysis Using Excel- the Basis of Signal Analysis- , Shoko-do Publishing, 2011 April (155 pages)
4. Hitoshi Kiya, Digital Image Processing, ,Shoko-do Publishing 2008 Oct.(190 pages)
5. Hitoshi Kiya, Essence of Digital Signal Processing, Shoko-do Publishing, 2007 April(185 pages)
6. Hitoshi Kiya, Intoroduction to Image and Video Processing, CQ Publishing, 2004 September(196 pages)
7. Hajime Maeda, Akira Sano, Hitoshi Kiya and Shinsuke Hara, Wavelet Trasforms and Their Applications, Asakura Publishing, 2001 November (161 pages)
8. Hitoshi kiya and Shogo Muramatsu, Introduction to DCT Transforms, CQ Publishing, 1997 July(236 pages)
9. Hitoshi Kiya, Digital Signal Processing, Shoko-do Publishing, 1997 June (195 pages)
10. Hitoshi Kiya, Intoroduction to Digital Image Processing, CQ Publishing, 1996 Feb.(230 pages)
11. Hitoshi Kiya, Multirate Signal Processing, Shoko-do Publishing, 1995 Oct. (202 pages)
12. Hitoshi Kiya, Intoroduction to Digital Signal Processing, Science and Industry Publishing, 1993 Feb. (243 pages)
13. Masahiko Sagawa and Hitoshi Kiya, Fast Fourier Transform, Shoko-do Publishing,1992 May (218 pages)
14. Ryuichi Yokoyama, Masahito Sagawa and Hitoshi Kiya Translators, Digital Control System Analysis and Design-by Charles L. Phillips and H.Troy Nagle, Jr., Nikkan Kogyo Shinbun, 1990 Oct.(689 pages)
15. Masahiko Sagawa and Hitoshi Kiya Translators, Fast Fourier Transform and Covolution Algorithms-by Henri J. Nussbaumer, Kagaku Gijyutsu Publishing, 1989 April (306 pages)
Edited Books
1. Guoping Qiu, Kin Man Lam, Hitoshi Kiya, Xiang-Yang Xue, C.-C. Jay Kuo, and Michael S. Lew Eds., ”Advances in Multimedia Information Processing ― PCM 2010,” Lecture Notes in Computer Science,
vol.6297 and 6298, Springer, September 2010(739 pages(Part1), 754 pages(Part2))
2. Hitoshi Kiya Edit, Image Coding ，Korona Publishing, 2008 April(244 pages)
Contributed Chapters in Books
1. Masahiro Iwahashi and Hitoshi Kiya, “Discrete Wavelet Transforms – A Compendium of New Approaches and Recent Applications, Edited by Awad Kh. Al – Asmari,” InTech, ISBN: 978-953-51-0940-2,
February 2013
2. Masaaki Fujiyoshi and Hitoshi Kiya, “Multimedia Information Hiding Technologies and Methodologies for Controlling Data,” IGI Global, ISBN: 978-146-662-217-3, October 2012.
3. Shouko Imaizumi, Masaaki Fujiyoshi and Hitoshi Kiya, “Multimedia – A Multidisciplinary Approach to Complex Issues, Edited by Ioannis Karydis,” InTech, ISBN: 978-953-51-0216-8, March 2012
4. Masahiro Iwahashi and Hitoshi Kiya,”Discrete Wavelet Transforms – Algorithms and Applications, Edited by Hannu Olkkonen,” InTech, ISBN:978-953-307-482-5, August 2011
5. Hitoshi Kiya and Izumi Ito ,”Signal Processing, Edited by Sebastian MIRON,” InTech, ISBN: 978-953-7619-91-6, March 2010
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Every countable model of set theory embeds into its own constructible universe
[bibtex key=Hamkins2013:EveryCountableModelOfSetTheoryEmbedsIntoItsOwnL]
In this article, I prove that every countable model of set theory $\langle M,{\in^M}\rangle$, including every well-founded model, is isomorphic to a submodel of its own constructible universe $\
langle L^M,{\in^M}\rangle$. Another way to say this is that there is an embedding
$$j:\langle M,{\in^M}\rangle\to \langle L^M,{\in^M}\rangle$$
that is elementary for quantifier-free assertions in the language of set theory.
Main Theorem 1. Every countable model of set theory $\langle M,{\in^M}\rangle$ is isomorphic to a submodel of its own constructible universe $\langle L^M,{\in^M}\rangle$.
The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading eventually to what I call the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, which is closely
connected with the surreal numbers. The proof shows that $\langle L^M,{\in^M}\rangle$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$, and so in fact this model is
universal for all countable acyclic binary relations of this rank. When $M$ is ill-founded, this includes all acyclic binary relations. The method of proof also establishes the following, thereby
answering a question posed by Ewan Delanoy.
Main Theorem 2. The countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory $\langle M,{\in^M}\rangle$ and $\langle N,{\in^N}\rangle$,
either $M$ is isomorphic to a submodel of $N$ or conversely. Indeed, the countable models of set theory are pre-well-ordered by embeddability in order type exactly $\omega_1+1$.
The proof shows that the embeddability relation on the models of set theory conforms with their ordinal heights, in that any two models with the same ordinals are bi-embeddable; any shorter model
embeds into any taller model; and the ill-founded models are all bi-embeddable and universal.
The proof method arises most easily in finite set theory, showing that the nonstandard hereditarily finite sets $\text{HF}^M$ coded in any nonstandard model $M$ of PA or even of $I\Delta_0$ are
similarly universal for all acyclic binary relations. This strengthens a classical theorem of Ressayre, while simplifying the proof, replacing a partial saturation and resplendency argument with a
soft appeal to graph universality.
Main Theorem 3. If $M$ is any nonstandard model of PA, then every countable model of set theory is isomorphic to a submodel of the hereditarily finite sets $\langle \text{HF}^M,{\in^M}\rangle$ of
$M$. Indeed, $\langle\text{HF}^M,{\in^M}\rangle$ is universal for all countable acyclic binary relations.
In particular, every countable model of ZFC and even of ZFC plus large cardinals arises as a submodel of $\langle\text{HF}^M,{\in^M}\rangle$. Thus, inside any nonstandard model of finite set theory,
we may cast out some of the finite sets and thereby arrive at a copy of any desired model of infinite set theory, having infinite sets, uncountable sets or even large cardinals of whatever type we
The proof, in brief: for every countable acyclic digraph, consider the partial order induced by the edge relation, and extend this order to a total order, which may be embedded in the rational order
$\mathbb{Q}$. Thus, every countable acyclic digraph admits a $\mathbb{Q}$-grading, an assignmment of rational numbers to nodes such that all edges point upwards. Next, one can build a countable
homogeneous, universal, existentially closed $\mathbb{Q}$-graded digraph, simply by starting with nothing, and then adding finitely many nodes at each stage, so as to realize the finite pattern
property. The result is a computable presentation of what I call the countable random $\mathbb{Q}$-graded digraph $\Gamma$. If $M$ is any nonstandard model of finite set theory, then we may run this
computable construction inside $M$ for a nonstandard number of steps. The standard part of this nonstandard finite graph includes a copy of $\Gamma$. Furthermore, since $M$ thinks it is finite and
acyclic, it can perform a modified Mostowski collapse to realize the graph in the hereditary finite sets of $M$. By looking at the sets corresponding to the nodes in the copy of $\Gamma$, we find a
submodel of $M$ that is isomorphic to $\Gamma$, which is universal for all countable acyclic binary relations. So every model of ZFC isomorphic to a submodel of $M$.
The article closes with a number of questions, which I record here (and which I have also asked on mathoverflow: Can there be an embedding $j:V\to L$ from the set-theoretic universe $V$ to the
constructible universe $L$, when $V\neq L$?) Although the main theorem shows that every countable model of set theory embeds into its own constructible universe $$j:M\to L^M,$$ this embedding $j$ is
constructed completely externally to $M$ and there is little reason to expect that $j$ could be a class in $M$ or otherwise amenable to $M$. To what extent can we prove or refute the possibility
that $j$ is a class in $M$? This amounts to considering the matter internally as a question about $V$. Surely it would seem strange to have a class embedding $j:V\to L$ when $V\neq L$, even if it is
elementary only for quantifier-free assertions, since such an embedding is totally unlike the sorts of embeddings that one usually encounters in set theory. Nevertheless, I am at a loss to refute the
hypothesis, and the possibility that there might be such an embedding is intriguing, if not tantalizing, for one imagines all kinds of constructions that pull structure from $L$ back into $V$.
Question 1. Can there be an embedding $j:V\to L$ when $V\neq L$?
By embedding, I mean an isomorphism from $\langle V,{\in}\rangle$ to its range in $\langle L,{\in}\rangle$, which is the same as a quantifier-free-elementary map $j:V\to L$. The question is most
naturally formalized in Gödel-Bernays set theory, asking whether there can be a GB-class $j$ forming such an embedding. If one wants $j:V\to L$ to be a definable class, then this of course implies
$V=\text{HOD}$, since the definable $L$-order can be pulled back to $V$, via $x\leq y\iff j(s)\leq_L j(y)$. More generally, if $j$ is merely a class in Gödel-Bernays set theory, then the existence of
an embedding $j:V\to L$ implies global choice, since from the class $j$ we can pull back the $L$-order. For these reasons, we cannot expect every model of ZFC or of GB to have such embeddings. Can
they be added generically? Do they have some large cardinal strength? Are they outright refutable?
It they are not outright refutable, then it would seem natural that these questions might involve large cardinals; perhaps $0^\sharp$ is relevant. But I am unsure which way the answers will go. The
existence of large cardinals provides extra strength, but may at the same time make it harder to have the embedding, since it pushes $V$ further away from $L$. For example, it is conceivable that the
existence of $0^\sharp$ will enable one to construct the embedding, using the Silver indiscernibles to find a universal submodel of $L$; but it is also conceivable that the non-existence of $0^\
sharp$, because of covering and the corresponding essential closeness of $V$ to $L$, may make it easier for such a $j$ to exist. Or perhaps it is simply refutable in any case. The first-order
analogue of the question is:
Question 2. Does every set $A$ admit an embedding $j:\langle A,{\in}\rangle \to \langle L,{\in}\rangle$? If not, which sets do admit such embeddings?
The main theorem shows that every countable set $A$ embeds into $L$. What about uncountable sets? Let us make the question extremely concrete:
Question 3. Does $\langle V_{\omega+1},{\in}\rangle$ embed into $\langle L,{\in}\rangle$? How about $\langle P(\omega),{\in}\rangle$ or $\langle\text{HC},{\in}\rangle$?
It is also natural to inquire about the nature of $j:M\to L^M$ even when it is not a class in $M$. For example, can one find such an embedding for which $j(\alpha)$ is an ordinal whenever $\alpha$ is
an ordinal? The embedding arising in the proof of the main theorem definitely does not have this feature.
Question 4. Does every countable model $\langle M,{\in^M}\rangle$ of set theory admit an embedding $j:M\to L^M$ that takes ordinals to ordinals?
Probably one can arrange this simply by being a bit more careful with the modified Mostowski procedure in the proof of the main theorem. And if this is correct, then numerous further questions
immediately come to mind, concerning the extent to which we ensure more attractive features for the embeddings $j$ that arise in the main theorems. This will be particularly interesting in the case
of well-founded models, as well as in the case of $j:V\to L$, as in question , if that should be possible.
Question 5. Can there be a nontrivial embedding $j:V\to L$ that takes ordinals to ordinals?
Finally, I inquire about the extent to which the main theorems of the article can be extended from the countable models of set theory to the $\omega_1$-like models:
Question 6. Does every $\omega_1$-like model of set theory $\langle M,{\in^M}\rangle$ admit an embedding $j:M\to L^M$ into its own constructible universe? Are the $\omega_1$-like models of set theory
linearly pre-ordered by embeddability?
19 thoughts on “Every countable model of set theory embeds into its own constructible universe”
1. Pingback: Every countable model of set theory embeds into its own constructible universe, Fields Institute, Toronto, August 2012 | Joel David Hamkins
2. Considering Conway’s system ‘No’ and Philip Ehrlich’s assertion that ‘No’ is an “absolute arithmetic continuum” containing “all numbers great and small”, could ‘No’ act as a foundation (so to
speak) of the naturalistic account of forcing?
□ Thomas, I don’t quite follow your suggestion. I believe that Ehrlich is using the word “absolute” in a way that recall’s Cantor’s term of “absolute infinity”, which is often understood to
refer to the class of all ordinals. But the truth remains that different models of set theory have different non-isomorphic surreals, and so No is not absolute in that sense. For example,
forcing creates new surreal numbers. Indeed, for models V and W of ZFC, I claim that V is isomorphic to W if and only if No^V is isomorphic to No^W, and in this sense, the surreals of a model
of ZFC capture the whole model. This is because the surreal numbers correspond to the transfinite binary sequences, which in turn corresponds to a set of ordinals, and two models of ZFC set
theory with the same ordinals and the same sets of ordinals are the same.
3. I will quote Ehrlich (quote found on page 240) from his paper “The Absolute Arithmetic Continuum and Its Peircean Counterpart”, “No not only exhibits [the rest of the sentence is italicized for
emphasis] all possible types of algebraic and set-theoretically [the word ‘set’ is underlined for further emphasis] defined order -theoretic gradations consistent with its structure as an ordered
field, it is to within isomorphism the unique structure that does.” I leave it to you to make of that what you will. Another (at least to me) telling remark of Ehrlich’s can be found in another
article “Universally extending Arithmetic Continua” where he entitles the section in which he defines the term “absolutely homogeneous universal” for a model ‘A’ “Maximal Models”. I would be
interested in seeing how forcing creates new surreal numbers since Conway ‘constructs’ No from the notation {L|R} which seems to me at least independent of any axiomatization of set theory
□ Forcing can create new real numbers, and therefore also new surreal numbers. So different models of set theory can have different surreal numbers. (This is clear already from the fact that
different models of set theory can have different ordinals.) The argument I gave in my comment above shows more: different models of ZFC necessarily have different No’s, since the surreal
numbers code up the entire universe in which they are constructed. So the surreal numbers No are not absolute in the sense of being-the-same-in-all-models. What Ehrlich is talking about is
the fact that we can prove in ZFC that No is the unique homogeneous and universal class linear order. This can be viewed in analogy with Cantor’s proof that the rational line $\mathbb{Q}$ is
the unique homogeneous universal countable linear order.
☆ Would the fact that in ZFC (though Ehrlich speaks of No as the absolute arithmetic continuum “modulo NBG” or by extension, modulo Ackermann Set Theory) No is the unique homogeneous and
universal class linear order be sufficient to use it as the basis for generating, say, Cohen and/or Random reals to affix to models of set theory, therefore liberating forcing from
countable, ‘toy’ models (since at least as I understand it, forcing needed to use countable models so that the extra sets of natural numbers not contained in the model in question could
act as generic sets)? In fact does No contain all possible interpretations of No relative to models of ZFC?
○ The referece to NGB is due to two facts, first, the class universality claim is explicitly a second-order claim, which is not expressible except case-by-case in the first-order
language; and second, one proves that No is universal for class linear orders in NGB, and the proof I know uses the global choice principle (which goes beyond ZFC) in order to guide
the back-and-forth argument to find the embedding of a given class order into No. I don’t really see how to use No in the way that you suggest. The No of one model will not generally
be universal for the linear orders that exist in another model of set theory.
4. Pingback: The countable models of ZFC, up to isomorphism, are linearly pre-ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic
to a submodel of its own $L$, New York, 2012 | Joel David Hamkins
5. Pingback: Every countable model of set theory is isomorphic to a submodel of its own constructible universe, Barcelona, December, 2012 | Joel David Hamkins
6. Pingback: The countable models of set theory are linearly pre-ordered by embeddability, Rutgers, November 2012 | Joel David Hamkins
7. Pingback: Embeddability amongst the countable models of set theory, plenary talk for ASL / Joint Math Meetings in Baltimore, January 2014 | Joel David Hamkins
8. Pingback: On embeddings of the universe into the constructible universe, partial results on an open question, CUNY Set Theory Seminar, March 2015 | Joel David Hamkins
9. Pingback: The hypnagogic digraph, with applications to embeddings of the set-theoretic universe, JMM Special Session on Surreal Numbers, Seattle, January 2016 | Joel David Hamkins
10. Pingback: Universality and embeddability amongst the models of set theory, CTFM 2015, Tokyo, Japan | Joel David Hamkins
11. Does the lack of elementarity in this type of embeddings mean the lack of any relation between them and large cardinal axioms?
A review on the original construction of a non-trivial elementary embedding derived from the existence of a measurable cardinal and in fact derived from the existence of a special type of filters
on a cardinal may suggest that by weakening the original properties of a non-principal $\kappa$ – complete ultrafilter, one can get the “embeddings” rather than “elementary embeddings” of the
universe through the same construction.
Though, I’m not sure how to interpret the existence of non-elementary embeddings between each two countable models of ZFC and the difference between the countable and the uncountable cases.
□ Large cardinals generally give you elementary embeddings, which are therefore also embeddings in this weaker model-theoretic strength, but the converse is not generally the case. For example,
in the paper I point out that it is a ZFC theorem that there is a nontrivial embedding $j:V\to V$; and so since every model of ZFC has such embeddings, their existence can carry no large
cardinal strength.
☆ Regarding the question 32 in your paper:
Can there be an embedding $j : V \rightarrow L$ when $V \neq L$?
You continued as follows:
… More generally, if $j$ is merely a class in Godel-Bernays set theory, then the existence of an embedding $j : V \rightarrow L$ implies global choice, since from the class $j$ we can
pull back the L-order. For these reasons, we cannot expect every model of $ZFC$ or of $GB$ to have such embeddings. Can they be added generically? …
My question is:
Did you examine the existence of such an embedding in a universe which $V=L[R]$?
I ask this in connection with Jensen’s coding theorem which allows us to always go to code the entire universe with a real in a generic extension made by a class forcing.
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What the parameter in PerturbedAdditive means
There’s a function in InferOpt package named PerturbedAdditive.
I’m curious about a parameter.
co_layer = PerturbedAdditive(knapsack_maximizer; ε=10.0, nb_samples=10)
For codes above, what does “ε=10.0” mean?
And I know that it will give 10 random numbers from the distribution N(0,1). If I want to fetch from N(0,2), what should I do?
Could you kindly provide me with the ε meaning and example code?
Thank you
In this function, ε is the scaling of the perturbation, which corresponds to the standard deviation for a Gaussian variable. So your code will actually perturb the knapsack_maximizer with nb_samples=
10 random numbers from the distribution N(0, ε^2).
Ping @BatyLeo
1 Like
Thank you for replying!
I understood. But I tried with the code.
And I got print like this:
PerturbedAdditive(pctsp, 10.0, 10, MersenneTwister, nothing, Normal(0, 1))
It indicates that it produced a distribution N(0,1) rather than N(0,10) ?
You should read the documentation ?PerturbedAdditive. ε is the standard deviation in the normal random variable case. The perturbation keyword, which is what you are seeing printed with Normal(0, 1),
is the base sampling distribution for that perturbation (which defaults to perturbation=nothing or equivalent Normal(0, 1)).
2 Likes
Thank you !
I see.
It means we fetch one random number z from distribution N(0,1) first. Then, we mutiple z with ε as our final perturbation?
I think I fully understood now.
1 Like
Hi, here is the full formula behind the hood:
\text{co_layer}(\theta) = \dfrac{1}{\text{nb_samples}} \sum\limits_{i=1}^{\text{nb_samples}} \text{knapsack_maximizer}(\theta + \varepsilon Z_i),
where Z_i are sampled from Normal(0, 1).
2 Likes
Thank you! I understand!
This is an execellent answer!
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Leakage resilient one-way functions: The auxiliary-input setting
Most cryptographic schemes are designed in a model where perfect secrecy of the secret key is assumed. In most physical implementations, however, some form of information leakage is inherent and
unavoidable. To deal with this, a flurry of works showed how to construct basic cryptographic primitives that are resilient to various forms of leakage. Dodis et al. (FOCS ’10) formalized and
constructed leakage resilient one-way functions. These are one-way functions f such that given a random image f(x) and leakage g(x) it is still hard to invert f(x). Based on any one-way function,
Dodis et al. constructed such a one-way function that is leakage resilient assuming that an attacker can leak any lossy function g of the input. In this work we consider the problem of constructing
leakage resilient one-way functions that are secure with respect to arbitrary computationally hiding leakage (a.k.a auxiliary-input). We consider both types of leakage — selective and adaptive — and
prove various possibility and impossibility results. On the negative side, we show that if the leakage is an adaptivelychosen arbitrary one-way function, then it is impossible to construct leakage
resilient one-way functions. The latter is proved both in the random oracle model (without any further assumptions) and in the standard model based on a strong vector-variant of DDH. On the positive
side, we observe that when the leakage is chosen ahead of time, there are leakage resilient one-way functions based on a variety of assumption.
Original language English
Title of host publication Theory of Cryptography - 14th International Conference, TCC 2016-B, Proceedings
Editors Adam Smith, Martin Hirt
Publisher Springer Verlag
Pages 139-158
Number of pages 20
ISBN (Print) 9783662536407
State Published - 2016
Externally published Yes
Event 14th International Conference on Theory of Cryptography, TCC 2016-B - Beijing, China
Duration: 31 Oct 2016 → 3 Nov 2016
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 9985 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 14th International Conference on Theory of Cryptography, TCC 2016-B
Country/Territory China
City Beijing
Period 31/10/16 → 3/11/16
Bibliographical note
Publisher Copyright:
© International Association for Cryptologic Research 2016.
Dive into the research topics of 'Leakage resilient one-way functions: The auxiliary-input setting'. Together they form a unique fingerprint.
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Complex Number to Trigonometric Form Calculator
Last updated:
Complex Number to Trigonometric Form Calculator
This complex number to trigonometric form calculator will help if you're still not quite sure how to find the trigonometric form of a complex number a + bi. In the brief article below, we explain
what the trig form of a complex number is and how to quickly convert from rectangular to trigonometric form.
What is the trigonometric form of a complex number?
The trigonometric form of a complex number z reads
z = r × [cos(φ) + i × sin(φ)],
• r is the modulus, i.e., the distance from (0,0) to z; and
• φ is the argument, i.e., the angle between the x-axis and the radius between (0,0) and z.
You can see r and φ in the picture below. Using basic trigonometry, we see that
cos(φ) = a × r and sin(φ) = b × r, exactly as claimed by the formula above.
How do I find the trigonometric form of a complex number?
To convert a complex number z = a + bi from rectangular to trigonometric form, you need to determine both the magnitude and the argument of z:
1. Compute the magnitude as r = √(a² + b²).
2. Compute the phase as φ = atan(b / a). If needed, correct by ±π to end up in the correct quadrant of the plane.
3. Write down the trigonometric form as z = r × [cos(φ) + i × sin(φ)].
How to use this complex number to trigonometric form calculator
To use the power of our tool, you need to feed it with the real and imaginary parts of your complex number, so with a and b from the a + bi form.
Our complex number to trigonometric form calculator will immediately compute the magnitude r and phase (argument) φ of your number and then display the trigonometric form in two ways: using radians
and degrees.
🙋 For some particular complex numbers you will get a third form as well: it will use π × rad as the angle unit. In this way, angles like π/4 have a chance of appearing.
Omni calculators and complex numbers
Satisfied with the complex number to trigonometric form calculator? Omni Calculator features a rich collection of calculators dedicated to complex numbers! Here's a full list in case you want to
learn more:
What is the trigonometric form of i+1?
The answer is i = √2 × [cos(π/4) + i × sin(π/4)]. To arrive at this result, observe that the magnitude of i equals r = √(1² + 1²) = √2. Then observe that the argument of i equals π/4 since atan(1/1)
= π/4.
How do I get the trigonometric form from polar form?
If you have a complex number in the polar form r × exp(iφ), it suffices to take r and φ from the polar form and write down r × [cos(φ) + i × sin(φ)] to get the trig form of your complex number. No
calculations required!
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SOR: Estimation using Sequential Offsetted Regression
Estimation for longitudinal data following outcome dependent sampling using the sequential offsetted regression technique. Includes support for binary, count, and continuous data. The first
regression is a logistic regression, which uses a known ratio (the probability of being sampled given that the subject/observation was referred divided by the probability of being sampled given that
the subject/observation was no referred) as an offset to estimate the probability of being referred given outcome and covariates. The second regression uses this estimated probability to calculate
the mean population response given covariates.
Version: 0.23.1
Depends: Matrix
Imports: methods, stats
Published: 2018-04-25
DOI: 10.32614/CRAN.package.SOR
Author: Lee McDaniel [aut, cre], Jonathan Schildcrout [aut]
Maintainer: Lee McDaniel <lmcda4 at lsuhsc.edu>
License: GPL-3
NeedsCompilation: no
CRAN checks: SOR results
Reference manual: SOR.pdf
Package source: SOR_0.23.1.tar.gz
Windows binaries: r-devel: SOR_0.23.1.zip, r-release: SOR_0.23.1.zip, r-oldrel: SOR_0.23.1.zip
macOS binaries: r-release (arm64): SOR_0.23.1.tgz, r-oldrel (arm64): SOR_0.23.1.tgz, r-release (x86_64): SOR_0.23.1.tgz, r-oldrel (x86_64): SOR_0.23.1.tgz
Old sources: SOR archive
Please use the canonical form https://CRAN.R-project.org/package=SOR to link to this page.
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2016 Fiscal Year Final Research Report
2016 Fiscal Year Final Research Report
Study on actions of discrete groups on graphs and metrics on groups
Project/Area 25800033
Research Grant-in-Aid for Young Scientists (B)
Allocation Type Multi-year Fund
Research Field Geometry
Research Tohoku University
Principal Mimura Masato 東北大学, 理学研究科, 助教 (10641962)
Project Period 2013-04-01 – 2017-03-31
Keywords 離散群 / 剛性 / エクスパンダー / 粗い幾何
For a fixed positive integer k, "the space of k-generated group" in certain sense is equipped with a natural topology that is metrizable and compact. For an infinite sequence of
Outline of finite k-generated groups, we establish correspendence between group property of elements in this topological space appearing as an accumulation point and coarse geometric property of
Final Research the infinite sequence. Moreover, we study on generalization of the Kazhdan constant, which is associated with Kazhdan's property (T). This quantity may be seen as a function on the
Achievements space of k-generated group. We generalize it to a general setting of fixed point property on a metric space, and under certain condition we prove that the defined function is lower
semi-continuous with respect to the convergence in the aforementioned topology.
Free Research 幾何学
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Source code for pyro.ops.contract
# Copyright (c) 2017-2019 Uber Technologies, Inc.
# SPDX-License-Identifier: Apache-2.0
import itertools
import warnings
from collections import OrderedDict, defaultdict
import opt_einsum
import torch
from opt_einsum import shared_intermediates
from pyro.ops.rings import BACKEND_TO_RING, LogRing
from pyro.util import ignore_jit_warnings
def _check_plates_are_sensible(output_dims, nonoutput_ordinal):
if output_dims and nonoutput_ordinal:
raise ValueError(
"It is nonsensical to preserve a plated dim without preserving "
"all of that dim's plates, but found '{}' without '{}'".format(
output_dims, ",".join(nonoutput_ordinal)
def _check_tree_structure(parent, leaf):
if parent == leaf:
raise NotImplementedError(
"Expected tree-structured plate nesting, but found "
"dependencies on independent plates [{}]. "
"Try converting one of the vectorized plates to a sequential plate (but beware "
"exponential cost in the size of the sequence)".format(
", ".join(getattr(f, "name", str(f)) for f in leaf)
def _partition_terms(ring, terms, dims):
Given a list of terms and a set of contraction dims, partitions the terms
up into sets that must be contracted together. By separating these
components we avoid broadcasting.
This function should be deterministic and free of side effects.
# Construct a bipartite graph between terms and the dims in which they
# are enumerated. This conflates terms and dims (tensors and ints).
neighbors = OrderedDict([(t, []) for t in terms] + [(d, []) for d in sorted(dims)])
for term in terms:
for dim in term._pyro_dims:
if dim in dims:
# Partition the bipartite graph into connected components for contraction.
components = []
while neighbors:
v, pending = neighbors.popitem()
component = OrderedDict([(v, None)]) # used as an OrderedSet
for v in pending:
component[v] = None
while pending:
v = pending.pop()
for v in neighbors.pop(v):
if v not in component:
component[v] = None
# Split this connected component into tensors and dims.
component_terms = [v for v in component if isinstance(v, torch.Tensor)]
if component_terms:
component_dims = set(
v for v in component if not isinstance(v, torch.Tensor)
components.append((component_terms, component_dims))
return components
def _contract_component(ring, tensor_tree, sum_dims, target_dims):
Contract out ``sum_dims - target_dims`` in a tree of tensors in-place, via
message passing. This reduces all tensors down to a single tensor in the
minimum plate context.
This function should be deterministic.
This function has side-effects: it modifies ``tensor_tree``.
:param pyro.ops.rings.Ring ring: an algebraic ring defining tensor
:param OrderedDict tensor_tree: a dictionary mapping ordinals to lists of
tensors. An ordinal is a frozenset of ``CondIndepStack`` frames.
:param set sum_dims: the complete set of sum-contractions dimensions
(indexed from the right). This is needed to distinguish sum-contraction
dimensions from product-contraction dimensions.
:param set target_dims: An subset of ``sum_dims`` that should be preserved
in the result.
:return: a pair ``(ordinal, tensor)``
:rtype: tuple of frozenset and torch.Tensor
# Group sum dims by ordinal.
dim_to_ordinal = {}
for t, terms in tensor_tree.items():
for term in terms:
for dim in sum_dims.intersection(term._pyro_dims):
dim_to_ordinal[dim] = dim_to_ordinal.get(dim, t) & t
dims_tree = defaultdict(set)
for dim, t in dim_to_ordinal.items():
# Recursively combine terms in different plate contexts.
local_terms = []
local_dims = target_dims.copy()
local_ordinal = frozenset()
min_ordinal = frozenset.intersection(*tensor_tree)
while any(dims_tree.values()):
# Arbitrarily deterministically choose a leaf.
leaf = max(tensor_tree, key=len)
leaf_terms = tensor_tree.pop(leaf)
leaf_dims = dims_tree.pop(leaf, set())
# Split terms at the current ordinal into connected components.
for terms, dims in _partition_terms(ring, leaf_terms, leaf_dims):
# Eliminate sum dims via a sumproduct contraction.
term = ring.sumproduct(terms, dims - local_dims)
# Eliminate extra plate dims via product contractions.
if leaf == min_ordinal:
parent = leaf
pending_dims = sum_dims.intersection(term._pyro_dims)
parent = frozenset.union(
*(t for t, d in dims_tree.items() if d & pending_dims)
_check_tree_structure(parent, leaf)
contract_frames = leaf - parent
contract_dims = dims & local_dims
if contract_dims:
term, local_term = ring.global_local(
term, contract_dims, contract_frames
local_dims |= sum_dims.intersection(local_term._pyro_dims)
local_ordinal |= leaf
term = ring.product(term, contract_frames)
tensor_tree.setdefault(parent, []).append(term)
# Extract single tensor at root ordinal.
assert len(tensor_tree) == 1
ordinal, (term,) = tensor_tree.popitem()
assert ordinal == min_ordinal
# Perform optional localizing pass.
if local_terms:
assert target_dims
term = ring.sumproduct(local_terms, local_dims - target_dims)
ordinal |= local_ordinal
return ordinal, term
def contract_tensor_tree(tensor_tree, sum_dims, cache=None, ring=None):
Contract out ``sum_dims`` in a tree of tensors via message passing.
This partially contracts out plate dimensions.
This function should be deterministic and free of side effects.
:param OrderedDict tensor_tree: a dictionary mapping ordinals to lists of
tensors. An ordinal is a frozenset of ``CondIndepStack`` frames.
:param set sum_dims: the complete set of sum-contractions dimensions
(indexed from the right). This is needed to distinguish sum-contraction
dimensions from product-contraction dimensions.
:param dict cache: an optional :func:`~opt_einsum.shared_intermediates`
:param pyro.ops.rings.Ring ring: an optional algebraic ring defining tensor
:returns: A contracted version of ``tensor_tree``
:rtype: OrderedDict
assert isinstance(tensor_tree, OrderedDict)
assert isinstance(sum_dims, set)
if ring is None:
ring = LogRing(cache)
ordinals = {term: t for t, terms in tensor_tree.items() for term in terms}
all_terms = [term for terms in tensor_tree.values() for term in terms]
contracted_tree = OrderedDict()
# Split this tensor tree into connected components.
for terms, dims in _partition_terms(ring, all_terms, sum_dims):
component = OrderedDict()
for term in terms:
component.setdefault(ordinals[term], []).append(term)
# Contract this connected component down to a single tensor.
ordinal, term = _contract_component(ring, component, dims, set())
contracted_tree.setdefault(ordinal, []).append(term)
return contracted_tree
def contract_to_tensor(
tensor_tree, sum_dims, target_ordinal=None, target_dims=None, cache=None, ring=None
Contract out ``sum_dims`` in a tree of tensors, via message
passing. This reduces all terms down to a single tensor in the plate
context specified by ``target_ordinal``, optionally preserving sum
dimensions ``target_dims``.
This function should be deterministic and free of side effects.
:param OrderedDict tensor_tree: a dictionary mapping ordinals to lists of
tensors. An ordinal is a frozenset of ``CondIndepStack`` frames.
:param set sum_dims: the complete set of sum-contractions dimensions
(indexed from the right). This is needed to distinguish sum-contraction
dimensions from product-contraction dimensions.
:param frozenset target_ordinal: An optional ordinal to which the result
will be contracted or broadcasted.
:param set target_dims: An optional subset of ``sum_dims`` that should be
preserved in the result.
:param dict cache: an optional :func:`~opt_einsum.shared_intermediates`
:param pyro.ops.rings.Ring ring: an optional algebraic ring defining tensor
:returns: a single tensor
:rtype: torch.Tensor
if target_ordinal is None:
target_ordinal = frozenset()
if target_dims is None:
target_dims = set()
assert isinstance(tensor_tree, OrderedDict)
assert isinstance(sum_dims, set)
assert isinstance(target_ordinal, frozenset)
assert isinstance(target_dims, set) and target_dims <= sum_dims
if ring is None:
ring = LogRing(cache)
ordinals = {term: t for t, terms in tensor_tree.items() for term in terms}
all_terms = [term for terms in tensor_tree.values() for term in terms]
contracted_terms = []
# Split this tensor tree into connected components.
modulo_total = bool(target_dims)
for terms, dims in _partition_terms(ring, all_terms, sum_dims):
if modulo_total and dims.isdisjoint(target_dims):
component = OrderedDict()
for term in terms:
component.setdefault(ordinals[term], []).append(term)
# Contract this connected component down to a single tensor.
ordinal, term = _contract_component(ring, component, dims, target_dims & dims)
target_dims.intersection(term._pyro_dims), ordinal - target_ordinal
# Eliminate extra plate dims via product contractions.
contract_frames = ordinal - target_ordinal
if contract_frames:
assert not sum_dims.intersection(term._pyro_dims)
term = ring.product(term, contract_frames)
# Combine contracted tensors via product, then broadcast.
term = ring.sumproduct(contracted_terms, set())
assert sum_dims.intersection(term._pyro_dims) <= target_dims
return ring.broadcast(term, target_ordinal)
[docs]def einsum(equation, *operands, **kwargs):
Generalized plated sum-product algorithm via tensor variable elimination.
This generalizes :func:`~pyro.ops.einsum.contract` in two ways:
1. Multiple outputs are allowed, and intermediate results can be shared.
2. Inputs and outputs can be plated along symbols given in ``plates``;
reductions along ``plates`` are product reductions.
The best way to understand this function is to try the examples below,
which show how :func:`einsum` calls can be implemented as multiple calls
to :func:`~pyro.ops.einsum.contract` (which is generally more expensive).
To illustrate multiple outputs, note that the following are equivalent::
z1, z2, z3 = einsum('ab,bc->a,b,c', x, y) # multiple outputs
z1 = contract('ab,bc->a', x, y)
z2 = contract('ab,bc->b', x, y)
z3 = contract('ab,bc->c', x, y)
To illustrate plated inputs, note that the following are equivalent::
assert len(x) == 3 and len(y) == 3
z = einsum('ab,ai,bi->b', w, x, y, plates='i')
z = contract('ab,a,a,a,b,b,b->b', w, *x, *y)
When a sum dimension `a` always appears with a plate dimension `i`,
then `a` corresponds to a distinct symbol for each slice of `a`. Thus
the following are equivalent::
assert len(x) == 3 and len(y) == 3
z = einsum('ai,ai->', x, y, plates='i')
z = contract('a,b,c,a,b,c->', *x, *y)
When such a sum dimension appears in the output, it must be
accompanied by all of its plate dimensions, e.g. the following are
assert len(x) == 3 and len(y) == 3
z = einsum('abi,abi->bi', x, y, plates='i')
z0 = contract('ab,ac,ad,ab,ac,ad->b', *x, *y)
z1 = contract('ab,ac,ad,ab,ac,ad->c', *x, *y)
z2 = contract('ab,ac,ad,ab,ac,ad->d', *x, *y)
z = torch.stack([z0, z1, z2])
Note that each plate slice through the output is multilinear in all plate
slices through all inptus, thus e.g. batch matrix multiply would be
implemented *without* ``plates``, so the following are all equivalent::
xy = einsum('abc,acd->abd', x, y, plates='')
xy = torch.stack([xa.mm(ya) for xa, ya in zip(x, y)])
xy = torch.bmm(x, y)
Among all valid equations, some computations are polynomial in the sizes of
the input tensors and other computations are exponential in the sizes of
the input tensors. This function raises :py:class:`NotImplementedError`
whenever the computation is exponential.
:param str equation: An einsum equation, optionally with multiple outputs.
:param torch.Tensor operands: A collection of tensors.
:param str plates: An optional string of plate symbols.
:param str backend: An optional einsum backend, defaults to 'torch'.
:param dict cache: An optional :func:`~opt_einsum.shared_intermediates`
:param bool modulo_total: Optionally allow einsum to arbitrarily scale
each result plate, which can significantly reduce computation. This is
safe to set whenever each result plate denotes a nonnormalized
probability distribution whose total is not of interest.
:return: a tuple of tensors of requested shape, one entry per output.
:rtype: tuple
:raises ValueError: if tensor sizes mismatch or an output requests a
plated dim without that dim's plates.
:raises NotImplementedError: if contraction would have cost exponential in
the size of any input tensor.
# Extract kwargs.
cache = kwargs.pop("cache", None)
plates = kwargs.pop("plates", "")
backend = kwargs.pop("backend", "torch")
modulo_total = kwargs.pop("modulo_total", False)
Ring = BACKEND_TO_RING[backend]
except KeyError as e:
raise NotImplementedError(
["Only the following pyro backends are currently implemented:"]
+ list(BACKEND_TO_RING)
) from e
# Parse generalized einsum equation.
if "." in equation:
raise NotImplementedError("ubsersum does not yet support ellipsis notation")
inputs, outputs = equation.split("->")
inputs = inputs.split(",")
outputs = outputs.split(",")
assert len(inputs) == len(operands)
assert all(isinstance(x, torch.Tensor) for x in operands)
if not modulo_total and any(outputs):
raise NotImplementedError(
"Try setting modulo_total=True and ensuring that your use case "
"allows an arbitrary scale factor on each result plate."
if len(operands) != len(set(operands)):
operands = [x[...] for x in operands] # ensure tensors are unique
# Check sizes.
with ignore_jit_warnings():
dim_to_size = {}
for dims, term in zip(inputs, operands):
for dim, size in zip(dims, map(int, term.shape)):
old = dim_to_size.setdefault(dim, size)
if old != size:
raise ValueError(
"Dimension size mismatch at dim '{}': {} vs {}".format(
dim, size, old
# Construct a tensor tree shared by all outputs.
tensor_tree = OrderedDict()
plates = frozenset(plates)
for dims, term in zip(inputs, operands):
assert len(dims) == term.dim()
term._pyro_dims = dims
ordinal = plates.intersection(dims)
tensor_tree.setdefault(ordinal, []).append(term)
# Compute outputs, sharing intermediate computations.
results = []
with shared_intermediates(cache) as cache:
ring = Ring(cache, dim_to_size=dim_to_size)
for output in outputs:
sum_dims = set(output).union(*inputs) - set(plates)
term = contract_to_tensor(
if term._pyro_dims != output:
term = term.permute(*map(term._pyro_dims.index, output))
term._pyro_dims = output
return tuple(results)
[docs]def ubersum(equation, *operands, **kwargs):
Deprecated, use :func:`einsum` instead.
"'ubersum' is deprecated, use 'pyro.ops.contract.einsum' instead",
if "batch_dims" in kwargs:
"'batch_dims' is deprecated, use 'plates' instead", DeprecationWarning
kwargs["plates"] = kwargs.pop("batch_dims")
kwargs.setdefault("backend", "pyro.ops.einsum.torch_log")
return einsum(equation, *operands, **kwargs)
def _select(tensor, dims, indices):
for dim, index in zip(dims, indices):
tensor = tensor.select(dim, index)
return tensor
class _DimUnroller:
Object to map plated dims to collections of unrolled dims.
:param dict dim_to_ordinal: a mapping from contraction dim to the set of
plates over which the contraction dim is plated.
def __init__(self, dim_to_ordinal):
self._plates = {
d: tuple(sorted(ordinal)) for d, ordinal in dim_to_ordinal.items()
self._symbols = map(opt_einsum.get_symbol, itertools.count())
self._map = {}
def __call__(self, dim, indices):
Converts a plate dim + plate indices to a unrolled dim.
:param str dim: a plate dimension to unroll
:param dict indices: a mapping from plate dimension to int
:return: a unrolled dim
:rtype: str
plate = self._plates.get(dim, ())
index = tuple(indices[d] for d in plate)
key = dim, index
if key in self._map:
return self._map[key]
normal_dim = next(self._symbols)
self._map[key] = normal_dim
return normal_dim
def naive_ubersum(equation, *operands, **kwargs):
Naive reference implementation of :func:`ubersum` via unrolling.
This implementation should never raise ``NotImplementedError``.
This implementation should agree with :func:`ubersum` whenver
:func:`ubersum` does not raise ``NotImplementedError``.
# Parse equation, without loss of generality assuming a single output.
inputs, outputs = equation.split("->")
outputs = outputs.split(",")
if len(outputs) > 1:
return tuple(
naive_ubersum(inputs + "->" + output, *operands, **kwargs)[0]
for output in outputs
(output,) = outputs
inputs = inputs.split(",")
backend = kwargs.pop("backend", "pyro.ops.einsum.torch_log")
# Split dims into plate dims, contraction dims, and dims to keep.
plates = set(kwargs.pop("plates", ""))
if not plates:
result = opt_einsum.contract(equation, *operands, backend=backend)
return (result,)
output_dims = set(output)
# Collect sizes of all dimensions.
sizes = {}
for input_, operand in zip(inputs, operands):
for dim, size in zip(input_, operand.shape):
old = sizes.setdefault(dim, size)
if old != size:
raise ValueError(
"Dimension size mismatch at dim '{}': {} vs {}".format(
dim, size, old
# Compute plate context for each non-plate dim, by convention the
# intersection over all plate contexts of tensors in which the dim appears.
dim_to_ordinal = {}
for dims in map(set, inputs):
ordinal = dims & plates
for dim in dims - plates:
dim_to_ordinal[dim] = dim_to_ordinal.get(dim, ordinal) & ordinal
for dim in output_dims - plates:
_check_plates_are_sensible({dim}, dim_to_ordinal[dim] - output_dims)
# Unroll by replicating along plate dimensions.
unroll_dim = _DimUnroller(dim_to_ordinal)
flat_inputs = []
flat_operands = []
for input_, operand in zip(inputs, operands):
local_dims = [d for d in input_ if d in plates]
offsets = [input_.index(d) - len(input_) for d in local_dims]
for index in itertools.product(*(range(sizes[d]) for d in local_dims)):
unroll_dim(d, dict(zip(local_dims, index)))
for d in input_
if d not in plates
flat_operands.append(_select(operand, offsets, index))
# Defer to unplated einsum.
result = torch.empty(
torch.Size(sizes[d] for d in output),
local_dims = [d for d in output if d in plates]
offsets = [output.index(d) - len(output) for d in local_dims]
for index in itertools.product(*(range(sizes[d]) for d in local_dims)):
flat_output = "".join(
unroll_dim(d, dict(zip(local_dims, index)))
for d in output
if d not in plates
flat_equation = ",".join(flat_inputs) + "->" + flat_output
flat_result = opt_einsum.contract(
flat_equation, *flat_operands, backend=backend
if not local_dims:
result = flat_result
_select(result, offsets, index).copy_(flat_result)
return (result,)
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Steady Stresses and Variable Stresses In Machine Members
It is defined as any external force acting upon a machine part. The following four types of the load are important from the subject point of view:
Dead or steady load. A load is said to be a dead or steady load, when it does not change in magnitude or direction.
Live or variable load.A load is said to be a live or variable load, when it changes continually.
Suddenly applied or shock loads. A load is said to be a suddenly applied or shock load, when it is suddenly applied or removed.
Impact load. A load is said to be an impact load, when it is applied with some initial velocity.
When some external system of forces or loads act on a body, the internal forces (equal and opposite) are set up at various sections of the body, which resist the external forces. This internal force
per unit area at any section of the body is known as unit stress or simply a stress. It is denoted by a Greek letter sigma (σ).
Stress, σ = P/A
where P = Force or load acting on a body, and A = Cross-sectional area of the body.
When a system of forces or loads act on a body, it undergoes some deformation. This deformation per unit length is known as unit strain or simply a strain. It is denoted by a Greek letter epsilon
Strain, ε = δl / l or δl = ε.l where δl = Change in length of the body, and
l = Original length of the body.
Tensile Stress and Strain
When a body is subjected to two equal and opposite axial pulls P (also called tensile load) as shown in Fig. (a), then the stress induced at any section of the body is known as tensile stress as
shown in Fig. (b). A little consideration will show that due to the tensile load, there will be a decrease in cross-sectional area and an increase in length of the body. The ratio of the increase in
length to the original length is known as tensile strain.
Let P = Axial tensile force acting on the body,
A = Cross-sectional area of the body,
l = Original length, and
δl = Increase in length.
Tensile stress, σt = P/A Tensile strain, εt = δl / l
Compressive Stress and Strain
When a body is subjected to two equal and opposite axial pushes P (also called compressive load) as shown in Fig.(a), then the stress induced at any section of the body is known as compressive stress
as shown in Fig.(b). A little consideration will show that due to the compressive load, there will be an increase in cross-sectional area and a decrease in length of the body. The ratio of the
decrease in length to the original length is known as compressive strain
Let P = Axial compressive force acting on the body,
A = Cross-sectional area of the body,
l = Original length, and
δl = Decrease in length.
Compressive stress, σc = P/A Compressive strain, εc = δl /l
Young's Modulus or Modulus of Elasticity
Hooke's law states that when a material is loaded within elastic limit, the stress is directly proportional to strain, i.e.
∝ ε
= E.ε
E = σ / ε
= P l / (A×δ l)
Where E is a constant of proportionality known as Young's modulus or modulus of elasticity. In S.I. units, it is usually expressed in GPa i.e. GN/m^2 or kN/mm^2. It may be noted that Hooke's law
holds good for tension as well as compression.
Shear Stress and Strain
When a body is subjected to two equal and opposite forces acting tangentially across the resisting section, as a result of which the body tends to shear off the section, then the stress induced is
called shear stress.
The corresponding strain is known as shear strain and it is measured by the angular deformation accompanying the shear stress. The shear stress and shear strain are denoted by the Greek letters tau
(τ) and phi (φ) respectively. Mathematically,
Shear stress, τ =Tangential force / Resisting area
Consider a body consisting of two plates connected by a rivet as shown in Fig. (a). In this case, the tangential force P tends to shear off the rivet at one cross-section as shown in Fig.(b). It may
be noted that when the tangential force is resisted by one cross-section of the rivet (or when shearing takes place at one cross-section of the rivet), then the rivets are said to be in single shear.
In such a case, the area resisting the shear off the rivet,
A = (π/4) × d^2
and shear stress on the rivet cross-section,
= P/A
= (4P/ π d^2)
Shear Modulus or Modulus of Rigidity
It has been found experimentally that within the elastic limit, the shear stress is directly proportional to shear strain. Mathematically
∝ φ
= C . φ
/ φ = C
τ = Shear stress,
φ = Shear strain, and
C = Constant of proportionality, known as shear modulus or modulus of rigidity. It is also denoted by N or G.
Working Stress
When designing machine parts, it is desirable to keep the stress lower than the maximum or ultimate stress at which failure of the material takes place. This stress is known as the working
Factor of Safety
It is defined, in general, as the ratio of the maximum stress to the working stress. Mathematically,
Factor of safety = Maximum stress / Working or design stress
In case of ductile materials e.g. mild steel, where the yield point is clearly defined, the factor of safety is based upon the yield point stress. In such cases,
Factor of safety = Yield point stress / Working or design stress
In case of brittle materials e.g. cast iron, the yield point is not well defined as for ductile materials. Therefore, the factor of safety for brittle materials is based on ultimate stress.
Factor of safety = Ultimate stress / Working or design stress
This relation may also be used for ductile materials
Poisson's Ratio
It has been found experimentally that when a body is stressed within elastic limit, the lateral strain bears a constant ratio to the linear strain, Mathematically,
Lateral strain / Linear strain = Constant
This constant is known as Poisson's ratio and is denoted by 1/m or µ.
Bulk Modulus
When a body is subjected to three mutually perpendicular stresses, of equal intensity, then the ratio of the direct stress to the corresponding volumetric strain is known as bulk modulus. It is
usually denoted by K. Mathematically, bulk modulus,
K = Direct stress / Volumetric strain
Relation Between Bulk Modulus and Young’s Modulus
The bulk modulus (K) and Young's modulus (E) are related by the following relation,
E= 3K (1 - 2 µ)
Relation Between Young’s Modulus and Modulus of Rigidity
The Young's modulus (E) and modulus of rigidity (G) are related by the following relation,
E= 2G (1 + µ)
When a body is loaded within elastic limit, it changes its dimensions and on the removal of the load, it regains its original dimensions. So long as it remains loaded, it has stored energy in itself.
On removing the load, the energy stored is given off as in the case of a spring. This energy, which is absorbed in a body when strained within elastic limit, is known as strain energy. The strain
energy is always capable of doing some work.
The strain energy stored in a body due to external loading, within elastic limit, is known as resilience and the maximum energy which can be stored in a body up to the elastic limit is called proof
resilience. The proof resilience per unit volume of a material is known as modulus of resilience.
It is an important property of a material and gives capacity of the material to bear impact or shocks. Mathematically, strain energy stored in a body due to tensile or compressive load or resilience,
U= (σ^2 ×V) / 2E
Modulus of resilience = σ^2 / 2E
where σ = Tensile or compressive stress, V = Volume of the body, and
= Young's modulus of the material of the body
Torsional Shear Stress
When a machine member is subjected to the action of two equal and opposite couples acting in parallel planes (or torque or twisting moment), then the machine member is said to be subjected to
torsion. The stress set up by torsion is known as torsional shear stress. It is zero at the centroidal axis and maximum at the outer surface.
Consider a shaft fixed at one end and subjected to a torque (T) at the other end as shown in Fig. As a result of this torque, every cross-section of the shaft is subjected to torsional shear stress.
We have discussed above that the torsional shear stress is zero at the centroidal axis and maximum at the outer surface. The maximum torsional shear stress at the outer surface of the shaft may be
obtained from the following equation:
(τ /r) = (T/J) = (Cθ/ l )
τ = Torsional shear stress induced at the outer surface of the shaft or maximum shear stress,
= Radius of the shaft,
= Torque or twisting moment,
= Second moment of area of the section about its polar axis or polar moment of inertia,
C = Modulus of rigidity for the shaft material, l = Length of the shaft, and
= Angle of twist in radians on a length l.
Shafts in Series and Parallel
When two shafts of different diameters are connected together to form one shaft, it is then known as composite shaft. If the driving torque is applied at one end and the resisting torque at the other
end, then the shafts are said to be connected in series as shown in Fig. (a). In such cases, each shaft transmits the same torque and the total angle of twist is equal to the sum of the angle of
twists of the two shafts.
Mathematically, total angle of twist,
If the shafts are made of he same material, then C1 = C2 = C.
When the driving torque (T) is applied at the junction of the two shafts, and the resisting torques T1 and T2 at the other ends of the shafts, then the shafts are said to be connected in parallel, as
shown in Fig. 5.2 (b). In such cases, the angle of twist is same for both the shafts,
If the shafts are mad^e of the same material, then C1 = C2.
Bending Stress in Straight Beams
In engineering practice, the machine parts of structural members may be subjected to static or dynamic loads which cause bending stress in the sections besides other types of stresses such as
tensile, compressive and shearing stresses. Consider a straight beam subjected to a bending moment M as shown in Fig. The following assumptions are usually made while deriving the bending formula.
The material of the beam is perfectly homogeneous (i.e. of the same material throughout) and isotropic (i.e. of equal elastic properties in all directions).
The material of the beam obeys Hooke’s law.
The transverse sections (i.e. BC or GH) which were plane before bending, remain plane after bending also.
Each layer of the beam is free to expand or contract, independently, of the layer, above or below it.
The Young’s modulus (E) is the same in tension and compression.
The loads are applied in the plane of bending.
A little consideration will show that when a beam is subjected to the bending moment, the fibres on the upper side of the beam will be shortened due to compression and those on the lower side will be
elongated due to tension. It may be seen that somewhere between the top and bottom fibres there is a surface at which the fibres are neither shortened nor lengthened. Such a surface is called neutral
surface. The intersection of the neutral surface with any normal cross-section of the beam is known as neutral axis. The stress distri bution of a beam is shown in Fig. The bending equation is given
[where] M = Bending moment acting at the given section, σ = Bending stress,
I = Moment of inertia of the cross-section about the neutral axis, y = Distance from the neutral axis to the extreme fibre,
E = Young’s modulus of the material of the beam, and R = Radius of curvature of the beam.
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Iain Moffatt
Royal Holloway, University of London Royal Holloway, University of London
From graph duals to matrix pivots: a tour through delta-matorids
Combinatorics Seminar
12th November 2019, 11:00 am – 12:00 pm
Fry Building, 2.04
This talk will consider a variety of everyday graph theoretical notions --- duals, circle graphs, pivot-minors, Eulerian graphs, and bipartite graphs --- and will survey how they appear in the theory
of delta-matroids. The emphasis will be on exposing the interplay between the graph theoretical and matroid theoretical concepts, and no prior knowledge of matroids will be assumed.
Matroids are often introduced either as objects that capture linear independence, or as generalisations of graphs. If one likes to think of a matroid as a structure that captures linear independence
in vector spaces, then a delta-matroid is a structure that arises by retaining the (Steinitz) exchange properties of vector space bases, but dropping the requirement that basis elements are all of
the same size. On the other hand, if one prefers to think of matroids as generalising graphs, then delta-matroids generalise graphs embedded in surfaces. There are a host of other ways in which
delta-matroids arise in combinatorics. Indeed, they were introduced independently by three different groups of authors in the 1980s, with each definition having a different motivation (and all
different from the two above). In this talk I'll survey some of the ways in which delta-matroids appear in graph theory. The focus will be on how the fundamental operations on delta-matroids appear,
in different guises, as familiar and well-studied concepts in graph theory. In particular, I'll illustrate how apparently disparate pieces of graph theory come together in the world of
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How Many Minutes In a Year?How Many Minutes In a Year?
There are around 525,600 minutes in a year. This intends that there are around 8,760 hours in a year. There is an hour in 60 minutes, 24 hours in a day, and 365 days a year.
This truly intends that there are 8760 hours in a year or 525600 minutes. Thus, there are approximately 525600 minutes in a year. There is an hour in 60 minutes, 24 hours in a day, and 365 days a
year. This intends that there are 8760 hours in a year or 525600 minutes. In this way, there are 525600 minutes in a year.
How to Calculate How Many Minutes are there In a Year
There are 525,600 minutes in a year. This truly intends that there are 8,760 hours in a year or 365 days. Every day has 24 hours, so in the event that you duplicate 24 by 365, you get the number of
hours in a year. To get the number of minutes, you simply have to duplicate the number of hours by 60.
The most effective method to Ascertain how many minutes in leap Years
To compute how many minutes are in a leap year, you want to know how many minutes are in a leap year. A leap year has 366 days, rather than the standard 365. This intends that there are 24 extra
hours, or 1,440 extra minutes, in a leap year. To work out this number, just increase 24 by 60.
How many minutes in the Gregorian year?
There are 8,766,400 minutes in a Gregorian year. This is what amount of time it requires for the Earth to complete one circle around the sun. The year is parceled into a year, and consistently has a
typical of 30 days. There are 24 hours in a day and an hour in an hour. Many minutes these lines, there are 8,766,400 minutes in a Gregorian year.
How many minutes in a year by Julian and normal calendars?
There is a sum of 525,949 minutes in a year as per the Julian calendar. This is the calendar that was utilized before the normal calendar. The normal calendar has a sum of 525,760 minutes in a year.
How many minutes are in a year? It relies upon which calendar you use. The Julian calendar, utilized before the Gregorian calendar, had 365.25 days in a year. This works out to be around 8,784 hours
or 525,949 minutes in a Julian year. The Gregorian calendar, utilized by a large portion of this present reality, has 365.2425 days in a year. This is around 8,765 hours or 525,760 minutes in a
Gregorian year. In this way, there are around 18 minutes fewer in a typical year than there are in a Julian year.
How many minutes in a year precisely?
Here we will find exactly minutes in the many minutes haul since 1 sun-based year is 365 days 5 hours 48 minutes and 46 seconds. So according to this, we will talk about nuances.
To start with, we will change all characteristics in a many minutes time like
• 365 days = 365 x 24 x an hour = 525,600 minutes
• 5 hours = 5 x 60 = 300 minutes
• 48 minutes = 48 minutes
• 46 seconds = 46/60 minutes=0.76666 minutes
By and by we will put all of the characteristics that we change in minutes
• 1 calendar year = 365 days 5 hours 48 minutes 46 seconds = (525,600 + 300 + 48 + 0.76666) minutes
• = 525948.76666 minutes
• =8765.8128 hours (525948.76666/60)
The most effective method to compute the number of minutes in a year
Prior to digging into why and in what circumstance 525,600 minutes could be mistaken, let us initially lay out how to stop by this outcome.
It is important to express that on the off chance that you are not a science lover or find computation by and large troublesome, you don’t need to stress on the grounds that the estimation
interaction is truly worked on here. As a matter of fact, you don’t require more than essential math information to entirely go through and grasp it. Simply follow the below-mentioned steps.
Step ONE: Breakdown the units of time in a year
A year is comprised of 12 months, every month is comprised of days, a day has 24 hours, and a single hour is comprised of an hour. You could go on down to seconds, however, it isn’t required until
further notice.
Step TWO: Figure out the total number of days in a year
This is on the grounds that the number of days in the months of the year isn’t something very similar. In this way, it is more straightforward to decide and work with the total number of days in a
year, which is 365.
Step THREE: Figure out how many minutes there are in a single day
To show up at the answer, all you really want to do is to duplicate the number of minutes in a solitary hour with the complete number of hours in a day. for example
an hour x 24 = 1440 minutes (in a day)
Step FOUR: Duplicate the number of minutes in a day with the total number of days in a year
1440 (minutes) x 365 = 525,600 minutes (in a year).
Showing up at the answer above is very simple and straightforward. Aside from the simplicity and certainty of the methodology, the response is likewise the most by and large acknowledged one in light
of the fact that as a rule, when the quantity of minutes in a year is asked, that is implied.
The unpopular response for the number of minutes in a year
While you might score 100% in the event that you present the response above in a general setting, that may not be the situation assuming that you wind up in a basic setting. A basic setting might
resemble an advanced geography class or a basic teacher of science. You most likely might be asking what precisely would make the famous and by and large acknowledged answer unique. Indeed, it is
right here.
Working out the real number of minutes in a year (basically talking)
The number of days in a year is gotten from the number of days it takes the earth to spin around the sun. You most likely knew that or had a thought. Yet, did you have any idea that a year isn’t
precisely comprised of 365 days but of 365 ¼ days.
This implies that it takes the earth around 365 days and one-fourth of a day (that is 6 hours) to finish a revolution. This is the motivation behind why once like clockwork, there is a leap year
where an additional day is added to the calendar on the period of February. In this way, in a basic setting, the overall minutes in a year would be given by
Total minutes in a day x 365 ¼ or
Total minutes in a day x (365 days + 6 hours)
Number of minutes in 6 hours = 60 x 6 = 360minutes
Number of minutes in 365 days = 1440 x 365 = 525,600 minutes
Absolute number of minutes = 525,600 + 360 = 525,960 minutes.
It is vital to take note that you would be viewed as wrong assuming that you give the above answer when you are not where this sort of criticality is required.
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10cm in inches
To convert 10cm to inches, we can use the conversion factor of 2.54, where 1 inch is equal to 2.54cm. This means that for every 2.54cm, there is 1 inch.
Step 1: Begin by writing down the given value, which is 10cm. This will be our starting point for the conversion.
Step 2: Next, write down the conversion factor, which is 2.54. This tells us how many centimeters are in 1 inch.
Step 3: To convert, we need to divide the given value (10cm) by the conversion factor (2.54).
10cm รท 2.54 = 3.937 inches
This means that 10cm is equal to 3.937 inches.
Step 4: Since we are converting from centimeters to inches, we need to make sure our final answer has the correct unit. In this case, our answer is in inches, so we can write 3.937 inches as our
final answer.
Therefore, 10cm is equal to 3.937 inches. This concludes the conversion process.
Step 5: If you want to double check your answer, you can also use an online unit converter or a calculator that has a conversion function to confirm that your answer is correct.
Congratulations, you have successfully converted 10cm to inches! Remember to use the conversion factor of 2.54 whenever you need to convert from centimeters to inches.
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Unitarity Bound on Dark Matter: Dark Matter Annihilation and Unitarity Bound
This paper is available on arxiv under CC 4.0 license.
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory.
Table of Links
3. Dark Matter Annihilation and Unitarity Bound
Now, the Boltzmann equation boils down to the following simplified form
Now, we can obtain the maximum thermally averaged rate for the r → 2 process using Eq. (3.2), as
where g stands for the internal degrees of freedom of the DM. Therefore, the maximum value of the thermally averaged s-wave annihilation cross section for 3 → 2 can be written as
Likewise, one can also express the maximum value of the thermally average s-wave annihilation cross section for 4 → 2 as
[3] Although the expression of Eq. (3.5) is derived considering equal mass for all the particles involved in the interaction; it provides maximum thermally-averaged cross section for WIMPs in the
case r = 2 where the mass of the final-state particles can be different from the same of the initial-state particles.
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Properties of Real & Complex Numbers Mathematics for Nurses - Nurses Educator
Properties of Real & Complex Numbers Mathematics for Nurses
Properties of Real & Complex Numbers: Properties of Real Numbers
Major Mathematical Operations and Properties of Real and Complex Numbers.
Numerical operations are mathematical operations that we perform with complex numbers. Operations with complex numbers are more or less like operations with normal numbers, except for division.
In this article you will learn about complex number operations, addition and subtraction of complex numbers, properties of addition and subtraction of complex numbers, multiplication properties of
complex number multiplication, division, conjugation, modulus of a complex number solved , examples.
Leave a Comment
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Math for 6th Grade Homeschoolers - BJU Press Homeschool Blog
I will never forget when my sixth grade math teacher presented the class with the largest multiplication problem we had ever encountered. He gave us a huge sheet of paper and asked us to multiply
123,456,789 by itself. We had a week to work on it at specific times of each day, and the first person in the class to finish correctly would win a prize. This was the perfect exercise to finish our
year. Math for 6th grade homeschoolers should take all the basic operations students have learned in elementary school and apply them to larger numbers, fractions, decimals, and even negative
numbers. Sixth grade is a year of transition from elementary to middle school. As such, it is a pivotal year for mastering basic math facts and starting to apply them to algebra, geometry,
statistics, and even some physics.
What do 6th graders learn in math?
• Number system fluency. In sixth grade math, students should become fluent with the number system. They learn to use the basic operations in order with multiple-digit whole numbers, decimal
numbers, fractions, integers (which includes negative numbers) and percentages. Problems with multiple steps require students to know the order of operations (parentheses, exponents,
multiplication/division, and addition/subtraction). This number system fluency prepares them to begin considering variables in pre-algebra.
• Expressions and equations. They will learn about expressions and equations and how to write and use them based on word problems. They also add ratios and rate calculations to their skillset.
• Geometry. They will expand their knowledge of lines and shapes. Sixth graders will measure angles, and learn how to solve area and volume of irregular shapes by deconstructing them into shapes
that have an equation.
• Basic Statistics. Sixth grade students begin their study of statistics and learning how to find mean, mode, and median. They also learn some important graphing techniques.
• Other. Sixth grade math students learn how to estimate and round numbers. They learn the metric system of measurement. Some basic physics concepts are also included in sixth grade, such as speed
and distance. They learn Roman numerals and squares and square roots.
What math should a 6th grader know by the end of the year?
By the end of the year, sixth graders should have the foundational knowledge that will prepare them for seventh and eighth grade math, including pre-algebra and beyond. Number system fluency is the
key skill that sixth grade students must master. They should be very comfortable with basic multiplication and division facts. And they should be able to use these facts to solve problems with
different types of numbers, including fractions, decimals and negative numbers. Many of the other topics in sixth grade math are a foundation to build upon later. In a spiral curriculum, they will
see these topics again and again in the following years. Starting students in sixth grade with a strong foundation in geometry, statistics, and writing, and solving expressions and equations will
make future math studies more productive.
Do I need to teach my child 6th grade math?
Whether your homeschooling style is traditional or not, you will want to be involved in your child’s sixth grade math instruction. This is a pivotal year, and you need to ensure your child has
mastered the number system. If math is not your strong suit, consider using an online school or investing in a curriculum with video lessons included, such as you find available from on BJU Press.
Child-led learning might work for this course if your child is especially motivated and self-driven in mathematics. Otherwise it is important that your sixth grader receives guidance to ensure that
they learn key skills by the end of the year.
6th Grade Math Activities for Different Learning Styles
Whatever your child’s learning style preference, he should practice speed drills to ensure that basic facts are quick to recall. Review games are also fun for all learning styles. Verbal and auditory
learners might want to recite equations they are solving during the game out loud.
6th Grade Math Learning Games
Sixth grade is the perfect time to introduce a game like 24. There is a branded game with special cards, but you can also play with any deck of numbered cards. In 24, you have 4 integers and you need
to make an equation that equals 24 by adding, subtracting, multiplying and/or dividing. If you have multiple children who are close in age, they can race to come up with an answer the fastest.
Otherwise, have your sixth grader race against herself and try to beat her previous time. Or, she can play against you! In the branded game, you have the assurance that each set of numbers will have
a solution. If you use a deck of cards and pull 4 random numbers, it is possible that the solution will not exist. Your child will still learn a lot by trying.
If you have a deck of numbered cards, you can play numerous other games with them in sixth grade. Some lower or higher grade children might be able to join in, depending on the rules you decide on.
There is a basic game of war, in which you divide the cards evenly among two players, each player draws the top card and the one with the higher card wins both cards. The final winner is the person
with the most cards after both piles have been exhausted. Or you can add cards you win to the bottom of your pile and play until someone has all the cards. You can modify this game by drawing
multiple cards and arranging them to make the highest number you can. The player with the higher multi-digit number wins all the cards.
Adding/Subtracting Cards
You could draw two cards and add, subtract, multiply or divide to see whose answer is higher. You could also draw two cards and consider the lower of the two to be a numerator of a fraction, and see
whose fraction is higher than the other. Or, you can use black cards as positive numbers and red cards as negative numbers, then add, subtract, multiply or divide.
Statistics Practice
You can also use a deck of cards to practice statistics. Draw 7 or more cards and find the mean, median, mode and range of the numbers you drew. For an added challenge, practice basic operations and
then do statistics on the results. Draw one card at a time and decide whether to add, subtract, multiply or divide to try to get to the answer of 1. Once you reach 1, write down the number of cards
you used to do it. Repeat this several times, until you have enough data to find mean, median, mode and range. You can also have your child graph the results of each round.
Other 6th Grade Math Activities
Take your sixth grader to the store with you to calculate the sales tax you will be charged. Look for other real world opportunities to practice what he is learning. A variety of learning experiences
helps make your children well-rounded.
Include some field trip experiences in your sixth grade plan that highlight what they are learning in math. Visit a mini-golf course at a time when they are not very busy, and let your student
measure the angles needed for a hole in one. Take a trip to an amusement park and talk about speed and distance. Some amusement parks even sponsor math and science days for students. Schedule a visit
to a construction site or tall building to see some geometric engineering in action. Go to a baseball game and calculate some statistics for the players in fractions, decimals and percentages. Tour a
factory and ask questions about the math involved in completing the final product. Visit a farm and measure the size of the field or calculate the yield it can produce using ratios and rates.
Planning for 6th Grade Math
Create a homeschool schedule that puts core subjects like math at ideal times for your children to learn. Create a master plan for 6^th grade math by adding the units you will cover to your
homeschool calendar. Leave plenty of time for review. Follow that up with individual lesson plans for each day. If you have decided not to teach your child 6^th grade math, you will still want to be
involved to make sure she sticks to a schedule to get it all done. An alternative to making your own plan is to use a homeschool planner that does some of the work for you. If you are using the
Homeschool Hub, learn how to use the included planner.
How do I choose my homeschool math curriculum?
Choosing a homeschool math curriculum with a biblical worldview should be your top priority. Consider whether you need a homeschool curriculum package. A homeschool curriculum package includes
student and teacher editions you will need to teach all the subjects for one grade. Look for the appropriate level of mastery vs. spiral review. Decide whether you will want to use any online content
or video lessons, such as are included in the BJU Press Homeschool Hub. Consider other factors that will help you choose the best homeschool curriculum for your family.
• • • • •
Valerie is a wife and a mother to a very busy preschooler. In her free time she enjoys reading all kinds of books. She earned a B.S. in Biology from Bob Jones University, minoring in Mathematics, and
a Ph.D. in Molecular Genetics from Ohio State University. Valerie has 15 years of experience working in research laboratories and has coauthored 8 original research articles. She has also taught
several classes and laboratories at the high school and college levels. She currently works as a Data Analyst and a freelance writer.
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Path integral formulation of quantum mechanics
I'm a mathematics student with not much background in physics. I'm interested in learning about the path integral formulation of quantum mechanics. Can anyone suggest me some books on this topic with
minimum prerequisite in physics?
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user user774025
Sources for the path integral
You can read any standard source, so long as you supplement it with the text below. Here are a few which are good:
• Feynman and Hibbs
• Kleinert (although this is a bit long winded)
• An appendix to Polchinski's string theory vol I
• Mandelstam and Yourgrau
There are major flaws in other presentations, these are good ones. I explain the major omission below.
Completing standard presentations
In order for the discussion of the path integral to be complete, one must explain how non-commutativity arises. This is not trivial, because the integration variables in the path integral for bosonic
fields or particle paths are ordinary real valued variables, and these quantities cannot be non-commutative themselves.
Non-commutative quantities
The resolution of this non-paradox is that the path integral integrand is on matrix elements of operators, and the integral itself is reproducing the matrix multiplication. So it is only when you
integrate over all values at intermediate times that you get a noncommutative order-dependent answer. Importantly, when noncommuting operators appear in the action or in insertions, the order of
these operators is dependent on exactly how you discretize them--- whether you put the derivative parts as forward differences or backward differences or centered differences. These ambiguities are
extremely important, and they are discussed only in a handful of places (Negele/Orland Yourgrau/Mandelstam Feynman/Hibbs, Polchinski, Wikipedia) and hardly anywhere else.
I will give the classic example of this, which is enough to resolve the general case, assuming you are familiar with simple path integrals like the free particle. Consider the free particle Euclidean
$$ S= -\int {1\over 2} \dot{x}^2 $$
and consider the evaluation of the noncommuting product $x\dot{x}$. This can be discretized as
$$ x(t) {x(t+\epsilon) - x(t)\over \epsilon} $$
or as
$$ x(t+\epsilon) {x(t+\epsilon) - x(t)\over \epsilon}$$
The first represents $p(t)x(t)$ in this operator order, the second represents $x(t)p(t)$ in the other operator order, since the operator order is the time order. The difference of the second minus
the first is
$$ {(x(t+\epsilon) - x(t))^2\over \epsilon} $$
Which, for the fluctuating random walk path integral paths has a fluctuating limit which averages to 1 over any finite length interval, when $\epsilon$ goes to zero. This is the Euclidean canonical
commutation relation, the difference in the two operator orders gives 1.
For Brownian motion, this relation is called "Ito's lemma", not dX, but the square of dX is proportional to dt. While dX is fluctuating over positive and negative values with no correlation and with
a magnitude at any time of approximately $\sqrt{dt}$, dX^2 is fluctuating over positive values only, with an average size of dt and no correlations. This means that the typical Brownian path is
continuous but not differentiable (to prove continuity requires knowing that large dX fluctuations are exponentially suppressed--- continuity fails for Levy flights, although dX does scale to 0 with
Although discretization defines the order, not all properties of the discretization matter--- only which way the time derivative goes. You can understand the dependence intuitively as follows: the
value of the future position of a random walk is (ever so slightly) correlated with the current (infinite) instantaneous velocity, because if the instantaneous velocity is up, the future value is
going to be bigger, if down, smaller. Because the velocity is infinite however, this teensy correlation between the future value and the current velocity gives a finite correlator which turns out to
be constant in the continuum limit. Unlike the future value, the past value is completely uncorrelated with the current (forward) velocity, if you generate the random walk in the natural way going
forward in time step by step, by a Markov chain.
The time order of the operators is equal to their operator order in the path integral, from the way you slice the time to make the path integral. Forward differences are derivatives displaced
infinitesimally toward the future, past differences are displaced slightly toward the past. This is is important in the Lagrangian, when the Lagrangian involves non-commuting quantities. For example,
consider a particle in a magnetic field (in the correct Euclidean continuation):
$$ S = - \int {1\over 2} \dot{x}^2 + i e A(x) \cdot \dot{x} $$
The vector potential is a function of x, and it does not commute with the velocity $\dot{x}$. For this reason, Feynman and Hibbs and Negele and Orland carefully discretize this,
$$ S = - \int \dot{x}^2 + i e A(x) \cdot \dot{x}_c $$
Where the subscript c indicates infinitesimal centered difference (the average of the forward and backward difference). In this case, the two orders differ by the commutator, [A,p], which is $\nabla\
cdot A$, so that there is an order difference outside of certain gauges. The correct order is given by requiring gauge invariance, so that adding a gradiant $\nabla \alpha$ to A does nothing but a
local phase rotation by $\alpha(x)$.
$$ ie \int \nabla\alpha \dot{x}_c = ie \int {d\over dt} \alpha(x(t))$$
Where the centered differnece is picked out because only the centered difference obeys the chain rule. That this is true is familiar from the Heisenberg equation of motion:
$$ {d\over dt} F(x) = i[H,F] = {i\over 2} [p^2,F] = {i/2}(p[p,F] + [p,F]p) = {1\over 2}\dot{x} F'(x) + {1\over2} F'(x) \dot{x}$$
Where the derivative is a sum of both orders. This holds for quadratic Hamiltonians, the ones for which the path integral is most straightforward. The centered difference is the sum of both orders.
The fact that the chain rule only works for the centered difference means that people who do not understand the ordering ambiguities 100% (almost everybody) have a center fetishism, which leads them
to use centered differences all the time.
THe centered difference is not appropriate for certain things, like for the Dirac equation discretization, where it leads to "Fermion doubling". The "Wilson Fermions" are a modification of the
discretized Dirac action which basically amounts to saying "Don't use centered derivatives, dummy!"
Anyway, the order is important. Any presentation of the path integral which gives the Lagrangian for a particle in a magnetic field without specifying whether the time derivative is a forward
difference or a past difference, is no good. That's most discussions.
A good formalism for path integrals thinks of things on a fine lattice, and takes the limit of small lattice spacing at the end. Feynman always secretly thought this way (and often not at all
secretly, as in the case above of a particle in a magnetic field), as does everyone else who works with this stuff comfortably. Mathematicians don't like to think this way, because they don't like
the idea that the continuum still has got new surprises in the limit.
The other thing that is hardly ever explained properly (except for Negele/Orland, David John Candlin's Neuvo Cimento original article of 1956, and Berezin) is the Fermionic field path integral. This
is a separate discussion, the main point here is to understand sums over Fermionic coherent states.
Downvoted, this doesn't answer his question.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user Benjamin Horowitz
"Mathematicians don't like to think this way"? Obtaining the continuum path integral measure from a limit of lattice measures is utterly standard. See Glimm & Jaffe.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user user1504
Jaffe. Not a mathematician. That's a clear case of choosing your definitions to make your theorems easy to prove.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user user1504
He was the chairman of Harvard's math department, and past President of the AMS. This is something you should have googled before spouting off about. Ron, I really enjoy your posts. I think you're
doing something that desperately needs doing: attempting to write well and accessibly about path integrals. (Hell, I'd be pleased to read something longer from you.) But let's not pretend that the
ideas are completely unknown to the math community. ps. I'd be happy to see measure theory be first against the wall when the revolution comes.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user user1504
I came to physics.SE today planning to ask the question "In the path integral formulation of QM, where do we choose a quantization?" So this is exactly what I was looking for. Thanks!
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user David Speyer
Apparently someone wants to discuss your answer, see here.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user Manishearth
From the perspective of a mathematician, the books written by physicists may be somewhat imprenetrable. However, path integral techniques are used by mathematicians as well; I recommend the book
Barry Simon. "Functional Integration and Quantum Physics"
The author mainly focuses on the single-particle Schrödinger equation with an external electric field, as this case can be treated with mathematical rigor. It does not go into quantum field theory at
To get you started here are some lecture notes I like on the path integral: http://bohr.physics.berkeley.edu/classes/221/1011/notes/pathint.pdf (from the page http://bohr.physics.berkeley.edu/classes
/221/1011/221.html )
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user Steve B
These lecture notes are missing the crucial derivation of the Ito-Lemma/Canonical commutation relation. Without this, no discussion of path integrals is complete. I have seen even great physicists
confused regarding the commutation relation in the path integral, and the dependence on discretization is crucial for applications. This derivation is in Feynman/Hibbs Yourgrau/Mandelstam Negele/
Orland Polchinski and Wikipedia. It is found almost nowhere else. It is the sign of a good presentation: no commutation relations, no comprehension.
This post imported from StackExchange Physics at 2014-03-24 04:48 (UCT), posted by SE-user Ron Maimon
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January 2022
Science seeks the basic laws of nature. Mathematics seeks new theorems based on old ones. Engineering builds systems to solve human needs. The three disciplines are interdependent but distinct. It is
very rare for one person to simultaneously make fundamental contributions to all three, but Claude Shannon was a rare person.
Despite being the subject of the recent documentary The Bit Player – and his research work and philosophy having inspired my own career – Shannon is not exactly a household name. He never won a Nobel
Prize and was not a celebrity like Albert Einstein or Richard Feynman, neither before nor after his death in 2001. But more than 70 years ago, in a single groundbreaking paper, he laid the foundation
for the entire communication infrastructure underlying the modern information age.
Shannon was born in Gaylord, Michigan, in 1916, the son of a local businessman and a teacher. After receiving bachelor’s degrees in electrical engineering and mathematics from the University of
Michigan, he wrote a master’s thesis at the Massachusetts Institute of Technology that applied a mathematical discipline called Boolean algebra to the analysis and synthesis of switching circuits. It
was a transformative work, turning circuit design from an art to a science, and is now considered to be the starting point of digital circuit design.
Next, Shannon set her sights on an even bigger goal: communication.
Claude Shannon wrote a master’s thesis that launched digital circuit design, and a decade later he wrote his seminal paper on information theory, “A Mathematical Theory of Communication.”
Communication is one of the most basic human needs. From smoke signals to carrier pigeons to telephones and televisions, humans have always sought ways to communicate further, faster and more
reliably. But the engineering of communication systems has always been linked to the source and the specific physical medium. Shannon instead wondered, “Is there a grand unified theory for
communication?” In a 1939 letter to his mentor, Vannevar Bush, Shannon outlined some of his early ideas on “the fundamental properties of general systems for the transmission of intelligence.” After
working on the problem for a decade, Shannon finally published his masterpiece in 1948: “A Mathematical Theory of Communication.”
The core of his theory is a simple but very general model of communication: A transmitter encodes information into a signal, which is corrupted by noise and decoded by the receiver. Despite its
simplicity, Shannon’s model incorporates two key ideas: isolate the sources of information and noise from the communication system to be designed, and model both sources probabilistically. He
imagined that the information source generated one of many possible messages to communicate, each of which had a certain probability. Probabilistic noise added more randomness for the receiver to
Before Shannon, the problem of communication was seen primarily as a problem of deterministic signal reconstruction: how to transform a received signal, distorted by the physical medium, to
reconstruct the original as accurately as possible. Shannon’s genius lies in his observation that the key to communication is uncertainty. After all, if you knew in advance what I was going to tell
you in this column, what would be the point of writing it?
Schematic diagram of Shannon’s communication model, excerpted from his paper.
This single observation moved the communication problem from the physical to the abstract, allowing Shannon to model uncertainty using probability. This was a complete shock to the communication
engineers of the time.
In this framework of uncertainty and probability, Shannon set out to systematically determine the fundamental limit of communication. His answer is divided into three parts. The concept of the “bit”
of information, used by Shannon as the basic unit of uncertainty, plays a fundamental role in all three. A bit, which is short for “binary digit,” can be either a 1 or a 0, and Shannon’s paper is the
first to use the word (although he said mathematician John Tukey used it first in a memo).
First, Shannon devised a formula for the minimum number of bits per second to represent information, a number he called the entropy rate, H. This number quantifies the uncertainty involved in
determining which message the source will generate. The lower the entropy rate, the lower the uncertainty, and therefore the easier it is to compress the message into something shorter. For example,
texting at a rate of 100 English letters per minute means sending one of 26,100 possible messages each minute, each represented by a sequence of 100 letters. All of these possibilities could be
encoded in 470 bits, since 2470 ≈ 26100. If the sequences were equally likely, Shannon’s formula would say that the entropy rate is effectively 470 bits per minute. In reality, some sequences are
much more likely than others, and the entropy rate is much lower, allowing for more compression.
Second, he provided a formula for the maximum number of bits per second that can be reliably communicated in the face of noise, which he called the system capacity, C. This is the maximum rate at
which the receiver can resolve the uncertainty in the message. , which makes it the communication speed limit.
Finally, he showed that reliable communication of source information versus noise is possible if and only if H < C. Thus, information is like water: If the flow rate is less than the capacity of the
pipe, the current happens reliably.
Although it is a theory of communication, it is at the same time a theory of how information is produced and transferred: an information theory. That is why Shannon is considered today “the father of
information theory”.
His theorems led to some counterintuitive conclusions. Suppose you are talking in a very noisy place. What’s the best way to make sure your message gets through? Maybe repeat it many times? That’s
certainly anyone’s first instinct in a noisy restaurant, but it turns out it’s not very effective. Surely the more times you repeat yourself, the more reliable the communication will be. But you’ve
sacrificed speed for reliability. Shannon showed us that we can do much better. Repeating a message is an example of using a code to convey a message, and by using different and more sophisticated
codes, you can communicate quickly – up to the speed limit, C – while maintaining any degree of reliability.
Another unexpected conclusion that emerges from Shannon’s theory is that, whatever the nature of the information (a Shakespeare sonnet, a recording of Beethoven’s Fifth Symphony, or a Kurosawa film),
it is always more efficient to encode it into bits. before transmitting it. Thus, in a radio system, for example, although both the initial sound and the electromagnetic signal sent through the air
are analog waveforms, Shannon’s theorems imply that it is optimal to first digitize the sound wave into bits, and then map those bits in the electromagnetic wave. This surprising result is the
cornerstone of the modern digital information age, in which the bit reigns as the universal currency of information.
Shannon also had a playful side, which she often brought to her work. Here, he poses with a maze he built for an electronic mouse, named Theseus.
Shannon’s general theory of communication is so natural that it’s as if he discovered the communication laws of the universe, instead of inventing them. His theory is as fundamental as the physical
laws of nature. In that sense, he was a scientist.
Shannon invented new mathematics to describe the laws of communication. He introduced new ideas, such as the entropy rate of a probabilistic model, which have been applied in far-reaching
mathematical branches, such as ergodic theory, the study of the long-term behavior of dynamical systems. In that sense, Shannon was a mathematician.
But above all, Shannon was an engineer. His theory was motivated by practical engineering problems. And while it was esoteric to engineers of his day, Shannon’s theory has become the standard
framework on which all modern communication systems are based: optical, submarine, and even interplanetary. Personally, I have been fortunate to be part of a worldwide effort to apply and extend
Shannon’s theory to wireless communication, increasing communication speeds by two orders of magnitude over multiple generations of standards. In fact, the 5G standard currently being rolled out uses
not one, but two proven codes of practice to reach the Shannon speed limit.
Although Shannon died in 2001, her legacy lives on in the technology that makes up our modern world and in the devices she created, like this remote-controlled bus.
Shannon discovered the basis for all this more than 70 years ago. How did he do it? Relentlessly focusing on the essential feature of a problem and ignoring all other aspects. The simplicity of his
communication model is a good illustration of this style. He also knew how to focus on what is possible, rather than what is immediately practical.
Shannon’s work illustrates the true role of high-level science. When I started college, my advisor told me that the best job was to prune the tree of knowledge, instead of growing it. So I didn’t
know what to make of this message; I always thought my job as a researcher was to add my own twigs. But throughout my career, when I had the opportunity to apply this philosophy in my own work, I
began to understand it.
When Shannon began studying communication, engineers already had a large collection of techniques. It was his work of unification that pruned all these twigs of knowledge into a single charming and
coherent tree, which has borne fruit for generations of scientists, mathematicians, and engineers.
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In 1981, the cost of tuition at a certain private high school was $4790. By 1995, it had risen approximately 78%. What was the approximate cost in 1995? | HIX Tutor
In 1981, the cost of tuition at a certain private high school was $4790. By 1995, it had risen approximately 78%. What was the approximate cost in 1995?
Answer 1
Approximate cost in 1995 is #$8526.20#
#78%# of #$4790# is #78/100xx4790#
= #(78xx4790)/100#
= #373620/100#
= #$3736.20#
Hence rise is by #$3736.20# and hence
Approximate cost in 1995 is #$4790+$3736.20# or #$8526.20#
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Answer 2
To find the approximate cost in 1995, you would calculate 78% of $4790 and then add that amount to the original cost of$4790.
78% of $4790 =$3736.20 Adding this to the original cost: $4790 +$3736.20 = $8526.20
So, the approximate cost in 1995 was $8526.20.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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NCERT Solutions for Class 8 Maths Chapter 15 Introduction to Graphs
NCERT Solutions for Class 8 Maths Chapter 15 Introduction to Graphs| PDF Download
Chapter 15 Introduction to Graphs NCERT Solutions for Class 8 Maths is available on this page which is very helpful solving the difficult problems in a given in a exercise. Also, you can download PDF
of Chapter 15 Introduction to Graphs NCERT Solutions which can be used to recall various questions any time. These NCERT Solutions will help an individual to increase concentration and will tell you
about the difficulty of questions.
Class 8 Maths NCERT Solutions will help in developing your problem solving skills and be aware of the concepts. There are variety of concepts given in this Chapter 15 Class 9 Maths textbook that will
develop your necessary skill to solve more and more questions.
Page No: 236 Exercise 15.1 1. The following graph shows the temperature of a patient in a hospital, recorded every hour:
(a) What was the patient’s temperature at 1 p.m.? (b) When was the patient’s temperature 38.5° C? (c) The patient’s temperature was the same two times during the period given. What were these two
times? (d) What was the temperature at 1.30 p.m.? How did you arrive at your answer? (e) During which periods did the patients’ temperature showed an upward trend? Answer
(a) The patient’s temperature was 36.5o C at 1 p.m.
(b) The patient’s temperature was 38.5o C at 12 noon.
(c) The patient’s temperature was same at 1 p.m. and 2 p.m.
(d) The temperature at 1.30 p.m. is 36.5o C. The point between 1 p.m. and 2 p.m., x-axis is equidistant from the two points showing 1 p.m. and 2 p.m. So, it represents 01.30 p.m. Similarly, the point
on y-axis, between 36o C and 37o C will represent 36.5o C.
(e) The patient’s temperature showed an upward trend from 9 a.m. to 11 a.m., 11 a. m. to 12 noon and 2 p.m. to 3 p.m.
2. The following line graph shows the yearly sales figures for a manufacturing company.
(a) What were the sales in (i) 2002 (ii) 2006? (b) What were the sales in (i) 2003 (ii) 2005? (c) Compute the difference between the sales in 2002 and 2006. (d) In which year was there the greatest
difference between the sales as compared
to its previous year? Answer
(a) The sales in: (i) 2002 was Rs.4 crores and (ii) 2006 was Rs.8 crores.
(b) The sales in: (i) 2003 was Rs.7 crores (ii) 2005 was Rs.10 crores.
(c) The difference of sales in 2002 and 2006 = Rs.8 crores – Rs.4 crores = Rs.4 crores
(d) In the year 2005, there was the greatest difference between the sales as compared to its previous year, which is (Rs.10 crores – Rs.6 crores) = Rs.4 crores.
3. For an experiment in Botany, two different plants, plant A and plant B were grownunder similar laboratory conditions. Their heights were measured at the end of each week for 3 weeks. The results
are shown by the following graph. (a) How high was Plant A after (i) 2 weeks (ii) 3 weeks? (b) How high was Plant B after (i) 2 weeks (ii) 3 weeks? (c) How much did Plant A grow during the 3rd week?
(d) How much did Plant B grow from the end of the 2nd week to the end of the 3rd week? (e) During which week did Plant A grow most? (f) During which week did Plant B grow least? (g) Were the two
plants of the same height during any week shown here? Specify. Answer
(a) (i) The plant A was 7 cm high after 2 weeks and
(ii) after 3 weeks it was 9 cm high.
(b) (i) Plant B was also 7 cm high after 2 weeks and
(ii) after 3 weeks it was 10 cm high.
(c) Plant A grew = 9 cm – 7 cm = 2 cm during 3rd week.
(d) Plant B grew during end of the 2nd week to the end of the 3rd week = 10 cm – 7 cm = 3 cm.
(e) Plant A grew the highest during second week.
(f) Plant B grew the least during first week.
(g) At the end of the second week, plant A and B were of the same height.
4. The following graph shows the temperature forecast and the actual temperature for each day of a week. (a) On which days was the forecast temperature the same as the actual temperature? (b) What
was the maximum forecast temperature during the week? (c) What was the minimum actual temperature during the week? (d) On which day did the actual temperature differ the most from the forecast
temperature? Answer
(a) On Tuesday, Friday and Sunday, the forecast temperature was same as the actual temperature.
(b) The maximum forecast temperature was 35o C.
(c) The minimum actual temperature was 15o C.
(d) The actual temperature differed the most from the forecast temperature on Thursday.
5. Use the tables below to draw linear graphs.
│Days│8 │10 │5 │12 │
(a) The number of days a hill side city received snow in different years.
│Year │2003│2004│2005│2006│2007│
│No. of Men │12 │12.5│13 │13.2│13.5│
│No. of Women │11.3│11.9│13 │13.6│12.8│
(b) Population (in thousands) of men and women in a village in different years. Answer
6. A courier-person cycles from a town to a neighbouring suburban area to deliver a parcel to a merchant. His distance from the town at different times is shown by the following graph.
(a) What is the scale taken for the time axis? (b) How much time did the person take for the travel? (c) How far is the place of the merchant from the town? (d) Did the person stop on his way?
Explain. (e) During which period did he ride fastest? Answer
(a) 4 units = 1 hour.
(b) The person took 3.1/2 hours for the travel.
(c) It was 22 km far from the town.
(d) Yes, this has been indicated by the horizontal part of the graph. He stayed from 10 am to 10.30 am.
(e) He rode the fastest between 8 am and 9 am.
7. Can there be a time-temperature graph as follows? Justify your answer. Answer
(i) It is showing the increase in temperature.
(ii) It is showing the decrease in temperature.
(iii) The graph figure (iii) is not possible since temperature is increasing very rapidly which is not possible.
(iv) It is showing constant temperature.
Page No. 243 Exercise 15.2 1. Plot the following points on a graph sheet. Verify if they lie on a line (a) A(4,0), B(4,2), C(4,6), D(4, 2.5) (b) P(1,1), Q(2,2), R(3,3), S(4,4) (c) K(2, 3), L(5, 3), M
(5, 5), N(2, 5) Answer
(a) All points A, B, C and D lie on a vertical line.
(b) P, Q, R and S points also make a line. It verifies that these points lie on a line.
(c) These points do not lie in a straight line.
2. Draw the line passing through (2, 3) and (3, 2). Find the coordinates of the points at which this line meets the x-axis and y-axis.
The coordinates of the points at which this line meets the x-axis at (5, 0) and y-axis at (0, 5).
3. Write the coordinates of the vertices of each of these adjoining figures.
Vertices of figure OABC
O (0, 0), A (2, 0), B (2, 3) and C (0, 3)
Vertices of figure PQRS
P (4, 3), Q (6, 1), R (6, 5) and S (4, 7)
Vertices of figure LMK
L (7, 7), M (10, 8) and K (10, 5)
4. State whether True or False. Correct that are false. (i) A point whose x coordinate is zero and y-coordinate is non-zero will lie on the y-axis. (ii) A point whose y coordinate is zero and
x-coordinate is 5 will lie on y-axis. (iii) The coordinates of the origin are (0, 0). Answer
(i) True
(ii) False, it will lie on x-axis.
(iii) True
Page No. 247 Exercise 15.3 1. Draw the graphs for the following tables of values, with suitable scales on the axes. (a) Cost of apples
│No. of apples │1│2 │3 │4 │5 │
│Cost (in Rs.) │5│10│15│20│25│
(b) Distance travelled by a car
│Time (in hours) │6 a.m.│7 a.m.│8 a.m.│9 a.m.│
│Distance (in km) │40 │80 │120 │160 │
(i) How much distance did the car cover during the period 7.30 a.m. to 8 a.m?
(ii) What was the time when the car had covered a distance of 100 km since it’s start? (c) Interest on deposits for a year.
│Deposit(inRs.) │1000│2000│3000│4000│5000│
│Simple Interest (inRs.) │80 │160 │240 │320 │400 │
(i) Does the graph pass through the origin?
(ii) Use the graph to find the interest on Rs 2500 for a year.
(iii) To get an interest of Rs 280 per year, how much money should be deposited? Answer
(b) (i) The car covered 20 km distance.
(ii) It was 7.30 am, when it covered 100 km distance.
(c) (i) Yes, the graph passes through the origin.
(ii) Interest on Rs. 2500 is Rs. 200 for a year.
(iii) Rs. 3500 should be deposited for interest of Rs. 280.
2. Draw a graph for the following. (i)
│Side of Square(in cm) │2│3 │3.5 │5 │6 │
│Perimeter │8│12│14 │20│24│
│(in cm) │ │ │ │ │ │
Is it a linear graph? (ii)
│ Side of Square(in cm) │ 2 │ 3 │ 4 │ 5 │ 6 │
│ Area (in cm2) │ 4 │ 9 │ 16 │ 25 │ 36 │
Is it a linear graph? Answer
(i) Yes, it is a linear graph.
(ii) No, it is not a linear graph because the graph does not provide a straight line.
NCERT Solutions for Class 8 Maths Chapter 15 Introduction to Graphs
Chapter 15 NCERT Solutions will prove useful guide in making a student confident and developing you understanding of the chapter and make a student confident. The graphs are visual representations of
data collected.
• Pictorial and graphical representation are easier of understand.
• Bar graphs and pie graphs are used for comparing categories and parts respectively.
• A histogram is a bar graph that shows data in intervals.
Exercisewise NCERT Solutions for Class 8 Maths can be really helpful if anyone want to understand the detailed solutions and minimise the errors wherever possible. Studyrankers experts have prepared
accurate and detailed NCERT Solutions through which you can be able to solve more questions related to chapter.
NCERT Solutions for Class 8 Maths Chapters:
FAQ on Chapter 15 Introduction to Graphs
Why we should solve NCERT Solutions for Chapter 15 Introduction to Graphs Class 8?
Chapter 15 Introduction to Graphs Class 8 NCERT Solutions are very essential in revising the chapter properly and build your own answers for homework and get good marks in the examination.
Are the coordinates (4, 5) and (5, 4) the same? Why?
No, (4, 5) and (5, 4) are not the same. In (4, 5) the x-coordinate is 4 whereas in (5, 4) it is 5. The same is with the y-coordinate too. As a result, they are different and they represent the
different sets of points.
What do you mean by Line Graph?
A line graph displays data that change continuously over periods of time. This graph being wholly an unbroken line is called a linear graph.
What do you mean by Pie Graph?
A pie-graph is used to compare various parts of a whole using a circle.
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144 MHz Long Yagi
144 MHz Long Yagi
43 element long yagi for the low end of the 2m band, 98.5' (32.349 m) long portable antenna wuth a high gain 19 dbd
post 03 Feb 2024
Antenna for UHF VHF Links
Stacking Transformation
The ideal impedance transformation ratio are actualy ZL = 50 Ohm and ZI = 100 Ohm, but to get this ideal impedance transformation means that for TR, we must use a coaxial cabel with a Characteristic
impedance Zo = 70.71 Ohm Super facil antena vhf 1/4 onda
Antenna Ground plane (GP) has the property of receiving and transmitting in an omnidirectional way, that is to say everywhere equally Super J-Pole Collinear
The antenna J is very simple in construction and has good results T-match for vhf -uhf Antennas
The T-Match with its balanced feed point is one method of feeding a Balanced Dipole Antenna Yagi 11 Elements
After using the charts according to DL6WU, the following values have been encoded for the calculation
Antenna splitter per VHF-UHF per quattro ricevitori Chi dispone di una sola antenna e vuole collegarvi più di un ricevitore necessità di uno splitter
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How to calculate percentage - Canaltech
How to calculate percentage – Canaltech
3 min read
Percentage arithmetic can be used in math problems such as the three bases and proportionality as well as for calculating interest and discounts. But precisely because it involves mathematics, it may
seem to many people like a seven-headed beast. Next, learn how to calculate the percentage of a value.
To make it easier, we’ve broken down two simple ways to do percentage calculations: using your cell phone’s calculator (or any other physical or digital calculator) and Microsoft excel.
How do you calculate the percentage
Calculate the percentage on the calculator
It is very easy to do the percentage calculation on the calculator, whether it is physical, digital or your mobile application. This is because almost all of these options provide their own key for
this operation, which is represented by the “%” button. Just multiply the total value in percentage and then use the “%” button.
Here, we will calculate 5% of 200. First, enter the integer number you want to calculate a part of. Now, press the “X” key, to double the value, enter the percentage number and go to the equals (“=”)
button to show the result. Finally, complete the use of the percent key (%).
The calculation will look like this: 200 x 5 = 1000. Then just use the percentage button. In this example, the calculator will show that 5% of 200 equals 10.
When the result of multiplying the total value by the percentage value appears, click on the “%” icon to make the calculator display the desired percentage (screenshot: Caio Carvalho / Canaltech)
Calculate percentage in Excel
In Microsoft Excel, the process of calculating a percentage is quite simple. Enter the total amount in the first column of the table, and in the second, enter the percentage that should be
calculated. Then divide them using the asterisk (“*”) symbol.
In the example below, let’s calculate 23% of $6,500. In cell B2 we put $6500 and in C2 the percentage we want to find (23%). In the third column, in cell D2, we insert the equal symbol (“=”),
followed by the number of the initial cell (B2), the asterisk (“*”) and the number of the second cell (C2). At the end, press “Enter” (Windows) or “Back” (Mac).
It would look like this: “= B2 * C2” (without the quotes). In this example, 23% of $6,500 equals $1,495. You can perform the calculation using any Excel cell, either horizontally or vertically, as
long as you put it in the correct numbering for each line.
You can also calculate the percentage of any value using Excel spreadsheets (screenshot: Caio Carvalho/Canaltech)
And ready! Calculating the percentage on mobile and Microsoft Excel is very simple, which avoids the agony of many people who will do this calculation using other mathematical forms.
Source: genuinely
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How to Convert Months into Days: Quick Calculations
Table of Contents :
Converting months into days can often seem challenging due to the variation in days across different months. However, with a few quick calculations and methods, you can efficiently make these
conversions for various purposes, whether for planning events, calculating age, or understanding durations in contracts. In this blog post, we will explore how to convert months into days, discuss
the average days in a month, and provide practical tips and examples. Letβ s dive in! π
Understanding the Basics of Month-to-Day Conversion
When converting months into days, the first thing to note is that not all months have the same number of days. Here's a quick breakdown of the days in each month:
Month Days
January 31
February 28/29 (leap year)
March 31
April 30
May 31
June 30
July 31
August 31
September 30
October 31
November 30
December 31
Note: February typically has 28 days, but every four years, it has 29 days (leap year).
Average Days in a Month
For easier calculations, it's often useful to use an average value for the number of days in a month. The average can be calculated based on the total days in a year divided by the number of months:
[ \text{Average days per month} = \frac{365 \text{ days}}{12 \text{ months}} \approx 30.42 \text{ days} ]
Key Point: This means that when you are converting months to days for rough estimates, you can use 30 or 30.5 as a simple average.
Quick Calculation Methods
1. Using Fixed Days
For most practical scenarios, you can round the average days as follows:
• 30 days per month for simple calculations.
• 31 days for months like January, March, May, July, August, October, December.
• 30 days for April, June, September, November.
• 28 or 29 days for February.
To convert 3 months into days, you can do the following:
• If considering 30 days per month:
[ 3 \text{ months} \times 30 \text{ days} = 90 \text{ days} ]
2. Specific Month Calculation
If you know the specific months involved, sum the days according to the month:
• Example: Convert 2 months (January and February) into days:
[ 31 \text{ (January)} + 28 \text{ (February)} = 59 \text{ days} ]
• For a leap year, it would be 60 days.
3. Using a Conversion Formula
You can also apply a simple formula:
• General Formula:
[ \text{Total Days} = (\text{Number of Months}) \times (\text{Average Days}) ]
Using 30.42 days as the average:
• For example, converting 4 months:
[ 4 \text{ months} \times 30.42 \text{ days} \approx 121.68 \text{ days} \approx 122 \text{ days} ]
Real-Life Applications of Month to Day Conversion
Understanding how to convert months to days can be incredibly useful in various scenarios, including:
Project Management
When planning project timelines, converting durations from months to days can provide a more precise measurement of deadlines and milestones.
Financial Calculations
Calculating interest or financial commitments often requires converting months into days for accurate results.
Personal Planning
When organizing events or vacations, knowing the exact number of days can help in budgeting and scheduling.
Age Calculation
To find the exact age of a person in days, knowing how to convert their age in months to days can provide a precise figure.
Important Note: Always remember to check for leap years when dealing with February or when a specific year is mentioned in your calculations!
Converting months into days is a straightforward process once you understand the variations in month lengths and the average values you can use. Whether you need a quick estimate or a precise
calculation, the methods outlined above will help you make your conversions effortlessly. By mastering these techniques, you'll find it easier to manage your time and commitments effectively. Now
that you know how to convert months into days, put this knowledge into practice and simplify your planning today! π
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Leaving Certificate Examination 1966 Honours Applied Mathematics
26 Nov 2018
Leaving Certificate Examination 1966 Honours Applied Mathematics
Question 1
(a) Represent any two velocities $\vec{v_1}$ and $\vec{v_2}$ by a vector diagram, and illustrate the vectors $(\vec{v_1}+\vec{v_2})$ and $(\vec{v_1}-\vec{v_2})$.
(b) When the sun, S, is $30^\circ$ above the horizon, an aeroplane $A$ glides in for landing with the sun behind, moving with a speed of $80$ m.p.h. along a path sloping $15^\circ$ down from the
horizontal, as in the diagram. Find the speed of the plane’s shadow on level ground ($PQ$).
Question 2
Write a short note on linear momentum.
A uniform cube of wood of mass $10$ lbs. hangs vertically by a long light string fixed in the centre of one face of the cube. A bullet of mass half an ounce moving horizontally with velocity $\vec{v}
$ strikes the centre of another face in a direction perpendicular to the face, embeds itself in the wood and causes the combined mass to swing. If in the first swing the centre of gravity of the
combined mass rises $6$ inches, calculate the magnitude of $\vec{v}$.
Question 3
For a body moving in a straight line with constant acceleration $\vec{f}$, initial velocity $\vec{u}$ and velocity $\vec{v}$ at time $t$, show that $v=u+ft$ and $a=ut+\frac{1}{2}ft^2$.
A cage travels down a vertical mine-shaft $750$ yards deep in $45$ seconds, starting from rest. For the first quarter of the distance only the velocity undergoes a constant acceleration, and for the
last quarter of the distance the velocity undergoes a constant retardation, the acceleration and retardation having equal magnitudes. If the velocity, $\vec{v_1}$, is uniform while traversing the
central portion of the shaft, find the magnitude of $\vec{v_1}$.
Question 4
The co-ordinates $(x,y)$ of a moving particle at any time $t$ are given by:
$x=2\cos\frac{t}{2}$, $y=2\sin\frac{t}{2}$.
Deduce that its motion is circular.
A particle is moving along the circumference of a circle with constant speed. Show that that components of the motion of the particle along any two perpendicular diameters are Simple Harmonic
Question 5
A particle is projected horizontally from a height with velocity $\vec{v}$ and falls under gravity. If $(x,y)$ are the co-ordinates of the particle at any time, and $(0,0)$ the co-ordinates of the
point of projection, show that $y=-\frac{gx^2}{2v^2}$ us the equation of its path, the x-axis being horizontal and the y-axis vertical.
A marble rolls off the top of a stairway horizontally and in a direction perpendicular to the edge of each step, with a speed of $20$ feet per second. Each step is $6$ in. high and $8$ in. wide.
Which is the first step struck by the marble (assuming the stairway is sufficiently high)?
Question 6
Masses of $1$ lb., $2$ lbs., $3$ lbs., and $4$ lbs. are placed at points the co-ordinate of which are $(4,1)$, $(-3,2)$, $(2,-3)$, and $(-1,4)$ respectively. Locate the centre of gravity (or the
centre of mass) of the system.
Two discs of radii $r$ and $\frac{r}{2}$ are placed flat on a horizontal plane with a distance $d>2r$ between their centres. If $X$ is their centre of gravity and $O$ the centre of gravity of the
large disc calculate the length of $OX$.
A square hole is punched out of a circular disc (radius $\alpha$), a diagonal of the square being a radius of the circle. Show that the centre of gravity of the remainder of the disc is at a distance
$\frac{\alpha}{4\pi-2}$ from the centre of the circle.
Question 7
Explain “normal reaction” and “angle of friction”.
Deduce $\theta \leq \lambda$ where $\lambda$ is the angle of friction and $\theta$ the angle between the normal reaction and the reaction (resultant reaction).
A uniform rod $AB$ $4$ feet long leans against the smooth edge of a table three feet in height with one end of the rod on a rough horizontal floor (as in diagram). If the rod is one the point of
slipping when inclined at an angle of $60^\circ$ to the horizontal, find the coefficient of friction.
Question 8
Two particles $P$ and $Q$ move with velocities $\vec{v_1}$ and $\vec{v_2}$ respectively. Use diagrams to illustrate clearly the velocity of $P$ relative to $Q$ when $\vec{v_1}$ and $\vec{v_2}$ are
(i) parallel and in the same direction,
(ii) parallel and in opposite directions,
(iii) non-parallel.
A steamer going north-east at $14$ knots observes a cruiser which is $10$ miles south-east from the steamer and which is travelling in a direction $30^\circ$ east of north at $25$ knots.
Illustrate by diagram the cruiser’s course as i appears to the steamer. Find the velocity of the cruiser relative to the steamer, in magnitude and direction, and find also the shortest distance
between the two ships.
Question 9
Establish an expression for the pressure at a depth $d$ in a liquid of density $\rho$ at rest under gravity. Hence show that the resultant thrust on a plane surface immersed vertically in the liquid
is the product of the area of the plane surface and the depth of its centre of gravity.
Find the resultant horizontal thrust on a rectangular lockgate $20$ feet wide when the water stands at a depth of $8$ feet on one side and $4$ feet on the other.
(1 cu. ft. of water weighs $62\frac{1}{2}$ lbs.).
State Examinations Commission (2023). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2023). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.
The EU Directive 2003/98/EC on the re-use of public sector information, its amendment EU Directive 2013/37/EC, its transposed Irish Statutory Instruments S.I. No. 279/2005, S.I No. 103/2008, and S.I.
No. 525/2015, and related Circulars issued by the Department of Finance (Circular 32/05), and Department of Public Expenditure and Reform (Circular 16/15 and Circular 12/16).
Note. Circular 12/2016: Licence for Re-Use of Public Sector Information adopts CC-BY as the standard PSI licence, and notes that the open standard licence identified in this Circular supersedes PSI
General Licence No: 2005/08/01.
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GCD and LCM
See also: numbers, prime numbers
Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of two positive integers is the largest positive integer that divides both numbers without remainder. It is useful for reducing fractions
to be in its lowest terms. See below on methods to find GCD.
Least Common Multiple (LCM) of two integers is the smallest integer that is a multiple of both numbers. See below on methods to find LCM.
You can find the GCD and LCM of two or three integers using the calculator below.
How to find Greatest Common Divisor (GCD)
There are few ways to find greatest common divisor. Below are some of them.
For example, let's try to find the HCM of 24 and 60.
Prime Factorization method
Using this method, first find the prime factorization of each number. Check the prime factors page to learn how to find prime factors of an integer.
24 = 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
Then we find the common prime factors of these two numbers.
24 = 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
The common prime factors are 2, 2 and 3. The greatest common divisor of 24 and 60 is the product of these common prime factors, i.e. 2 × 2 × 3 = 12.
Division by primes
First divide all the numbers by the smallest prime that can divide all of them. The smallest prime that divides 24 or 60 is 2.
Continue the steps until we can't find any prime number that can divide all the number on the right side.
The GCD is 2 × 2 × 3 = 12.
Euclidean Algorithm
This algorithm finds GCD by performing repeated division starting from the two numbers we want to find the GCD of until we get a remainder of 0.
For our example, 24 and 60, below are the steps to find GCD using Euclid's algorithm.
• Divide the larger number by the small one. In this case we divide 60 by 24 to get a quotient of 2 and remainder of 12.
• Next we divide the smaller number (i.e. 24) by the remainder from the last division (i.e. 12). So 24 divide by 12, we get a quotient of 2 and remainder of 0.
• Since we already get a remainder of zero, the last number that we used to divide is the GCD, i.e 12.
Let's look at another example, find GCD of 40 and 64.
• 64 ÷ 40 = 1 with a remainder of 24
• 40 ÷ 24 = 1 with a remainder of 16
• 24 ÷ 16 = 1 with a remainder of 8
• 16 ÷ 8 = 2 with a remainder of 0.
We stop here since we've already got a remainder of 0. The last number we used to divide is 8 so the GCD of 40 and 64 is 8
How to find Least Common Multiple (LCM)
Some methods to find least common multiple are
For example, let's try to find the LCM of 24 and 60.
Prime Factorization method
Using this method, first find the prime factorization of each number and write it in index form. Check the prime factors page to learn how to find prime factors of an integer.
24 = 2^3 × 3
60 = 2^2 × 3 × 5
The least common multiple is the product of the each prime factors with the greatest power. So for the above example, the LCM is 2^3 × 3 × 5 = 120.
Division by primes
First divide all the numbers by the smallest prime that can divide any of them. The smallest prime that divides 24 or 60 is 2.
Continue the steps until we have all prime numbers on the left side and at the bottom.
The LCM is 2 × 2 × 3 × 2 × 5 = 120.
If we know the greatest common divisor (GCD) of integers a and b, we can calculate the LCM using the following formula.
Using back the same example above, we can find the LCM of 24 and 60 as follows.
Of course, you can use this formula to find GCD of two integers if you already know the LCM.
Confused and have questions? We’ve got answers. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field.
See also: numbers, prime numbers
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Counting Passengers || CodeTo.win
Problem Setter: safayat jakir rifat
In a dreamland, there is a road, which have some lanes. If there is n lanes in that road, they are marked by 1 to n. i-th lane is wider than (i-1)-th lane, so cars of i-th lane contain one more wheel
from the previous lane. Every i-th lane can afford cars which have lowest number of wheels and number of wheels in cars are distinct. Car which have more wheels is bigger in size so it can take more
passengers. Every lane contain one more car than the previous lane and a car can hold double number of passengers form its wheels.
Note: Minimum number of wheels in a car is 2.
Input Format
The only line of the input file contain n and x(1<=n,x<= 50000) denote number of lanes of that road and number of car in first lane.
Output Format
The only line of output contains the number of passengers.
Language Time Memory
GNU C 11 1s 512MB
GNU C++ 14 1s 512MB
GNU C++ 11 1s 512MB
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Write a sample research question where the chi-square test of
independence would be the most appropriate...
Write a sample research question where the chi-square test of independence would be the most appropriate...
Write a sample research question where the chi-square test of independence would be the most appropriate statistical technique to use?
Sample research question where the chi - square test of independance would be the most approprite statistical technique to use:
Research question:
Is the Party Affiliation and Opinion on Tax Reform are independent?
Here, the chi-square test of indepence is most appropriate bcause:
The variables are categorical.
The Population parameters are unknown,. So, the test should be a non-paramtric test.
H0: In thepopulation, the two categorical variables are indpendent
HA:In the population, the two categorical variablesare dependent.
The Observed Data is given in the form of Contigency Table as follows:
│ │Favor│Indifferent │Oppose│Total│
│Democrat │ │ │ │ │
│Republican │ │ │ │ │
│Total │ │ │ │ │
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How Do You Calculate A Trifecta? - Winners Wire
A trifecta is a wager in which the bettor must select the top three finishers in a race in the exact order of finish. It is becoming an increasingly popular bet due to the potential for huge payouts.
How to Calculate a Trifecta Payout
Calculating a trifecta payout is relatively simple. The payout is determined by the number of horses you have selected to finish first, second, and third and the amount of money that has been bet.
The formula looks like this:
• Payout = (Number of Combinations x Bet Amount) / (Number of Winners)
For example, if you are betting a $1 trifecta with three horses, and the horses you have chosen finish first, second, and third, then the payout would be calculated as follows:
The payout in this case would be $3.
Cashing Out Your Trifecta
Once you have calculated the payout for your trifecta, you can cash out your winnings at the track or at an off-track betting facility. The payout will be in the form of a voucher that can be
redeemed for cash or a bank deposit.
Calculating the Cost of a Trifecta Bet
When calculating the cost of a trifecta bet, you must first determine how many combinations you are betting. For example, if you are betting a $1 trifecta on three horses, then the cost of the bet
would be the following:
• Cost = Number of Combinations x Bet Amount
In this case, the cost would be (3 x $1) = $3.
Odds of Winning a Trifecta
The odds of winning a trifecta are dependent on the number of horses in the race and the number of combinations you are betting. Generally speaking, the more horses in the race and the more
combinations you have bet, the lower the odds of winning.
Calculating the Exacta and Superfecta
In addition to the trifecta, there are two other wager types that are similar but with different payouts. The exacta requires you to pick the first two finishers in the exact order and the superfecta
requires you to pick the first four finishers in the exact order. The formulas for calculating the payouts for these two wagers are the same as the trifecta.
Calculating the Boxed Trifecta
A boxed trifecta allows you to select the top three finishers in a race without having to pick the exact order. This type of wager is more expensive than the straight trifecta but it increases your
chances of winning. The formula for calculating the cost of a boxed trifecta is as follows:
• Cost = Number of Combinations x Bet Amount x Number of Horses Selected
For example, if you are betting a $1 trifecta on three horses, then the cost of the bet would be the following:
Calculating the Key Trifecta
A key trifecta is similar to a boxed trifecta but with a twist. In a key trifecta, you must select one horse to finish in first place and then select the other two finishers in any order. The formula
for calculating the cost of a key trifecta is as follows:
• Cost = Number of Combinations x Bet Amount
For example, if you are betting a $1 trifecta on three horses, then the cost of the bet would be the following:
Calculating the Pick Three/Pick Six
The pick three and pick six are two popular wagers in which the bettor must select the first three or six finishers in a race in the exact order. The cost of the bet is determined by the number of
horses in the race and the number of combinations you are betting. The formula for calculating the cost of a pick three or pick six is as follows:
• Cost = Number of Combinations x Bet Amount x Number of Horses Selected
For example, if you are betting a $2 pick three on three horses, then the cost of the bet would be the following:
Calculating a trifecta payout is relatively simple. The payout is determined by the number of horses you have selected to finish first, second, and third and the amount of money that has been bet.
The cost of the bet is determined by the number of horses in the race and the number of combinations you are betting. Once you have calculated the payout for your trifecta, you can cash out your
winnings at the track or at an off-track betting facility.
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Auflistung Workshops 2008 nach Erscheinungsdatum
[OWR-2008-37] Workshop Report 2008,37 (2008) - (17 Aug - 23 Aug 2008)
The theory of C*-algebras plays a major role in many areas of modern mathematics, like Non-commutative Geometry, Dynamical Systems, Harmonic Analysis, and Topology, to name a few. The aim of the
conference “C*-algebras” ...
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Gaussian elimination
Gaussian elimination is direct method of solving systems of linear equations. Recall that a system of linear equations can be expressed as a simple equation Ax = b, where A is an nxn square matrix,
and the given b, and the unknown x are both vectors of size n (nx1 matrices).
Gaussian elimination works by turning to zero (eliminating) every element in the ith column below the ith row of A. The elimination step is done by adding a multiple of the ith row of A to the i+kth
row of A.
If A is an nxn matrix, the time complexity of Gaussian elimination is O(n^3). This makes it an impractical algorithm for a lot of scientific applications, where the matrix may be sparse (ie composed
mostly of zeroes), but n can be in the order of millions.
A potentially cheaper way of solving systems of linear equations is using the so called "iterative methods."
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Marmousi model
Smoothing the Marmousi model for Gaussian-packet migrations
This is an example how to fit the gridded velocities by a smooth model, without interfaces, suitable for ray tracing. The models with interfaces will require further study.
We take the gridded velocities of the "Marmousi model and dataset" and build the smooth velocity model suitable for ray tracing.
Notes on history file ' mar-vel.h'
History file 'mar-vel.h' is not designed to be executed with data of subdirectory 'mar' of package DATA. It requires file 'velocity.h@' of the "Marmousi model and dataset", containing the gridded
velocities. Since file 'velocity.h@' is large, history file 'mar-vel.h' has been used to create file 'mar-vel.dat' used by history files 'mar-inv.h' and 'mar-test.h'.
Data and model parametrization
Velocities in the Marmousi model are discretized on the grid of 751*2301 points with spacing 4m*4m. This is too large number of data for inversion. Since the bicubic B-splines in the model are to be
spaced much more sparsely, the set of slownesses inside a cell small with respect to the B-spline cell act nearly like a single average slowness.
We choose the data grid of 76*116=8816 points with spacing 40m*80m and the B-spline grid of 16*24=384 points with spacing 200m*400m.
Notes on history file ' mar-inv.h'
Choosing the maximum Lyapunov exponent
The length of synthetic seismograms is Tmax=2.9s. This means that the maximum sum of travel times from the source and receiver is limited by Tmax,
For average Lyapunov exponent lambda, the maximum number of arrivals from a source times the number of arrivals from a receiver may be roughly approximated by
exp(lambda*TS) * exp(lambda*TR) . (2)
If we choose
lambda .LE. 2/Tmax = 0.690 /s , (3)
the product of the numbers of arrivals from a source and a receiver probably does not exceed exp(2), i.e., probably does not exceed 7.
Choosing the maximum Sobolev norm of the model
We assume in equation (51) of Klimes(1999) at least one shift of -ln2 for a source and one for a receiver. We thus approximate the Lyapunov exponent by
lambda = average{sqrt[neg(V*v)]} - 2 ln2 / Tmax , (4)
average{sqrt(neg[V*v)]} .LE. 2 * (1+ln2) / Tmax , (5)
where neg(f)=(f-|f|)/2 is the negative part of f, V is the second velocity derivative perpendicular to a ray and v is the velocity. We make several other rough approximations,
average[sqrt(|V*v|)] .LE. 4 * (1+ln2) / Tmax , (6)
average[(V*v)^2] .LE. [ 4 * (1+ln2) / Tmax ]^4 , (7)
average[(U*v^3)^2] .LE. [ 4 * (1+ln2) / Tmax ]^4 , (8)
where U is the second slowness derivative perpendicular to a ray.
average(U^2) .LE. [ 4 * (1+ln2) / Tmax ]^4 / average(v)^6 , (9)
||u||^2 * 3/8 .LE. [ 4 * (1+ln2) / Tmax ]^4 / average(v)^6 , (10)
||u|| .LE. sqrt(8/3) * [ 4 * (1+ln2) / Tmax ]^2 / average(v)^3 , (11)
where ||u|| is the 2-D Sobolev norm given by file 'sob22.dat',
||u|| = sqrt[(u[,11])^2 +(u[,22])^2 +(2/3)u[,11]u[,22] +(4/3)u[,12])^2] . (11a)
For average(v)=3000m/s, we wish the maximum Sobolev norm of the model
||u||[max] = 0.330E-9s/m^3 . (12)
Inversion and smoothing
Objective function is taken in the form of
y = [|u-u[0]|/(given slowness deviation)]^2 + (SOBMUL*||u||)^2 , (13)
where |u-u[0]| is the standard slowness deviation of the model from gridded slownesses and ||u|| is the 2-D Sobolev norm given by file 'sob22.dat'. We specify a unit given slowness deviation,
(given slowness deviation)=1s/m . (13a)
(a) First iteration:
We first perform inversion with SOBMUL=0 (i.e., without smoothing) and obtain the standard slowness deviation of the model
|u-u[0]| = 0.290E-4 s/m (14)
stored in file 'mar-ud1.out'. Note that the standard slowness deviation from the densely sampled grid 4m*4m is larger, 0.340E-4s/m. The Sobolev norm of the model smoothed just by the projection onto
the B-splines is large in comparison with (12),
||u|| = 3.735E-9 s/m^3 , (15)
see file 'mar-un1.out'. The value of the objective function is
(b) Second iteration:
We choose
SOBMUL = |u-u[0]| / (given slowness deviation) / ||u||[max] , (17)
where we specified (given slowness deviation)=1s/m. Then
SOBMUL = 0.290E-4 / 0.330E-9 s/m^3 = 87878 m^3/s (18)
and we enter
for the second iteration. After the inversion, we obtain the standard slowness deviation of
|u-u[0]| = 0.408E-4 s/m (20)
stored in file 'mar-ud1.out', and the Sobolev norm
||u|| = 0.134E-9 s/m^3 , (21)
stored in file 'mar-un1.out'. The value of the objective function is
y = 1.66E-9 + 0.145E-9 . (22)
(c) Third iteration:
We choose
SOBMUL = SOBMUL * ||u|| / ||u||[max] , (23)
SOBMUL = 90000 m^3/s * 0.1336E-9 s/m^3 / 0.330E-9 s/m^3 , (24)
which is approximately
SOBMUL = 36000 m^3/s . (25)
After the inversion, we obtain the standard slowness deviation of
|u-u[0]| = 0.377E-4 s/m (26)
stored in file 'mar-ud1.out', and the Sobolev norm
||u|| = 0.328E-9 s/m^3 , (27)
stored in file 'mar-un1.out'. The value of the objective function is
y = 1.42E-9 + 0.139E-9 . (28)
(d) Subsequent iterations:
The iterations may be repeated until we arrive at the desired value of ||u||. Then, we may repeat the inversion with sparser or denser B-spline grid. If the B-spline grid is sufficiently dense, a
denser B-spline grid does not influence the resulting model. If the resulting model does not change with a sparser grid, we may switch to the sparser B-spline grid. We performed the following tests
of the B-spline grid density:
│B-spline grid │200m*400m │250m*400m │300m*400m │375m*400m │
│|u-u[0]| / (s/m) │0.376725E-04│0.377243E-04│0.379596E-04│0.384356E-04│
│||u|| / (s/m^3) │0.328061E-09│0.325954E-09│0.316512E-09│0.296923E-09│
│B-spline grid │200m*400m │200m*460m │200m*511m │200m*575m │
│|u-u[0]| / (s/m) │0.376725E-04│0.377246E-04│0.377508E-04│0.378320E-04│
│||u|| / (s/m^3) │0.328061E-09│0.326122E-09│0.325090E-09│0.322291E-09│
We might use a B-spline grid up to 1.25 times sparser in both directions than 200m*400m. Coarser grids than 250m*500m are smoothed by insufficiently sampled splines. Since the spline smoothing is
much worse than smoothing with the Sobolev scalar products, we should avoid sparser B-spline grids than 250m*500m for the smoothed Marmousi model.
We stop the iterations, rename the resulting model 'mar-mod.out' to 'mar-mod.dat' and proceed to more detailed tests of the smoothed model.
Notes on history file ' mar-test.h'
Statistical properties of the model
We discretize the velocity and its first and second derivatives in the smoothed model 'mar-mod.dat'. To test the code, we calculate the standard slowness deviation once more from the gridded velocity
and obtain the value of
|u-u[0]|[grid 40*80] = 0.377E-4 s/m (29)
stored in file 'mar-ud2.out'. Similarly, we calculate the Sobolev norm of the slowness from the gridded velocity and its derivatives and obtain the value of
||u||[grid 40*80] = 0.330E-9 s/m^3 , (30)
stored in file 'mar-un2.out'. Note that for denser grid 8m*16m we obtain
|u-u[0]|[grid 8*16] = 0.413E-4 s/m (31)
||u||[grid 8*16] = 0.328E-9 s/m^3 . (32)
The difference between ||u||[grid 8*16] and the value of stored ||u|| in file 'mar-un1.out' is 0.0001E-9 s/m^3. For original dense grid 4m*4m we have
|u-u[0]|[grid 4*4] = 0.418E-4 s/m . (33)
The standard relative slowness deviation, which equals the standard relative travel-time deviation for very short rays, is
|u/u[0]-1|[grid 40*80] = 0.130 , (34)
see file 'mar-ur2.out', and
|u/u[0]-1|[grid 4*4] = 0.145 . (35)
We may also plot gridded slowness 'mar-u0.ps' having been fit, and slowness 'mar-u.ps' in the smoothed model.
Average Lyapunov exponent for the model
Average Lyapunov exponent for the 2-D model, calculated by program 'modle2d.for', has the value of
and is stored in file 'mar-lem.out' in the form of line in order to be plotted by program 'pictures.for' into the graph of the angular dependence of the directional Lyapunov exponents 'mar-led.out',
see figure 'mar-le.ps'. The frame of the figure is stored in file 'mar-lef.out'. The Lyapunov exponent is averaged over the angles with a uniform weight, in accordance with the isotropic Sobolev
scalar products used for smoothing. Although the Lyapunov exponent is little bit better than we formerly required, it is not as small as to decrease the smoothness of the model without testing ray
tracing and the widths of Gaussian beams in the model. The product of the numbers of arrivals from a source and a receiver thus probably does not exceed exp(.519*2.9)=exp(1.5), i.e., probably does
not exceed 5.
Ray tracing and widths of Gaussian beams
The source and receiver points in the Marmousi data set are equally spaced with the interval of 25m. The leftmost point at x[2]=425m corresponds to receiver 1 of shot 1. The rightmost point at x[2]=
8975m corresponds to shot 240. There are thus 343 different source and receiver points from 425m to 8975m. We index them by integers from 1 to 343. File 'mar-srp.dat' contains position of formal
point 0 and the interval vector between the points. Program 'srp.for' may be used to generate positions of points 1 to 343.
Ray tracing and the calculation of the gridded numbers of arrivals and the widths of Gaussian beams for a specified source point is performed by history file 'mar-crt.h'. We selected 18 source
positions 1, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 335. The results for all specified individual sources are accumulated in common grid files (data cubes). The
calculation of rays and travel times is terminated at the travel time of 2.3s. For greater travel times from a source or a receiver, the sum of both travel times would exceed the maximum travel time
of 2.9s for the measurement configuration under consideration. The numbers of arrivals are small and the travel-time triplications occur only at the ends of rays terminated at travel time 2.3s. The
numbers of arrivals are thus fine for calculations, as we expected from the mean Lyapunov exponent for the model.
Eighteen rows in the figure below (3.5kB) correspond to source positions 1, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 335. Travel times up to 2.3s are calculated.
Areas with no travel times are grey. Areas with 1 arrival are yellow, areas with 3 arrivals are green.
We now calculate the standard halfwidths of Gaussian beams to check the suitability of the model for Gaussian-beam and Gaussian-packet calculations.
Standard halfwidth S of Gaussian beam of crossection
multiplied by the square root of omega=2*pi*f, is interpolated between rays and displayed in colours,
W = S * sqrt(2*pi*f) . (38)
The yellow colour corresponds to W[yellow]=0km/s^0.5, the green colour to
W[green]= 3000 m/s^0.5 . (39)
Trapezoidal frequency filter applied to seismograms is determined by frequencies 0Hz, 10Hz, 35Hz and 55Hz. Gaussian beam halfwidth corresponding to the green colour is at lower corner frequency of
S[green] = 378 m , S[cyan] = 756 m ; (40)
at central frequency of 25 Hz
S[green] = 239 m , S[cyan] = 478 m ; (41)
and at the upper corner frequency of 35 Hz
S[green] = 202 m , S[cyan] = 404 m . (42)
Since the halfwidth should be smaller than the B-spline intervals, acceptable halfwidth are between yellow and green. The values between the cyan and red are awfully bad (2 to 5 times greater values
than for the green).
Three columns in the figure below (331kB) correspond to three different initial conditions for Gaussian beams, constant along the surface of the model. Eighteen rows correspond to source positions 1,
20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 335.
Unlike all previous tests, the standard halfwidths of Gaussian beams indicate that the model is at the edge of applicability of Gaussian beams and Gaussian packets, because of too small frequencies
for the high-frequency asymptotic methods. We also see that the initial conditions for the Gaussian beams should not be constant along the model surface and should be carefully optimized. Until doing
that, we cannot be sure whether the smoothed model is sufficiently smooth or must be smoothed more.
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Design Concepts for the REopt Module
At a high level each REopt model consists of four major components:
The REopt Model is built via the build_reopt! method. However, the run_reopt method includes build_reopt! within it so typically a user does not need to directly call build_reopt! (unless they wish
to modify the model before solving it, eg. by adding a constraint).
run_reopt is the main interface for users.
The max_kw input value for any technology is considered to be the maximum additional capacity that may be installed beyond the existing_kw. Note also that the Site space constraints (roof_squarefeet
and land_acres) for PV technologies can be less than the provided max_kw value.
The min_kw input value for any technology sets the lower bound on the capacity. If min_kw is non-zero then the model will be forced to choose at least that system size. The min_kw value is set equal
to the existing_kw value in the Business As Usual scenario.
In order to calculate the Net Present Value of the optimal solution, as well as other baseline metrics, one can optionally run the Business As Usual (BAU) scenario. When an array of JuMP.Models is
provided to run_reopt the BAU scenario is also run. For example:
m1 = Model(optimizer_with_attributes(Xpress.Optimizer, "OUTPUTLOG" => 0))
m2 = Model(optimizer_with_attributes(Xpress.Optimizer, "OUTPUTLOG" => 0))
results = run_reopt([m1,m2], "./scenarios/pv_storage.json")
The BAU scenario is created by modifying the REoptInputs (created from the user's Scenario). In the BAU scenario we have the following assumptions:
• Each existing technology has zero capital cost, but does have operations and maintenance costs.
• The ElectricTariff, Financial, Site, ElectricLoad, and ElectricUtility inputs are the same as the optimal case.
• The min_kw and max_kw values are set to the existing_kw value.
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Shear Design of Reinforced Concrete Beams - CivilEngineeringBible.com
Shear Design of Reinforced Concrete Beams
Courses > Reinforced Concrete Design > Design of Concrete Members > Shear Design of Reinforced Concrete Beams
Design shear stresses generally act along planes perpendicular to the longitudinal axis of a member. These stresses, either alone or in combination with normal stresses, create principal tensile
stresses that can lead to sudden failure. ACI design criteria alleviate the risk of such failures by applying more conservative theories and resistance factors than are used for more ductile modes of
In the following sections, the ACI 318 shear provisions are summarized and illustrated for some of the more commonly encountered design cases.
Shear Strength of Slender Reinforced Concrete Beams
The basic strength requirement for shear design is
$V_u \leq \phi V_n$
$V_u \leq \phi (V_c+V_s)$
Vu is the shear caused by the factored loads, Vn is the nominal shear strength of the member, Vc is the contribution of concrete to shear resistance, Vs is the contribution of shear reinforcement to
shear resistance, and φ is the capacity reduction factor, which is 0.75.
The following step-by-step guide summarizes the ACI 318 shear design provisions that apply to the most commonly encountered case, in which the slender reinforced concrete beam is subject to the
following restrictions.
• The span-to-depth ratio is greater than or equal to four.
• The beam is supported and loaded such that the reaction induces compression in the region near the support.
• The beam is transversely loaded by static forces so that the torsion and axial forces caused are negligible.
• Only vertical leg stirrups are used, and their total cross-sectional area is Av.
• The simpler expression for the contribution of the concrete to shear resistance is used.
What is the step-by-step procedure for shear design of concrete members?
1- compute design shear force, Vu, at appropriate location
2- compute Vc using simple equation ACI 11-3:
$V_c=2\lambda \sqrt{f'_c}b_wd$
where l = 1 for normal weight concrete; 0.85 sand-lightweight concrete ; 0.75 for all-lightweight concrete.
3- Check
$V_u\leq\frac{\phi V_c}{2}$
3-1- if Yes â–º no stirrups required
3-2- if No â–º
$\alpha \geq\left\{\begin{matrix} 0.75\sqrt{f'_c}\\ 50 \end{matrix}\right.$
4- Check
$\frac{\phi V_c}{2}<V_u<\phi V_c$
4-1- if Yes â–º STOP and use min. stirrups required:
$s\leq \left\{\begin{matrix} d/2\\ 24\;in\\ \frac{A_vf_y}{\alpha b_w} \end{matrix}\right.$
4-2- if No â–º
$V_s=V_u/\phi - V_c$
5- Check
$V_s\geq 8\sqrt{f'_c}b_wd$
5-1- if Yes â–º web crushes: STOP and redesign beam
5-2- if No â–º Check
$V_s\geq 4\sqrt{f'_c}b_wd$
5-2-1 if Yes â–º
$s_{max}\leq\left\{\begin{matrix} d/4\\ 12\;in \end{matrix}\right.$
$s\leq\left\{\begin{matrix} s_{max}\\ \frac{A_vf_y}{\alpha b_w}\\ \frac{A_vf_y d}{V_s} \end{matrix}\right.$
5-2-2 if No â–º
$s_{max}\leq\left\{\begin{matrix} d/2\\ 24\;in \end{matrix}\right.$
$s\leq\left\{\begin{matrix} s_{max}\\ \frac{A_vf_y}{\alpha b_w}\\ \frac{A_vf_y d}{V_s} \end{matrix}\right.$
Comprehensive textbooks on reinforced concrete treat more general cases involving deep beams, beams with axial forces or torsion, or inclined shear reinforcement.
Shear Friction
There are many situations in which shear force is transferred from one concrete element to another or between a concrete element and another material. A model has been developed that has been shown
to correlate with the nominal strength of this force transfer. This model is analogous to static friction with a normal force provided by properly developed reinforcement that crosses the shear
plane, modified by an appropriate coefficient of static friction, μ.
ACI Sec. 11.6.4 defines the coefficient of friction as follows:
• For concrete placed monolithically, μ = 1.4λ.
• For concrete placed against hardened and deliberately roughened concrete, with surface clean and roughened to a minimum amplitude of 1/4 in, μ = 1.0λ.
• For concrete placed against hardened concrete cleaned of laitance, but not roughened, μ = 0.6λ.
• For concrete placed against as-rolled, unpainted structural steel, μ = 0.7λ.
The constant λ is 1.0 for normal weight concrete, 0.85 for sand-lightweight concrete, and 0.75 for all lightweight concrete.
When shear friction reinforcement with an area of A[vf] crosses the shear plane at right angles, which is the usual case, the nominal shear strength is
$V_n\leq\left\{\begin{matrix} \mu A_{vf} f_y\\ 0.2f'_cA_c\\ (800lbf/in^2)A_c \end{matrix}\right.$
fy, the yield strength of reinforcement, is limited to a maximum of 60,000 psi. Ac is the area of the concrete section resisting shear. ACI 318 also gives a more general expression for shear transfer
when tensile shear friction reinforcement crosses the shear plane at an acute angle.
Brackets and Corbels
A bracket or corbel is a reinforced concrete element that projects from a wall or column and supports a reaction from another element. In most cases, the reaction is primarily a vertical force whose
resultant acts at a short distance av from the face of the supporting wall or column, as shown in the figure below. Unless special provisions are made to alleviate horizontal forces (for example, by
using very low friction-bearing pads or roller supports), there will also be a horizontal tensile force, Nuc, transmitted into the support. ACI Sec. 11.8.3 requires that the horizontal force be
treated as a live load with a magnitude of at least 20% of the vertical force, Vu. For most practical cases, when the ratio av/d is less than 1, the prescriptive procedure in Sec. 11.8 of ACI 318 is
used to design the bracket or corbel.
The following restrictions apply:
• The yield strength of reinforcement, fy, must not exceed 60,000 psi.
• The horizontal tensile force, Nuc, must not be less than 0.2Vu or greater than Vu.
• The effective depth, d, is determined at face of support.
• The main reinforcement, with an area of Asc, must fully develop at face of support. This usually requires mechanical anchorage of the main steel, either by welding a cross bar of the same
diameter or by welding to steel plates.
• The depth at the outside edge of bearing must be not less than 0.5d.
• The capacity reduction factor, φ, is 0.75 in all calculations.
• Shear transfer occurs through shear friction.
• Supplementary shear friction steel in the form of closed ties is distributed over the upper 2d/3 of the member.
What is the step-by-step prescriptive design provisions in ACI 318 for a bracket or corbel?
1- Given V[u], N[uc], f[y], f'[c], unit wieght λ,h, b=b[w], d, a[v], φ=0.75, μ:
2- Check if it is normal weight concrete
2-1- if Yes â–º
$V_n\leq\left\{\begin{matrix} 0.2f'_cb_wd\\ (480+0.08f'_c)b_wd\\ 1600b_wd \end{matrix}\right.$
2-2- if No â–º
$V_n\leq\left\{\begin{matrix} (0.2-\frac{0.07a_v}{d})f'_cb_wd\\ (800-\frac{280a_v}{d})f'_cb_wd \end{matrix}\right.$
3- Check
3-1- if Yes â–º
3-2- if No â–º continue to step 4
$A_f=\frac{M_u}{\phi f_y(0.9d)}$
$A_n=\frac{N_{uc}}{\phi f_y}$
$A_{vf}=\frac{V_u}{\phi \mu f_y}$
main (top) steel:
$A_{sc}\geq \left\{\begin{matrix} A_f+A_n\\ 0.67A_{vf+A_n}\\ (\frac{0.04f'_c}{f_y})bd \end{matrix}\right.$
supplemental shear friction steel
Most reinforced concrete members are loaded either axially or transversely in such a manner that twisting of the cross section is negligible. ACI Sec. 11.5 sets threshold limits for torsional moments
beyond which their effects must be included in designs. From ACI Sec. 11.5.1, for nonprestressed members, the threshold limit is
$T_u<\phi \lambda \sqrt{f'_c}(\frac{A^2_{cp}}{p_{cp}})$
T[u] is the factored torsional moment, A[cp] is the area of the outside perimeter of section resisting torsion, p[cp] is the outside perimeter of the section resisting torsion.
If the applied factored torsional moment exceeds the threshold, the member must be reinforced with additional transverse ties and longitudinal steel to resist the torsional moment. ACI 318 considers
two cases: equilibrium torsion and compatibility torsion.
Equilibrium torsion applies to situations where redistribution of loads cannot occur and the torsional resistance is necessary to maintain equilibrium. A common example is a precast concrete spandrel
beam. Compatibility torsion describes a situation in which loads can redistribute after torsional cracking. In such a case, a reduction is permitted in the design value of torsional moment. In the
case of compatibility torsion, ACI Sec. 11.5.2 allows the maximum torsional moment that the member must be designed to carry to be limited to four times the threshold value.
For the precast beam, full torsional resistance is needed to maintain equilibrium and this member must be designed for equilibrium torsion. In contrast, for the cast-in-place spandrel, a reduction in
torsional resistance reduces the negative bending moment transferred from the slab with a corresponding redistribution of moment to the positive region of the slab. This is a case of compatibility
Design for torsion is performed using a space truss model and assuming an equivalent hollow tube, compression diagonals developing in the concrete at 45â—¦ to the longitudinal axis, and longitudinal
and transverse steel resisting all tension. The ties must be closed, with details defined in ACI Sec. 11.5.4. The transverse steel needed to resist torsion is additive to the stirrups resisting
shear. Because only the outer legs of the stirrups provide torsional resistance, the transverse steel required is
s is the spacing, Av is the area of outer stirrup legs resisting shear, and At is the required area of one outer leg to resist torsion. The space truss analogy also requires longitudinal steel, Al,
that is generally additive to any other longitudinal steel required to resist axial forces or bending. ACI Sec. 11.5.6.2 requires that the longitudinal torsional steel must be distributed around the
perimeter resisting torsion at a spacing of no more than 2 in. The design procedure for both transverse and longitudinal reinforcement for members with torsional moment, Tu, exceeding the threshold
is summarized below:
What is step-by-step design prodecure for both transverse and longitudinal reinforcement for members with torsional moment, Tu, exceeding the threshold of Sec 11.5.1?
1- take θ=45, A[0]=0.85A[oh]
2- Check
$\sqrt{(\frac{V_u}{b_wd})^2+(\frac{T_uP_h}{1.7A^2_{oh}})^2}\leq\phi (\frac{V-c}{b_wd}+8\sqrt{f'_c})$
2-1- if No â–º compression struts control; STOP and redesign section
2-2- if Yes â–º
$\frac{A_t}{s}=\frac{T_u}{\phi (2A_o)f_{yt}cot45}=\frac{T_u}{\phi (2A_o)f_{yt}}$
3- Check
$\frac{A_v}{s}+\frac{2A_t}{s}\geq max\left\{\begin{matrix} \frac{0.75\sqrt{f'_c}b_w}{f_{yt}}\\ \frac{50b_w}{f_{yt}} \end{matrix}\right.$
3-1- if No â–º
$\frac{A_v}{s}+\frac{2A_t}{s}= max\left\{\begin{matrix} \frac{0.75\sqrt{f'_c}b_w}{f_{yt}}\\ \frac{50b_w}{f_{yt}} \end{matrix}\right.$
3-2- if Yes â–º continue to step 4
4- A[vo] = total area in outer legs of closed stirrups
5- Check
$s\leq min\left\{\begin{matrix} p_h/8\\ 12\;in \end{matrix}\right.$
4-1- if No â–º
$s= min\left\{\begin{matrix} p_h/8\\ 12\;in \end{matrix}\right.$
4-2- if Yes â–º continue to step 6
7- Check
$\frac{A_t}{s}\geq \frac{25b_w}{f_y}$
7-1- if No â–º
$\frac{A_t}{s}= \frac{25b_w}{f_y}$
7-2- if Yes â–º continue to step 8
8- Check
$a_l\geq \frac{5\sqrt{f'_c}A_{cp}}{f_y}-(A_t/s)p_h(f_{yt}/f_y)$
8-1- if No â–º
$a_l = \frac{5\sqrt{f'_c}A_{cp}}{f_y}-(A_t/s)p_h(f_{yt}/f_y)$
8-2- if Yes â–º End
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Continuity of Functions of Several Variables
Continuity of Functions of Several Variables
Recall that a function of a single variable $y = f(x)$ is continuous at $c \in D(f)$ if $\lim_{x \to c} f(x) = f(c)$. We will now extend the concept of continuity of a function of a single variable
to a function of several variables.
Definition: A two variable real-valued function $z = f(x, y)$ is said to be Continuous at $(a, b) \in D(f)$ if $\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)$. If $f$ is not continuous at $(a,b)$, then we
say that $f$ is Discontinuous at $(a, b)$. We say that $f$ is a Continuous Two Variable Function if $f$ is continuous for all $(a, b) \in D(f)$.
Geometrically, a two variable function $z = f(x, y)$ is continuous if the graph of $f$ does not contain any holes, breaks, jumps, or asymptotes.
In determining discontinuities of a two variable real-valued function, we need to only consider points of the function that would normally cause discontinuities in single variable real-valued
functions. For example, consider the following two variable function:
f(x, y) = \frac{x^2 + y^2}{1 + x}
We note that both the numerator and denominator is continuous for all $x, y \in \mathbb{R}$ as they're both polynomial functions. However, notice that if $x = -1$ then $f$ is not defined. Therefore,
the points $(-1, y) \in \mathbb{R}^2$ where $y \in \mathbb{R}$ are the only discontinuities to $f$, and so $f$ is continuous elsewhere.
Of course, we can extend to concept of continuity to functions of three or more variables.
Definition: A three variable real-valued function $w = f(x, y, z)$ is said to be Continuous at $(a, b, c) \in D(f)$ if $\lim_{(x,y,z) \to (a,b,c)} f(x,y,z) = f(a,b,c)$. An $n$ variable real-valued
function $z = f(x_1, x_2, ..., x_n)$ is said to be Continuous at $(x_1, x_2, ..., x_n) \in D(f)$ if $\lim_{(x_1, x_2, ..., x_n) \to (a_1, a_2, ..., a_n)} f(x_1, x_2, ..., x_n) = f(a_1, a_2, ..., a_n)
For example, consider the following three variable real-valued function:
f(x,y,z) = \frac{x^2 + y^2 + z^2}{\sqrt{x + y}}
We see that both the numerator and denominator are continuous on their respective domains, so we will look at where $f$ is undefined. If $x + y < 0$, then $f$ is not defined, and so $f$ is
discontinuous for $y > -x$.
Let's look at some more examples.
Example 1
Determine any points of discontinuity for the function $f(x, y) = x^2y + 2xy + y^2 - 4y + 3$.
We note that $f$ represents a polynomial, and we know that polynomials are defined for all real numbers. Therefore, $f$ is discontinuous nowhere, i.e, $f$ is continuous on all of $\mathbb{R}^2$.
To show this, let $(a, b) \in \mathbb{R}^2$. Then:
\quad \lim_{(x,y) \to (a,b)} f(x,y) = \lim_{(x,y) \to (a,b)}x^2y + 2xy + y^2 - 4y + 3 = a^2b + 2ab + b^2 - 4b + 3 = f(a, b)
So $f$ is continuous for all $(a, b) \in \mathbb{R}^2$.
Example 2
Determine any points of discontinuity for the function $f(x, y) = \tan x + \sin y$.
We note that $\tan x$ is undefined if $x = (2k-1)\frac{\pi}{2}$ where $k \in \mathbb{Z}$, and so $f$ is discontinuous for $(x,y) = \left ([2k-1]\frac{\pi}{2}, y \right )$ where $k \in \mathbb{Z}$.
The graph of $f(x,y) = \tan x + \sin y$ is given below.
Example 3
Determine any points of discontinuity for the function $f(x,y) = \left\{\begin{matrix} x^2 + y^2& \mathrm{if} \: (x,y) \neq (1, 1) \\ 3 & \mathrm{if} (x, y) = (1, 1) \end{matrix}\right.$.
Clearly $f$ is defined for all $(x, y) \in \mathbb{R}^2$, however, $f$ is not continuous at $(1, 1)$. Notice that $\lim_{(x, y) \to (1,1)} f(x,y) = 2 \neq 3 = f(1, 1)$.
Example 4
Determine any points of discontinuity for the function $f(x, y, z) = \sqrt{e^x + \frac{2e^y}{\sin x} + \frac{3}{z}}$.
Notice that if $\sin x = 0$ or $z = 0$ then $f$ is undefined. We note that $\sin x = 0$ if $x = k\pi$ where $k \in \mathbb{Z}$, and so $\{ (x, y, z) : x = k\pi \: \mathrm{or} \: z = 0 \}$ is the set
of points of discontinuity to $f$.
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CourseNana | COMP2022 Models of Computation - Assignment 3: Turing Machine and Morphett notation
This question has been solved
comp2022 Assignment 3 (70 marks) s2 2024 This assignment is due in Week 10 and should be submitted to Gradescope.
All work must be done individually without consulting anyone else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism” policies.
Go to the last page of this document and read the Submission Instructions. For clarifications and updates, monitor “Assignment FAQ”.
Problem 1. (10 marks) Consider the following deterministic Turing Machine M over input alphabet Σ = {a, b}:
0__L1 0**R0
1b_L2 2a_L3
1 _ _ * halt_accept
3__R0 3**L3
1. (5 marks) State five strings that are in L(M), and five that are not. The strings should be over Σ.
2. (5 marks) Provide a low level description in Morphett notation of a (1-tape deterministic) Turing Machine for the language that has time complexity at most 5n + 5.
Problem 2. (10 marks) Consider the following nondeterministic Turing Machine N over input alphabet Σ = {a, b}:
0 _ _ * halt-reject 0aar0
1xxl1 1axr2 1bxr2 1__r4
comp2022 Assignment 3 (70 marks) s2 2024
2 _ _ * halt-reject
3 _ _ * halt-reject
4 a a * halt-reject 4 b b * halt-reject 4 _ _ * halt-accept
1. (5 marks) State five strings that are in L(N), and five that are not. The strings should be over Σ.
2. (5 marks) Provide a low level description in Morphett notation of a (1-tape deterministic) Turing Machine for the language.
Note: Morphett’s simulator of nondeterministic TMs uses randomness to resolve nondeterminism. This is not the semantics of NTMs.
Problem 3. (30 marks) For each of the following languages over the input al- phabet Σ = {a, b, c}, provide a low level description in Morphett notation of a (1-tape deterministic) TM for the
1. The language of non-empty strings where the final character appears at most 3 times in the string (including the final character).
E.g., abccaba is in the language, while abcbcbab is not.
2. The language of strings of the form anbncnan for n ≥ 1.
E.g., aabbccaa is in the language, while abc is not.
3. The language of strings that can be turned into a palindrome by replacing
at most two characters by other characters.
E.g., aba is in the language because it is a palindrome, abb is in the lan- guage because we can change one character to get a palindrome (e.g., aba), and aabc is in the language because we can
change two characters to get a palindrome (e.g., aaaa); however aabbccc is not in the language.
4. The language of strings for which the longest substring that matches a∗ is longer than the longest substring that matches b∗.
E.g., caaaccbbaabaaac, baaacbbcaaabb and aaaa are in the language, while aabbbcacacacaca is not.
comp2022 Assignment 3 (70 marks) s2 2024
5. The language of strings of the form uvcvu where u, v ∈ {a, b}∗.
E.g., aabbacbaaab is in the language (take u = aab, v = ba), while aabbcabab is not.
6. The language of strings of the form uvw where v is a non-empty string with the same number of as, bs, and cs. E.g., bbaabbbccaccbc is in the language, while bbaabbbcc is not.
Problem 4. (5 marks + 5 bonus marks)
Your robot buddy GNPT-4 has come up with a revolutionary new strategy to prove that it is in fact equal in computational power to its more well-known cousin. It has a simple yet brilliant proof
strategy: it will start by proving that P in fact equals the set of Turing-decidable languages, by showing that every decider runs in polynomial time. Once it has done this, it will obtain as a
corol- lary that NP is also equal to this set, and the result will follow. GNPT-4 would like you to check its generated proof, and has generously offered you half of the million dollar bounty for
doing so.
Unfortunately, you’re starting to have some concerns about the claim that every decider runs in polynomial time. GNPT-4’s proof of this claim is 2123 pages long, so you don’t really feel like
checking it in detail for a flaw. Instead, you have a much better idea: you’ll provide an explicit counterexample of a machine that does not run in polynomial time.
1. (5 marks) Provide a low level description in Morphett notation of a (1-tape deterministic) TM over input alphabet Σ = {a} that accepts every string, has at most 20 states, and has time complexity
f (n) such that 2n ≤ f (n) ≤ 22n+1 for all n.
2. (5 bonus marks) Provide a low level description in Morphett notation of a (1-tape deterministic) TM over input alphabet Σ = {a} that accepts every string, has at most 40 states, and has time
complexity exactly 2n.
Problem 5. (15 marks)
You’re a budding cartoonist, trying to create the next great TV animation. You’ve come up with the perfect idea, but now you need to pitch it to the executives. You know from your experience in the
industry how the process works: you make a proposal with a string over Σ = {a, b} and the network runs a Turing machine Q on it. If Q accepts, your show will be ready for broadcast, but if it
doesn’t, you will be shown the door, filled with eternal regret at what could have been. Of course, as Q is a Turing machine, there is also the possibility that Q will diverge. (For example, this is
what happened after season 7 of Futurama.)
One of your shady contacts (apparently they’re a secret agent who uses finite au- tomata, or something?) has managed to obtain a copy of the network’s machine Q for you. You now want to analyse Q to
figure out how to pitch your show
comp2022 Assignment 3 (70 marks) s2 2024
so it will be accepted. Furthermore, you’ve heard that it’s considered especially fortuitous if Q runs in a number of steps that is a multiple of 77, and such shows will be given air during the
network’s prime timeslots. So you’d like a machine that will analyse Q and your proposal to see if that will be the case.
1. (5 marks) Prove that the language {M, x: M halts on x in exactly 77n steps for some integer n > 0} is undecidable.
Okay, so that was a bust. You’ve set your sights lower: at this point you just want any description that will be accepted, and you’re willing to retool your proposal to make it work. Rather than
focusing on your specific string, you’d like a machine that will analyse just Q, and find some string, any string, that it will accept. There is, however, the possibility that Q doesn’t accept any
string. (That would explain why there are no decent new shows these days.) In this event, your endeavour is doomed and you don’t care about the output, but you’d like the analysing machine to at
least halt, so you’re not stuck waiting forever.
2. (10 marks) Consider the following specification. The inputs are Turing ma- chines over input alphabet Σ = {a, b}.
1. (a) If the input is a Turing machine M that accepts some input, the output should be any string x that M accepts.
2. (b) If the input is a Turing machine M that does not accept any input, the output should be any string x. (There still must be an output, ie. the machine satisfying this specification must halt.)
Prove or disprove whether there exists a Turing Machine that halts on every input and satisfies this specification.
comp2022 Assignment 3 (70 marks) s2 2024 Submission Instructions
You will submit answers to all the problems on Gradescope.
Problems 1, 2, 3 and 4 are autograded.
It is essential that you ensure that your submission is formatted so that the autograder can understand it. Upon submitting your responses, you should wait for the autograder to provide feedback on
whether your submission format was correct. An incorrectly formatted submission for a question will receive zero marks for that question. A scaffold will be provided on Ed with the file names the
autograder expects.
Problem 1.1, 2.1 format:
The first line of each answer should contain a comma separated sequence of five strings that are in the language, and the second line should contain a comma separated sequence of five strings that
are not in the language. For example, if the language consists of all strings that only contain b’s, an example of a correct text file would be:
Problem 1.2, 2.2, 3, 4 format (TMs):
All TMs that you are required to provide in this assignment are deterministic and have a single tape, and that tape is doubly-infinite. When asked to give a low-level description use Morphett’s
format. The initial state must be 0
Note that your machine should use an explicit transition to halt-reject when rejecting a string. If the machine has no transition on a (state, input) pair, this will be treated as an error, and will
not be treated as rejecting the string. You may wish to include the following line in your machines, to treat all undefined transitions as rejects: * * * * halt-reject
Problem 5 format:
Problem 5 is handgraded. You will submit a single typed pdf (no pdf containing text as images, no handwriting). Start by typing your student ID at the top of the first page of each pdf. Do not type
your name. Do not include a cover page. Submit only your answers to the questions. Do not copy the questions. Your pdf must be readable by Turnitin.
epsilon, b, bb, bbb, bbbb
a, aa, aaa, aaaa, aaaaa
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Symbolic Computation Seminar
North Carolina State University
The Symbolic Computation Seminar takes place in
335 Harrelson Hall
unless otherwise noted.
Fall 2004 Schedule
Wednesday, September 1, 5:00 - 6:00pm (NOTE SPECIAL TIME )
• Ruriko Yoshida, Duke University
• Short Rational Functions for Toric Algebra and Applications
• Abstract: In 1993 Barvinok gave an algorithm that counts lattice points in convex rational polyhedra in polynomial time when the dimension of the polytope is fixed via short rational functions.
The main theorem on this talk concerns the computation of the toric ideal I_A of an integral n x d matrix A. We encode the binomials belonging to the toric ideal I_A associated with A using short
rational functions. If we fix d and n, this representation allows us to compute a universal Groebner basis and the reduced Groebner basis of the ideal I_A, with respect to any term order, in
polynomial time. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions of the division algorithm. This is joint work
with J. De Loera, D. Haws, R. Hemmecke, P. Huggins and B. Sturmfels.
Wednesday, September 8
• Jack Perry, NCSU
• When can we skip S-polynomial reduction?
• Abstract: Gröbner basis computation usually requires many reductions of S-polynomials. These reductions are computationally expensive. Bruno Buchberger discovered that sometimes we can skip the
reduction of some S-polynomials. Specifically, he gave two criteria on their leading terms. A natural question is: are there more?
This talk will give an affirmative answer. Further, we will provide the necessary and sufficient conditions on the leading terms when there are at most three polynomials. Generalizing the result
to an arbitrary number of polynomials remains an open problem.
Wednesday, September 15
• David Saunders and Information Sciences, University of Delaware
• Integer matrix determinants and formulas for pi
• Abstract: Gosper gave a formula for Pi which Bellard has used to design an algorithm for computing the n-th digit of pi without computing the earlier ones. Almkvist, Krattenthaler, and Petersson
have developed a scheme for generating a family of similar formulas which involves computing determinants of certain integer matrices. I will present a summary of their approach and use it as
motivation to discuss the state of the art of determinant computation. Can there be a fully automatic determinant program which, for all integer matrices, is (nearly) the best?
Wednesday, October 6
• George E. Collins
• Coefficients of Factor Polynomials
• Abstract: The max norm of an integral polynomial is the maximum of the absolute values of its coefficients. Let d be the max norm of an integral polynomial A(x) and let e be the minimum of the
max norms of its irreducible factors. How large can e/d be? Can it be greater than 1, greater than 2, arbitrarily large? We explore these questions experimentally and obtain some answers.
Wednesday, October 13
• Teresa Krick, Universidad de Buenos Aires
• Sparse univariate polynomial interpolation
• Abstract: The motivation of this work comes from interesting discussions with Wen-shin Lee on her work on sparse multivaritate interpolatin. She pointed out to me the paper " A deterministic
algorithm for sparse multivariate polynomial interpolation", 1988, where M.Ben-Or and P. Tiwari raised the question wheter one can efficiently reconstruct a univariate $t$-sparse polynomial from
its values in any set of $2t$ (positive) points. This is work in progress , joint with Martin Avendano and Ariel Pacetti , from the University of Buenos Aires.
The talk consists of two parts.
1) $t$-sparse polynomials represented by evaluation programs: This representation allows to evaluate derivatives as well. Using this, there is a very simple matrix which eigenvalues are exactly
the exponents arising in the $t$-sparse polynomial. The coefficients are then recovered in the usual way, by solving the corresponding Vandermonde linear system.
2) Integer $t$-sparse polynomials: I' ll present a Hensel type lemma that allows to lift an exact $t$-sparse polynomial modulo $p$, of degree bounded by $p-1$, to a $t$- sparse polynomial modulo
$p^k$, of degree bounded by $(p-1)p^{k-1}$, from its values in $2t$ points of size bounded by $2p$. This Hensel interpolation allows in some cases to reconstruct an integer sparse polynomial from
its values in more general and smaller height families of points than the ones used by Ben-Or and Tiwari.
Whil making some progress on the general question raised by Ben-Or and Tiwari, these results still do not give a definitive answer.
Wednesday, Ocotber 20, 4.30 pm, HA 335 (NOTE SPECIAL TIME)
• Carlos D'Andrea, University of California, Berkeley
• The jacobian scheme of the discriminant
• Abstract: The discriminant of a binary form of degree d is a polynomial which appears naturally in algebraic geometry and invariant theory. Geometrically, it encodes the variety of univariate
polynomials with multiple roots. The jacobian ideal of the discriminant contains more precise information about the nature of these multiple roots.
In this talk, I will give a brief introduction to this topic and show the algebraic structure of the jacobian ideal.
This is a joint work with Jaydeep Chipalkatti.
Thursday, October 21, Computer Science Colloquium, 4:00pm, Withers 402A (NOTE SPECIAL TIME AND PLACE)
• Erich Kaltofen, NCSU
• The Art of Symbolic Computation
• Abstract: Computation in the past 40 years has brought us remarkable theory: Groebner bases, the Risch integration algorithm, polynomial factorization lattice basis reduction, hypergeometric
summation algorithms, sparse polynomial interpolation, etc. And it produced remarkable software like Mathematica and Maple, which supplies implementations of these algorithms to the masses.
Several system designers even became rich by selling their software. As it turned out, a killer application of computer algebra is high energy physics, where a special purpose computer algebra
system, SCHOONSHIP, helped in work worthy of a Nobel Prize in physics in 1999.
In my talk I will give glimpses what the descipline of Symbolic Computation is and what problems it tackles. In particular, I will discuss the use of heuristics (numerical, randomized, and
algorithmic) that seem necessary to solve some of today's problems in geometric modeling and equation solving. Thus, we seem to have come full cycle (the discipline may have started in the 1960s
at the MIT AI Lab), but with a twist that I shall explain.
Wednesday, October 27
• Zhendong Wan, and Information Sciences, University of Delaware
• Exactly Solve Integer Linear Systems Using Numerical Methods
• Abstract: We present a new fast way to exactly solve non-singular linear systems with integer coefficients using numerical methods. This is important both in practice and in theory. Our method is
to find an initial approximate solution by using a fast numerical method, then approximate the solution amplified by a scalar, with integers. Update the amplified residual, and repeat these steps
to refine the solution until high enough accuracy is achieved, and finally reconstruct the rational solution. We will expose the theoretical cost and show some experimental results.
Friday, October 29, 4:00pm, Harrelson 330 (NOTE SPECIAL TIME AND PLACE)
• Alicia Dickenstein, Universidad de Buenos Aires
• Binomial Complete Intersections (joint work with Eduardo Cattani)
• Abstract: A binomial ideal in the polynomial ring R:=k[x_1,...,x_n], is an ideal generated by binomials: a x^A - b x^B, where A,B are nonnegative integer n-tuples and a,b in k are non zero.
Binomial ideals are quite ubiquitous in very different contexts particularly those involving toric geometry and its applications, and in the study of semigroup algebras. While binomial ideals are
quite amenable to Groebner and standard bases techniques, they also provide some of the "worst-case" examples in computational algebra, such as the Mayr-Meyer ideals. Thus, we are interested in
algorithms that allow us to obtain information about binomial ideals purely in terms of the data defining them.
In this talk we will discuss how to determine when a binomial ideal is a zero-dimensional complete intersection and, if so, how we may compute the total number of solutions and the total
multiplicity of solutions in the coordinate axes. These problems arise naturally in the study of sparse discriminants and in the study of hypergeometric systems of differential equations.
11 October 2004
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JUPEB Mathematics 2024 Questions And Answers Expo
Welcome to Answerhub.com.ng No.1 JUPEB runz website where we shared with you the procedures to get JUPEB Mathematics 2024 Questions And Answers Expo midnight before exam.
According to the Joint Universities Preliminary Examinations Board (JUPEB), 2024 JUPEB Mathematics Questions And Answers Expo paper will hold on friday 23rd august 2024.
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JUPEB Mathematics Area Of Concentration 2024
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2. Group Theory
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4. Vector Spaces
5. Calculus
6. Differential Equations
7. Real Analysis
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13. Analytic Geometry
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#> Loading required package: fda
#> Loading required package: splines
#> Loading required package: fds
#> Loading required package: rainbow
#> Loading required package: MASS
#> Loading required package: pcaPP
#> Loading required package: RCurl
#> Loading required package: deSolve
#> Attaching package: 'fda'
#> The following object is masked from 'package:graphics':
#> matplot
#> Loading required package: rgl
#> Loading required package: geometry
The Meuse data analyzed in this vignette are locations along the bank of the river Meuse in Belgium where the amount of zinc in the soil was recorded. The river loops sharply in this location, and
the soil pollution by heavy metals used in nearby industry was of concern.
The list object MeuseData.rda contains the locations of soil samples, and these are also used as nodes in the mesh that will define the estimated surface displaying soil pollution. The mesh was built
by using Delaunay mesh generation software, such as R functions geometry::delaunayn or RTriangle::triangulate. It should be appreciated that Delaunay meshes are often not unique.
The Delaunay mesh generated a number of triangles that were outside of the desired boundary because of the non-convexity of this boundary. These triangles were removed by hand to define the final
mesh. The 155 points, the 52 edges and the 257 triangles are retrieved as follows:
p <- MeuseData$pts
e <- MeuseData$edg
t <- MeuseData$tri
np <- dim(p)[[1]]
ne <- dim(e)[[1]]
nt <- dim(t)[[1]]
covdata <- MeuseData$covdata
The edge data are not needed in this analysis.
The mesh is displayed by
The data used to define the surface is the log of the zinc level, and preliminary plots indicated that this variable was strongly correlated with the shorted distance from the river bank. The first
variable is treated in the analysis as what is to be fit by the surface, and distance was treated as a covariate that provides further information about the estimated height of the log-zinc surface.
These data are in the two-column matrix MeuseData$covdata, with distance in the first column and zinc level in the second.
data <- matrix(0,nrow=np,ncol=2)
data[,1] <- covdata[,1]
data[,2] <- log(covdata[,2])
Dist <- as.matrix(covdata[,1])
Zinc <- as.matrix(log(covdata[,2]))
LogZinc <- log10(Zinc)
We can conduct a preliminary assessment of the log-zinc/distance covariation using
lsList <- lsfit(Dist, LogZinc)
print(paste("Regression coefficient =",round(lsList$coef[1],2)))
#> [1] "Regression coefficient = 0.81"
print(paste("Intercept =",round(lsList$coef[2],2)))
#> [1] "Intercept = -0.2"
and displayed by
plot(Dist, LogZinc, type="p", xlim=c(0,0.9), ylim=c(0.65, 0.90),
xlab = "Distance from shore",
ylab = "Log10 Zinc Concentration")
abline(lsList$int, lsList$coef)
The first step in our surface estimation process is to define an FEM basis by using the “pet” architecture of the mesh as arguments:
A hazard when the observations are all at the nodes is that the linear equation that must be solved in smoothing function cannot be solved because the left side is not of full rank. A simple way
around that is to impose some penalty on the roughness of the solution by setting roughness penalty lambda to a positive value. Here is the command that produces a nice smooth surface by using lambda
= 1e-4:
smoothList <- smooth.FEM.basis(p, LogZinc, MeuseBasis, lambda=1e-4, covariates=Dist)
LogZincfd <- smoothList$fd
df <- smoothList$df
gcv <- smoothList$gcv
beta <- as.numeric(smoothList$beta)
SSE <- smoothList$SSE
The following commands plot the fitted surface by using the powerful graphics capabilities of the rgl package.
Xgrid <- seq(-2.2, 2.2, len=21)
Ygrid <- seq(-2.2, 2.2, len=21)
op <- par(no.readonly=TRUE)
plotFEM.fd(LogZincfd, Xgrid, Ygrid,
xlab="Latitude", ylab="Longitude", zlab="log10 Zinc Concentration")
The coefficient for the covariate Dist is 1.83 and error sum of squares is 49.9. We can rerun the analysis without the contribution from Dist:
Now we get the error sum of squares 59.8. The squared multiple correlation coefficient is R^2 = (59.8 - 49.9)/58.8 = 0.165, which indicates a relatively uninteresting contribution of the Dist
variable to the fit to the data.
There are many ways to improve this analysis. The mesh has a lot of small triangles due to a large number of sampling positions in the lower left corner of the mesh. We can leave the nodes in as
locations of the data, and replace p in the argument by loc which is identical to p. Then we might then remove a number of nodes in this densely sampled area so that the triangles that are defined
have roughly the same areas as those elsewhere in the mesh. The number of rows i p will decrease, and we will rebuild the FEM basis using the create.FEM.basis function. This will define a smooth
surface using smaller values of lambda.
In any case, the triangles needn’t be as small they are in this analysis for the purposes of examining the variation in the zinc deposits. Reducing the mesh density is a good way to go.
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Singapore School Math | Online Practice, Questions, Tests, Worksheets, Quizzes, Assignments | Edugain Singapore
Grade 1
Shapes and Solids Around Us, Spatial Understanding, Numbers from 1 to 10, Numbers from 11 to 20, Numbers from 21 to 100, Numbers from 1 to 100, Addition (1 to 20), Subtraction (1 to 20), Addition
and Subtraction (1 to 20), Addition and Subtraction of Numbers up to 100, Multiplication, Skip Counting, Patterns and Shapes, Time, Probability, Data Handling
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Numbers (3-Digit), Addition, Subtraction, Multiplication, Division, Fractions, Measurement, Geometry, Time, Patterns, Data Handling
Grade 3
Numbers (4-Digit), Roman Numerals, Addition, Subtraction, Multiplication, Division, Fractions, Measurement, Time, Geometry, Symmetry, Data Handling
Grade 4
Numbers (5-Digit and 6-Digit), Roman Numerals (up to 100), Addition, Subtraction, Addition and Subtraction, Multiplication, Division, Factors and Multiples, Patterns, Rounding Numbers, Fractions,
Decimals, Money, Measurement, Unitary Method, Geometry, Perimeter of Rectilinear Figures, Solid Shapes (3-D shapes), Symmetry, Time, Data Handling
Grade 5
5 Large Numbers, Roman Numerals (up to 500), Simplification, Factors and Multiples, Fractions, Decimals, Rounding Numbers, Measurement, Average, Percentage, Time, Money, Geometry, Perimeter, Area,
Volume, Data Handling, Exponents, Problem solving, Number sequences, Coordinate plane, Variable expressions, Probability and statistics, Symmetry and transformations, Two-dimensional figures,
Three-dimensional figures, Financial literacy
Grade 6
Large Numbers (7-Digit to 9-Digits), Roman Numerals, Whole Numbers, Integers, Factors and Multiples, Fractions, Decimals, Simplification, Ratio and Proportion, Unitary Method, Algebraic
Expressions, Linear Equations in One Variable, Mensuration, Geometry, Three Dimensional (3-D) Shapes, Linear Symmetry, Data Handling
Grade 7
Integers, Fractions, Rational Numbers, Decimals, Ratio and Proportion, Unitary Method, Percentage, Profit and Loss, Simple Interest, Exponents, Algebraic Expressions, Linear Equations in One
Variable, Mensuration, Geometry, Reflection and Rotational Symmetry, Three-Dimensional Shapes, Probability, Data Handling
Grade 8
Rational Numbers, Square and Square Roots, Cube and Cube Roots, Exponents, Direct and Inverse Proportions, Time and Work, Percentage, Profit and Loss, Compound Interest, Algebraic Expressions,
Linear Equations in One Variable, Factorization, Playing with Numbers, Geometry, Graphs, Mensuration, Volume and Surface Area, Three-Dimensional Figures, Probability, Data Handling
Grade 9
Number System, Polynomials, Geometry, Coordinate Geometry, Linear Equations in Two Variables, Mensuration, Statistics and Probability
Grade 10
10 Real numbers, Polynomials, Linear Equations in Two Variables, Triangles, Trigonometry, Statistics and Probability, Quadratic Equations, Arithmetic Progression (AP), Circles, Coordinate Geometry,
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CFD Modelling Of Heat Exchanger Assignment - Native Assignment Help
CFD Modelling Of Heat Exchanger Assignment
Need an Assignment Helper in the UK? Native Assignment Help is here to support you every step of the way. Our skilled experts specialize in a wide range of subjects and are committed to delivering
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Introduction - CFD Modelling Of Heat Exchanger Assignment
The report is about to explore the theory of how heat is transferred through a parallel plate heat exchanger. One of the important pieces of equipment that are used in the modern era both in
residential and industrial areas is the heat exchanger. The heat exchanger is based on the principle of transfer of heat from a higher temperature (source) to a lower temperature (sink). The heat
exchanger is a device that transfers heat energy from one fluid to another or between two media. The fluid could be gas or liquid or a combination of liquid and gas. These mediums can be separated by
a solid wall or just by the air. The energy efficiency of a system can be improved by the heat exchanger because it transfers heat from a system where there is no need for this energy and the energy
can be useful for the second system. The type and the shape of the heat exchanger vary depending on the application of the equipment. The fluid flows in a doubled line in most of the heat exchangers
in which the total heat lost by the fluid with higher is gained by the cold fluid. Here the heating element is a solid, resistance coil used in the heating system of the air conditioning for
industrial, medical application, and residential. The control and design of a heat exchanger are dependent on the need of the system where it will be used. Any heat exchanger has many requirements
and among them, the most important requirements are the temperature at which the fluid in the control panel exists (set point) and the time required to catch this temperature (time of settling). Any
output/input equipment or the control of the heat exchanger required a control algorithm that can control the output variable by taking specific decisions. The achievement of the reach point
accurately at the small-time without any overshooting is the research topic for many researchers nowadays. In this coursework, the parallel plate heat exchanger is used. This type of connection
allows forming a series of channels to flow fluid between them. The place between two plates forms a channel that allows flow of the fluid. In a parallel plate heat exchanger, a series of parallel
plates are connected parallels one after another. The ratio of the inertia force to friction force or viscous within a flow of fluid is called the Reynolds number. It is a dimensionless quantity as
it is the ratio of the forces. The characteristics of the flow of fluid are shown by the Reynolds number whether it is laminar, turbulent, or translational. The equation of continuity stated that the
mass entered in a static volume must be equal to the mass left the volume. This also provides information about the increase in the velocity of the liquid within a specific volume. Different fluid
mechanics’ principles’ are the basis of Computational Fluid Dynamic (CFD) modeling. It utilizes algorithms and numerical processes to solve any problem which involves the flow of fluid. Different
chemical reaction processes such as combustion with the flow of fluid are integrated into the model to provide a 3-dimensional understanding of the performance of the boiler. The simulation of the
interaction between the gases and liquid where boundary conditions are defined by the surfaces are attempted by the CFD model. The model also tracks the flow of solid particles by a system. The
results of the model are analyzed by the Navier-Stokes equations either a transient condition or a steady-state condition.
Aims and Objectives
The main aim of this coursework is to develop different key capabilities that are required in the innovation process of different mechanical products which especially includes management, creativity,
method of modeling, synthesis, and analysis, and the uses of CFD application across the world with different tools such as ANSYS software for simulation and to find the results of typical problems of
engineering. To meet the aim of the modeling, a case study has to be selected to perform all the practical work. All the requirements related to heat transfer and fluid mechanics have to be satisfied
by the application of engineering problems.
The objectives of this laboratory are as follows:
• To analyze the parallel plate heat exchanger by using software of CFD simulation and different laboratory equipment
• To Conduct a convergence study of mesh, analyzing the effect of size of mesh on the result on the output
• Expansion of knowledge of the transfer of heat and fluid mechanics through practical investigation
• Model of fluid flow in the heat exchanger by using software of ANSYS
• Analyzation of all the results through hand calculations, simulations, and laboratory equipment to assimilate why and how they differ
• Discussing the disadvantages of using ANSYS CFD software
Heat exchangers are devices that transfer heat energy from one medium to another. The fluid could be gas or liquid or a combination of liquid and gas. A complete wall or simply the air can divide
these mediums. The heat exchanger can increase a system's energy efficiency by transferring heat from one system to another that doesn't require it. This energy can then be used by the second system
(Yoon et al. 2017). Computational Fluid Dynamic (CFD) modeling is based on several fluid mechanics principles. It solves any problem involving fluid flows with the help of algorithms and numerical
procedures. Different chemical reaction processes, such as combustion with fluid movement, are incorporated into the model to provide a three-dimensional view of the boiler's performance
(Hernández-Parra et al. 2018). The CFD model attempts to simulate the interaction between gases and liquids where boundary conditions are defined by surfaces. A system's flow of solid particles is
also tracked by the model. In fluid dynamics, incompressible flow defines a constant density within a particular fluid packet. The fast current from the hose assembly is an example of an
incompressible flow. Incompressible flow is a property of the material. All liquids are marked as compressible, but many liquids are incompressible due to liquid density fluctuations and can be
ignored in applications. Incompressibility is a property of liquids under certain conditions. All heat exchangers have many requirements; most importantly the temperature at which the liquid is in
the control panel (setting point) and the time it takes for the liquid to reach that temperature (setting time). Controlling the output/input device or heat exchanger required a control algorithm
that could control the output variables through targeted decisions. Reaching range points on a small scale with pinpoint accuracy without overshooting is a research topic for many researchers today
(La Madrid et al. 2017). This course uses a parallel plate heat exchanger. This type of connection allows the fluid to form a set of channels for it to flow between them. The points between the two
plates form a channel through which the liquid can flow.
The connection in the parallel plate heat exchanger allows forming a series of channels to flow fluid between them. The place between two plates forms a channel that allows flow of the fluid. In a
parallel plate heat exchanger, a series of parallel plates are connected parallels one after another. There are two outlet flows and two inlet flows in the parallel plate heat exchanger. Every
configuration of outlet and inlet has a separate flow of cold and hot fluid. Figure two shows the assembling of the parallel plate heat exchanger and has the makings of placement of thermocouple. In
this experiment, the flow direction of the flow was the same for the cold and hot fluid as shown in figure 2. This type of flow is called parallel flow. The various limitations and the errors of this
experiment are explained in the next chapter of the experiment’s work (El et al. 2017). The flow boundary in this experiment is 3.4 mm, for that reason, the flow between the parallel plates will be
turbulent and the boundary lamina formed in the walls of the steel plate will be interacting with each other. The flow disrupter will increase the turbulent flow through the bounded wall. This is
observed in figure 3.
Figure 2: The parallel flow of heat exchanger
(Source: Given)
Figure 3: Overhead image of steel plate with flow
(Source: Given)
ANSYS Fluent 2018 software will be used to simulate the model that does not limit the size of the mesh whether students constraints the limit or not the model to elements of 512000. This software is
used to carry out the numerical simulation with a steady-state solver of 3D double accuracy. A symmetry plane is used to cut the reactor to save the time of calculation in the longitudinal direction
and the conduits’ length feeds the Y-mixer was reduced to 0. For this purpose, the velocity of the inlet did not provide a constant value. The non-slip condition is also known as the novelty offset
condition. The velocity of a liquid or gas layer associated with a fluid boundary is believed to be similar to the velocity of the boundary. This means that there is no relative velocity between the
liquid layer and the boundary that caused the slip in these areas. Velocity offset or slips conditions where velocity continuity is assumed to be lacking in this region. This means that there is a
relative velocity between the liquid layer and the boundary, and slippage has occurred in that area (De et al. 2017). The virtual boundary length between the fluid and the boundary where the fluid
reaches the effective velocity condition is expressed as the slip length.
The key challenges that will be faced by the students are the creation of the geometry that has to mesh and the convergence of the mesh to get the representative values. The simplified models such as
3D and 2D and another complex model of 3D. The important equation that helps in the calculation of the findings of the value is stated in table 1 below;
Definition Equation Units
Temperature for cold and hot flow K, or ^0C
Equation 1
The difference of temperature between outlet and inlet for cold and hot fluids > K, or ^0C
Equation 2
Equation 3
The mean temperature for cold and hot flows K, or ^0C
Coefficient of theoretical energy balance Equation 4
Equation 5
Thermal efficiency for cold and hot flow (for hot and cold respectively) * 100
Equation 6
(- ) * 100
The efficiency of mean temperature (η mean) Equation 7
Logarithmic Mean Temperature Difference (LMTD) Equation 8 K, or ^0C
- - )]/
Equation 9
Coefficient of heat transfer (U) W/m^2K
(A * LMTD)
Table 1: Sample equation
(Source: Self-created)
All of the above-mentioned equations will help to compare the qualitative values of the different heat exchangers. For example, the Coefficient of heat transfer can compare the efficiency of the heat
exchangers through numerical ways. All heat exchangers have many requirements; most importantly the temperature at which the liquid is in the control panel (setting point) and the time it takes for
the liquid to reach that temperature (setting time). Controlling the output/input device or heat exchanger required a control algorithm that could control the output variables through targeted
decisions. Reaching range points on a small scale with pinpoint accuracy without overshooting is a research topic for many researchers today. This course uses a parallel plate heat exchanger (Abbasi
et al. 2020). This type of connection allows the fluid to form a set of channels for it to flow between them. The points between the two plates form a channel through which the liquid can flow.
In the experiment model, due to the precision of the answer, the equation of motion can be found by using the k-epsilon simulation solver (Khalil et al. 2019). The laws for this type of flow are
dictated by the Navier-Stokes equations being it is the flow of viscous fluid, incompressible, a solver is used as all of these equations are very difficult to equate by hand.
Equation 10,
X [ δu/δt + u δu/δx + v δu/δy + w δu/δz]
The above equation 10, shows the fluid flow in the x-direction where the majority of the fluid flows. The z and y components can also be calculated for every element. This is a reason for which the
hand calculation is not efficient due to the time required for labor.
5.1 Reynolds number
The ratio of the inertia force to friction force or viscous within a flow of fluid is called the Reynolds number. It is a dimensionless quantity as it is the ratio of the forces. The characteristics
of the flow of fluid are shown by the Reynolds number whether it is laminar, turbulent, or translational (Malekan et al. 2019). The symbol of the Reynolds number is Re. It is defined as
Equation 11,
/μ of the liquid (Kg/m
U (or velocity of the liquid or gas in the specific channel (m/s)
diameter (m) of the liquid or gas (Kg/ms)
Equation 12,
Figure 4: Position where Reynolds number calculated theoretically
(Source: Given)
If the value of Reynolds number is less than 2300, then the flow of liquid is laminar, i.e. Re < 2300. When the value of Reynolds number is between 2300 and 4000, the flow will be transient, i.e.
2300 < Re < 4000. When the value of Reynolds number is greater than 4000 then the flow will be Turbulent, i.e. Re > 4000. But in actual cases, laminar flow can be seen only in viscous fluids such as
fuel oil, crude oil, and other different oils (Biçer et al. 2020). The determination of the characteristics of the fluid is used in the different models, wherein the 3D complex model is determined as
turbulent or laminar. All the non-measured values can be determined from relevant properties of linear interpolation provided on the tables or can be calculated from the stated equations.
5.2 Assumption
There are a set of assumptions that need to be set for this experiments to apply the theory throughout the report, these assumptions are -
• Have to satisfy the continuity equation - the mass entered in a static volume must be equal to the mass leaving the volume.
• The flow of fluid should be incompressible - incompressible flow defines the density which is constant in a specific parcel of fluid.
• There should be a no-slip condition
5.2.1 Continuity Equation
The statement of the conservation of mass is the Equation of continuity for the liquid flow within the pipeline. It is simply a balance of mass that flows through a static volume element. It stated
that the mass entered in a static volume must be equal to the mass leaving the volume (Sallam et al. 2019). This also provides information about the increase in the velocity of the liquid within a
specific volume.
Figure 5: Continuity Equation
(Source: dusling.github.io)
The continuity equation helps to understand the decrease and increase in the velocity of the fluid of a point in a specific cross sectional area through equation 13.
Since no mass deletion or creation can occur in the pipe then the variation of the mass can be written as
Where is the mass of fluid in section 1 of the pipe and the mass of fluid in section 2.
Equation 13,
Where the density of the fluid, v the velocity of the fluid, and A is the cross-sectional area of the fluid.
5.2.2 Incompressible Flow
In fluid mechanics, incompressible flow defines the density which is constant in a specific parcel of fluid. The flow of stream at the high speed from the hose pipe is an example of incompressible
flow. Incompressible flow is the characteristics of the material. All the fluids are characterized to be compressible but many fluids are incompressible due to the variation of the density of the
fluid and it is negligible for the applications (Zhu, X., & Haglind, F. 2020). Incompressibility is the features of the liquid exhibited to certain conditions.
Equation 14,
5.2.3 No-slip condition
The no-slip condition is also known as the no-velocity offset condition. It is assumed that the velocity of the liquid or gas layer connected to the boundary of the fluid is similar to the speed of
the boundary. This means there is no relative velocity between the fluid layer and the boundary which resulted in no-slip in those regions. Velocity offset condition or the slip condition assumes
that there is a lack of continuity in the velocity in that region. This means there is the relative velocity between the fluid layer and the boundary which resulted in slip in that region (Zhang et
al. 2017). The hypothetical boundary length between the fluid and the boundary where the fluid reaches the effective velocity condition is denoted by the slip length. The differences between the
velocity of the boundary and the last layer of the fluid can be determined by the below equation.
The slip length only depends on the solid and the pair of fluids thus it can be determined experimentally (Ishaque et al. 2020). The boundary condition has an important role in different applications
of physics. There are different historical perspectives of boundary conditions such as no-slip conditions. The continuity equation along with the Navier-Stokes equation gives the governing and
fundamental equation of motion for the viscous fluids. The Navier-Stokes equation for the constant velocity in the incompressible fluid is stated as follows:
ρ [ δu/δt + u δu/δx + v δu/δy + w δu/δz]
And the equation of continuity is
But this equation cannot be used immediately for a specific calculation. This needs the specific situation to use this equation. This condition must be created or structured to use this equation. The
boundary conditions are set forth for other things. In fluid dynamics, a common type of idealized boundary is found (Özdemir, K., & Serincan, M. F. 2018). To justify the no-slip condition there must
be one type of consideration that has to be taken care of. This is the experiment has to be involved in the direct variation with the physical surface,
In this experiment, the no-slip conditions are assumed and it is assumed that the velocity of the liquid or gas layer connected to the boundary of the fluid is similar to the speed of the boundary.
This means there is no relative velocity between the fluid layer and the boundary which resulted in no slip in that region, which is shown in figure 6.
Figure 6: Profile of velocity in no-slip condition
(Source: Given)
The 4 steel plates of the parallel plate heat exchanger cause a profile of velocity analogous to shown in figure 7, the boundary layer is formed due to the no-slip condition between the fluid and
boundary wall (Sutanto, B., & Indartono, Y. S. 2019). The center velocity flow of the fluid has a chance to increase due to the area of the inlet in comparison to the length of the plate to develop a
fully create flow.
Figure 7: Profile of velocity of turbulent flow between two boundaries
(Source: Given)
The connection of parallel plate heat exchangers allows the formation of a series of channels for fluid to flow between them. The points between the two plates form a channel through which the liquid
can flow. In the parallel plate heat exchanger, multiple parallel plates are connected in parallel one after another (Islam et al. 2020). The parallel plate heat exchanger has two outlet flows and
two inlet flows. Each outlet and inlet configuration has separate flows of cold and hot fluids. Figure 2 shows the assembly of a parallel plate heat exchanger and shows the requirements for placing a
thermocouple. In this experiment, the flow direction was the same for cold and hot fluids. This type of flow is known as parallel flow. There are a hard and fast of assumptions that want to be set
for this experiment to use the concept for the duration of the report, those assumptions are the fluid need to fulfill the continuity equation - the mass entered in a static extent ought to be the
same to the mass leaving the extent (Mitra et al. 2018). The go with the drift of fluid needs to be incompressible - incompressible go with the drift defines the density that is steady in a
particular parcel of fluid. There need to be a no-slip circumstance
The various limitations and errors of this experiment are discussed in the next chapter of the experimental work. The flow limit in this experiment is 3.4 mm. As a result, the flow between parallel
plates is turbulent and the boundary layers formed on the walls of the steel sheet interact with each other. Flow interrupters increase turbulence through confined walls.
Abbasi, H. R., Sedeh, E. S., Pourrahmani, H., & Mohammadi, M. H. (2020). Shape optimization of segmental porous baffles for enhanced thermo-hydraulic performance of shell-and-tube heat exchanger.
Applied Thermal Engineering, 180, 115835. Retrieved From https://www.researchgate.net/profile/Hamid-Abbasi-3/publication/
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Biçer, N., Engin, T., Ya?ar, H., Büyükkaya, E., Ayd?n, A., & Topuz, A. (2020). Design optimization of a shell-and-tube heat exchanger with novel three-zonal baffle by using CFD and taguchi method.
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El Maakoul, A., El Metoui, M., Abdellah, A. B., Saadeddine, S., & Meziane, M. (2017). Numerical investigation of thermohydraulic performance of air to water double-pipe heat exchanger with helical
fins. Applied Thermal Engineering, 127, 127-139. Retrieved From https://www.researchgate.net/profile/Said-Saadeddine/publication/
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Hernández-Parra, O. D., Plana-Fattori, A., Alvarez, G., Ndoye, F. T., Benkhelifa, H., & Flick, D. (2018). Modeling flow and heat transfer in a scraped surface heat exchanger during the production of
sorbet. Journal of Food Engineering, 221, 54-69. Retrieved From https://www.researchgate.net/profile/Artemio-Plana-Fattori/publication/
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Kim, S., Song, H., Yu, K., Tserengombo, B., Choi, S. H., Chung, H., ... & Jeong, H. (2018). Comparison of CFD simulations to experiment for heat transfer characteristics with aqueous Al2O3 nanofluid
in heat exchanger tube. International Communications in Heat and Mass Transfer, 95, 123-131. Retrieved From https://www.researchgate.net/profile/Sedong-Kim/publication/
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management, 204, 622-636. Retrieved From https://www.researchgate.net/profile/Daniel-Marcelo-Aldana/publication/
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Lee, M. S., Li, Z., Ling, J., & Aute, V. (2018). A CFD assisted segmented control volume based heat exchanger model for simulation of air-to-refrigerant heat exchanger with air flow mal-distribution.
Applied Thermal Engineering, 131, 230-243. Retrieved From https://www.researchgate.net/profile/Md-Washim-Akram/post/Heat-Exchanger-Simulation/attachment/5ac21f7f4cde260d15d64f68/
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Mitra, S., Muttakin, M., Thu, K., & Saha, B. B. (2018). Study on the influence of adsorbent particle size and heat exchanger aspect ratio on dynamic adsorption characteristics. Applied Thermal
Engineering, 133, 764-773. Retrieved From https://www.researchgate.net/profile/Mahbubul-Muttakin/publication/
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Özdemir, K., & Serincan, M. F. (2018). A computational fluid dynamics model of a rotary regenerative heat exchanger in a flue gas desulfurization system. Applied Thermal Engineering, 143, 988-1002.
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The table to the right contains observed values and expected
values in parentheses for two categorical...
The table to the right contains observed values and expected values in parentheses for two categorical...
The table to the right contains observed values and expected values in parentheses for two categorical variables, X and Y, where variable X has three categories and variable Y has two categories.
Use the table to complete parts (a) and (b) below. Upper X 1 Upper X 2 Upper X 3 Upper Y 1 34 left parenthesis 35.67 right parenthesis 43 left parenthesis 44.77 right parenthesis 51 left
parenthesis 47.56 right parenthesis Upper Y 2 17 left parenthesis 15.33 right parenthesis 21 left parenthesis 19.23 right parenthesis 17 left parenthesis 20.44 right parenthesis (a) Compute the
value of the chi-square test statistic. chi Subscript 0 Superscript 2equals nothing (Round to three decimal places as needed.) (b) Test the hypothesis that X and Y are independent at the alpha
equals 0.01 level of significance. A. Upper H 0: The Upper Y category and Upper X category have equal proportions. Upper H 1: The proportions are not equal. B. Upper H 0: The Upper Y category and
Upper X category are independent. Upper H 1: The Upper Y category and Upper X category are dependent. C. Upper H 0: mu Subscript xequals Upper E Subscript x and mu Subscript yequals Upper E
Subscript y Upper H 1: mu Subscript xnot equals Upper E Subscript x or mu Subscript ynot equals Upper E Subscript y D. Upper H 0: The Upper Y category and Upper X category are dependent. Upper H
1: The Upper Y category and Upper X category are independent. What is the P-value? P-valueequals nothing (Round to three decimal places as needed.) Should the null hypothesis be rejected? A.
No, do not reject Upper H 0. There is not sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the P-valuegreater thanalpha. B. No, do
not reject Upper H 0. There is sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the P-valueless thanalpha. C. Yes, reject Upper H 0.
There is not sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the P-valuegreater thanalpha. D. Yes, reject Upper H 0. There is not
sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the P-valueless thanalpha. Click to select your answer(s).
_ X1 X2 X3
Y1 34 43 51
Y2 17 21 17
Solution -
Hypotheses would be-
H0 - There is independence between X and Y values
Ha - There is no independence between X and Y values
we will use contingency table here:-
Table showing Expected frequencies is-
Expected frequency in each cell is calculated using - Row total * Column Total / Grand total
Value of chi-square test statistic is calculated as
= 1.321
P-value will be calculated as P (
= 0.5167 [ Calculated using chi-square tables ]
As p-value is less than 0.01 then null hypothesis is not rejected at 1% level of significance.
Hence we can conclude that there is independence between X and Y values
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Pia - Math Tutor - Learner - The World's Best Tutors
I am passionate about learning, teaching, and developing. My tutoring experience started in middle school when my friends asked for my help whenever we had an exam. I have a bachelor's degree in
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proofs whenever students are interested.
Success story:
One of my favorite success stories is when I worked with a student for 3 months on a linear algebra class. Not only was he able to perform much better at the end of the quarter and feel more
confident, but he was able to go from hating math to accepting it. It was his last quarter before graduation, and on the last day I tutored him, he said "I cannot believe you were able to make me
like math in my last quarter in college after having despised it for my entire 4 years of education."
Hobbies and interests:
My number one passion is traveling! I love discovering new countries, cultures, and lifestyles. My favorite city is Paris, and my most recent trip was Copenhagen, which I found to be mesmerizing.
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American Mathematical Society
We consider solutions of a system of refinement equations written in the form \begin{equation*}\phi = \sum _{\alpha \in \mathbb {Z}} a(\alpha )\phi (2\cdot -\alpha ),\end{equation*} where the vector
of functions $\phi =(\phi ^{1},\ldots ,\phi ^{r})^{T}$ is in $(L_{p}(\mathbb {R}))^{r}$ and $a$ is a finitely supported sequence of $r\times r$ matrices called the refinement mask. Associated with
the mask $a$ is a linear operator $Q_{a}$ defined on $(L_{p}(\mathbb {R}))^{r}$ by $Q_{a} f := \sum _{\alpha \in \mathbb {Z}} a(\alpha )f(2\cdot -\alpha )$. This paper is concerned with the
convergence of the subdivision scheme associated with $a$, i.e., the convergence of the sequence $(Q_{a}^{n}f)_{n=1,2,\ldots }$ in the $L_{p}$-norm. Our main result characterizes the convergence of a
subdivision scheme associated with the mask $a$ in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its
relation to the subdivision scheme are discussed. In particular, the $L_{2}$-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator
restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector
subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry. References
• Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
• C. K. Chui and J. A. Lian, A study of orthonormal multi-wavelets, J. Applied Numerical Math. 20 (1996), 273–298.
• A. Cohen, I. Daubechies, and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. 3 (1997), 295-324.
• A. Cohen, N. Dyn, and D. Levin, Stability and inter-dependence of matrix subdivision schemes, in Advanced Topics in Multivariate Approximation, F. Fontanella, K. Jetter and P.-J. Laurent (eds.),
1996, pp. 33-45.
• W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293–328.
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1166574, DOI 10.1137/0523059
• George C. Donovan, Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996), no. 4,
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• Nira Dyn, John A. Gregory, and David Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991), no. 2, 127–147. MR 1101059, DOI 10.1007/BF01888150
• T. N. T. Goodman, R. Q. Jia, and C. A. Micchelli, On the spectral radius of a bi-infinite periodic and slanted matrix, Southeast Asian Bull. Math., to appear.
• T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press,
Boston, MA, 1994, pp. 335–360. MR 1244611
• B. Han and R. Q. Jia, Multivariate refinement equations and subdivision schemes, SIAM J. Math. Anal., to appear.
• Christopher Heil and David Colella, Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl. 2 (1996), no. 4, 363–377. MR 1395770
• Christopher Heil, Gilbert Strang, and Vasily Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), no. 1, 75–94. MR 1379281, DOI 10.1007/s002110050185
• Loïc Hervé, Multi-resolution analysis of multiplicity $d$: applications to dyadic interpolation, Appl. Comput. Harmon. Anal. 1 (1994), no. 4, 299–315. MR 1310654, DOI 10.1006/acha.1994.1017
• T. A. Hogan, Stability and linear independence of the shifts of finitely many refinable functions, J. Fourier Anal. Appl. 3 (1997), 757–774.
• Rong Qing Jia, Subdivision schemes in $L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309–341. MR 1339166, DOI 10.1007/BF03028366
• Rong-Qing Jia, Shift-invariant spaces on the real line, Proc. Amer. Math. Soc. 125 (1997), no. 3, 785–793. MR 1350950, DOI 10.1090/S0002-9939-97-03586-7
• Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI
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• W. Lawton, S. L. Lee, and Z. W. Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal. 28 (1997), 999–1014.
• W. Lawton, S. L. Lee, and Z. W. Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998), 427–438.
• R. L. Long, W. Chen, and S. L. Yuan, Wavelets generated by vector multiresolution analysis, Appl. Comput. Harmon. Anal. 4 (1997), no. 3, 293–316.
• R. L. Long and Q. Mo, $L^{2}$-convergence of vector cascade algorithm, manuscript.
• Charles A. Micchelli and Hartmut Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841–870. MR 986909, DOI 10.1016/0024-3795(89)90495-3
• G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (1997), 221–244.
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• Z. W. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), 235–250.
• Lars F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), no. 5, 1433–1460. MR 1289147, DOI 10.1137/S0036141092228179
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Additional Information
• Rong-Qing Jia
• Email: jia@xihu.math.ualberta.ca
• S. D. Riemenschneider
• Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
• Email: sherm@approx.math.ualberta.ca
• Ding-Xuan Zhou
• Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
• Email: mazhou@math.cityu.edu.hk
• Received by editor(s): December 12, 1996
• Additional Notes: Research supported in part by NSERC Canada under Grants # OGP 121336 and A7687.
• © Copyright 1998 American Mathematical Society
• Journal: Math. Comp. 67 (1998), 1533-1563
• MSC (1991): Primary 39B12, 41A25, 42C15, 65F15
• DOI: https://doi.org/10.1090/S0025-5718-98-00985-5
• MathSciNet review: 1484900
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Bespoke CloudFront Edge Metrics - Speedrun
Bespoke CloudFront Edge Metrics
When I migrated the Speedrun API from API Gateway to CloudFront, I lost an important latency metric; End-to-End (E2E) API latency. Detailed CloudFront metrics provide this latency metric and a few
others, but I was curious to see if I could get more granular. Could I get realtime origin latency by edge location, software version and whether there was a coldstart? Kind of. In this post, I'll
show how I used CloudFront Functions and the Embedded Metrics Format (EMF) to obtain more insight into my end user latency.
green box above). This setup provided latency metrics, but since it was a regional API Gateway, it masked most of the network latency between my API and my users. If the end user was geographically
far from my API, they would experience much higher latency than what the metrics showed. With CloudFront (shown in the orange box above), I could measure the latency from the edge location to the
API. Since the edge location is geographically close to the end user, I get a more accurate view of the latency my end users experience. As you can see above, with CloudFront the measured latency
(blue) captures more of the latency than what is hidden (red).
CloudFront allows you to run little bits of JavaScript on the edge with CloudFront Functions. Here we'll use them to read headers and emit logs on both the viewer request and response. To get the
origin latency, we'll inject a header called x-request-time and calculate the delta using the current time in the response. To get the software version, coldstart and other metadata, we'll set them
in our API as x-meta-* headers in the response. We'll combine those headers with latency and location information from CloudFront and emit it to the log in the Embedded Metrics Format (EMF). That
will allow us to query them using CloudWatch Logs Insights and set alarms using CloudWatch Alarms.
Why not just use CloudFront access or real-time logs?
In addition to those logs requiring a more involved workflow to process, they don't include response header information. Consequently, I can't enrich them with metadata like software version, API
request id or whether my API had a coldstart. Without this enrichment, I can't slice the data to extract the insights I want.
The Code
The code to do this is fairly simple. I've created a CDK project that you can use to get started. Most of the interesting parts are in the CloudFront Function. Here's the code:
1 async function handler(event) {
2 if (event.context.eventType == "viewer-request") {
3 // inject edge request time header
4 event.request.headers["x-request-time"] = { value: Date.now().toString() };
5 return event.request;
6 } else if (event.context.eventType == "viewer-response") {
7 // build EMF metric from API response headers
8 let metrics = {
9 _aws: {
10 Timestamp: Date.now(),
11 CloudWatchMetrics: [
12 {
13 Namespace: getHeader(
14 event.response,
15 "x-meta-namespace",
16 "OriginMetrics/Default"
17 ),
18 Dimensions: [["functionVersion", "coldstart"]],
19 Metrics: [
20 {
21 Name: "originLatency",
22 Unit: "Milliseconds",
23 StorageResolution: 60,
24 },
25 ],
26 },
27 ],
28 },
29 originLatency:
30 Date.now() - new Date(+getHeader(event.request, "x-request-time")),
31 requestId: getHeader(event.response, "x-meta-requestid"),
32 functionVersion: getHeader(event.response, "x-meta-version", 0),
33 coldstart: getHeader(event.response, "x-meta-coldstart", "false"),
34 cfCity: getHeader(event.request, "cloudfront-viewer-city"),
35 cfCountry: getHeader(event.request, "cloudfront-viewer-country"),
36 };
37 console.log(JSON.stringify(metrics));
38 // strip headers that have x-meta- prefix
39 for (let key in event.response.headers) {
40 if (key.startsWith("x-meta-")) {
41 delete event.response.headers[key];
42 }
43 }
44 return event.response;
45 }
46 }
48 function getHeader(source, header, defaultValue) {
49 return (
50 source.headers[header] || {
51 value: defaultValue,
52 }
53 ).value;
54 }
And the Lambda function code:
1 let counter = 0;
3 export async function handler(event, context) {
4 return {
5 statusCode: 200,
6 body: { message: "OK" },
7 headers: {
8 "x-meta-coldstart": `${counter++ === 0}`,
9 "x-meta-namespace": `OriginMetrics/${process.env.AWS_LAMBDA_FUNCTION_NAME}`,
10 "x-meta-version": process.env.AWS_LAMBDA_FUNCTION_VERSION,
11 "x-meta-requestid": context.awsRequestId,
12 },
13 };
14 }
Any header that starts with x-amz like x-amzn-requestid or x-amz-cf-pop isn't available in a CloudFront function viewer response even if they are returned to the viewer. To get request id of the API,
I had to use a different header name. I used x-meta-requestid to get the requestid as a response header from my API.
Metrics and Slicing and Dicing the Data
Now that I have metrics, I can view them in CloudWatch Metrics and set Alarm thresholds.
But because they are just logs, I can slice and dice them in dimensions I didn't emit a metric for. Using a CloudWatch Insights query like this, I can get average origin latency by viewer location
and lambda deployment version and compare coldstarts to non-coldstarts:
filter strcontains(@message, '_aws') |
parse @message "\"originLatency\":*," as originLatency |
parse @message "\"requestId\":\"*\"" as requestId |
parse @message "\"functionVersion\":\"*\"" as functionVersion |
parse @message "\"coldstart\":\"*\"" as coldstart |
parse @message "\"cfCity\":\"*\"" as cfCity |
parse @message "\"cfCountry\":\"*\"" as cfCountry |
stats avg(originLatency) by functionVersion, coldstart,cfCity,cfCountry
functionVersion coldstart cfCity cfCountry avg(originLatency)
5 true Seattle US 1037
5 false Seattle US 462
This is useful for understanding whether I have latency regressions across code deployments.
The metrics and logs are always published in us-east-1 for CloudFront functions. If you can't find them, double check your region.
Beyond your standard CloudFront and Lambda costs, this approach will incur costs for CloudWatch Logs, CloudWatch Metrics and CloudFront Function Invocations.
Logs are priced at $0.50 per GB ingested and $0.03 per GB archived. Metrics are priced at $0.30 per metric per month and this produces 2. CloudFront Functions are priced at $0.10 per million
invocations. The CloudFront invocations will be 2x the number of requests to your API because each request triggers a viewer request and response.
If I may be so bold
There are a couple of AWS limitations that make this slightly less effective than I'd like:
1. The CloudFront POP location isn't available to the CloudFront function. Instead of having a nice clean origin latency for each edge location, I have to use the viewer city and country. This isn't
as accurate as using the edge location and sometimes city isn't available. If it was available in the context, that would be very useful (and if request time was also there I'd only need to do
work in the viewer response).
2. CloudWatch Logs Insights doesn't natively parse the json out of logs from CloudFront Functions. It chokes on the CloudFront request id format. Instead of referring to the fields by field name, I
have to manually extract them using parse. This makes the queries harder to write.
If these get fixed, it will simplify the effort and make this approach more useful.
I've demonstrated a method for emitting realtime granular metrics and logs from CloudFront. As long as your API returns information as response headers, you can use this approach with CloudFront to
gain insights into your API performance and usage. I'm using a slightly modified version of this to get per route latency metrics for the Speedrun API. If you have questions or issues with the sample
code, reach out on Twitter or cut an issue in the GitHub repository
Further reading
1. The bespoke-edge-metrics GitHub repository is a working CDK project you can use to experiment with this approach.
2. Read about the Embedded Metrics format if you want to change the dimensions or other aspects of the metrics you are emitting.
3. I am emitting the metrics as a structured log (sometimes called a request log or canonical log). If you aren't familiar with this type of log, read Logging for Scale to learn about its benefits.
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The Golden Ticket: P, NP, and the Search for the Impossible
The Golden Ticket: P, NP, and the Search for the Impossible
Princeton University Press
“. . . high praise.”
The title The Golden Ticket is taken from Roald Dahl’s Charlie and the Chocolate Factory.
Dr. Fortnow explains his reference: Dahl’s book has a machine that can tell if there is a golden ticket inside the wrapper of a candy bar without opening the wrapper. The intent of The Golden Ticket
is to consider the possibility of a real-world algorithmical equivalent of Dahl’s fictional machine to solve certain classes of mathematical problems.
Whether or not quick solutions exist for all algorithmical problems is in computer science referred to as the P versus NP problem. “P” indicates the class of problems that can be solved (relatively)
quickly by computer and “NP” indicates the class where today’s fastest computers could take longer than the age of the universe to complete.
The P versus NP problem has been previously addressed in popular science texts, a few have been reviewed here: Visions of Infinity by Ian Stewart and In Pursuit of the Traveling Salesman by William
J. Cook. In fact so many academic papers have been written on this topic that it has strained the ability of reviewers to keep up.
The journal Communications of the ACM has even put in a rule: if you submit a paper on the topic of P versus NP, you are not permitted to submit another for 24 months. Lance Fortnow’s article on P
versus NP, published in Communications of the ACM in 2009 quickly became that journal’s most downloaded article ever and led directly to the creation of this book.
The P versus NP problem remains an unanswered question, whether every problem whose solution can be quickly checked can also be quickly solved. If P equals NP then every problem in NP can be computed
quickly. If P doesn’t equal NP then there will always remain mathematical problems that can’t be solved quickly.
The P versus NP problem is also a way of reasoning about computational complexity. Alan Turing first identified computational complexity with the “halting problem,” the most familiar example
involving the hourglass or ball (also known as the spinning beach ball from hell) you get when your computer does not immediately respond. You ask yourself: Is the computer taking a long time to
think or is it stuck?
The solution to the P versus NP problem is so valuable that there is a million-dollar prize offered by the Clay Mathematics Institute. The P versus NP problem is just one of the institution’s seven
Millennium Prize problems, though if you solve P versus NP, you have a head start on the others.
One example of a practical problem in NP is the “travelling salesman” problem. Given a number of cities, calculate the shortest path that goes through all the cities. To solve the travelling salesman
problem you would have to determine all possible paths (a hard problem) before you could begin to choose the shortest path (an easy problem). The result of calculating all possible paths for
traversing the capitals of the 48 contiguous states has 62 digits! And BTW, solving the traveling salesman problem wins you another Millennium Prize.
The nature of the P versus NP problem has been hinted at by many mathematicians and philosophers including William of Ockham, memorialized for Occam’s razor: The simplest solution is usually the
best. And Einstein who said, ”Make everything as simple as possible, but not simpler.” The limitation with sayings like these is that knowing simple is good isn’t good enough. They don’t tell us how
to make things simple or whether a solution is the simplest of all possible solutions.
The honor of identifying P versus NP goes back to the mathematicians Kurt Gödel and John Von Neumann, determined from a personal letter written in 1956 that was lost and not rediscovered until the
The P versus NP problem was publicly announced in 1971 with a paper written by the mathematician Stephen Cook. In 1972 the computer scientist Richard Karp identified 21 important computationally
complex problems that capture the essence of P versus NP. After Karp’s paper was published, computer scientists acknowledged the significance of P versus NP.
The discovery of P versus NP also followed independent paths in the East and the West. At the same time that Steve Cook and Richard Karp were creating its foundations in the West, so were Leonid
Levin, Sergey Yablonsky, and Andrey Kolmorgorov on the other side of the Iron Curtain. The Soviet study of P versus NP went under a branch of science called Theoretical Cybernetics.
Dr. Fortnow playfully imagines the consequences if P were proven to equal NP. The future holds a cure for cancer, perfect weather prediction, perfect voice recognition, perfect language translation,
and real-time 3D graphics rendering. Computers would be so fast and accurate that software programs replace human umpires for sporting events (though that is happening right now in soccer—without
having to solve P versus NP).
Dr. Fortnow also addresses machine learning by asking the question, how can a computer read handwriting when everyone’s is different? He points out another machine learning advance, “recommender”
systems used by Amazon and Pandora for suggesting books and music. Dr. Fortnow extends what can be done today into the future: having a computer write a novel specifically for you (and not by simply
slipping your name into a template) by creating a unique story tailored to your interests. Computers of the future could also write speeches uniquely tailored to audiences, and fight crime by
generating physical sketches of suspects based on DNA found at crime scenes.
Today our inability to quickly solve NP problems is a two-edged sword. We have secure transactions on the Internet today because of the difficulty in factoring large primes. In this respect, the
possibility that P could eventually be proven to equal NP offers its dark side—by factoring primes quickly it becomes very easy to crack public key cryptography, breaking security on the Internet.
The author also makes a beginner’s mistake however by saying private key cryptography could be used as a backstop, not realizing that distribution of private keys over the Internet is only practical
through secure public key cryptography.
Dr. Fortnow also offers the reader a game based on an imaginary world called “Frenemy.” In Frenemy people are grouped by chains of associations of friends and enemies. The purpose of the game is to
determine the effort needed to discover new associations. The reader learns that the discovery of simple associations can be algorithmically complex though matches could be solved in reasonable time
if the cost of computation were low, that is, if P equaled NP.
The application of Frenemy to the real world is that the algorithms for finding relationships correspond to reverse engineering anonymized data used to assure privacy while sharing redacted medical,
telecom, insurance, merchant, and social network data. (And again, redacted associations can be efficiently recovered today without solving the P versus NP problem).
Quick algorithms for discovering associations or “matchmaking” can also be applied to games such as Sudoku, Minesweeper and Tetris, as well as be used to solve important problems in science.
Dr. Fortnow points out the value for speedier matchmaking in biology, for example, quicker DNA sequencing and protein folding. For physics, speedier finding of minimum energy states in complex
systems. Economics would be served through a speedup in linear programming algorithms, and problems in number theory could be solved by faster proof finding.
What is a computer scientist to do if P is never proven to equal NP? Sometimes you just have to keep trying. When there’s no best solution to be had, non-optimal solutions can be found by heuristics
or rules of thumb. An example of heuristics is given for determining the minimum number of colors needed to color an arbitrary map. Here Dr. Fortnow’s explanation is intricate and a reader might be
better served if the illustrations that were provided had been in color—the irony of the penny-wise, pound-foolish publisher.
Dr. Fortnow also considers problems that lie in-between, that is, problems that are hard to solve but don’t quite hit the threshold of NP. In many instances NP problems get easier to solve as the
number of elements to calculate decrease (for example, fewer cities to traverse in the traveling salesman problem).
For problems below a certain size (that can be parallelized), one can just throw on more computers. Getting a solution by simplifying the problem or by brute force may not be elegant but may be good
enough until something better comes along.
The end chapters cover current challenges for computing, including quantum computing, quantum cryptography, and quantum teleportation. Quantum computing, if (or when) the technical issues can be
solved would speed up some but not all NP problems. As to which depends on the algebraic structure.
Dr. Fortnow expects P to eventually be proven to not equal NP though he also expects this question to remain unresolved for a very long time. He notes the difficulty might hinge on a paradox first
identified by Gödel that there are mathematically true statements that cannot be proven true.
Though Dr. Fortnow explains P versus NP clearly and simply, The Golden Ticket gets fairly involved in two sections: zero-knowledge proofs and heuristics for determining the minimum number of colors
(illustrations handicapped by a lack of color).
Dr. Fortnow has a playful writing style reminiscent of Douglas R. Hofstadter’s Gödel, Escher, Bach, which should be taken as high praise. Looking forward to Dr. Fortnow’s next book.
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[Solved] The following model was used to relate E( | SolutionInn
The following model was used to relate E(y) to a single qualitative variable with four levels: E(y)
The following model was used to relate E(y) to a single qualitative variable with four levels:
E(y) = β0 + β1x1 + β2x2 + β3x3
This model was fit to n = 30 data points, and the following result was obtained:
= 10.2 - 4x1 + 12x2 + 2x3
a. Use the least squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable.
b. Specify the null and alternative hypotheses you would use to test whether E(y) is the same for all four levels of the independent variable.
Fantastic news! We've Found the answer you've been seeking!
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Mathematics Education (BA)
2021-2022 Undergraduate Catalog [ARCHIVED CATALOG]
Mathematics Education (BA)
Return to Department.
University Requirements:
College Requirements:
College Breadth Requirements:
The College Breadth requirements are in addition to the University Breadth requirement. Up to three credits from each of the University Breadth Requirement categories may be used to
simultaneously satisfy these College of Arts and Sciences Breadth Requirements. Minimum grade C- required for courses used to satisfy College Breadth.
*If the grade earned is sufficient, a course may be applied toward more than one requirement (e.g., breadth and major requirements), but the credits are counted only once toward the total
credits for graduation. If all but one course in a group has been taken in one department or program, a course cross-listed with that program will not satisfy the distribution requirement.
Foreign Language:
• Completion of the intermediate-level course (107 or 202) in an ancient or modern language with minimum grades of D-.
□ The number of credits (0-12) needed and initial placement will depend on the number of years of high school study of foreign language.
☆ Students with four or more years of high school work in a single foreign language, or who have gained proficiency in a foreign language by other means, may attempt to fulfill the
requirement in that language by taking an exemption examination through the Languages, Literatures and Cultures Department.
The math requirement must be completed by the time a student has earned 60 credits. Students who transfer into the College of Arts and Sciences with 45 credits or more must complete this
requirement within two semesters.
Complete one of the following four options (minimum grade D-):
Option One:
Option Two:
One of the following:
Option Four:
• Successful performance on a proficiency test in mathematics administered by the Department of Mathematical Sciences (0 credits awarded).
Second Writing Requirement:
A Second Writing Requirement approved by the College of Arts and Sciences. This course must be taken after completion of 60 credit hours, completed with a minimum grade of C-, and the
section enrolled must be designated as satisfying the requirement in the academic term completed.
Major Requirements:
Minimum grade of C- required for major courses and Professional Studies courses. Students lacking preparation for MATH 242 should begin with MATH 241 .
Modeling Elective:
One of the following:
• MATH 518 - Mathematical Models and Applications
• MATH 512 - Contemporary Applications of Mathematics
Computer Science:
One of the following:
To be eligible to student teach, Mathematics Education students must have a GPA of 2.5 in their mathematics major and an overall GPA of 2.5. They must also pass a teacher competency test as
established by the University Council on Teacher Education. Remaining in the program is subject to periodic review of satisfactory progress and, to be admitted to EDUC 400 - Student Teaching
, students must have completed all the mathematics courses required in the secondary mathematics education program. Students should consult the teacher education program coordinator to
obtain the student teaching application and other information concerning student teaching policies.
After required courses are completed, sufficient elective credits must be taken to meet the minimum credit requirement for the degree.
Credits to Total a Minimum of 124
Last Revised 2017-2018 Academic Year
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SBAC Calculator | Techfonist
SBAC Calculator
Introduction to SBAC
The SBAC, or Smarter Balanced Assessment Consortium, is an important standardized testing system used in various U.S. states to measure students’ proficiency in subjects like math and English
language arts (ELA). It helps determine whether students are meeting grade-level expectations and are prepared for future academic challenges.
The SBAC plays a vital role in the education system, providing insights into students’ strengths and areas for improvement. For students, doing well on the SBAC is important for progressing through
their education with confidence.
Table of Contents
What is an SBAC Calculator?
An SBAC calculator is a specific tool approved for use during the SBAC math assessments. Its purpose is to help students perform mathematical calculations that may be too complex to solve manually
during the test. The calculator is often built into the test’s digital platform, ensuring that only the approved functions are available to students.
This calculator is designed to ensure a level playing field, allowing students to focus on problem-solving without getting bogged down by lengthy calculations.
Understanding SBAC Testing
The SBAC test is divided into two main areas: Math and English Language Arts (ELA). For the math section, calculators are only allowed in specific parts of the test, depending on the grade level and
the complexity of the problems.
In SBAC testing, students are tested on their ability to reason mathematically and solve complex problems. The calculator provides support by handling more difficult arithmetic, but students still
need a solid understanding of mathematical concepts to succeed.
Features of an SBAC Calculator
An SBAC calculator is not the same as a regular scientific or graphing calculator. It is specifically programmed to offer only the functions allowed on the SBAC test, including:
• Basic arithmetic operations (addition, subtraction, multiplication, division)
• Square roots and exponentiation
• Some trigonometric functions (depending on grade level)
However, certain features, like advanced graphing capabilities, are disabled to prevent students from using shortcuts or tools that could give them an unfair advantage.
How to Use an SBAC Calculator Effectively
To use an SBAC calculator effectively during the test, follow these steps:
1. Familiarize yourself with the calculator before test day: Practice using the specific functions that will be available during the test.
2. Use the calculator only when necessary: While it’s tempting to use the calculator for every problem, reserve it for calculations that you can’t easily do in your head.
3. Double-check your work: The calculator is a tool, but it’s always wise to verify the result to avoid simple input errors.
For example, if you need to calculate 25² during the test, enter 25 and then press the exponentiation key (often marked as ^ or similar). The result will be 625, which is the correct answer.
Calculator Policies During SBAC Testing
Students are only allowed to use the SBAC calculator in certain sections of the math test. Typically, these are more advanced parts of the test that require complex calculations. In contrast, basic
operations like addition and subtraction must be done manually during other parts of the test.
Moreover, the calculator function is built into the digital test platform, so students do not need to bring their own. If you bring an external calculator, it will not be allowed unless it complies
with strict SBAC guidelines.
SBAC Math Tests: Calculator vs. No Calculator Sections
The SBAC math test is divided into two sections:
1. Calculator-Allowed Section: In this section, you can use the calculator for more challenging problems.
2. No-Calculator Section: Here, you must solve problems without the aid of a calculator.
Being able to switch between these modes is key to succeeding in the SBAC test.
Why SBAC Calculators are Important for Students
The SBAC calculator is an essential aid, particularly for students tackling advanced math problems. It allows them to focus on problem-solving strategies rather than being weighed down by tedious
calculations. This can lead to better test performance, as students are able to allocate more mental energy to understanding the question.
Common Mistakes to Avoid with SBAC Calculators
While the calculator is a helpful tool, many students make avoidable mistakes. Some common errors include:
• Relying too heavily on the calculator: Don’t use it for simple math problems that can be done mentally.
• Entering the wrong values: Make sure to carefully input the numbers; even a small mistake can lead to incorrect answers.
Types of SBAC Calculators
There are different types of SBAC calculators available for students:
• Online Calculators: These are integrated into the digital SBAC platform.
• Physical Calculators: Some students may be allowed to use physical calculators, but they must comply with SBAC regulations.
It’s essential to know which type will be provided during your test.
Best Practices for Using SBAC Calculators
Here are some tips for using your SBAC calculator effectively:
• Know when to use it: Don’t waste time using the calculator for simple calculations.
• Practice with the calculator: Before test day, familiarize yourself with the functions so you’re comfortable using them during the exam.
SBAC and Technological Integration
The SBAC test is administered digitally, and the built-in calculator is just one way technology is shaping modern assessments. These digital tools help ensure that testing is fair and standardized
across all students.
SBAC Test Prep: Practice with the Calculator
Many resources are available to help students practice using the SBAC calculator, including:
• SBAC practice tests: These tests mimic the real test environment, complete with the calculator.
• Math websites: Various educational websites offer tools to practice with an SBAC-approved calculator.
Alternative Tools and Resources
In cases where calculators are not allowed, students need to rely on mental math and estimation strategies. Other resources include:
• Math flashcards: To improve quick-recall skills.
• Math practice apps: Many apps provide practice problems that help sharpen math skills without a calculator.
Understanding the SBAC calculator is critical for students preparing for the SBAC math test. This tool can help students solve more complex problems efficiently, but it’s essential to practice using
it before test day. With the right preparation and understanding of the calculator’s features, students can improve their performance and boost their confidence during the test.
1. What functions are allowed on the SBAC calculator?
Basic arithmetic, exponentiation, and some trigonometric functions are allowed, but advanced features like graphing are restricted.
2. Can I bring my own calculator to the SBAC test?
No, the SBAC provides an integrated calculator, and external devices are generally not permitted.
3. Is the SBAC calculator different from a regular calculator?
Yes, it has specific functions allowed by the SBAC and may not have all the capabilities of a regular scientific calculator.
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Fluid Mechanics - Pressure at a point pt1
Lecture 4: Fluid Mechanics - Pressure at a point pt1
Lecture Details
The pressure at a point in a fluid is the same regardless of the direction the pressure is directed to.
Gaussian Math Fluid Mechanics module, situable for those studying it as an undergraduate module.
Check out www.gaussianmath.com for an indepth study with downloadable notes or for more math related content.
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how to know if a function is continuous
Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value c in its domain: f(c) must be defined. How To Check for The Continuity of a
Function. Active 25 days ago. If the point was represented by a hollow circle, then the point is not included in the domain (just every point to the right of it, in this graph) and the function would
not be right continuous. If it is, your function is continuous. In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities.More
precisely, sufficiently small changes in the input of a continuous function result in arbitrarily small changes in its output. We say that the function f(x) has a global maximum at x=x 0 on the
interval I, if for all .Similarly, the function f(x) has a global minimum at x=x 0 on the interval I, if for all .. Continuity. Technically (and this is really splitting hairs), the scale is the
interval variable, not the variable itself. In other words, if your graph has gaps, holes or is a split graph, your graph isn’t continuous. For example, the difference between 10°C and 20°C is the
same as the difference between 40°F and 50° F. An interval variable is a type of continuous variable. Retrieved December 14, 2018 from: http://www.math.psu.edu/tseng/class/Math140A/
Notes-Continuity.pdf. For example, you could convert pounds to kilograms with the similarity transformation K = 2.2 P. The ratio stays the same whether you use pounds or kilograms. Vector Calculus in
Regional Development Analysis. Continuous. When a function is differentiable it is also continuous. Bogachev, V. (2006). its domain is all R.However, in certain functions, such as those defined in
pieces or functions whose domain is not all R, where there are critical points where it is necessary to study their continuity.A function is continuous at Video Discussing The Continuity And
Differentiability Of A. Before we look at what they are, let's go over some definitions. The theory of functions, 2nd Edition. New York: Cambridge University Press, 2000. The left and right limits
must be the same; in other words, the function can’t jump or have an asymptote. The way this is checked is by checking the neighborhoods around every point, defining a small region where the function
has to stay inside. Titchmarsh, E. (1964). Note that the point in the above image is filled in. I need to define a function that checks if the input function is continuous at a point with sympy.
There are two “matching” continuous derivatives (first and third), but this wouldn’t be a C2 function—it would be a C1 function because of the missing continuity of the second derivative. Your first
30 minutes with a Chegg tutor is free! Contents (Click to skip to that section): If your function jumps like this, it isn’t continuous. A continuous function, on the other hand, is a function that
can take on any number within a certain interval. The uniformly continuous function g(x) = √(x) stays within the edges of the red box. Greatest integer function (f (x) = [x]) and f (x) = 1/x are not
continuous. If a function is continuous at every value in an interval, then we say that the function is continuous in that interval. If the same values work, the function meets the definition. But a
function can be continuous but not differentiable. Sine, cosine, and absolute value functions are continuous. However, 9, 9.01, 9.001, 9.051, 9.000301, 9.000000801. Example For a function to be
continuous at a point, the function must exist at the point and any small change in x produces only a small change in `f(x)`. lim x-> x0- f (x) = f (x 0 ) (Because we have filled circle) lim x-> x0+
f (x) â f (x 0 ) (Because we have unfilled circle) Hence the given function is not continuous at the point x = x 0. For example, the difference between a height of six feet and five feet is the
same as the interval between two feet and three feet. This kind of discontinuity in a graph is called a jump discontinuity . 10 hours ago. Retrieved December 14, 2018 from: https://math.dartmouth.edu
//archive/m3f05/public_html/ionescuslides/Lecture8.pdf is continuous at x = 4 because of the following facts: f(4) exists. The proof of the extreme value theorem is beyond the scope of this text. The
label “right continuous function” is a little bit of a misnomer, because these are not continuous functions. What are your thoughts? The initial function was differentiable (i.e. Ratio scales (which
have meaningful zeros) don’t have these problems, so that scale is sometimes preferred. Step 2: Figure out if your function is listed in the List of Continuous Functions. (n.d.). For example, sin(x)
* cos(x) is the product of two continuous functions and so is continuous. Below is a graph of a continuous function that illustrates the Intermediate Value Theorem.As we can see from this image if we
pick any value, MM, that is between the value of f(a)f(a) and the value of f(b)f(b) and draw a line straight out from this point the line will hit the graph in at least one point. A uniformly
continuous function on a given set A is continuous at every point on A. (B.C.!). A discrete variable can only take on a certain number of values. f <- sapply(foo, is.factor) will apply the is.factor
() function to each component (column) of the data frame.is.factor() checks if the supplied vector is a factor as far as R is concerned. Need help with a homework or test question? For example, the
roll of a die. For example, 0 pounds means that the item being measured doesn’t have the property of “weight in pounds.”. Close. Continuity. Function f is said to be continuous on an interval I if f
is continuous at each point x in I. Note here that the superscript equals the number of derivatives that are continuous, so the order of continuity is sometimes described as “the number of
derivatives that must match.” This is a simple way to look at the order of continuity, but care must be taken if you use that definition as the derivatives must also match in order (first, second,
third…) with no gaps. The point doesn’t exist at x = 4, so the function isn’t right continuous at that point. The function’s value at c and the limit as x approaches c must be the same. Larsen, R.
Brief Calculus: An Applied Approach. 3 comments. Sin(x) is an example of a continuous function. Example Ask Question Asked 1 year, 8 months ago. The only way to know for sure is to also consider the
definition of a left continuous function. In order for a function to be continuous, the right hand limit must equal f(a) and the left hand limit must also equal f(a). The following image shows a
right continuous function up to point, x = 4: This function is right continuous at point x = 4. Possible continuous variables include: Heights and weights are both examples of quantities that are
continuous variables. For example, the range might be between 9 and 10 or 0 to 100. Well, to check whether a function is continuous, you check whether the preimage of every open set is open. The SUM
of continuous functions is continuous. Dates are interval scale variables. The function f(x) = 1/x escapes through the top and bottom, so is not uniformly continuous. However, some calendars include
zero, like the Buddhist and Hindu calendars. In calculus, a continuous function is a real-valued function whose graph does not have any breaks or holes. On a graph, this tells you that the point is
included in the domain of the function. The function might be continuous, but it isn’t uniformly continuous. Springer. And the general idea of continuity, we've got an intuitive idea of the past, is
that a function is continuous at a point, is if you can draw the graph of that function at that point without picking up your pencil. How to know whether a function is continuous with sympy? And if a
function is continuous in any interval, then we simply call it a continuous function. Many functions have discontinuities (i.e. Image: By Eskil Simon Kanne Wadsholt – Own work, CC BY-SA 4.0, https://
commons.wikimedia.org/w/index.php?curid=50614728 Then. A discrete function is a function with distinct and separate values. Dartmouth University (2005). Posted by. In most cases, it’s defined over a
range. Theorem 4.1.1: Extreme Value Theorem If f is a continuous function over the closed, bounded interval [a, b], then there is a point in [a, b] at which f has an absolute maximum over [a, b] and
there is a point in [a, b] at which f has an absolute minimum over [a, b]. The ratio f(x)/g(x) is continuous at all points x where the denominator isn’t zero. In simple English: The graph of a
continuous function can be drawn without lifting the pencil from the paper. CRC Press. Ratio data this scale has measurable intervals. Reading, MA: Addison-Wesley, pp. It’s represented by the letter
X. X in this case can only take on one of three possible variables: 0, 1 or 2 [tails]. A function f : A → ℝ is uniformly continuous on A if, for every number ε > 0, there is a δ > 0; whenever x, y ∈
A and |x â y| < δ it follows that |f(x) â f(y)| < ε. This function (shown below) is defined for every value along the interval with the given conditions (in fact, it is defined for all real
numbers), and is therefore continuous. In this lesson, we're going to talk about discrete and continuous functions. The definition for a right continuous function mentions nothing about what’s
happening on the left side of the point.
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Ways to implement factor analysis in research with SPSS help
How to Implement Factor Analysis in Your Research
The factor analysis technique used with SPSS help specially conducts a process of extraction of all of the maximum common variances from the group of all the included variables and then further puts
them into an analysis procedure of the common score. Many forms of data analysis are there when it comes to studying survey data and making a report on it. When it is about shortening complex sets of
data with many variables, then it can be considered Factor Analysis as the best method. This whole procedure is known by the name of data reduction through the factor analysis technique. When the
researcher assumes that all the variables of the survey, they are then turned into one or more of such factors. Research assignment is the process to collect information about a specific topic or
subject. After collecting all the data, you need to analysis it minutely so that you can make a report out of it.
How to implement Factor analysis in your research
Factor analysis mainly allows the user to conduct a process of simplifying a particular set of few complex variables or things through the application of statistical analysis procedures in it for
exploring all the underlying dimensions present in that group later which will explain the relationships among all the multiple variables/things. Factor analysis is typically a kind of statistical
process that minimises the variables set for extracting their unities into a reduced count of factors which can refer it as data reduction. There are a few conventions that are used by factor
analysis –
1. Application of the variables
2. The linear relationships of variables’
3. Existence of true correlation of factors and variables
4. Nonappearance of multi-collinearity
Some mutual patterns appear while perceiving vast numbers of variables; they are known to be a factor. These are used as indexes related to the variables that are involved in the process and further
this can be applied for the analysis. The factor analysis statistical method mainly works for reducing the variables from the group or set of them all by extracting all the similarities and common
matches in them which are then divided into a smaller number of groups of factors. The factor analysis technique is also a huge portion of the prime General Linear Model (GLM), so this factor
analysis method primarily assumes various types of assumptions. Further, it also includes all the variables that are relevant to a process of analysis, and lastly, there exist all the very true
correlations among the factors and variables.
There is quite a connected relationship between research assignment formation and factor analysis since the factor analysis technique has lots of applications in the research-related aspects and SPSS
data analysis of variable groups. The factor analysis forecasts the movement towards a wide range of areas and provides fruitful insights based on the factors under the sensor to ensure a diverse
portfolio. From a business perspective, it can use factor analysis to enlighten all complex types of variables by applying a matrix of association. It reads and understands the interdependencies of
relational data to assume the complex variables that can be abridged to a few significant extents. Whereas, in academic field, the hiring procedure and curriculum decision-making for an academic year
is facilitated by the factor analysis where it acts majorly in this matter. It is utilised to govern classroom sizes, salary distribution, recruitment limits, and an extensive range of other
essential requirements that are a must to run the institution smoothly. Some questions have consisted of variables concerning the product’s usability, features, visual appeal, pricing, and so on.
These are measured on a numeric scale but from the researchers’ perception, it notices the underlying dimensions or factors related to customer satisfaction. These are generally fluctuating due to
the emotional or psychological factors directed toward the product which can’t be measured directly. The extraction of the primary factor is to determine the maximum variance which is then vanished
by the factor. The second factor is determined by the next highest level of variance where the procedure continues till there are zero variances left.
Factor analysis is a popular and widely used way to reduce a large number of numbers into smaller ones. There are many techniques or methods available for factor extraction, but factor analysis is
the one most commonly used. In the method of Common Factor Analysis, the factors are taken from the most common variances and are also not included in the unique variances. The next is the Image
Factoring technique which is based on a correlation matrix and this particular process is applied for predicting variables such as the OLS regression method. Factor analysis is a statistical
technique that is used to observe how the groups share a common variance. Although, it is mostly applied in case of the assignment research, and can also apply it in the fields such as business
analytics and market research, understanding customer satisfaction levels and employees’ job satisfaction. The prime activity of the factor analysis is to minimise or reduce the huge number of
variables into smaller or fewer amounts of factors through the method of extraction finding out the common variances and gathering all of them into a platform of the common score of those numbers.
There is quite a connected relationship between research assignment formation and factor analysis since the factor analysis technique has lots of applications in the research-related aspects and
analysis of variable groups. The factor analysis forecasts the movement towards a wide range of areas and provides fruitful insights based on the factors under the sensor to ensure a diverse
portfolio. It is utilised to govern classroom sizes, salary distribution, recruitment limits, and an extensive range of other essential requirements that are a must to run the institution smoothly.
The factor analysis statistical method mainly works for reducing the variables from the group or set of them all by extracting all the similarities and common matches in them which are then divided
into a smaller number of groups of factors. The prime activity of the factor analysis is to minimise or reduce the huge number of variables into smaller or fewer amounts of factors through the method
of extraction finding out the common variances and gathering all of them into a platform of the common score of those numbers.
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Abstract algebra/Related Articles
From Citizendium
Jump to navigation Jump to search
Main Article Discussion Related Articles ^[?] Bibliography ^[?] External Links ^[?] Citable Version ^[?]
See also changes related to Abstract algebra, or pages that link to Abstract algebra or to this page or whose text contains "Abstract algebra".
Parent topics
• Algebra [r]: A branch of mathematics concerning the study of structure, relation and quantity. ^[e]
Disciplines within abstract algebra
• Category theory [r]: Loosely speaking, a class of objects and a collection of morphisms which act upon them; the morphisms can be composed, the composition is associative and there are identity
objects and rules of identity. ^[e]
• Commutative algebra [r]: Branch of mathematics studying commutative rings and related structures. ^[e]
• Field theory [r]: A subdiscipline of abstract algebra that studies fields, which are mathematical constructs that generalize on the familiar concepts of real number arithmetic. ^[e]
• Galois theory [r]: Algebra concerned with the relation between solutions of a polynomial equation and the fields containing those solutions. ^[e]
• Group theory [r]: Branch of mathematics concerned with groups and the description of their properties. ^[e]
• Linear algebra [r]: Branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformations. ^[e]
• Ring theory [r]: The mathematical theory of algebraic structures with binary operations of addition and multiplication. ^[e]
• Universal algebra [r]: Add brief definition or description
Algebraic structures
• Field [r]: An algebraic structure with operations generalising the familiar concepts of real number arithmetic. ^[e]
• Group [r]: Set with a binary associative operation such that the operation admits an identity element and each element of the set has an inverse element for the operation. ^[e]
• Lattice (order) [r]: An ordered set in which any two element set has a supremum and an infimum. ^[e]
• Module [r]: Mathematical structure of which abelian groups and vector spaces are particular types. ^[e]
• Monoid [r]: An algebraic structure with an associative binary operation and an identity element. ^[e]
• Ring [r]: Algebraic structure with two operations, combining an abelian group with a monoid. ^[e]
• Scheme [r]: Topological space together with commutative rings for all its open sets, which arises from 'glueing together' spectra (spaces of prime ideals) of commutative rings. ^[e]
• Semigroup [r]: An algebraic structure with an associative binary operation. ^[e]
• Vector space [r]: A set of vectors that can be added together or scalar multiplied to form new vectors ^[e]
Other related topics
• Algebraic geometry [r]: Discipline of mathematics that studies the geometric properties of the objects defined by algebraic equations. ^[e]
• Algebraic topology [r]: Add brief definition or description
• Combinatorics [r]: Branch of mathematics concerning itself, at the elementary level, with counting things. ^[e]
• Number theory [r]: The study of integers and relations between them. ^[e]
Articles related by keyphrases (Bot populated)
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Is Bitcoin’s Merkle tree a binary search tree?Is Bitcoin’s Merkle tree a binary search tree?
This post originally appeared on ZeMing M. Gao’s website, and we republished with permission from the author. Read the full piece here.
Is Bitcoin’s Merkle tree a binary search tree?
Dr. Craig Wright says Bitcoin’s Merkle tree is a binary search tree. BTC devs disagree and question his qualifications. The questioning by BTC and Dr. Wright’s answers show a typical example of Dr.
Wright talking about it at a level and depth that others haven’t thought of.
Ironically, some even question Dr. Wright’s understanding of technology at the undergraduate level, not in spite of this but because of this.
(And it is this same kind of ongoing questioning during the last eight years that dominated the Wright Satoshi subject. Frankly, it is shameful and unjust. It is also what often compels me to explain
to others.)
Anyone who might have had an impression that Dr. Wright doesn’t even know the basic stuff should reexamine their basic mindset.
No one should trust what another person says blindly, but a much more common mistake people make about Dr. Wright is to reject him presumptively. It’s time to learn from what has happened again and
again. Judging from past repetitive experiences, the basic sense of prudence would require one to think more deliberately and deeply about what Dr. Wright says to just avoid embarrassment.
If you truly wish to investigate the truth in Bitcoin, resist the temptation of focusing on insulated technicalities too much. It is a system. A system that exists in a computational, economic, and
legal environment, to be more particular.
Your understanding of these matters highly depends on whether you want a techno-political system that is primarily for censorship resistance or an economic system that is primarily for productivity.
Each view can have its own respectful rationales, but one must understand the difference before concluding that others don’t know a thing.
I understand both views very well. I just happen to prefer the scalable and productive economic system to a techno-political law-resistance system. The real world may have room to allow both systems
to exist. My preference has to do with the values I believe in, and I understand others have their own values and beliefs.
So please hear me out on this particular question of Bitcoin’s binary search tree. I discuss it not only because it is of unique technical significance but also because it’s a good example to show
why one should pay attention to what Dr. Wright says.
Merkle tree and binary tree
Dr. Wright said the Merkle tree structure that was designed by Satoshi originally and has now been implemented in BSV along with SPV (Simplified Payment Verification) which is a binary search tree.
His opponents suggest that it is ridiculously wrong because everyone who has a basic understanding of computer science would know that a Merkle tree is not a binary search tree.
His opponent is right in the sense that the Merkle tree, in its base application taught in school, is not a binary search tree.
But what they do not know is that Satoshi had designed the Bitcoin Merkle tree in such a way that, although it is a Merkle tree, it is also a search tree, even a binary search tree.
Dr. Wright is just unbelievably right, once again.
Let me explain the technical details below.
First of all, the key concept about a ‘binary tree’ refers to a feature that cuts the tree (together a corpus of data) in two halves (hence ‘binary’) in every step. It determines the direction to
proceed for the next step based on a match or mismatch and discards the half in the other direction. This is repeated until the target item is located. This binary fashion makes the process very fast
and scalable. Because every step is a binary, it progresses exponentially, resulting in a logarithmic scalability.
The fact that half of the data can be instantly disregarded with no risk of losing the target is what creates this remarkable effect. But this necessarily requires that the nodes and leaves in the
tree are ordered and balanced in a particular way.
Therefore, calling a tree “binary” means that the tree is constructed in a way that an act (e.g., a search) can be performed in a binary fashion. If the structure does not have this binary feature,
then calling it binary would be incorrect. But if the structure does have this binary feature, but no one else knows such a structure and has never called it binary, the one who first calls it would
be not only correct but also innovative.
The binary tree can be structured to perform various functions and achieve various goals. When the tree is structured to perform a binary search, it is a binary search tree. But binary trees can be
used for other purposes, such as for ensuring data integrity.
So the question is, why does Dr. Wright call Bitcoin’s Merkle tree a binary search tree?
First, if Bitcoin’s Merkle tree is structured in a way to find the location of a certain piece of data and confirm its existence, it is a search tree. This is plain, regardless of what others say.
But is it ‘binary’ and hence a binary search tree? That is a more elaborate consideration.
The answer is yes. Bitcoin’s Merkle tree is a binary search tree because the Bitcoin data is deliberately structured as a key-value database, meaning that each item (a transaction or a hash) is
associated with (labeled by) a key. The keys are sequential. There are different ways to structure the keys. It depends on the design of the tree structure and the algorithm (see below for an
example). Note, here, the word “key” is a standard database technology, like a label of an item, and has nothing to do with cryptography keys.
Because Bitcoin transactions are timestamped according to the order of time each transaction is received, this sequential order is natural. Now, with the sequential keys built in, each corresponding
to a timestamped sequential transaction, you have an ordered and balanced tree of hashes of leaves.
Figure 1. A binary search tree based on a Merkle tree
Figure 1 shows an example of such a Merkle tree that is also a binary search tree.
Transactions are denoted by Di. Transaction Di corresponds to hash (i, i), which is a special case of key (i, j) where i=j.
Each node is denoted by a key consisting of a pair of numbers (i, j), such as (1, 4). The meaning of the numbers is quite simple: a node with a key (i, j) means that the data in the node and the
child nodes under it cover transactions in the range of i to j. For example, (1, 4) means that the node and the child nodes under it cover transactions 1, 2, 3, and 4. Implicitly, it also means that
they are unrelated to transactions 5, 6, 7, and 8, which are covered by the other branch under (5,8).
According to this sequential numbering, (i, k) and (k+1, j) are two neighboring nodes, as indicated by k and k+1. When two neighboring hashes, H (i, k) and H(k+1, j), are concatenated and hashed, it
generates a new hash H (i, j). The resulting keys (i, j) are logical because concatenating two neighbor hashes concatenates the two ranges, namely i to k and k+1 to j, to create a new range i to j,
hence its resulting hash is denoted as H (i, j). Note that the middle numbers k and k+1 do not appear in the new key (i, j). This is because the range i to j denotes an extended range which
necessarily covers the middle numbers k and k+1.
Figure 1 illustrates a simple case where i=1, 2,…8. In reality, i=1, 2,…n, where n is the total number of transactions in the block and can be a large number.
Merkle tree is formed as follows:
1. Hash each transaction Di to get H(i, i). This is the first level of the Merkle tree, in which transaction data items and their hashes have a 1-to-1 correspondence.
2. Hash the concatenation of every pair of neighboring H(i, i) and H(i+1, i+1) to get H(i, i+1). This gets the second level. The second level will have n/2 nodes.
3. Repeat the same process in step 2 above until we reach the final level where only one node remains. This is the Merkle root.
Because the number of nodes in each next level is halved, this process reaches the root (top) quickly. The total number of levels m=log2(n). The more transactions the block has, the greater advantage
it enjoys.
For example, when n = 8, m=3. When n= 1 million, m=20; when n=1 billion, m=30; when n=1 trillion, m=40, and so on. You can see that, from n=8 to n=1 billion, the total number of transactions in a
block has increased more than 100 million times, and the total number of levels in the tree increases only 10 times.
The above is the basic characteristics of the Merkle tree. But it will be clear that labeling the nodes using sequential keys transforms the Merkle tree into a binary search tree.
In the example shown in Figure 1, suppose we need to search for transaction D5. which corresponds to key (5, 5). We start from the top (root) H(1, 8), which indicates that the block has a total of 8
transactions from 1 to 8).
First, we compare our target (5, 5) with the present node (1,8). Because 5 > 8/2, we know our target is on the right side. We go to the right side to arrive at node H(5, 8) and disregard the entire
left branch, which is half of the tree. If this is not obvious in this small example, imagine that if the block has 1 billion transactions, this one step skips 500 million transactions.
Next, we compare our target (5, 5) with the present node (5, 8). Because 5=5, we know we already got this part of the key correct, which is the lower boundary of the transaction range. But because 5
<8, we go to the left and reach H(5,5), which corresponds to transaction D5, the exact transaction we are looking for.
Thus, the Merkle tree structured above enables binary search and is thus a binary search tree. There can be no debate over this because it is precisely how a binary search tree is defined.
Such a binary search tree combined with an elaborate design of SPV makes Bitcoin, as implemented in BSV, extremely powerful.
Binary search tree and SPV
SPV itself is another subject matter. Although it has been disclosed in the Bitcoin white paper, SPV has only been properly understood and further developed on Bitcoin SV (BSV). I will skip an
explanation of SPV itself, but would like to emphasize it’s relevant to Bitcoin’s binary search tree.
SPV is about scalability. If you only look at an individual incidence of how SPV works with a user submission of the Merkle path of his transaction, it would seem that it has nothing to do with
search because the user readily produces the Merkle path, which specifically locates not only the transaction ID but also its location, leaving nothing else to be searched on the blockchain. The only
thing that remains is that the recipient needs to compute a Merkle proof to verify the existence of the transaction using the Merkle path.
But remember, SPV is about scalability. It means that it’ll work on a blockchain that has billions or even trillions of transactions in a single block. On such a scaled blockchain, scalable and
efficient search becomes absolutely essential, for otherwise, everyone, including the sender, the recipient, and the service providers, will be at a loss.
So we need scalable search along with SPV. But what else can do this better than a binary search tree?
The binary search tree and SPV thus work together like a hand-in-glove. That is what BSV has done. The upcoming Teranode under testing is going to showcase these features. If you don’t care about a
scalable system, you may not be excited to hear this. But that doesn’t mean you should puff up to claim that Dr. Wright is ignorant because he claims the Bitcoin Merkle tree is a binary search tree.
To the contrary, you should find your own ignorance in failing to understand what he said.
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Equations devs & users
So I am trying to add inductive types to my toy language, and the constructor for recursor looks as follows:
Inductive Term : Type :=
| var (idx: nat): Term
| rec
(* name of the inductive type we are recursing on *)
(n: string)
(params: seq Term)
(* output type of the recursion branch *)
(C: Term)
(* what do to for every constructor*)
(branches: seq Term)
(* the main argument to provide to the rec function *)
(arg: Term).
The problem is that any recursive function has to also recurse over params and branches as well, and many o them fail for arbitrary reasons:
I was able to define a length function which simply counts nodes:
Equations term_nodes (t: Term) : nat := {
term_nodes (var _) := 1;
term_nodes (rec _ p c b a) := 1 + (sumn (map term_nodes p)) + term_nodes c + (sumn (map term_nodes b)) + term_nodes a
Definition term_lt t1 t2 : Prop := term_nodes t1 < term_nodes t2.
Then I could do something like:
Equations rename (ξ : nat -> nat) (t: Term): Term := {
rename ξ (var idx) := var (ξ idx);
rename ξ (rec n p c b a) := let ξ' := uprenn ξ (length p) in
rec n (rename_params ξ p)
(rename ξ' c)
(rename_branches ξ' b)
(rename ξ a);
where rename_params (ξ: nat -> nat) (s: seq Term) : seq Term := {
rename_params ξ [::] := [::];
rename_params ξ (t::s') := (rename (uprenn ξ (length s')) t) :: (rename_params ξ s');
where rename_branches (ξ: nat -> nat) (s: seq Term) : seq Term := {
rename_branches ξ [::] := [::];
rename_branches ξ (t::s') := (rename ξ t):: rename_branches ξ s'
Which works perfectly, and coq can actually see that everything is structurally decreasing, but when I do a similar function
Equations subst (σ: nat -> Term) (t: Term) : Term := {
subst σ (var idx) := σ idx;
subst σ (rec n p c b a) := rec n (subst_params σ p) c b a;
with subst_params (σ: nat -> Term) (s: seq Term) : seq Term := {
subst_params σ [::] := [::];
subst_params σ (t'::s') := (subst σ t')::(subst_params σ s');
Coq gives:
Error: Cannot guess decreasing argument of fix.
It is not clear what is the problem, I tried using
Equations subst (σ: nat -> Term) (t: Term) : Term by wf t term_lt
and I get the error: Error: Mutual well-founded definitions are not supported
So my question is how can I just get this to work ?
Also, I am worried that if writing code is hard, then writing proofs is even harder so any general tips will also be appreciated
It might be another case than rec/params where your definition is not structurally recursive
this is not possible, the other cases are simple and they always worked and I never had to make any changes to them,
Let me show it here:
Equations subst (σ: nat -> Term) (t: Term) : Term := {
subst σ (var idx) := σ idx;
subst σ (univ l) := univ l;
subst σ (pi A B) := pi (subst σ A) (subst (up σ) B);
subst σ (lam y) := lam (subst (up σ) y);
subst σ (appl m n) := appl (subst σ m) (subst σ n);
subst σ (sigma A B) := sigma (subst σ A) (subst (up σ) B);
subst σ (pair m n) := pair (subst σ m) (subst σ n);
subst σ (proj1 m) := proj1 (subst σ m);
subst σ (proj2 m) := proj2 (subst σ m);
subst σ (gvar t) := gvar t;
subst σ (induct t) := induct t;
subst σ (constr t) := constr t;
subst σ (rec n p c b a) := rec n (subst_params σ p) c b a;
with subst_params (σ: nat -> Term) (s: seq Term) : seq Term := {
subst_params σ [::] := [::];
subst_params σ (t'::s') := (subst σ t')::(subst_params σ s');
again this works with rename which takes (nat -> nat) but fails with subst which takes (nat -> Term)
if you use a section like
Section S.
Variable (σ: nat -> Term).
Equations subst (t: Term) : Term := {
subst (var idx) := σ idx;
the error might be more explicit, because there is only 1 argument so there is no guessing about which might be the structural argument
or you could add an explicit by struct t annotation
Gaëtan Gilbert said:
if you use a section like
Section S.
Variable (σ: nat -> Term).
Equations subst (t: Term) : Term := {
subst (var idx) := σ idx;
the error might be more explicit, because there is only 1 argument so there is no guessing about which might be the structural argument
I am not sure if that will work, As you see there cases where \sigma is not passed as is as in subst σ (sigma A B) := sigma (subst σ A) (subst (up σ) B);
Kenji Maillard said:
or you could add an explicit by struct t annotation
I did that as follows:
Equations subst (σ: nat -> Term) (t: Term) : Term by struct t := {
subst σ (var idx) := σ idx;
subst σ (univ l) := univ l;
subst σ (pi A B) := pi (subst σ A) (subst (up σ) B);
subst σ (lam y) := lam (subst (up σ) y);
subst σ (appl m n) := appl (subst σ m) (subst σ n);
subst σ (sigma A B) := sigma (subst σ A) (subst (up σ) B);
subst σ (pair m n) := pair (subst σ m) (subst σ n);
subst σ (proj1 m) := proj1 (subst σ m);
subst σ (proj2 m) := proj2 (subst σ m);
subst σ (gvar t) := gvar t;
subst σ (induct t) := induct t;
subst σ (constr t) := constr t;
subst σ (rec n p c b a) := rec n (subst_params σ p) c b a;
with subst_params (σ: nat -> Term) (s: seq Term) : seq Term by struct s := {
subst_params σ [::] := [::];
subst_params σ (t'::s') := (subst σ t')::(subst_params σ s');
And now I get the full error:
Error: Recursive definition of subst_params is ill-formed.
In environment
subst : (nat -> Term) -> Term -> Term
subst_params : (nat -> Term) -> seq Term -> seq Term
σ : nat -> Term
s : seq Term
t : Term
l : seq Term
Recursive call to subst has principal argument equal to
"t" instead of "l".
Recursive definition is:
"fun (σ : nat -> Term) (s : seq Term) =>
match s with
| [::] => [::]
| t :: l => subst σ t :: subst_params σ l
I am not sure what the problem is, it says subst is recusing over t instead of l and apparently this is a problem here
note that the rename function is almost the same structure, ad there was no problems regarding being ill-formed.
what is truly puzzling me, is that all functions are working except for this subst function:
I created a function:
Equations term_nodes (t: Term) : nat := {
term_nodes (var _) := 1;
term_nodes (univ _) := 1;
term_nodes (pi A B) := 1 + term_nodes A + term_nodes B;
term_nodes (lam y) := 1 + term_nodes y;
term_nodes (appl m n) := 1 + term_nodes m + term_nodes n;
term_nodes (sigma A B) := 1 + term_nodes A + term_nodes B;
term_nodes (pair m n) := 1 + term_nodes m + term_nodes n;
term_nodes (proj1 m) := 1 + term_nodes m;
term_nodes (proj2 m) := 1 + term_nodes m;
term_nodes (gvar _) := 1;
term_nodes (induct _) := 1;
term_nodes (constr _) := 1;
term_nodes (rec _ p c b a) := 1 + (sumn (map term_nodes p)) + term_nodes c + (sumn (map term_nodes b)) + term_nodes a
Which is almost the same structure for recursion and I tried to show that this is the thing structurally decreasing on every recursive call, but coq Equations is does not support wf for mutual
recursion as in subst
also when I tried to use use the same map trick on subst instead of subst_params it was accepted, because the correct subst_paramwill be more complex than the one shown, and this is just the mimimum
bug reproducer.
I guess mutual fixpoint doesn't work well with nested types, instead you need separately defined fixpoints
you may have to define subst_params first with an abstract subst, ie
Section S.
(* IIRC the "subst" argument needs to be a lambda not a fix argument *)
Variable (subst : (nat -> Term) -> Term -> Term).
Equations subst_params (σ: nat -> Term) (s: seq Term) : seq Term by struct s := {
subst_params σ [::] := [::];
subst_params σ (t'::s') := (subst σ t')::(subst_params σ s');
End S.
Equations subst (σ: nat -> Term) (t: Term) : Term by struct t := {
subst σ (rec n p c b a) := rec n (subst_params subst σ p) c b a;
this is basically how map works, its function argument is a lambda not fix argument iemap = fun A B f => fix map l := ... not map = fix map A B f l := ...
so your suggestion worked, but I don't fully understand it
so I kept making small modifications till it get broken, and this is the last version that works:
Section SubstBranch.
Variable subst : (nat -> Term) -> Term -> Term.
Fixpoint map_rev_index (σ: nat -> Term) (s: seq Term) : seq Term :=
match s with
| [::] => [::]
| t::s' => let l := size s' in
(subst (upn l σ) t)::map_rev_index σ s'
End SubstBranch.
Equations subst (σ: nat -> Term) (t: Term) : Term := {
subst σ (var idx) := σ idx;
subst σ (univ l) := univ l;
subst σ (pi A B) := pi (subst σ A) (subst (up σ) B);
subst σ (lam y) := lam (subst (up σ) y);
subst σ (appl m n) := appl (subst σ m) (subst σ n);
subst σ (sigma A B) := sigma (subst σ A) (subst (up σ) B);
subst σ (pair m n) := pair (subst σ m) (subst σ n);
subst σ (proj1 m) := proj1 (subst σ m);
subst σ (proj2 m) := proj2 (subst σ m);
subst σ (gvar t) := gvar t;
subst σ (induct t) := induct t;
subst σ (constr t) := constr t;
subst σ (rec n p c b a) := let σ' := upn (length p) σ in
rec n (map_rev_index subst σ p)
(subst σ' c)
(map (subst σ') b)
(subst σ a);
Which is exactly what I want, but I make this tiny modification to the branch function:
Fixpoint map_rev_index (subst : (nat -> Term) -> Term -> Term) (σ: nat -> Term) (s: seq Term) : seq Term :=
match s with
| [::] => [::]
| t::s' => let l := size s' in
(subst (upn l σ) t)::map_rev_index subst σ s'
I get the exact same error of "cannot figure the decreasing argument of fix" for in the real subst equation
Gaëtan Gilbert said:
this is basically how map works, its function argument is a lambda not fix argument iemap = fun A B f => fix map l := ... not map = fix map A B f l := ...
I guess it has something to do with this but I don't understand what you are trying to say
@walker Fixpoint foo with bar can _only_ consume types defined as Inductive foo with bar (mutual inductives), _not_ Inductive foo := ... seq foo (EDIT: that's what's called a nested inductive)
it'd be nice if Coq were more liberal (and Agda is, but they have a _very_ different story for soundness)
said otherwise, the recursion structure of your fixpoints must match closely the recursion structure of your inductives
that would make sense, alright. my next question is, is there general pattern for dealing with nested induction ?
yes! for a more extensive explanation, my favorite Google query is "cpdt coq nested inductive", which leads to http://adam.chlipala.net/cpdt/html/Cpdt.InductiveTypes.html; look at "Nested Inductive
but there's a second point about how map must be implemented, and I don't remember if that's explained there
I think I checked that book in the past and if I remember correctly, my conclusion was it was mostly creativity, but I will check it again.
(I don't think so and I've never fully understood it — I've just seen it come up often)
generally, I'd say creativity is the problem; being _less_ creative and following the inductive structure more closely is the fix. Since seq is a separate parametric inductive, then functions on it
(like map or subst_params) must be separate fixpoints
noted, I will reread this section again while keeping this in mind, to be honest I am scared how the proofs of type safety will end up looking like given that I was struggling with just basic
there is a symmetry between how the inductives are defined and how the fixpoints are defined
with the inductives there the inside type which takes a parameter, and the outside type which instantiates the inside type's parameter with itself (ie with the outside type)
with the fixpoints the fixpoint for the inside type needs to take a "parametric" argument which will b instantiated with the fixpoint for the outside type, after reduction this produces something
like fix outside_fix x := ... (fix inside_fix y := ... (outside_fix something) ...) ... ie a nested fixpoint
importantly it does NOT produce fix outside_fix x := ... ((fix inside_fix f y := ...) outside_fix ..., because the guard checking will not identify the f argument and outside_fix so it only sees that
you are using the unapplied outside_fix in some abstract way, which is not allowed
for a concrete example look at
Require Import List.
Inductive rose := Rose (_:list rose).
Fixpoint size r := let '(Rose l) := r in S (fold_left (fun acc r => acc + (size r)) l 0).
Eval cbv [size fold_left] in size.
Fixpoint bad_fold_left A B (f:A->B->A) l acc :=
match l with
| nil => acc
| x :: tl => bad_fold_left A B f tl (f acc x)
Fail Fixpoint bad_size r := let '(Rose l) := r in S (bad_fold_left _ _ (fun acc r => acc + (bad_size r)) l 0).
Eval cbv [bad_fold_left] in (fun bad_size r => let '(Rose l) := r in S (bad_fold_left _ _ (fun acc r => acc + (bad_size r)) l 0)).
BTW I don't think this is necessarily a fundamental restriction, after all Coq does track that the r in fun acc r => is related to l
it's just how it works currently
You can use where instead of with for nested fixpoints
I haven't tried it, but what will be the point of where if the two functions are no longer mutual?
Equations lets you write them as if they were mutual but in reality you can think of the the where definition being expanded at each use, as required for nested fixpoints by Coq's kernel.
I think I will need more help here,
So here is the current progress:
I was able to define this function:
Section MapIndexRev.
Variable T U: Type.
Variable f: nat -> T -> U.
Fixpoint map_index_rev (s: seq T) : seq U :=
match s with
| [::] => [::]
| t::s' => let l := size s' in
(f l t)::map_index_rev s'
it is decreasing on s and I used it to create the following as previously described:
Section RenameParams.
Variable (rename: (nat -> nat) -> Term -> Term).
Definition map_rename_params (ξ: nat -> nat) (s: seq Term) : seq Term :=
map_index_rev (fun l t => (rename (uprenn l ξ) t)) s.
End RenameParams.
Equations rename (ξ : nat -> nat) (t: Term): Term := {
rename ξ (var idx) := var (ξ idx);
rename ξ (univ l) := univ l;
rename ξ (pi A B) => pi (rename ξ A) (rename (upren ξ) B);
rename ξ (rec n p c b a) := let ξ' := uprenn (size p) ξ in
rec n
(map_rename_params rename ξ p) (* The part that will act funny *)
(rename ξ' c)
(map (rename ξ') b)
(rename ξ a);
Now in A different Coq user thread I got an equivalent but nicer map_index_rev implementation:
Section MapIndexRevParam.
Variable T U: Type.
Variable f: nat -> T -> U.
Definition map_index_rev_params (start count: nat) (s: seq T) :=
map (uncurry f) (zip (rev (iota start count)) s).
End MapIndexRevParam.
Section MapIndexRev.
Variable T U: Type.
Variable f: nat -> T -> U.
(* this one does not work with subst, and rename*)
Definition map_index_rev' (s: seq T) :=
map_index_rev_params f 0 (length s) s.
End MapIndexRev.
but when I use it in map_rename_params no problem happens, But then problems appear in rename itself, first it cannot guess the decreasing argument, then I did Equations rename (ξ : nat -> nat) (t:
Term): Term by struct t and this was the error
In environment
rename : (nat -> nat) -> Term -> Term
ξ : nat -> nat
t : Term
n : String.string
params : seq Term
C : Term
branches : seq Term
arg : Term
ξ' := uprenn (size params) ξ : nat -> nat
ξ0 := ξ : nat -> nat
s := params : seq Term
s0 := s : seq Term
start := 0 : nat
finish := length s0 : nat
s1 := s0 : seq Term
map : seq (nat * Term) -> seq Term
s2 : seq (nat * Term)
x : nat * Term
s' : seq (nat * Term)
p := x : nat * Term
x0 : nat
y : Term
l := x0 : nat
t0 := y : Term
Recursive call to rename has principal argument equal
to "t0" instead of
one of the following variables: "params"
"C" "branches" "arg" "s" "s0"
Recursive definition is:
"fun (ξ : nat -> nat) (t : Term) =>
match t with
| var idx => var (ξ idx)
| univ l => univ l
| pi A B => pi (rename ξ A) (rename (upren ξ) B)
| lam y => lam (rename (upren ξ) y)
| appl m n => appl (rename ξ m) (rename ξ n)
| sigma A B =>
sigma (rename ξ A) (rename (upren ξ) B)
| pair m n => pair (rename ξ m) (rename ξ n)
| proj1 m => proj1 (rename ξ m)
| proj2 m => proj2 (rename ξ m)
| gvar n => gvar n
| induct n => induct n
| constr n => constr n
| rec n params C branches arg =>
let ξ' := uprenn (size params) ξ in
rec n (map_rename_params rename ξ params)
(rename ξ' C) [seq rename ξ' i | i <- branches]
(rename ξ arg)
So what makes map_index_rev good function and what makes map_index_rev' bad one from. My code feels a bit unstable, because modification to one place will lead to compiler errors in a completely
different location
I was intending to just stick to map_index_rev but writing proofs for this one is very hard, for instance, I cannot even figure the lemma for map_index_rev (s1 ++ s2) but the version built with map,
iota, zip it is easier.
[DEL:Where is map_index_rev actually reversing the list? It seems just map_index?:DEL]
Paolo Giarrusso said:
[DEL:Where is map_index_rev actually reversing the list? It seems just map_index?:DEL]
well in the first iteration of development, I must admit, I tried so many things and only this version was the one that made the fix checker happy, but it is very hard to prove anything about :(
No the indexes are reversed, that's fine
At least, you can prove the two versions are equivalent and then do the actual proofs on the nice version
no I couldn't do that, but I can open a different thread for that
Ah, you'd probably need to lift out the size computation
It feels not straight forward to show that the two implementations are equal, because It is not clear what I am doing induction on, forward induction gets stuck at some point, backwards induction get
stuck at different point.
Paolo Giarrusso said:
Ah, you'd probably need to lift out the size computation
yes and it gets fuzzy from here, so I thought , perhaps not worth it
On your question: it's not clear that map (uncurry (rename ...)) is structurally recursive
yes as a I suspected, but what are my options, other than proving that both versions are equivalent, which I don't know how to do?
Instead of doing a recursive call on the sublist, you're zipping it and applying uncurry (rename ...) to it... I'm not sure the termination checker can expose this is structurally recursive by
Well! The problem for structural recursion is the zipping, and the problem for proofs is the size computation
is there a way to prove structural recursion manually in equations, like in program ? I mean I know it is not possible, but what is the closest alternative
That's not structural recursion. You can use well-founded induction but I'd try to avoid it here...
Paolo Giarrusso said:
That's not structural recursion. You can use well-founded induction but I'd try to avoid it here...
oh that is the by WF thing ... yeah I never understood it and never ever knew how to do it by myself for once
So one approach (untested) is to combine the working approaches in both cases: write a recursive function (like the one passing termination checking) and passes start and/or count like the
well-behaved function... Sketch in 5 minutes
Paolo Giarrusso said:
So one approach (untested) is to combine the working approaches in both cases: write a recursive function (like the one passing termination checking) and passes start and/or count like the
well-behaved function... Sketch in 5 minutes
I will think how to do it, because without iota + rev, I will have to do something similar to count -= 1 which was something I tried hard to avoid
at this point, the metacoq solution for this problem starts to feel nice (due to Meven):
Definition map_index_rev {A B : Type} (f : nat -> A -> B) (l : list A) : list B :=
mapi (fun n => f (length l - n)) 0 l.
the subtraction is ofcourse annoying but I guess it is the not that annoying when compared to dealing with unhappy termination checker.
Is there a fundamental reason why you need to present the list in this order? If you are going to be doing a lot of de Bruijn manipulations on your list (and it looks like you are), it is nice to
present it in the direction that makes these manipulations the least painful
(Btw, when I referred to MetaCoq I meant mostly for the usage of mapi, I believe we have very little length/subtraction yoga there, because we try and do things "in the right direction")
(And the one place I remember where we reverse lists around is a formalization nightmare and one where I wish I'll never have to get into again)
so this is in particular the parameter list for inductive types:
Inductive MyType (T : Type) (L: seq T).......
Then there is two directions here to store this list of T and L ... the direction that makes subtitution work with out subtraction (that will be T::L::[::] because then T will be lifted zero times,
and L will be lifted once ...etc
But there is the direction that makes type checking natural which will be the other direction L::T::[::] because than we can always pop elements from the left and we are still left with well formed
context .
Tell me that this is the one case where you had to deal with subtraction (checking metacoq code base is still a bit hard for me because everything is extra generalized)
If I recall correctly, this is a situation where we have two notions of a "closed context", one of them where we see the context as an extension of another one and want to start it from one side
(which would be T in your example I think), and one where we want to see a context by itself and want to start from the other side (the one on which we add new types)
So it is indeed more or less linked with the issue you describe
And in theory we have lemmas relating the two notions of closedness, but in practice it is really frustrating to juggle between them
what about doing rev (mapi f (rev l))?
but would rev l be a subterm when l is?
I guess not
how about
Definition map_index_rev T U (f : nat -> T -> U) s :=
snd (List.fold_right (fun x '(len, v) => let v' := f (S len) x in (S len, v' :: v)) (0, nil) s).
Last updated: Oct 13 2024 at 01:02 UTC
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On the combinational complexity of certain symmetric Boolean functions for Mathematical Systems Theory
Mathematical Systems Theory
On the combinational complexity of certain symmetric Boolean functions
View publication
A property of the truth table of a symmetric Boolean function is given from which one can infer a lower bound on the minimal number of 2-ary Boolean operations that are necessary to compute the
function. For certain functions of n arguments, lower bounds between roughly 2 n and 5 n/2 can be obtained. In particular, for each m ≥ 3, a lower bound of 5 n/2 -O(1) is established for the function
of n arguments that assumes the value 1 iff the number of arguments equal to 1 is a multiple of m. Fixing m = 4, this lower bound is the best possible to within an additive constant. © 1977
Springer-Verlag New York Inc.
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Introduction to Gradient Decent - DataScienceCentral.com
The gradient decent approach is used in many algorithms to minimize loss functions. In this introduction we will see how exactly a gradient descent works. In addition, some special features will be
pointed out. We will be guided by a practical example.
First of all we need a loss function. A very common approach is
where y are the true observations of the target variable and yhat are the predictions. This function looks lika a squared function – and yes, it is. But if we assume a non linear approach for the
prediction of yhat, we end up in a higher degree. In deep neural networks for example, it is a function of weights and input-features which is nonlinear.
So, lets assume that the function looks like the following polynom – as an example:
Where f is the loss and x the weights.
• f is the loss function
• f’ is the first derivation to x and
• f” is the second derivation to x.
We want to find the minimum. But there are two minima and one maximum. So, firstly we find out where the extreme values are. For this intention, we use the Newton iteration approach. If we set a
tangent to an arbitrary point x0 of the green function we can calculate where the linear tangent is zero instead. Through this procedure we iteratively approach the zero points of f’ and thus obtain
the extrema. This intercept with x is supposed to be x1.
When we have x1, then we start again. But now with x1.
To find the three points of intersection with x, we set x0 close to the suspected zero points which we can see in the function-plot.
x0 = c(1, -1, -4)
j = 1
for(x in x0) {
for (i in 1:20) {
fx = 6 * x ^ 5 + 20 * x ^ 4 + 8 * x ^ 3 + 15 * x ^ 2 + 40 * x + 1
dfx = 30 * x ^ 4 + 80 * x ^ 3 + 24 * x ^ 2 + 30 * x + 40
x = x – fx / dfx
assign(paste(“x”, j, sep = “”), x)
j = j + 1
Results are:
x1 = -0.03
x2 = -1.45
x3 = -2.9
Now that we know where f has its extrema, we can apply the gradient decent – by the way, for the gradient descent you don’t need to know where the extremes are, that’s only important for this
example. To search for the minima, especially the global minimum,we apply the gradient decent.
The gradient decent is an iterative approach. It starts from initial values and moves downhill. The initial values are usually set randomly, although there are other approaches.
Where x is for example the weight of a neuron relation and
is the partial derivation to a certein weight. The i is the iteration index.
Why the negative learning_rate times the derivation? Well, if the derivation is positive, it means that an increase of x goes along with an increase of f and vice versa. So if the derivative is
positive, x must be decreased and vice versa.
Lets start from:
x = 2
We set the learning rate to:
learning_rate = .001
for(i in 1:2000){
fx = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = x – fx*learning_rate
Here it becomes clear why the initial values (weights) are so important. The minimum found corresponds to x1 and not x3. This is due to the initial value of x = 2. Let us now change it to:
x = -2
for(i in 1:2000){
fx = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = x – fx*learning_rate
This is the global minimum we have been searching for.
Now, we change the learning rate to:
learning_rate = .007
And we start from:
x = 2
for(i in 1:2000){
fx = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = x – fx*learning_rate
How can such an obviously wrong result come about?
Well, let´s plot the first 100 x iterations:
This phenomenon is called oscillation and hinders the optimization. It happens because the absolute increase f’ is same for the two opposite sides at x = -3.03 and x = -2.67:
If we subtract the gradients from the opposite sides, we should get a difference of zero.
x = -2.67302
fx1 = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = -3.02809
fx2 = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
Difference is:
Learning Rate Decay
However, oscillation is not only a phenomenon that cannot be influenced, it occurs due to the chosen learning rate and the shape of the loss function. This learning rate causes the algorithm to jump
from one side to the other. To counteract this, the learning rate can be reduced. Again, we start from:
x = 2
Initial learning rate:
learning_rate = .007
Learning rate decay:
decay = learning_rate / 2500; decay
checkX = NULL
for(i in 1:2000){
fx = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = x – fx*(learning_rate – decay * i)
checkX = c(checkX, x)
This asymmetric wedge-shaped course shows that the shrinking learning rate fulfils its task. As we know, -2.9 is the correct value.
A frequently used approach is to add a momentum term. A momentum term is the change of the value x (weight) from one iteration before:
Where M is the weight of the momentum term. This approach is comparable to a ball rolling down the road. In the turns, it swings back with the swing it has gained. In addition, the ball rolls a bit
further on flat surfaces. Lets check what happens with the oscillation when we add a momentum term.
x = 2
Learning rate is again:
learning_rate = .007
Momentum weight is set to:
M = .2
checkX = NULL
for(i in 1:2000){
fx = 6*x^5+20*x^4+8*x^3+15*x^2+40*x+1
x = x – fx*learning_rate + M * x_1
x_1 = – fx*learning_rate
checkX = c(checkX, x)
As you can see, a minimum was reached, but unfortunately the wrong one. A Momentum term can protect against oscillation, but it is no guarantee for the global minimum.
We can see that the process of optimization of the gradiant decent is highly dependent from the initial values (weights). So, it is possible that the best solution you find when using a method based
on a gradient decent – such as a neural network – is not optimal. To prevent from oscillation, one can add a momentum term or change the learning rate. But even that is no guarantee of finding the
global minimum.
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Execution time limit is 1 second
Runtime memory usage limit is 122.174 megabytes
Find the Lowest Common Multiple of n positive integers.
The first line contains number n (1 < n < 21). The second line contains n positive integers, separated with a space. All integers are not greater than 100.
Print the Lowest Common Multiple of n given numbers.
Submissions 20K
Acceptance rate 12%
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Where's the Father? :: Transum Newsletter
Where's the Father?
Sunday 1st August 2021
This is the Transum Newsletter for the month of August 2021. It begins, as usual with the puzzle of the month.
A woman is 21 years older than her son.
In six years' time she will be 5 times as old as her son.
Where is the father?
While you think about that (the answer is at the end of this newsletter) here are some of the key resources added to the Transum website during the last month:
I am very grateful to White Rose Mathematics for allowing me to share their Schemes of Learning (known as schemes of work back in the day) on the Transum website. I am making each scheme into a
clickable page linking to the Transum resources for each of the Blocks (topics). For example here is Year 8:
www.transum.org > Enter > Curriculum > Year 8
I hope those of you using the White Rose schemes find this directory useful.
One of the new set of exercises I created while working on the Year 7 scheme is called Fraction of... It proved to be particularly popular as many pupils earned trophies from it in its first day.
Product Square is a randomly-generating puzzle presented in eight levels. The objective is to place the numbers one to nine on a three by three grid so that the row and column products are as given.
At level one all but two of the numbers are already in place on the grid but as you go forward through the levels the number of ready-placed numbers is less generous. At level eight no clues are
given and you will find that knowledge of the divisibility tests is really useful.
Bar Charts was suggested by the Conyers' School Head of Mathematics and went live in the middle of last month. It begins with reading simple bar charts then continues with dragging bars to the
correct height then comparative, composite and grouped bar charts.
Angles Mixed has a new Level 4 which includes circle theorems.
Equivalent Fractions has a new Level 4 dedicated to cancelling fractions.
Venn Diagrams of Sequences is one of the wonderful ideas from Craig Barton and he kindly gave me permission to make an interactive version of the activity. I've already tried it out on some students
and have been very pleased with the results. The best time to use it is just after the time when students can look at an arithmetic sequence and figure out the formula for the nth term in their
How can the game of Dots and Boxes be upgraded? By adding fractions as scores is the answer. I would like to introduce you to Boxed In Fractions.
Overloaded Fraction is yet another new resource (can you tell I'm in lockdown?). It is presented in the form of a ten-level puzzle in the drag and drop style. Use the digits provided to produce a
fraction equivalent to the simplified fraction given.
For those in the northern hemisphere it won't be too long until you are Back to School. I hope you find one or two of the resources I have selected useful.
I heard recently that each ear of corn has an even number of rows. I haven't checked and have yet to find a nice way to include that in a mathematical activity. Ideas welcome!
When you use Google it sometimes goes above and beyond the normal list of search results and provides instant search cards at the top of the page. These cards contain the information you searched
for, but also include interactive controls to let you experience the information in a more engaging manner. A canny pupil might use these cards as a short cut to answering their maths homework.
With this in mind I remembered a poster I had in my classroom when I first started teaching many years ago. I think it was produced by the Mathematical Association but I can't find it on their
website so I assume it is out of print. Here is the gist of the poster as best I can remember it.
Looking up the answer in the back of the book to check you have answered the question correctly is not cheating.
Asking a friend how to solve a problem is not cheating.
Using a calculator to check your work is not cheating.
Cheating is when you pretend you understand when you really don't. That's when you are cheating yourself.
So in 2021 we should add "Using Google search cards to help develop a better understanding of a concept is not cheating."
Transum subscriber Claudette suggested a modification to the teacher's Trophies page. She wanted to be able to be able to select which trophies she can view by selecting the activities in the key
below the table. That feature has now been included. It does make me realise just how many trophies pupils are earning and how easy it is for teachers to keep track of progress. Just think of all
that marking you are spared!
Don't forget you can listen to this month's podcast which is the audio version of this newsletter. You can find it on Spotify, Stitcher or Apple Podcasts. You can follow Transum on Twitter and 'like'
Transum on Facebook
Finally the answer to this month's puzzle:
W = S + 21
(W + 6) = 5(S + 6)
S + 27 = 5S +30
4S = -3
S = -3/4
That's strange. A negative amount! In terms of time this represents nine months before the son is to be born. I will leave the last part of this puzzle for you to work out yourself. Where was the
father nine months before his son was born?
That's all for now,
P.S. I'm a Maths teacher ... of course I have problems!
Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics
anywhere in the world. Click here to enter your comments.
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What Is Dsa In Computer Science? - Jamie Foster Science
What Is Dsa In Computer Science?
DSA or Data Structures and Algorithms form the fundamental building blocks of computer programming. If you’re short on time, here’s a quick answer: DSA refers to the key mathematical and
organizational structures like arrays, linked lists, trees, graphs as well as techniques like recursion, sorting/searching used to design efficient and optimized computer programs.
In this comprehensive guide, we explain what DSA is, why it is important in computer science, and provide an overview of the major data structures and algorithms concepts covered in this field.
Understanding Data Structures
Data Structures and Algorithms (DSA) is a fundamental concept in computer science. It involves the organization, management, and storage of data in a way that enables efficient access and
Data structures are essential for solving complex problems and optimizing the performance of software applications. In this article, we will explore some of the most common data structures used in
computer science.
An array is a collection of elements that are stored in contiguous memory locations. It is one of the simplest and most widely used data structures. Arrays allow efficient access to individual
elements using an index.
They are used to store homogeneous data types such as integers, characters, or floating-point numbers. Arrays are fixed in size and have a constant time complexity for accessing elements.
Linked Lists
A linked list is a dynamic data structure that consists of a sequence of nodes, where each node contains a value and a reference to the next node in the sequence. Unlike arrays, linked lists can grow
or shrink in size at runtime.
Linked lists are particularly useful when the size of the data is unknown or constantly changing. They have a time complexity of O(1) for inserting or deleting elements at the beginning or end of the
A stack is a last-in, first-out (LIFO) data structure. Elements can only be added or removed from the top of the stack. Stacks are used to implement algorithms that require backtracking or keeping
track of nested function calls. They are also used in compiler design and expression evaluation.
Stacks can be implemented using arrays or linked lists.
A queue is a first-in, first-out (FIFO) data structure. Elements are added to the back of the queue and removed from the front. Queues are commonly used in scheduling algorithms, breadth-first
search, and simulations. They can be implemented using arrays or linked lists.
Trees are hierarchical data structures consisting of nodes connected by edges. Each node can have zero or more child nodes. Trees are used to represent hierarchical relationships, such as file
systems, organization charts, or family trees.
Binary trees are the most common type of trees, where each node has at most two child nodes.
Graphs are a collection of nodes (vertices) connected by edges. They are used to represent relationships between objects, such as social networks, transportation networks, or computer networks.
Graphs can be directed or undirected, weighted or unweighted.
They are a fundamental data structure in graph algorithms, such as breadth-first search, depth-first search, and shortest path algorithms.
Hash Tables
A hash table is a data structure that maps keys to values using a hash function. It provides efficient insertion, deletion, and retrieval of elements. Hash tables are used to implement associative
arrays, databases, caches, and symbol tables.
They have an average time complexity of O(1) for search, insert, and delete operations.
Understanding data structures is crucial for any computer scientist or software engineer. They provide the foundation for efficient algorithm design and problem-solving. To learn more about data
structures and algorithms, you can visit websites like GeeksforGeeks or TutorialsPoint that provide comprehensive tutorials and examples.
Types of Algorithms
Searching Algorithms
Searching algorithms are used to find a specific element or a set of elements within a given data structure. They are commonly used in applications such as search engines, databases, and
recommendation systems. Popular searching algorithms include linear search, binary search, and hash-based search.
These algorithms have different time complexities and are chosen based on the specific requirements of the problem at hand.
Sorting Algorithms
Sorting algorithms are used to arrange elements in a specific order, typically in ascending or descending order. They are widely used in various applications such as data analysis, database
management, and information retrieval.
Some commonly used sorting algorithms include bubble sort, insertion sort, selection sort, merge sort, and quicksort. Each algorithm has its own advantages and disadvantages in terms of time and
space complexity.
Graph Algorithms
Graph algorithms are used to solve problems related to graphs, which are a collection of nodes (vertices) connected by edges. These algorithms are widely used in network optimization, transportation
planning, and social network analysis.
Some popular graph algorithms include breadth-first search (BFS), depth-first search (DFS), Dijkstra’s algorithm, and Kruskal’s algorithm. These algorithms help in finding the shortest path,
detecting cycles, and determining the connectivity of a graph.
Greedy Algorithms
Greedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum. They are often used in optimization problems where the goal is to
find the best possible solution.
Examples of greedy algorithms include the knapsack problem, the traveling salesman problem, and the minimum spanning tree problem. These algorithms may not always guarantee the optimal solution, but
they are efficient and easy to implement.
Divide and Conquer
Divide and conquer is a problem-solving technique where a problem is divided into smaller subproblems, which are then solved independently. The solutions to the subproblems are then combined to solve
the original problem.
This technique is commonly used in algorithms such as merge sort, quicksort, and binary search. Divide and conquer algorithms are known for their efficiency and are widely used in various
Dynamic Programming
Dynamic programming is a method for solving complex problems by breaking them down into simpler overlapping subproblems. The solutions to these subproblems are stored and reused to solve larger
problems. This technique is particularly useful when the same subproblems need to be solved multiple times.
Dynamic programming is commonly used in algorithms such as the Fibonacci sequence, the knapsack problem, and the longest common subsequence problem.
Backtracking is a technique used to find solutions to problems by exploring all possible paths and backtracking when a solution is not possible. It is commonly used in problems that involve searching
for a solution in a large solution space.
Examples of problems that can be solved using backtracking include the N-Queens problem, Sudoku, and the Hamiltonian cycle problem. Backtracking algorithms can be computationally expensive, but they
guarantee finding a solution if one exists.
Mathematical Foundations
In computer science, mathematical foundations play a crucial role in understanding and analyzing algorithms and data structures. By applying mathematical concepts and principles, computer scientists
can evaluate the efficiency and effectiveness of various computational processes.
This section will explore some of the key mathematical foundations used in the field of computer science.
Time and Space Complexity
Time and space complexity are fundamental concepts in computer science that help measure the efficiency of an algorithm. Time complexity refers to the amount of time an algorithm takes to run as a
function of the input size, while space complexity refers to the amount of memory or storage an algorithm requires.
These measures are important for determining the scalability and performance of algorithms, and understanding how they will perform as the input size grows.
Big O Notation
Big O notation is a mathematical notation used to describe the upper bound or worst-case scenario of an algorithm’s time or space complexity. It provides a standardized way to compare the efficiency
of different algorithms.
For example, if an algorithm has a time complexity of O(n), it means that the running time of the algorithm grows linearly with the input size. On the other hand, if an algorithm has a time
complexity of O(n^2), it means that the running time grows quadratically with the input size.
Big O notation allows computer scientists to analyze and compare algorithms without getting bogged down in specific implementation details.
Recursion is a mathematical concept that is widely used in computer science. It refers to the process of solving a problem by breaking it down into smaller instances of the same problem. In
programming, recursive functions call themselves within their own definition, allowing them to solve complex problems by reducing them to simpler subproblems.
Recursion is particularly useful when dealing with problems that can be naturally divided into smaller, identical subproblems. However, it is important to ensure that recursive functions have a
well-defined base case to prevent infinite recursion.
By understanding the principles of recursion, computer scientists can develop elegant and efficient solutions to a wide range of problems.
For more in-depth information on mathematical foundations in computer science, you can visit https://www.cs.cmu.edu/~fp/courses/15213-s07/schedule.html or https://www.topcoder.com/thrive/articles/
Applications of DSA
Database Systems
DSA (Data Structures and Algorithms) play a crucial role in the development and optimization of database systems. They provide efficient ways to store, retrieve, and manipulate data in databases. For
example, techniques like hashing and indexing, which are based on data structures, are used to improve the performance of database operations such as searching, sorting, and joining.
DSA also helps in designing efficient data models and query optimization algorithms, making databases more reliable and scalable.
Network Protocols
DSA is extensively used in the design and implementation of network protocols. Protocols like TCP/IP rely on efficient data structures and algorithms to ensure reliable and secure communication over
computer networks.
For instance, data structures like queues and stacks are employed to handle network packets, while algorithms like routing and congestion control algorithms are used to optimize data transfer.
Without the use of DSA, network protocols would struggle to deliver data efficiently and effectively.
Compiler Design
DSA has a significant impact on the field of compiler design. Compilers are responsible for translating high-level programming languages into machine code that can be executed by a computer. Data
structures such as symbol tables, syntax trees, and symbol tables are crucial components of a compiler.
Algorithms like lexical analysis, parsing, and code optimization heavily rely on efficient data structures and algorithms to generate optimized and error-free code. DSA plays a vital role in ensuring
the correct and efficient functioning of compilers.
Graphics Algorithms
DSA is essential in the field of computer graphics as it enables the creation and manipulation of visual images. Data structures like matrices, graphs, and trees are used to represent and transform
objects in computer graphics.
Algorithms like line drawing, rasterization, and shading are employed to render images efficiently. DSA provides the foundation for creating visually appealing and interactive graphics applications.
Artificial Intelligence
DSA is at the core of many artificial intelligence (AI) algorithms and techniques. AI involves simulating human intelligence in machines, and DSA enables efficient processing and manipulation of
large datasets.
Data structures like graphs and trees are used to represent knowledge and relationships in AI systems. Algorithms like search, optimization, and machine learning rely heavily on DSA to make
intelligent decisions and predictions.
The combination of DSA and AI has led to significant advancements in fields such as natural language processing, computer vision, and robotics.
Importance of Learning DSA
Learning Data Structures and Algorithms (DSA) is essential for computer science students and professionals alike. DSA forms the backbone of efficient programming and problem-solving techniques,
allowing developers to write optimized code that can handle large amounts of data and complex operations.
1. Efficient Problem Solving
DSA provides a systematic approach to problem-solving. By understanding various data structures such as arrays, linked lists, stacks, queues, trees, and graphs, along with algorithms like sorting,
searching, and traversal, programmers can develop efficient solutions to real-world problems.
DSA helps in breaking down complex problems into smaller, manageable components, making it easier to devise effective solutions.
2. Optimized Code
Efficiency is crucial in software development. DSA enables programmers to write optimized code that minimizes memory usage and execution time. For example, using the right data structure for a
particular scenario can dramatically improve the performance of an algorithm.
By learning DSA, developers can make informed decisions about choosing the most appropriate data structure and algorithm for a given problem, resulting in faster and more efficient code.
3. Interview Performance
DSA is a fundamental topic in technical interviews for software engineering positions. Employers often assess a candidate’s problem-solving skills and ability to write efficient code using DSA
A strong foundation in DSA can greatly enhance one’s chances of performing well in job interviews and securing a desirable position in the industry.
4. Career Advancement
Proficiency in DSA opens up numerous opportunities for career advancement. Companies value programmers who can handle complex data structures and algorithms, as they are essential for building
scalable and high-performance applications.
By mastering DSA, individuals can stand out in the competitive job market and increase their chances of landing lucrative job offers and promotions.
DSA provides the key knowledge required for efficiently storing, accessing, modifying and analyzing data to design optimized software solutions. Mastering data structures and algorithms is crucial
for successfully programming and problem solving in computer science roles.
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How to Trace Excel Errors
Regardless of the version of excel you are using, there will always be error messages if something is not quite right. All errors presented by excel are preceded by a (#) hashtag and will look like
the provided screen shot. Errors will always be shown with a red triangle in the upper left corner of a cell with the particular error message as the cell value.
Types of Errors
There are many types of errors within Excel and it is important to understand the differences between them and why they occur. Below are some error values, what they mean and what they are generally
caused from.
#DIV/0 – The #DIV/0 error will occur when the division operation in your formula is referring to an argument that contains a 0 or is blank.
#N/A – The #N/A error is, in fact, not really an error. This more so indicates the unavailability of a necessary value. The #N/A error can be manually thrown using =NA(). Some formulas will throw the
error message as well.
#NAME? – The #NAME? error is saying that excel can’t find or recognize the provided name in the formula. Most often this error will appear when your formulas contain unspecified name-elements.
Usually these would be named ranges or tables that don’t exist. This is mostly caused by misspellings or incorrect use of quotations.
#NULL! – Spaces in excel indicate intersections. It’s because of this that an error will occur if you use a space instead of a comma (union operator) between ranges used in function arguments. Most
of the time you will see this occur when you specify an intersection of two cell ranges, but the intersection never actually occurs.
#NUM! – Although there are many circumstances the #NUM! error can appear, it is generally produced by an invalid argument in an Excel function or formula. Usually one that produces a number that is
either too large or too small and cannot be represented in the worksheet.
#REF! – Commonly referred to as “reference”, #REF! errors can be attributed to any formulas or functions that reference other cells. Formulas like VLOOKUP() can throw the #REF! error if you delete a
cell that is referred to by a formula or possibly paste over the cells that are being referred too.
#VALUE! – Whenever you see a #VALUE! error, there is usually an incorrect argument or the incorrect operator is being used. This is commonly seen with “text” being passed to a function or formula as
an argument when a number is expected.
Error Tracing
Excel has various features to assist you in figuring out the location of your errors. For this section, we are going to talk about Tracing errors. On the Formulas tab in the “Formula Auditing”
section you will see “Trace Precedents” and “Trace Dependents”.
In order to use these, you first have to activate a cell containing a formula. Once the cell is active, select one of the trace options to assist you in resolving your issue. Tracing dependents shows
all cells that the active cell is influencing whereas Tracing precedents shows all cells whose values influence the calculation of the active cell.
The Error Alert
When formulas don’t work as we expect them to Excel helps us out by providing a green triangle indicator in the top left corner of a cell. An “alert options button” will appear to the left of this
cell when you activate it.
When hovering over the button, a ScreenTip will appear with a short description of the error value. In this same vicinity, a drop-down will be presented with the available options:
1. Help on this error: This will open Excel’s help window and provide information pertaining to the error value and give suggestions and examples on how to resolve the issue.
2. Show Calculation Steps: This will open the “Evaluate Formula” dialog box, also found on the Formulas Tab. This will walk you through each step of the calculation showing you the result of each
3. Ignore Error: Bypasses all error checking for the activated cell and removes the error notification triangle.
4. Edit in Formula Bar: This will active “Edit Mode” moving your insertion point to the end of your formula on the “Formula Bar”.
5. Error Checking Options: This will open Excels default options for error checking and handling. Here you can modify the way Excel handles various errors.
Additional Error Handling
Although the above error handling features of Excel are nice, depending on your level this may not be enough. For instance, say you have a large project where you are referencing various data sources
of information that is populated by multiple members of your team. It is likely that not all members are going to input the data the exact same all the time. This is when advanced error handling
within Excel comes in handy.
There are a few different ways to handle errors. IFERROR() is a valuable formula providing you two different processes depending on an error being present or not. Other options are using formula
combinations such as IF(ISNUMBER()). Most of the time this formula combination is used with SEARCH(). We know that when Excel returns something TRUE it can be represented by a 1. So, when you write =
IF(ISNUMBER(SEARCH(“Hello”,A2)),TRUE,FALSE) you are saying, “If you find Hello in A2 return a 1, otherwise return a 0.” Another formula that comes in handy for later versions of Excel is the
AGGREGATE() function. We’re going to go over some brief examples below.
Generating an error without the use of IFERROR()
1. In the below example you will see that we are trying to subtract “text” from the sum of another range. This could have occurred for various reasons but subtracting “text” from a number obviously
will not work too well.
2. In this example it generates a #VALUE! error because the formula is looking for a number but it is receiving text instead
3. In the same example, lets use the IFERROR() and have the function display “There was a problem”
4. You can see below that we wrapped the formula in the IFERROR() function and provided a value to show. This is a basic example but you can be creative and come up with many ways to handle the
error from this point forward depending on the scope of the formula and how complex it may be.
Using the AGGREGATE() function can be a little daunting if you have never used it before.
1. This formula is however versatile and flexible and is a great option for error handling depending on what your formulas are doing.
2. The first argument in the AGGREGATE() function is the formula you want to use such as SUM(), COUNT() and a few others as you see in the list.
3. The second part of the argument are options in which you can incorporate with the formula in the first argument. For the sake of this example and this lesson, you would select option 6, “Ignore
error values”.
4. Lastly you would simply but the range or cell reference that the formula is to be used for. An AGGREGATE() function will look similar to this:
5. In this example we are wanting to COUNT A2 through A6 and ignore any errors that may occur
6. To write the example formula as a different method it would look like this:
So as you can see there are many options and ways to handle errors within Excel. There are options for very basic methods such as using the Calculation Steps to assist you or more advanced options
like using the AGGREGATE function or combining formulas to handle various circumstances.
I encourage you to play around with the formulas and see what works best for your situation and your style.
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The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as
"if ... then ...". A material conditional formula ${\displaystyle P\rightarrow Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false. If natural language conditionals were
understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a
seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.^[1]
Paradox of entailment
As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction.
In natural language, an instance of the paradox of entailment arises:
It is raining
It is not raining
George Washington is made of rakes.
This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all.
This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true).
Validity is defined in classical logic as follows:
An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false.
For example a valid argument might run:
If it is raining, water exists (1st premise)
It is raining (2nd premise)
Water exists (Conclusion)
In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid.
But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid argument since there would be no possible situation in which all the premises are
true and therefore no possible situation in which all the premises are true and the conclusion is false.
For example an argument with inconsistent premises might run:
It is definitely raining (1st premise; true)
It is not raining (2nd premise; false)
George Washington is made of rakes (Conclusion)
As there is no possible situation where both premises could be true, then there is certainly no possible situation in which the premises could be true while the conclusion was false. So the argument
is valid whatever the conclusion is; inconsistent premises imply all conclusions.
The classical paradox formulae are closely tied to conjunction elimination, ${\displaystyle (p\land q)\to p}$ which can be derived from the paradox formulae, for example from (1) by importation. In
addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:
1. ${\displaystyle (p\to q)\land (r\to s)\ \vdash \ (p\to s)\lor (r\to q)}$
2. ${\displaystyle (p\land q)\to r\ \vdash \ (p\to r)\lor (q\to r)}$
but mapping these back to English sentences using "if" gives paradoxes.
The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is true that either (a) if John is in London then he is in France, or
(b) if he is in Paris then he is in England." Using material implication, if John is not in London then (a) is true; whereas if he is in London then, because he is not in Paris, (b) is true. Either
way, the conclusion that at least one of (a) or (b) is true is valid.
But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where
John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.
The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is
closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "whenever switch A is closed, the light is on," and "whenever switch B is
closed, the light is on." Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that comes on only when both switches are
See also
1. ^ von Fintel, Kai (2011). "Conditionals" (PDF). In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.). Semantics: An international handbook of meaning. de Gruyter Mouton.
pp. 1515–1538. doi:10.1515/9783110255072.1515. hdl:1721.1/95781. ISBN 978-3-11-018523-2.
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Credit Cards tools and calculators
One can review their closing balance and the “Minimum Monthly repayment” amount each month. If the credit card holder pays more than the minimum amount, one can save money on interest and clear the
balance amount faster. The credit card calculator enables the user to answer the question, “By how much can one save by paying more than the minimum?”
What is Credit Card Calculator?
The Credit Card Calculator allows the user to calculate the following:
• The credit card calculator enables users to estimate the tenure of the repayment of the card.
• One can also compute the amount saved on the interest in case one makes larger repayments.
• Individuals can also calculate the amount they might save on interest by switching to a different card.
How Does UM Oceania's Credit Card Calculator Work?
UM Oceania’s credit card calculator allows users to compute credit card repayments. One can also mention multiple values on the calculator that will enable them to compare and calculate the Credit
card repayment. One can input multiple values in the calculator to select and derive the amount that complements their budgetary goals.
How To Use A Credit Card Calculator?
The user can conveniently utilise the credit card calculator that features a straightforward interface to derive the following:
• The credit card calculator allows users to review the amount they must pay on their credit card. One can conveniently oversee the tenure of repayment based on the interest charged for the same.
To conclude the transaction, the user should input the amount they’re willing to pay in the “Monthly Payment” section.
• Users can also calculate the amount they’re required to pay off to obtain their budgetary goal. The credit card calculator features a “Payoff Goal” section that enables them to determine the
payoff amount that complements their budgetary goal.
• The calculator also allows the user to compute the interest amount the credit card holder can save. One is required to input a higher amount in the “Monthly Payment” section and select “Months to
Payoff” in the “Payoff Goal” section.
Features and Benefits of A Credit Card Calculator
The Credit Card Calculator provides the following features and benefits to the user:
• The credit card calculator allows you to input multiple values to review and derive the payoff goal from the comfort of your home.
• Credit Card calculators are straightforward to use and user-friendly. Even first-time users can utilise the calculator without any hassle.
• Credit Card Calculators are easily accessible as one can select from multiple online options to obtain the desired amount.
• Credit card calculators ensure accuracy and save time. An individual does not require to perform manual calculations, which are prone to inaccuracies.
• One can also calculate the interest one can save from the credit card repayment calculator, and it also helps the user to obtain more lucrative deals.
How to Calculate Interest Charges on Credit Cards
The Interest on the credit card is calculated by employing the compound interest method that is computed on a fixed budget from the credit card balance. It will include existing charges on interest.
The interest would build over some time. If the credit card holder pays off an amount close to the minimum balance, they would be required to pay higher interest. On the other hand, paying more than
the minimum balance amount would reduce the interest on credit card repayment.
Things You Should Know About Credit Card Calculator
The credit card Calculator is an AI-powered tool that enables the user to obtain information on the amount of time it would take for an individual to pay off their credit card. The Credit card
calculator also provides the user with all the information regarding the interest one can save on the payments and compare with different credit card lenders to find the deal that complements their
budgetary goals.
Minimum Amount To Be Paid On A Credit Card
The cardholder's credit card statement will showcase the minimum amount one must pay on their credit card and the due date before making the payments. Generally, the minimum percentage set by the
credit card issuer can range from anywhere between 2% to 3% of the closing balance. Let’s take an example for a better understanding.
If the credit card balance is $4,000, the minimum repayment on the credit card will range from $80 to $120. On the other hand, if the balance amount is $500, then the minimum repayment can range from
$10 to 15$.
Frequently Asked Questions
What happens if I fail to make the minimum monthly payment before the due date?
Any default in the credit card repayment can result in penalty fees and withdrawal of interest rate. It could also impact the credit score of the applicant.
How do I calculate my credit card payment?
The credit card payment can be calculated by calculating the interest on the repayments.
What's the minimum payment of $1000 on a credit card?
The minimum payment on the credit card generally ranges from 2% to 3% of the credit balance. In this case, the minimum payment on a credit card balance of $1,000 would be $20 to $30.
Where can I find out how much my minimum repayment is?
A cardholder can refer to their credit card statement to review the minimum repayment.
Will I still be charged interest if I make the minimum repayment?
Yes, if you make the minimum repayment on the credit card, you will be charged interest.
Is there anything I can do to reduce the interest charged on my debt?
The credit card holder reduces the interest rate by paying off more than the minimum repayment amount on the credit card.
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Sensor Fusion: Part 4
In this post, we’ll add the math and provide implementation for adding image based measurements. Let’s recap the notation and geometry first introduced in part 1 of this series
The state vector in equation 16 of ^1 consists of 21 elements.
This is a 21 element state vector with the camera-IMU offset as one of its elements. We’ll consider a simpler problem first where the IMU is aligned with the camera. Thus, our state vector consists
of the errors in the position, velocity and orientation of the IMU (or Camera) wrt the global frame and the gyro and accel bias errors. The imaged feature points provide measurements that will be
used to correct our estimates of position, velocity, orientation, gyro and accel bias.
In the previous posts, we already derived the state propagation equations for gyro bias and orientation errors. Let’s now derive the propagation equations for the accel bias, position and velocity.
As before, for a variable
and therefore,
I’m not using bold for
Taking the derivative,
Note that since there are now multiple DCMs, I’m explicitly stating the coordinate systems a given DCM acts on. For example,
From post 2 of this series,
Now, since
From the first post, we know that
Similar to the gyro bias, the accel bias changes very slowly and therefore the error in the accel bias can be assumed to be constant over the time step.
Combining the equations 1,2,3 with the state propagation equations for the orientation and gyro bias errors developed in the previous posts, the entire
As before, the state transition matrix
The covariance is propagated as:
Let’s now look at the measurement update equations. The simplified geometry given that the IMU and the camera are coincident is shown below:
Note that in our coordinate system, the
Let the 3D position of the feature point
The final image coordinates are obtained by applying the perspective divide.
The jacobian matrix consists of the partial derivatives of the image coordinates against the parameters we are trying to estimate, i.e.,
From straightforward differentiation,
Let’s denote the matrix represented by
Now let’s consider
From the vector calculus results derived in part 1,
The derivatives against the other elements of the state vector are 0. Therefore (after replacing
Simulation Setup
Our simulation set up consists of a camera and IMU located coincident to each other. In the initial position, the optical axis of the camera is pointed along positive
State estimation is performed by integrating noisy angular velocities and accelerations (after adjusting for estimated gyro and accel bias) to obtain estimated orientation, position and velocity. The
state covariance matrix is propagated according to eq (7). The noise image measurements are then used as a measurement source to correct for the errors in the state vector.
It is well known that integrating noisy accelerometer data to estimate velocity and double integrating to obtain position leads to quick accumulation of errors. The error in the position and
orientation will cause the estimated image measurements to drift away from the true measurements (the difference between the two acts as a source of measurement). The video below shows this drift.
The camera in the video was held stationary and thus the image projection of the feature points should not move (except for the small perturbation caused by pixel noise).
Let’s also take a look at our simulated trajectory. We use a cube as prop to show the movement of the camera +IMU along the simulated trajectory.
Now let’s look at some of the Matlab code that implements one iteration of the Kalman filter operation.
% Obtain simulated IMU measurements
[omega_measured, accel_measured, q_prev] = sim_imu_tick(sim_state, time_step, idx, sensor_model, q_prev);
% Obtain simulated camera measurements. p_f_in_G contains the feature
% locations and p_C_in_G the camera location in the global coordinate system
% sigma_image is the std. dev of the pixel noise, K the camera
% calibration matrix.
image_measurements = sim_camera_tick(dcm_true, p_f_in_G, p_C_in_G, K, numFeatures, sigma_image);
% Obtain estimated values by adjusting for current estimate of the
% biases
omega_est = omega_measured - sv.gyro_bias_est;
accel_est = accel_measured - sv.accel_bias_est;
% Multiply by time_delta to get change in orientation/position
phi = omega_est*time_step;
vel = accel_est*time_step;
vel3 = make_skew_symmetric_3(vel);
phi3 = make_skew_symmetric_3(phi);
% Generate new estimates for q, position, velocity using equations of motion
sv = update_state(sv, time_step, g, phi, vel);
% State transition matrix
F = eye(15) + [zeros(3,3) eye(3)*time_step zeros(3,3) zeros(3,3) zeros(3,3);
zeros(3,3) zeros(3,3) sv.dcm_est*vel3 zeros(3,3) -sv.dcm_est*time_step;
zeros(3,3) zeros(3,3) -phi3 eye(3)*time_step zeros(3,3);
% Propagate covariance
P = F*P*F' + Qd;
% Apply image measurement update
if (mod(i, image_update_frequency) == 0)
[sv, P] = process_image_measurement_update(sv, p_f_in_G, image_measurements, ...
numFeatures, P, K, imageW, imageH, sigma_image);
function image_measurements = sim_camera_tick(dcm_true, p_f_in_G, p_C_in_G, K, numFeatures, sigma_image)
for fid = 1: numFeatures
% feature position in camera frame
fic = dcm_true'*(p_f_in_G(fid,:) - p_C_in_G')';
% apply perspective projection and add noise
image_measurements(2*fid-1:2*fid) = K*[fic(2)/fic(1) fic(3)/fic(1) 1]' ...
+ normrnd(0, [sigma_image sigma_image]');
Let’s first see what happens if no image updates are applied.
We can see that the estimated position very quickly diverges from the true position. Let’s now look at the code for the image measurement update.
% Iterate until the CF < 0.25 or max number of iterations is reached
while(CF > 0.25 && numIter < 10)
numIter = numIter + 1;
estimated_visible = [];
measurements_visible = [];
residual = [];
for indx = 1:numVisible
% index of this feature point
fid = visible(indx);
% estimated position in the camera coordinate system
fi_e = dcm_est'*(p_f_in_G(fid,:)' - p_C_in_G_est); % feature in image, estimated
% focal lengths (recall that the x axis of our coordinate
% system points along the optical axis)
fy = K(1,1); fz = K(2,2);
J = [-fy*fi_e(2)/fi_e(1).^2 fy*1/fi_e(1) 0;
-fz*fi_e(3)/fi_e(1).^2 0 fz*1/fi_e(1)];
% Measurement matrix
H_image(2*indx-1: 2*indx,:) = [-J*dcm_est' zeros(2,3) ...
J*dcm_est'*make_skew_symmetric_3((p_f_in_G(fid,:)' - p_C_in_G_est)) zeros(2,3) zeros(2,3)];
% estimated image measurement
estimated_visible(2*indx-1: 2*indx) = K*[fi_e(2)/fi_e(1) fi_e(3)/fi_e(1) 1]';
% actual measurement
measurements_visible(2*indx-1: 2*indx) = measurements(2*fid-1: 2*fid)';
% vector of residuals
residual = estimated_visible - measurements_visible;
%show_image_residuals(f4, measurements_visible, estimated_visible);
% Kalman gain
K_gain = P*H_image'*inv([H_image*P*H_image' + R_im]);
% Correction vector
x_corr = K_gain*residual';
% Updated covariance
P = P - K_gain*H_image*P;
% Apply image update and correct the current estimates of position,
% velocity, orientation, gyro/accel bias
x_est = apply_image_update(KF_SV_Offset, x_corr, ...
[position_est; velocity_est; q_est; gyro_bias_est; accel_bias_est]);
position_est = x_est(KF_SV_Offset.pos_index:KF_SV_Offset.pos_index+KF_SV_Offset.pos_length-1);
velocity_est = x_est(KF_SV_Offset.vel_index:KF_SV_Offset.vel_index+KF_SV_Offset.vel_length-1);
gyro_bias_est = x_est(KF_SV_Offset.gyro_bias_index:KF_SV_Offset.gyro_bias_index + KF_SV_Offset.gyro_bias_length-1);
accel_bias_est = x_est(KF_SV_Offset.accel_bias_index:KF_SV_Offset.accel_bias_index + KF_SV_Offset.accel_bias_length-1);
q_est = x_est(KF_SV_Offset.orientation_index:KF_SV_Offset.orientation_index+KF_SV_Offset.orientation_length-1);
dcm_est = quat2dc(q_est);
p_C_in_G_est = position_est;
% Cost function used to end the iteration
CF = x_corr'*inv(P)*x_corr + residual*R_inv*residual';
The code follows the equation for the image measurement update developed earlier. One difference between image measurement update and accelerometer update discussed in the last post is that we run
the image measurement update in a loop until the cost function defined in equation (24) of ^1 is lower than a threshold or the maximum number of iterations is reached.
Let’s look at the trajectory with the image updates applied at 1/10 the IMU sampling frequency with two values of image noise sigma
image noise sigma = 0.1
image noise sigma = 0.5
As we can see, adding the image measurement update prevents any systematic error in position from developing and keeps the estimated position wandering around the true position. The amount of
wander is higher for higher value of image noise sigma, as expected. The wander around the starting point is likely due to the kalman filter in the process of estimating the gyro and accel biases and
converging to a steady state. Once the biases have been estimated and the covariance has converged, the estimated position tracks the true position closely.
The system also correctly estimates the accel and gyro biases. The true and estimated biases for the two different values of image noise sigma are shown below.
Gyro (x) Gyro (y) Gyro (z) Accel (x) Accel (y) Accel (z)
True 0.0127 -0.0177 -0.0067 -0.06 0 0
Estimated ( 0.0126 -0.0176 -0.0067 -0.0581 0.0053 0.0010
Estimated ( 0.0127 -0.0175 -0.0071 -0.0736 0.0031 -0.0092
The residual plots for the position (along y axis) and orientation (roll) are shown below. The residuals mostly lie between the corresponding 3 sigma bounds and thus the filter appears to be
performing well.
As suggested in ^2, it is better to maintain the covariance matrix in UD factorized form for better numerical stability and efficient calculation of matrix inverse. We have not implemented this
The full source code is available here.
Implementing a Real System
Our simulation results look quite promising, and thus it is time to implement a real, working system. A real system will present many challenges –
• Picking suitable feature points in the environment and computing their 3D positions is not easy. Some variant of epipolar geometry based scene geometry computation plus bundle adjustment will
have to be implemented.
• Some way of obtaining the correspondence between 3D feature points and their image projections will have to be devised.
• Image processing is typically a much slower operation than processing IMU updates. It is quite likely that in a real system, updates from the image processing module are stale by the time they
arrive and thus can’t be applied directly to update the current covariance matrix. This time delay will have to be factored in the implementation of the covariance update.
Mirzaei FM, Roumeliotis SI. A Kalman Filter-Based Algorithm for IMU-Camera Calibration: Observability Analysis and Performance Evaluation.
IEEE Trans Robot
. 2008;24(5):1143-1156. doi:
Maybeck P. Stochastic Models: Estimation and Control: Volume 2. Academic Press; 1982.
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Tuition Policies - Simpedia
Tuition Policies
Found in: Fees Rates & Cost
Carrie L., Michigan
How do you explain your tuition policies? I must be unclear because students are confused. This has happened 2-3 times in the past month so I need to change something I’m saying.
I charge a monthly rate, but I will tell students when they call for the first time that I charge an average of $22 per lesson. I feel like when people call and are ‘rate shopping’ they need to know
it’s an average of $22 a week to compare with the others around. Then they think it should be $88 a month instead of $95 a month which is the averaged amount for the year. I need to learn how to
explain this in a more clear way.
Dodie B., Washington
Here is the wording I use:
FEES: I have adjusted my fees according to the number of lessons I teach in a year, divided by 12 for the number of months in a year. Since I plan to teach 48 weeks out of the year, Your monthly fees
will be $80 a month for shared lessons & $120 a month for private lessons.
Your fees for each month are due at the last lesson of the preceding month. I will provide a reminder a week in advance. If you do not have your fees at the lesson at which they are due, please send
them through the mail to Dodie Brueggeman at the address at the bottom of this page
If I do not have your fees by the 1st, a $20 late payment fee will be assessed. If extenuating circumstances prevent you from paying on time, don’t hesitate to let me know and I will be happy to work
with you.
Laurie Richards, Nebraska
I explain it very clearly during the foundation session. I take 8 weeks off during the year, so they are paying for 44 lessons. The cost of the 44 lessons is divided evenly over the 12 months to make
it easier for everyone. So, regardless of whether there are 3, 4, or 5 lessons in any given month, the monthly tuition remains the same.
If I cancel a lesson for any reason, and it is in addition to the 8 weeks vacation, I prorate the monthly tuition and credit their account.
I ask if anyone has questions about it. I also detail this in my policies. Btw, I’ve learned the hard way!
Mark M., New York
I set a round-number per-lesson rate, and I tell them this is the cost of each lesson, but I bill it monthly, so I multiply by 52 weeks and divide by 12 months. Then they understand right away why
the monthly number isn’t just 4 x the lesson rate.
This is also where I like the benefit of having a lesson rate that’s a multiple of $3, because it means that the monthly rate is also a nice round, even number of dollars. This is true because of the
52 and the 12. For anyone who calculated a monthly rate based on some other number of lessons per year due to planned vacations or what have you, this multiple of $3 thing would not be true, but it
would be true for some other figure.
Cindy B., Illinois
I don’t explain my per week figure – I just give them the per month figure. If it comes up in a specific case, I will explain that I teach 40 lessons a year, 11 months, and averaged the monthly cost
based on that. If that’s not enough of an explanation (only twice have I needed to “do the math” for them) I go ahead and do it. I also explain that when the program is totally adhered to they will
be accomplished pianists in an average of 4 years, which is how long the first 9 levels of the foundation take when you only teach 40 lessons per year instead of 52, and that the results outweigh the
cost no matter which way you look at it.
Amber B., Michigan
I like the per lesson rate. I charge $25 for shared lessons and $30 for demanding a private. The monthly thing seems like more to say. Maybe you should give them a choice, your monthly rate or my
per lesson rate:) Don’t worry if they are shopping. You are worth more than any traditional teacher and these same people will probably be a pain in other areas as well down the road.
Anne S., Nebraska
I tell potential students that my tuition is a flat $xx per month. The only time I break it down into individual lesson charge is if I need to credit them for a cancelled lesson.
The most important thing I have learned about discussing my tuition rate is to do it matter-of-factly and unapologetically. When I first started teaching I felt like my tuition was very high compared
to traditional teachers, and my hesitation would show in my voice. Now, as Neil says, “it is what it is” and I feel confident in discussing my rates. Simply Music is just worth it.
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Bundle Map
Bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber
bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three
sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.
Bundle maps over a common base
Let π[E]:E→ M and π[F]:F→ M be fiber bundles over a space M. Then a bundle map from E to F over M is a continuous map φ:E→ F such that
should commute. Equivalently, for any point x in M, φ maps the fiber E[x] = π[E]^−1({x}) of E over x to the fiber F[x] = π[F]^−1({x}) of F over x.
General morphisms of fiber bundles
Let π[E]:E→ M and π[F]:F→ N be fiber bundles over spaces M and N respectively. Then a continuous map φ:E→ F is called a bundle map from E to F if there is a continuous map f:M→ N such that the
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How To Make Chart In Excel With Multiple Data 2024 - Multiplication Chart Printable
How To Make Chart In Excel With Multiple Data
How To Make Chart In Excel With Multiple Data – You could make a multiplication graph in Excel by using a format. You will discover several samples of web templates and learn to formatting your
multiplication chart making use of them. Below are a few tricks and tips to make a multiplication chart. Upon having a web template, all you want do is copy the formulation and paste it in the new
cell. You may then take advantage of this solution to flourish several amounts by an additional set up. How To Make Chart In Excel With Multiple Data.
Multiplication desk format
If you are in the need to create a multiplication table, you may want to learn how to write a simple formula. Initial, you need to locking mechanism row one of many header column, then flourish the
quantity on row A by cell B. An alternate way to create a multiplication desk is to use merged personal references. In cases like this, you might enter $A2 into line A and B$1 into row B. The result
is a multiplication kitchen table with a formulation that actually works both for columns and rows.
If you are using an Excel program, you can use the multiplication table template to create your table. Just open up the spreadsheet along with your multiplication table template and change the label
for the student’s name. You can even adjust the page to fit your person demands. It comes with an option to modify the hue of the cellular material to modify the look of the multiplication kitchen
table, too. Then, you can transform all the different multiples suitable for you.
Making a multiplication graph in Shine
When you’re utilizing multiplication kitchen table application, you can actually create a straightforward multiplication kitchen table in Stand out. Just build a page with rows and columns numbered
from a to thirty. The location where the rows and columns intersect will be the response. If a row has a digit of three, and a column has a digit of five, then the answer is three times five, for
example. The same thing goes for the opposite.
Very first, you may go into the numbers you need to multiply. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To create the figures
greater, select the cellular material at A1 and A8, then select the correct arrow to select a range of cellular material. You can then variety the multiplication method within the tissues in the
other rows and columns.
Gallery of How To Make Chart In Excel With Multiple Data
How To Plot Multiple Data Sets On The Same Chart In Excel 2010 YouTube
Working With Multiple Data Series In Excel Pryor Learning Solutions
Working With Multiple Data Series In Excel Pryor Learning Solutions
Leave a Comment
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Assembly Syscalls in 64-bit Windows
I am on windows 10, with Cygwin installed. I have been using Cygwin for the compilation/assembly of c and assembly programs using Cygwin-installed "gcc" and "nasm". From what I know, nasm has a -f
win64 mode, so it can assemble 64-bit programs. Now, for x64 assembly programming on windows, it seems youtube has a lack of tutorials. Most assembly programming tutorials on youtube are either for
x64 linux or x32 windows, and I need to be able to print a string to the console on x64 windows, without the use of any external functions such as C's "printf".
StackOverflow links that didn't work for me:
As far as I know, nasm does support 64-bit windows using the -f win64 extension. Also, the answers have nothing to do with how to write an actual program in assembly on x64 bit windows
How to write hello world in assembler under Windows?
All answers that give code give only code for outdated windows versions (32-bit), except one. The one answer that works for 64 bit I tried, but the command to link the object file gave an error for
me, that the system could not find the path specified.
This site does not contain any code, which is what I need. Also, I am trying to write the Hello World program in nasm.
64 bit assembly, when to use smaller size registers
The question actually contains code for a hello world program, but when executed under cygwin on windows 10 (my device), I get a segmentation fault.
Why does Windows64 use a different calling convention from all other OSes on x86-64?
I have tried disassembling a C Hello World Program with objdump -d, but this calls printf, a prebuilt C function, and I am trying to avoid using external functions.
I have tried other external functions (such as Messagebox), but they return errors because something does not recognize the functions. Also, I am trying not to use any extern functions if possible,
only syscall, sysenter, or interrupts.
Attempt #1:
section .data
msg db "Hello, World!", 10, 0
section .text
mov rax, 1
mov rdi, 1
mov rsi, msg
mov rdx, 14
mov rax, 60
mov rdi, 0
Problem: Assembles and links properly, but throws a segmentation fault when run.
Attempt #2:
section .text
mov ah, 0eh
mov al, '!', 0
mov bh, 0
mov bl, 0
int 10h
Problem: Does not assemble properly; says "test.asm:3: error: invalid combination of opcode and operands"
Attempt #3:
section .text
msg db 'test', 10, 0
mov rdi, 1
mov rsi, 1
mov rdx, msg
mov rcx, 5
mov rdi, 60
mov rsi, 0
Problem: Assembles and links properly, but throws a segmentation fault when run.
Attempt #4:
section .text
mov ah, 0eh
mov al, '!'
mov bh, 0
mov bl, 0
int 10h
Problem: Assembles and links properly, but throws a segmentation fault when run.
I just need an example of a hello world assembly program that works on 64-bit Windows 10. 32-bit programs seem to return errors when I run them. If it is not possible to print a string to the console
without using external functions, an example of a program that uses external functions and works on 64-bit Windows 10 would be nice.
I heard that syscalls in windows have far different addresses than those in linux.
Attempt #1:
Attempt #2:
Internally Windows uses either interrupts or system calls. However they are not officially documented and they may change anytime Microsoft wants to change them:
Only a few .dll files directly use the system call instructions; all programs are directly or indirectly accessing these .dll files.
A certain sequence (for example mov rax, 5, syscall) might have the meaning "print out a text" before a Windows update and Microsoft might change the meaning of this sequence to "delete a file" after
a Windows update.
To do this Microsoft simply has to replace the kernel and the .dll files which are officially "allowed" to use the syscall instruction.
The only way to call Windows operating system functions in a way that it is guaranteed that your program is still working after the next Windows update is to call the functions in the .dll files -
just the same way you would do this in a C program.
Example (32-bit code):
push 1
push msg
push 14
call _write
mov dword [esp], 0
call _exit
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The Lattice Structure of Weak Factorization Systems on Finite Lattices
This paper investigates the properties of weak factorization systems on finite lattices, demonstrating that they form a semidistributive, trim, and congruence uniform lattice with a close
relationship to transfer systems.
Bibliographic Information: Luo, Y., & Rognerud, B. (2024). On the lattice of the weak factorization systems on a finite lattice. arXiv preprint arXiv:2410.06182.
Research Objective: This paper aims to explore and characterize the lattice structure of weak factorization systems on finite lattices, drawing connections to transfer systems and highlighting their
relevance to model structures and G-equivariant topology.
Methodology: The authors utilize concepts and techniques from lattice theory, category theory, and combinatorial topology. They analyze cover relations, join-irreducible elements, and congruences
within the lattice of weak factorization systems. The paper also employs graph-theoretical interpretations, particularly through the concept of elevating graphs, to study these systems.
Main Conclusions: The study reveals a rich structure within the seemingly abstract concept of weak factorization systems on finite lattices. The findings provide valuable insights into the
organization and classification of these systems, with implications for areas such as model category theory and the study of N8-operads in G-equivariant topology.
Significance: This research contributes significantly to the field of lattice theory and its applications in other mathematical disciplines. The exploration of weak factorization systems on finite
lattices and their connection to transfer systems deepens our understanding of these structures and opens avenues for further investigation in related areas.
Limitations and Future Research: The paper primarily focuses on finite lattices. Exploring the properties of weak factorization systems on infinite lattices or more general categories could be a
potential direction for future research. Additionally, investigating the implications of the established lattice properties for specific applications, such as classifying model structures or
understanding the homotopy category of N8-operads, could be fruitful research avenues.
When transitioning from finite to infinite lattices, the properties of the lattice of weak factorization systems can change significantly. Here's a breakdown of the challenges and potential changes:
Finite Lattice Case: Existence of Factorizations: In finite lattices, we can always construct a weak factorization system by taking the pullback closure of a relation. This ensures every morphism
factors through a morphism in L followed by a morphism in R. Lattice Structure: The poset of weak factorization systems on a finite lattice forms a complete lattice. This is primarily due to the
finite nature allowing for intersections and closures to be well-defined. Semidistributivity, Trimness, Congruence Uniformity: As shown in the provided text, these properties hold for the lattice of
weak factorization systems on finite lattices. Infinite Lattice Case: Factorization Issues: The existence of weak factorizations for all morphisms becomes more intricate. We can't rely on simple
pullback closures as they might not "stabilize" after finitely many steps. Lattice Structure: The poset of weak factorization systems might not form a complete lattice. Infinite joins and meets might
not be well-defined. Even if it is a lattice, it might not be complete. Preservation of Properties: Semidistributivity, trimness, and congruence uniformity might not hold in general for infinite
lattices. These properties rely on finiteness arguments related to cover relations and join-irreducible elements. Further Considerations: Specific Classes of Infinite Lattices: It's possible that
certain well-behaved classes of infinite lattices (e.g., those satisfying certain chain conditions) might preserve some of the properties. Topological Lattices: Introducing a topology on the lattice
might provide a framework to handle limits and potentially recover some properties. In summary, the transition to infinite lattices introduces significant complexities. The existence of weak
factorization systems, the lattice structure itself, and the preservation of desirable properties like semidistributivity are no longer guaranteed and require careful examination.
Yes, exploring alternative representations of weak factorization systems on finite lattices could offer valuable insights. Here are a few avenues to consider: 1. Closure Operators: Idea: Instead of
focusing on the sets L and R directly, investigate closure operators associated with them. A closure operator on a poset P is a function cl: P -> P that is extensive (x ≤ cl(x)), monotone (x ≤ y
implies cl(x) ≤ cl(y)), and idempotent (cl(cl(x)) = cl(x)). Connection: Left saturated sets and transfer systems can be characterized by their closure properties. A closure operator viewpoint might
reveal deeper connections between the lattice structure of weak factorization systems and the underlying lattice L. 2. Graph-Theoretic Representations: Beyond Elevating Graphs: The paper already
introduces elevating graphs. Explore other graph constructions where nodes represent elements of the lattice L, and edges encode the relations in L or R. Analyzing Graph Properties: Study how
properties of these graphs (e.g., connectivity, chromatic number) relate to the properties of the corresponding weak factorization systems. 3. Categorical Language: Factorization Systems in Lattice
Categories: Formalize the notion of weak factorization systems within the category of finite lattices. This could lead to a more abstract understanding and connections to other categorical
structures. 4. Combinatorial Descriptions: Enumeration and Generating Functions: Develop combinatorial techniques to enumerate weak factorization systems on specific classes of finite lattices. Find
generating functions that encode this information. Bijections with Other Structures: Search for bijections between weak factorization systems and other combinatorial objects (e.g., certain types of
tableaux, lattice paths). Benefits of Alternative Representations: New Structural Insights: Different representations can highlight different aspects of weak factorization systems, leading to a more
comprehensive understanding. Connections to Other Areas: New representations might reveal unexpected connections to other areas of mathematics, such as graph theory, order theory, or category theory.
Computational Advantages: Some representations might be more amenable to algorithmic analysis, enabling more efficient computation and enumeration of weak factorization systems.
The properties of semidistributivity, trimness, and congruence uniformity of the lattice of weak factorization systems on finite lattices, while interesting in their own right, have subtle and
indirect implications for the broader study of model structures in homotopy theory. Here's a nuanced perspective: 1. Limited Direct Impact: Model Structures as Intervals: Remember that model
structures are specific intervals within the lattice of weak factorization systems, satisfying additional axioms related to weak equivalences. Properties Not Lifted Easily: The properties of the
entire lattice don't automatically transfer to its intervals. An interval within a semidistributive lattice might not itself be semidistributive, and so on. 2. Potential for Specialized Insights:
Finite Model Categories: In the niche case of model categories whose underlying category is a finite lattice, these properties might provide some constraints or structural insights. Inspiration for
New Concepts: The study of these lattice-theoretic properties could inspire the development of analogous concepts or techniques applicable to the study of model structures in more general settings.
3. Indirect Benefits: Deeper Understanding of Building Blocks: A thorough understanding of weak factorization systems is essential as they are fundamental components of model structures. Tools for
Exploration: The properties and representations (like elevating graphs) provide tools to explore the "space" of possible weak factorization systems, which might indirectly aid in understanding the
landscape of potential model structures. 4. Bridging Different Areas: Connections to Other Fields: The appearance of concepts like semidistributive lattices and trimness highlights connections
between homotopy theory and order theory. This cross-pollination of ideas can be fruitful. In conclusion, while the direct implications for the classification and understanding of general model
structures might be limited, these lattice-theoretic properties offer valuable insights into the structure of weak factorization systems. This deeper understanding, the potential for specialized
applications, and the bridging of different mathematical areas make these findings significant.
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How to use discrete choice experiments to capture stakeholder preferences in social work research
Data from: How to use discrete choice experiments to capture stakeholder preferences in social work research
Data files
May 27, 2024 version files 25.58 MB
The primary article (cited below under "Related works") introduces social work researchers to discrete choice experiments (DCEs) for studying stakeholder preferences. The article includes an online
supplement with a worked example demonstrating DCE design and analysis with realistic simulated data. The worked example focuses on caregivers' priorities in choosing treatment for children with
attention deficit hyperactivity disorder. This dataset includes the scripts (and, in some cases, Excel files) that we used to identify appropriate experimental designs, simulate population and sample
data, estimate sample size requirements for the multinomial logit (MNL, also known as conditional logit) and random parameter logit (RPL) models, estimate parameters using the MNL and RPL models, and
analyze attribute importance, willingness to pay, and predicted uptake. It also includes the associated data files (experimental designs, data generation parameters, simulated population data and
parameters, simulated choice data, MNL and RPL results, RPL sample size simulation results, and willingness-to-pay results) and images. The data could easily be analyzed using other software, and the
code could easily be adapted to analyze other data. Because this dataset contains only simulated data, we are not aware of any legal or ethical considerations.
README: Data from: How to Use Discrete Choice Experiments to Capture Stakeholder Preferences in Social Work Research
This dataset supports the worked example in:
Ellis, A. R., Cryer-Coupet, Q. R., Weller, B. E., Howard, K., Raghunandan, R., & Thomas, K. C. (2024). How to use discrete choice experiments to capture stakeholder preferences in social work
research. Journal of the Society for Social Work and Research. Advance online publication. https://doi.org/10.1086/731310
The referenced article introduces social work researchers to discrete choice experiments (DCEs) for studying stakeholder preferences. In a DCE, researchers ask participants to complete a series of
choice tasks: hypothetical situations in which each participant is presented with alternative scenarios and selects one or more. For example, social work researchers may want to know how parents and
other caregivers prioritize different aspects of mental health treatment when choosing services for their children. A DCE can explore this question by presenting scenarios that include different
types of mental health care providers, treatment methods, costs, locations, and so on. Caregivers’ stated choices in these scenarios can provide a lot of information about their priorities.
In a DCE, the scenarios presented to participants are called alternatives. The characteristics of the alternatives are called attributes, and the attribute values presented to participants are called
levels. Participants' choices are typically analyzed according to random utility theory, which holds that utility is a latent variable underlying people's observed choices. Utility has a systematic
component derived from the attributes of alternatives and the individual characteristics of participants, as well as a random component.
Our article includes an online supplement with a worked example demonstrating DCE design and analysis with realistic simulated data. The worked example focuses on caregivers' priorities in choosing
treatment for children with attention deficit hyperactivity disorder. This submission includes the scripts (and, in some cases, Excel files) that we used to create and analyze the worked example, as
well as the associated data files and images.
In the worked example, we used simulated data to examine caregiver preferences for 7 treatment attributes (medication administration, therapy location, school accommodation, caregiver behavior
training, provider communication, provider specialty, and monthly out-of-pocket costs) identified by dosReis and colleagues (2007, 2015) in a previous DCE. We employed an orthogonal design (so named
because all pairs of attribute levels appear with equal frequency, making the attributes independent of each other) with 18 choice tasks. Our design included 1 continuous attribute (cost) and 12
categorical attributes, which were dummy-coded. The online supplement shows all of the attribute levels and which combinations of levels were presented in the simulated DCE.
Using the parameter estimates published by dosReis et al., with slight adaptations, we simulated utility values for a population (N=100,000), then selected a sample (n=500) for analysis. Relying on
random utility theory, we used the multinomial logit (MNL, also known as conditional logit) and random parameter logit (RPL) models to estimate utility parameters. The MNL model assumes that the
systematic component of utility is distributed uniformly across individuals. The RPL model assumes that utility varies randomly across individuals according to a specified distribution.
In addition to estimating utility, we measured the relative importance of each attribute, estimated caregivers’ willingness to pay (WTP) for differences in attributes (e.g., how much they would be
willing to pay for their child to see one type of provider versus another) with bootstrapped 95% confidence intervals, and predicted the uptake of three treatment packages with different sets of
attributes. This dataset includes both the simulated source data and the processed results. The online supplement of the referenced article describes the methods in greater detail.
The next section of this document describes how the files in this submission relate to each other, and the two sections following that describe (a) the structure and content of the data files and (b)
the structure and content of the Excel files, which contain mixtures of data and formulas. The code included with this submission can be used to replicate the data. Before the R code is run, the
working directory must be set (near the top of each script, where it says "set your working directory here").
File structure
The following table describes the files in the order in which they were created. It also describes how they relate to each other.
File Description Source Program or Input File(s) Output File(s)
00_README.md this file
01_design.r R code to identify an appropriate experimental design 03_table_s2.csv, 04_design_wide_cost_dummy.csv,
02_design.sas SAS code to identify an alternate design
03_table_s2.csv experimental design (shown in Table S2 of the online 01_design.r
04_design_wide_cost_dummy.csv experimental design, dummy-coded 01_design.r
05_design_long_cost_continuous.csv experimental design, adjusted to make cost continuous 01_design.r
06_population_simulation_input.xlsx parameters used for population simulation adapted from dos Reis et al. (2017)
07_simulate_population.r R code to simulate the study population 06_population_simulation_input.xlsx 08_population_data.csv,
08_population_data.csv simulated population data 07_simulate_population.r
09_population_parameters.csv true values of the population parameters 07_simulate_population.r
10_sample_size_cl.r R code to estimate required sample size for multinomial 05_design_long_cost_continuous.csv,
logit (conditional logit) model 09_population_parameters.csv
11_simulate_sample.r R code to select sample and simulate choice data 05_design_long_cost_continuous.csv, 12_choice_data.csv
12_choice_data.csv choice data from the study sample 11_simulate_sample.r
13_analysis.r R code to run multinomial logit (conditional logit) and 12_choice_data.csv 14_results_cl.csv, 15_results_rpl.csv
random parameter logit models
14_results_cl.csv multinomial logit (conditional logit) results 13_analysis.r
15_results_rpl.csv random parameter logit results 13_analysis.r
16_sample_size_rpl_simulation.r simulation to estimate sample size requirements for random 05_design_long_cost_continuous.csv, 17_sample_size_rpl_results.csv
parameter logit model 08_population_data.csv, 14_results_cl.csv
17_sample_size_rpl_results.csv sample size simulation results 16_sample_size_rpl_simulation.r
18_sample_size_rpl_analysis.r R code to analyze sample size simulation results for random 09_population_parameters.csv,
parameter logit model 17_sample_size_rpl_results.csv
19_attribute_importance_analysis.xlsx Excel file to analyze attribute importance based on 14_results_cl.csv, 15_results_rpl.csv
attribute level range and parameter estimates
20_attribute_importance_plot.r R code to plot attribute importance results 19_attribute_importance_analysis.xlsx 21_attribute_importance.png
21_attribute_importance.png plot of attribute importance results 20_attribute_importance_plot.r
22_wtp_analysis.r R code to estimate willingness to pay with bootstrapped 12_choice_data.csv, 14_results_cl.csv 23_wtp_results_cl.csv, 24_wtp_results_rpl.csv
estimates and confidence intervals
23_wtp_results_cl.csv willingness-to-pay estimates for multinomial logit 22_wtp_analysis.r
(conditional logit) model
24_wtp_results_rpl.csv willingness-to-pay estimates for random parameter logit 22_wtp_analysis.r
25_wtp_summary.r R code to summarize willingness-to-pay results 23_wtp_results_cl.csv, 24_wtp_results_rpl.csv 26_wtp_plot.png
26_wtp_plot.png willingness-to-pay plot 25_wtp_summary.r
27_predicted_uptake.xlsx Excel file to predict uptake of specific alternatives 14_results_cl.csv
Description of data file contents
This section describes the contents of each individual data file, in the order in which the files are listed above.
experimental design (shown in Table S2 of the online supplement)
Each row in the file represents one of the 18 choice tasks. Each choice task had two alternatives. In the original study by dosReis et al., each attribute (including cost) had 3 levels (shown in the
online supplement of the paper referenced above). In this file, each attribute level is coded as 1, 2, or 3.
Column Description
meds1 level of the medication administration attribute for alternative 1
therapy1 level of the therapy location attribute for alternative 1
accommodation1 level of the school accommodation attribute for alternative 1
cgtraining1 level of the caregiver behavior training attribute for alternative 1
prvcomm1 level of the provider communication attribute for alternative 1
prvspec1 level of the provider specialty attribute for alternative 1
cost1 level of the monthly out-of-pocket costs attribute for alternative 1
meds2 level of the medication administration attribute for alternative 2
therapy2 level of the therapy location attribute for alternative 2
accommodation2 level of the school accommodation attribute for alternative 2
cgtraining2 level of the caregiver behavior training attribute for alternative 2
prvcomm2 level of the provider communication attribute for alternative 2
prvspec2 level of the provider specialty attribute for alternative 2
cost2 level of the monthly out-of-pocket costs attribute for alternative 2
experimental design, dummy-coded
Each row in the file represents one of the 18 choice tasks. Each choice task had two alternatives. In the original study by dosReis et al., each attribute (including cost) had 3 levels (shown in the
online supplement of the paper referenced above). In this file, each attribute level is coded using two dummy variables for levels 2 and 3, with level 1 as the reference category.
Column Description
meds1_2 =1 if medication administration = 2 for alternative 1, 0 otherwise
meds1_3 =1 if medication administration = 3 for alternative 1, 0 otherwise
therapy1_2 =1 if therapy location = 2 for alternative 1, 0 otherwise
therapy1_3 =1 if therapy location = 3 for alternative 1, 0 otherwise
accommodation1_2 =1 if school accommodation = 2 for alternative 1, 0 otherwise
accommodation1_3 =1 if school accommodation = 3 for alternative 1, 0 otherwise
cgtraining1_2 =1 if caregiver behavior training = 2 for alternative 1, 0 otherwise
cgtraining1_3 =1 if caregiver behavior training = 3 for alternative 1, 0 otherwise
prvcomm1_2 =1 if provider communication = 2 for alternative 1, 0 otherwise
prvcomm1_3 =1 if provider communication = 3 for alternative 1, 0 otherwise
prvspec1_2 =1 if provider specialty = 2 for alternative 1, 0 otherwise
prvspec1_3 =1 if provider specialty = 3 for alternative 1, 0 otherwise
cost1_2 =1 if monthly out-of-pocket costs = 2 for alternative 1, 0 otherwise
cost1_3 =1 if monthly out-of-pocket costs = 3 for alternative 1, 0 otherwise
meds2_2 =1 if medication administration = 2 for alternative 2, 0 otherwise
meds2_3 =1 if medication administration = 3 for alternative 2, 0 otherwise
therapy2_2 =1 if therapy location = 2 for alternative 2, 0 otherwise
therapy2_3 =1 if therapy location = 3 for alternative 2, 0 otherwise
accommodation2_2 =1 if school accommodation = 2 for alternative 2, 0 otherwise
accommodation2_3 =1 if school accommodation = 3 for alternative 2, 0 otherwise
cgtraining2_2 =1 if caregiver behavior training = 2 for alternative 2, 0 otherwise
cgtraining2_3 =1 if caregiver behavior training = 3 for alternative 2, 0 otherwise
prvcomm2_2 =1 if provider communication = 2 for alternative 2, 0 otherwise
prvcomm2_3 =1 if provider communication = 3 for alternative 2, 0 otherwise
prvspec2_2 =1 if provider specialty = 2 for alternative 2, 0 otherwise
prvspec2_3 =1 if provider specialty = 3 for alternative 2, 0 otherwise
cost2_2 =1 if monthly out-of-pocket costs = 2 for alternative 2, 0 otherwise
cost2_3 =1 if monthly out-of-pocket costs = 3 for alternative 2, 0 otherwise
experimental design, adjusted to make cost continuous
Each pair of rows in the file represents one of the 18 choice tasks, with one row for each alternative presented in the task. Each choice task had two alternatives. In the original study by dosReis
et al., each attribute (including cost) had 3 levels (shown in the online supplement of the paper referenced above). In this file, the cost variable is coded as continuous. Otherwise, each attribute
level is coded using two dummy variables for levels 2 and 3, with level 1 as the reference category.
Column Description
task index of the choice task (1 to 18)
alt index of the alternative within the choice task (1 or 2)
meds_2 =1 if medication administration = 2, 0 otherwise
meds_3 =1 if medication administration = 3, 0 otherwise
therapy_2 =1 if therapy location = 2, 0 otherwise
therapy_3 =1 if therapy location = 3, 0 otherwise
accommodation_2 =1 if school accommodation = 2, 0 otherwise
accommodation_3 =1 if school accommodation = 3, 0 otherwise
cgtraining_2 =1 if caregiver behavior training = 2, 0 otherwise
cgtraining_3 =1 if caregiver behavior training = 3, 0 otherwise
prvcomm_2 =1 if provider communication = 2, 0 otherwise
prvcomm_3 =1 if provider communication = 3, 0 otherwise
prvspec_2 =1 if provider specialty = 2, 0 otherwise
prvspec_3 =1 if provider specialty = 3, 0 otherwise
cost monthly out-of-pocket costs in hundreds of US dollars
simulated population data
This file contains the population data that we simulated. Each row represents an individual. There are two columns for each categorical attribute and one column for cost. The number in each cell is
the individual's utility for the specified attribute level, or, in the case of cost, for a $100 increase in cost.
Column Description
id ID of the individual (1 to 100,000)
meds_2 utility for medication administration = 2
meds_3 utility for medication administration = 3
therapy_2 utility for therapy location = 2
therapy_3 utility for therapy location = 3
accommodation_2 utility for school accommodation = 2
accommodation_3 utility for school accommodation = 3
cgtraining_2 utility for caregiver behavior training = 2
cgtraining_3 utility for caregiver behavior training = 3
prvcomm_2 utility for provider communication = 2
prvcomm_3 utility for provider communication = 3
prvspec_2 utility for provider specialty = 2
prvspec_3 utility for provider specialty = 3
cost utility for a $100 increase in monthly out-of-pocket costs
true values of the population parameters
This file contains the true values of the population parameters, obtained by calculating the mean and standard deviation of each of the 13 utility variables in the simulated population data. There
are 13 rows for the means, followed by 13 rows for the standard deviations.
Column Description
parameter name of the parameter
truth true calculated value of the parameter
choice data from the study sample
We selected a simple random sample of 500 individuals from the simulated population. We used each individual's simulated utility values to determine what choices the individual would make given the
18 choice tasks. For each individual and each choice task (500 x 18 = 9,000 rows), this file shows the attribute levels for the two alternatives in the choice task, as well as the choice that the
individual "made."
Column Description
id study ID of the individual (1 to 500)
case unique row ID (1 to 9,000)
task index of the choice task (1 to 18)
meds_2.1 =1 if medication administration = 2 for alternative 1, 0 otherwise
meds_3.1 =1 if medication administration = 3 for alternative 1, 0 otherwise
therapy_2.1 =1 if therapy location = 2 for alternative 1, 0 otherwise
therapy_3.1 =1 if therapy location = 3 for alternative 1, 0 otherwise
accommodation_2.1 =1 if school accommodation = 2 for alternative 1, 0 otherwise
accommodation_3.1 =1 if school accommodation = 3 for alternative 1, 0 otherwise
cgtraining_2.1 =1 if caregiver behavior training = 2 for alternative 1, 0 otherwise
cgtraining_3.1 =1 if caregiver behavior training = 3 for alternative 1, 0 otherwise
prvcomm_2.1 =1 if provider communication = 2 for alternative 1, 0 otherwise
prvcomm_3.1 =1 if provider communication = 3 for alternative 1, 0 otherwise
prvspec_2.1 =1 if provider specialty = 2 for alternative 1, 0 otherwise
prvspec_3.1 =1 if provider specialty = 3 for alternative 1, 0 otherwise
cost.1 monthly out-of-pocket costs in hundreds of US dollars for alternative 1
meds_2.2 =1 if medication administration = 2 for alternative 2, 0 otherwise
meds_3.2 =1 if medication administration = 3 for alternative 2, 0 otherwise
therapy_2.2 =1 if therapy location = 2 for alternative 2, 0 otherwise
therapy_3.2 =1 if therapy location = 3 for alternative 2, 0 otherwise
accommodation_2.2 =1 if school accommodation = 2 for alternative 2, 0 otherwise
accommodation_3.2 =1 if school accommodation = 3 for alternative 2, 0 otherwise
cgtraining_2.2 =1 if caregiver behavior training = 2 for alternative 2, 0 otherwise
cgtraining_3.2 =1 if caregiver behavior training = 3 for alternative 2, 0 otherwise
prvcomm_2.2 =1 if provider communication = 2 for alternative 2, 0 otherwise
prvcomm_3.2 =1 if provider communication = 3 for alternative 2, 0 otherwise
prvspec_2.2 =1 if provider specialty = 2 for alternative 2, 0 otherwise
prvspec_3.2 =1 if provider specialty = 3 for alternative 2, 0 otherwise
cost.2 monthly out-of-pocket costs in hundreds of US dollars for alternative 1
choice the number of the alternative that the individual chose (1 or 2)
multinomial logit (conditional logit) results
The multinomial logit (MNL) model estimates only the mean utility because it assumes constant utility across individuals with random error. This file contains the estimated mean utilities from the
MNL model.
Column Description
parameter name of the parameter
estimate estimated mean
stderr standard error of the estimated mean
random parameter logit results
The random parameter logit (RPL) model estimates mean utilities and deviations around the means. In this case, the deviations are standard deviations because we simulated, and assumed in the RPL
model, a normal distribution for each utility parameter. This file contains the estimated mean utilities and estimated standard deviations from the RPL model.
Column Description
parameter name of the parameter
estimate estimated mean or standard deviation
stderr standard error of the estimate
sample size simulation results
We ran simulations to estimate the sample size required to estimate the parameters in the RPL model with 80% statistical power and an alpha level of .05. This file contains the simulation results.
Column Description
n sample size (250 to 2,500)
sim simulation number (1 to 50 for each sample size)
parameter name of the parameter
estimate estimated mean or standard deviation
stderr standard error of the estimate
willingness-to-pay estimates for multinomial logit (conditional logit) model
This file contains estimates of participants' willingness to pay (WTP) for differences in attributes (for example, how much they would be willing to pay for their child to see one type of provider
versus another). The estimates in this file came from the MNL model. We used 200 bootstrap replications to estimate 95% confidence intervals for the WTP estimates. Each row in the file represents one
of the 200 bootstrap replications. Each column represents an attribute level (2 or 3). Each data cell contains a WTP estimate for a given attribute level, relative to the reference level (1) for that
Column Description
meds_2 WTP for medication administration = 2
meds_3 WTP for medication administration = 3
therapy_2 WTP for therapy location = 2
therapy_3 WTP for therapy location = 3
accommodation_2 WTP for school accommodation = 2
accommodation_3 WTP for school accommodation = 3
cgtraining_2 WTP for caregiver behavior training = 2
cgtraining_3 WTP for caregiver behavior training = 3
prvcomm_2 WTP for provider communication = 2
prvcomm_3 WTP for provider communication = 3
prvspec_2 WTP for provider specialty = 2
prvspec_3 WTP for provider specialty = 3
willingness-to-pay estimates for random parameter logit model
This file contains estimates of participants' willingness to pay (WTP) for differences in attributes (for example, how much they would be willing to pay for their child to see one type of provider
versus another). The estimates in this file came from the RPL model. We used 200 bootstrap replications to estimate 95% confidence intervals for the WTP estimates. Each row in the file represents one
of the 200 bootstrap replications. Each column represents an attribute level (2 or 3). Each data cell contains a WTP estimate for a given attribute level, relative to the reference level (1) for that
Column Description
meds_2 WTP for medication administration = 2
meds_3 WTP for medication administration = 3
therapy_2 WTP for therapy location = 2
therapy_3 WTP for therapy location = 3
accommodation_2 WTP for school accommodation = 2
accommodation_3 WTP for school accommodation = 3
cgtraining_2 WTP for caregiver behavior training = 2
cgtraining_3 WTP for caregiver behavior training = 3
prvcomm_2 WTP for provider communication = 2
prvcomm_3 WTP for provider communication = 3
prvspec_2 WTP for provider specialty = 2
prvspec_3 WTP for provider specialty = 3
Description of Excel file contents
This section describes the contents of each individual Excel file, in the order in which the files are listed above.
These files include formulas, so they are considered software and hosted on Zenodo.
parameters used for population simulation
This file contains 3 rows for each of the 7 attributes. Each set of 3 rows corresponds to the 3 levels used for an attribute in the original study by dosReis et al. (and shown in the online
supplement of our article). The attribute names and descriptions follow:
• meds = medication administration
• therapy = therapy location
• accomodation = school accommodation
• cgtraining = caregiver behavior training
• prvcomm = provider communication
• prvspec = provider specialty
• cost = monthly out-of-pocket cost
The levels of each attribute are numbered 1 through 3.
Column Description
attribute (dos Reis et al., 2017) name and level of the attribute
estimate (dosReis et al. 2017) estimated utility of the attribute as reported by dosReis et al.
SE (dosReis et al. 2017) standard error of the estimated utility as reported by dosReis et al.
means, converted to dummy coding estimated mean utility for levels 2 and 3 of each categorical parameter (relative to level 1) and for the cost parameter
SD, using abs(mean)/2 standard deviation of utility, set at half the absolute value of the mean
For each attribute, to estimate the mean utility for levels 2 and 3, we subtracted the level 1 parameter estimate from the level 2 and 3 parameter estimates, respectively.
To estimate the mean utility for each $100 increase in cost in the same way, we calculated the difference in utility between levels 1 and 2 ($50 and $150, respectively), calculated a third of the
difference in utility between levels 2 and 3 ($150 and $450, respectively), and took the average.
We used the means and standard deviations in this file to simulate the population data from which we selected our sample.
Excel file to analyze attribute importance based on attribute level range and parameter estimates
We used this file to assess the relative importance of each attribute. We measured importance using the utility range of each attribute. The utility range is the difference between the highest and
lowest estimated utilities associated with levels of the attribute. Because the cost attribute had the largest utility range, we expressed the importance of the other attributes relative to the
importance of the cost attribute.
This file contains two tabs, one labeled CL for the conditional logit (multinomial logit) model, and the other labeled RPL for the random parameter logit model.
Column Description
parameter name of the parameter
estimate estimated mean from the MNL (CL) or RPL model
attribute name of the attribute associated with the parameter
min minimum level of the attribute
max maximum level of the attribute
rng placeholder for calculating max-min
U(min) estimated utility of the attribute's minimum level
U(max) estimated utility of the attribute's maximum level
range absolute difference between U(min) and U(max)
range/max importance of the attribute relative to the most important attribute, expressed as a percentage
Excel file to predict uptake of specific alternatives
We designed three treatment packages (Alternatives A, B, and C) with different sets of attributes (described in the online supplement). We then used the MNL results to predict the uptake of each
package, that is, what percentage of the population would choose each package, given (a) a choice among the three or (b) a choice between the package and the status quo. We used this file to make the
predictions. We also estimated how much the cost of each package would have to change in order for that package to have the same uptake as each other package.
The top portion of the file contains the data used to predict uptake:
Column Description
parameter name of the parameter (attribute)
CL estimate utility estimate from the conditional logit (multinomial logit) model
Alternative A The level of the attribute under Alternative A, dummy coded
Alternative B The level of the attribute under Alternative B, dummy coded
Alternative C The level of the attribute under Alternative C, dummy coded
The bottom portion of the file contains the uptake predictions:
Label in Column A Label in Column B Value in Column Labeled Alternative A, B, or C
Uptake utility estimated utility of the alternative
Uptake exp(utility) exponentiated utility
Uptake pred. prob. of selecting (versus others) predicted probability of selecting the alternative, versus the others
Uptake pred. prob. of selecting (versus status quo) predicted probability of selecting the alternative, versus the status quo
Estimated change in cost to balance uptake utility, except for cost utility of the alternative minus the utility for cost
Estimated change in cost to balance uptake cost that would make uptake equal to that of Alternative A cost that would make the alternative's uptake the same as that of Alternative A
Estimated change in cost to balance uptake cost that would make uptake equal to that of Alternative B cost that would make the alternative's uptake the same as that of Alternative B
Estimated change in cost to balance uptake cost that would make uptake equal to that of Alternative C cost that would make the alternative's uptake the same as that of Alternative C
Source of data-generating parameters
We adapted the data-generating parameters from the following sources:
• dosReis, S., Mychailyszyn, M. P., Myers, M., & Riley, A. W. (2007). Coming to terms with ADHD: How urban African-American families come to seek care for their children. Psychiatric Services, 58
(5), 636-641. https://doi.org/10.1176/ps.2007.58.5.636
• dosReis, S., Ng, X., Frosch, E., Reeves, G., Cunningham, C., & Bridges, J. (2015). Using best-worst scaling to measure caregiver preferences for managing their child’s ADHD: A pilot study. The
Patient, 8(5), 423–431. https://doi.org/10.1007/s40271-014-0098-4
Most of the code in this submission was implemented in R version 4.3.2. The single SAS program (which we used to identify an alternate experimental design) was implemented in SAS Version 9.4. For
some analyses, we used Microsoft Excel Version 16.84.
In the worked example, we used simulated data to examine caregiver preferences for 7 treatment attributes (medication administration, therapy location, school accommodation, caregiver behavior
training, provider communication, provider specialty, and monthly out-of-pocket costs) identified by dosReis and colleagues in a previous DCE. We employed an orthogonal design with 1 continuous
variable (cost) and 12 dummy-coded variables (representing the levels of the remaining attributes, which were categorical). Using the parameter estimates published by dosReis et al., with slight
adaptations, we simulated utility values for a population of 100,000 people, then selected a sample of 500 for analysis. Relying on random utility theory, we used the mlogit package in R to estimate
the MNL and RPL models, using 5,000 Halton draws for simulated maximum likelihood estimation of the RPL model. In addition to estimating the utility parameters, we measured the relative importance of
each attribute, estimated caregivers’ willingness to pay (WTP) for differences in attributes (e.g., how much they would be willing to pay for their child to see one type of provider versus another)
with bootstrapped 95% confidence intervals, and predicted the uptake of three treatment packages with different sets of attributes. This submission includes both the simulated source data and the
processed results. The online supplement of the primary article describes the methods in greater detail.
Works referencing this dataset
|
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|
NOV.ABS[BIB,CSR] - www.SailDart.org
perm filename NOV.ABS[BIB,CSR] blob
filedate 1978-09-18 generic text, type C, neo UTF8
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C00006 00004 %3AIM-316 NATURAL LANGUAGE PROCESSING IN AN AUTOMATIC PROGRAMMING DOMAIN
C00010 00005 %3STAN-CS-78-673 A NUMERICAL LIBRARY AND ITS SUPPORT
C00013 00006 %3AIM-317 TAU EPSILON CHI, A SYSTEM FOR TECHNICAL TEXT
C00018 00007 %3STAN-CS-78-677: COMPREHENSIVE EXAMINATIONS IN COMPUTER SCIENCE, 1972-1978
C00021 00008 %3STAN-CS-78-679 STEPLENGTH ALGORITHMS FOR MINIMIZING A CLASS OF NONDIFFERENTIABLE
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%3MOST RECENT CS REPORTS - NOVEMBER 1978%1
@Listed below are abstracts of the most recent reports published by the Computer
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.begin nofill
%3AIM-316 NATURAL LANGUAGE PROCESSING IN AN AUTOMATIC PROGRAMMING DOMAIN
%3Author:%1 Jerrold Ginsparg ↔(Thesis)
@%3Abstract:%1@This paper is about communicating with computers
in English. In particular, it describes an interface system which allows
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@The interface consists of two parts. The first is a parser called Reader.
Reader was designed to facilitate writing English grammars which
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%3STAN-CS-78-672 COMPARISON OF NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS
%3Author:%1 Tony F. Chan ↔(Thesis)
@%3Abstract:%1@A framework is set up within which the efficiency and storage
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differential equations can be realistically and rigorously compared. The
approach is based on getting good error estimates for a one-dimensional model
equation u↓[t] = au↓[x] + bu↓[xx] + cu↓[xxx] . Results on the comparison
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%3STAN-CS-78-673 A NUMERICAL LIBRARY AND ITS SUPPORT
%3Authors:%1 Tony F. Chan, William M. Coughran, Jr., Eric H. Grosse & Michael T. Heath
@%3Abstract:%1@Reflecting on four years of numerical consulting at the Stanford
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%3STAN-CS-78-674 FINITE ELEMENT APPOXIMATION AND ITERATIVE SOLUTION OF A CLASS OF
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@%3Abstract:%1@We describe in this report the numerical analysis of a particular
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%3AIM-317 TAU EPSILON CHI, A SYSTEM FOR TECHNICAL TEXT
%3Author:%1 Donald E. Knuth
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%3STAN-CS-78-676 A METHOD FOR DETERMINING THE SIDE EFFECTS OF PROCEDURE CALLS
%3Author:%1 John Phineas Banning ↔(Thesis)
@%3Abstract:%1@The presence of procedures and procedure calls in a programming
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%3STAN-CS-78-677: COMPREHENSIVE EXAMINATIONS IN COMPUTER SCIENCE, 1972-1978
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%3AIM-314 REASONING ABOUT RECURSIVELY DEFINED DATA STRUCTURES
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%3STAN-CS-78-680 BIBLIOGRAPHY OF STANFORD COMPUTER SCIENCE REPORTS, 1963-1978
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by the Stanford Computer Science Department since 1963. Each report
is identified by Computer Science number, author's name, title, National
Technical Information Service (NTIS) retrieval number, date, and
number of pages. Complete listings of
Artificial Intelligence Memos and Heuristic Programming Reports are given
in the Appendix.
@Also, for the first time, each report has been marked as to its availability
for ordering and the cost if applicable.
.begin nofill
No. of pages: 100
Available in hardcopy only, free of charge
%3SLAC PUB-2006 A NESTED PARTITIONING PROCEDURE FOR NUMERICAL MULTIPLE INTEGRATION
AND ADAPTIVE IMPORTANCE SAMPLING
%3Author:%1 Jerome H. Friedman
@%3Abstract:%1@An algorithm for adaptively partitioning a multidimensional coordinate
space, based on a scalar function of the coordinates, is presented. The method
is based on multiparameter optimization. The goal is to adjust the size, shape
and number of resulting hyperrectangular regions so that the variation of function
values within each is relatively small. These regions can then be used as a basis
for a stratified sampling estimate of the definite integral of the function, or
to efficiently sample points with the function as their probability density.
Available by writing directly to: Harriet Canfield, Bin 88, Stanford Linear
Accelerator Center, P.O. Box 4349, Stanford, California 94305 U.S.A.
.begin nofill
%3SLAC PUB-2189 A SURVEY OF ALGORITHMS AND DATA STRUCTURES FOR RANGE SEARCHING
%3Authors:%1 Jon Louis Bentley and Jerome H. Friedman
@%3Abstract:%1@An important problem in database systems is answering queries
quickly. This paper surveys a number of algorithms for efficiently answering
range queries. First a set of "logical structures" is described and then their
implementation in primary and secondary memories is discussed. The algorithms
included are of both "practical" and "theoretical" interest. Although some new
results are presented, the primary purpose of this paper is to collect together
the known results on range searching and to present them in a common terminology.
Available by writing directly to: Harriet Canfield, Bin 88, Stanford Linear
Accelerator Center, P.O. Box 4349, Stanford, California 94305 U.S.A.
.begin nofill
|
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Cookie has to many searchcols
Cookie has to many searchcols
I'm having a question about the cookie that is stored.
I'm using individual filtering and global filtering together.
This is the value in my cookie:
{"iStart": 0,"iEnd": 1,"iLength": 10,"sFilter": "ad","sFilterEsc": true,"aaSorting": [],"aaSearchCols": [ ['sta',true],['',true],['',true],['',true],['',true],['',true],['',true],['',true],['',true],
Like you see is my aaSearchCols part to much. I've only 6 cols?
But when i do "oSettings.aoPreSearchCols.length" after the cookie loaded he's givin me 120 and everytime i refresh the number of cols is more. Now is already 150. When i delete the cookie the length
starts again from 6
Can somebody help me?
Thanks in advance
Yikes! Sorry about this - this is an error in DataTables 1.5 beta 6. The issue is that a new entry is always created when a new column is added - so the saved state always goes up by the number
of columns in your table.
I'll include the fix in the next release, but if you want to apply the fix immediatly, what to do is:
Replace line 1646:
oSettings.aoPreSearchCols[ oSettings.aoPreSearchCols.length++ ] = {
oSettings.aoPreSearchCols[ oSettings.aoColumns.length-1 ] = {
The down-side is that your huge cookie will be retained (although nothing will be done with the values with an index > than the number of columns). Ideally you should delete the cookie as well...
Allan i tried your solution but that doesn't do the trick for me.
Because then the individual inputfields are not being saved to the cookie!
I changed
in the funtion _fnSaveState.
And this looks like it does the trick!
Many thanx 4 your help the past days.
In the past i used several other filter/sort table scripts but this is the best in my opinion.
Good work
Hi whizzboy,
Oops - sorry. My fix was incomplete. The saved state was always being written over with a blank search... The fix to that is to check if it should add the new object or not. Just before the
changed line from before you can insert:
if ( typeof oSettings.aoPreSearchCols[ iLength ] == 'undefined' )
Or if you have a working fix, keep using that :-)
This discussion has been closed.
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Credit Scoring Using Logistic Regression and Decision Trees
Create and compare two credit scoring models, one based on logistic regression and the other based on decision trees.
Credit rating agencies and banks use challenger models to test the credibility and goodness of a credit scoring model. In this example, the base model is a logistic regression model and the
challenger model is a decision tree model.
Logistic regression links the score and probability of default (PD) through the logistic regression function, and is the default fitting and scoring model when you work with creditscorecard objects.
However, decision trees have gained popularity in credit scoring and are now commonly used to fit data and predict default. The algorithms in decision trees follow a top-down approach where, at each
step, the variable that splits the dataset "best" is chosen. "Best" can be defined by any one of several metrics, including the Gini index, information value, or entropy. For more information, see
Decision Trees.
In this example, you:
• Use both a logistic regression model and a decision tree model to extract PDs.
• Validate the challenger model by comparing the values of key metrics between the challenger model and the base model.
Compute Probabilities of Default Using Logistic Regression
First, create the base model by using a creditscorecard object and the default logistic regression function fitmodel. Fit the creditscorecard object by using the full model, which includes all
predictors for the generalized linear regression model fitting algorithm. Then, compute the PDs using probdefault. For a detailed description of this workflow, see Case Study for Credit Scorecard
% Create a creditscorecard object, bin data, and fit a logistic regression model
load CreditCardData.mat
scl = creditscorecard(data,'IDVar','CustID');
scl = autobinning(scl);
scl = fitmodel(scl,'VariableSelection','fullmodel');
Generalized linear regression model:
logit(status) ~ 1 + CustAge + TmAtAddress + ResStatus + EmpStatus + CustIncome + TmWBank + OtherCC + AMBalance + UtilRate
Distribution = Binomial
Estimated Coefficients:
Estimate SE tStat pValue
_________ ________ _________ __________
(Intercept) 0.70246 0.064039 10.969 5.3719e-28
CustAge 0.6057 0.24934 2.4292 0.015131
TmAtAddress 1.0381 0.94042 1.1039 0.26963
ResStatus 1.3794 0.6526 2.1137 0.034538
EmpStatus 0.89648 0.29339 3.0556 0.0022458
CustIncome 0.70179 0.21866 3.2095 0.0013295
TmWBank 1.1132 0.23346 4.7683 1.8579e-06
OtherCC 1.0598 0.53005 1.9994 0.045568
AMBalance 1.0572 0.36601 2.8884 0.0038718
UtilRate -0.047597 0.61133 -0.077858 0.93794
1200 observations, 1190 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 91, p-value = 1.05e-15
% Compute the corresponding probabilities of default
pdL = probdefault(scl);
Compute Probabilities of Default Using Decision Trees
Next, create the challenger model. Use the Statistics and Machine Learning Toolbox™ method fitctree to fit a Decision Tree (DT) to the data. By default, the splitting criterion is Gini's diversity
index. In this example, the model is an input argument to the function, and the response 'status' comprises all predictors when the algorithm starts. For this example, see the name-value pairs in
fitctree to the maximum number of splits to avoid overfitting and specify the predictors as categorical.
% Create and view classification tree
CategoricalPreds = {'ResStatus','EmpStatus','OtherCC'};
dt = fitctree(data,'status~CustAge+TmAtAddress+ResStatus+EmpStatus+CustIncome+TmWBank+OtherCC+UtilRate',...
PredictorNames: {'CustAge' 'TmAtAddress' 'ResStatus' 'EmpStatus' 'CustIncome' 'TmWBank' 'OtherCC' 'UtilRate'}
ResponseName: 'status'
CategoricalPredictors: [3 4 7]
ClassNames: [0 1]
ScoreTransform: 'none'
NumObservations: 1200
The decision tree is shown below. You can also use the view function with the name-value pair argument 'mode' set to 'graph' to visualize the tree as a graph.
Decision tree for classification
1 if CustIncome<30500 then node 2 elseif CustIncome>=30500 then node 3 else 0
2 if TmWBank<60 then node 4 elseif TmWBank>=60 then node 5 else 1
3 if TmWBank<32.5 then node 6 elseif TmWBank>=32.5 then node 7 else 0
4 if TmAtAddress<13.5 then node 8 elseif TmAtAddress>=13.5 then node 9 else 1
5 if UtilRate<0.255 then node 10 elseif UtilRate>=0.255 then node 11 else 0
6 if CustAge<60.5 then node 12 elseif CustAge>=60.5 then node 13 else 0
7 if CustAge<46.5 then node 14 elseif CustAge>=46.5 then node 15 else 0
8 if CustIncome<24500 then node 16 elseif CustIncome>=24500 then node 17 else 1
9 if TmWBank<56.5 then node 18 elseif TmWBank>=56.5 then node 19 else 1
10 if CustAge<21.5 then node 20 elseif CustAge>=21.5 then node 21 else 0
11 class = 1
12 if EmpStatus=Employed then node 22 elseif EmpStatus=Unknown then node 23 else 0
13 if TmAtAddress<131 then node 24 elseif TmAtAddress>=131 then node 25 else 0
14 if TmAtAddress<97.5 then node 26 elseif TmAtAddress>=97.5 then node 27 else 0
15 class = 0
16 class = 0
17 if ResStatus in {Home Owner Tenant} then node 28 elseif ResStatus=Other then node 29 else 1
18 if TmWBank<52.5 then node 30 elseif TmWBank>=52.5 then node 31 else 0
19 class = 1
20 class = 1
21 class = 0
22 if UtilRate<0.375 then node 32 elseif UtilRate>=0.375 then node 33 else 0
23 if UtilRate<0.005 then node 34 elseif UtilRate>=0.005 then node 35 else 0
24 if CustIncome<39500 then node 36 elseif CustIncome>=39500 then node 37 else 0
25 class = 1
26 if UtilRate<0.595 then node 38 elseif UtilRate>=0.595 then node 39 else 0
27 class = 1
28 class = 1
29 class = 0
30 class = 1
31 class = 0
32 class = 0
33 if UtilRate<0.635 then node 40 elseif UtilRate>=0.635 then node 41 else 0
34 if CustAge<49 then node 42 elseif CustAge>=49 then node 43 else 1
35 if CustIncome<57000 then node 44 elseif CustIncome>=57000 then node 45 else 0
36 class = 1
37 class = 0
38 class = 0
39 if CustIncome<34500 then node 46 elseif CustIncome>=34500 then node 47 else 1
40 class = 1
41 class = 0
42 class = 1
43 class = 0
44 class = 0
45 class = 1
46 class = 0
47 class = 1
When you use fitctree, you can adjust the Name-Value Arguments depending on your use case. For example, you can set a small minimum leaf size, which yields a better accuracy ratio (see Model
Validation) but can result in an overfitted model.
The decision tree has a predict function that, when used with a second and third output argument, gives valuable information.
% Extract probabilities of default
[~,ObservationClassProb,Node] = predict(dt,data);
pdDT = ObservationClassProb(:,2);
This syntax has the following outputs:
• ObservationClassProb returns a NumObs-by-2 array with class probability at all observations. The order of the classes is the same as in dt.ClassName. In this example, the class names are [0 1]
and the good label, by choice, based on which class has the highest count in the raw data, is 0. Therefore, the first column corresponds to nondefaults and the second column to the actual PDs.
The PDs are needed later in the workflow for scoring or validation.
• Node returns a NumObs-by-1 vector containing the node numbers corresponding to the given observations.
Predictor Importance
In predictor (or variable) selection, the goal is to select as few predictors as possible while retaining as much information (predictive accuracy) about the data as possible. In the creditscorecard
class, the fitmodel function internally selects predictors and returns p-values for each predictor. The analyst can then, outside the creditscorecard workflow, set a threshold for these p-values and
choose the predictors worth keeping and the predictors to discard. This step is useful when the number of predictors is large.
Typically, training datasets are used to perform predictor selection. The key objective is to find the best set of predictors for ranking customers based on their likelihood of default and estimating
their PDs.
Using Logistic Regression for Predictor Importance
Predictor importance is related to the notion of predictor weights, since the weight of a predictor determines how important it is in the assignment of the final score, and therefore, in the PD.
Computing predictor weights is a back-of-the-envelope technique whereby the weights are determined by dividing the range of points for each predictor by the total range of points for the entire
creditscorecard object. For more information on this workflow, see Case Study for Credit Scorecard Analysis.
For this example, use formatpoints with the option PointsOddsandPDO for scaling. This is not a necessary step, but it helps ensure that all points fall within a desired range (that is, nonnegative
points). The PointsOddsandPDO scaling means that for a given value of TargetPoints and TargetOdds (usually 2), the odds are "double", and then formatpoints solves for the scaling parameters such that
PDO points are needed to double the odds.
% Choose target points, target odds, and PDO values
TargetPoints = 500;
TargetOdds = 2;
PDO = 50;
% Format points and compute points range
scl = formatpoints(scl,'PointsOddsAndPDO',[TargetPoints TargetOdds PDO]);
[PointsTable,MinPts,MaxPts] = displaypoints(scl);
PtsRange = MaxPts - MinPts;
Predictors Bin Points
_______________ _____________ ______
{'CustAge' } {'[-Inf,33)'} 37.008
{'CustAge' } {'[33,37)' } 38.342
{'CustAge' } {'[37,40)' } 44.091
{'CustAge' } {'[40,46)' } 51.757
{'CustAge' } {'[46,48)' } 63.826
{'CustAge' } {'[48,58)' } 64.97
{'CustAge' } {'[58,Inf]' } 82.826
{'CustAge' } {'<missing>'} NaN
{'TmAtAddress'} {'[-Inf,23)'} 49.058
{'TmAtAddress'} {'[23,83)' } 57.325
fprintf('Minimum points: %g, Maximum points: %g\n',MinPts,MaxPts)
Minimum points: 348.705, Maximum points: 683.668
The weights are defined as the range of points, for any given predictor, divided by the range of points for the entire scorecard.
Predictor = unique(PointsTable.Predictors,'stable');
NumPred = length(Predictor);
Weight = zeros(NumPred,1);
for ii = 1 : NumPred
Ind = strcmpi(Predictor{ii},PointsTable.Predictors);
MaxPtsPred = max(PointsTable.Points(Ind));
MinPtsPred = min(PointsTable.Points(Ind));
Weight(ii) = 100*(MaxPtsPred-MinPtsPred)/PtsRange;
PredictorWeights = table(Predictor,Weight);
PredictorWeights(end+1,:) = PredictorWeights(end,:);
PredictorWeights.Predictor{end} = 'Total';
PredictorWeights.Weight(end) = sum(Weight);
Predictor Weight
_______________ _______
{'CustAge' } 13.679
{'TmAtAddress'} 5.1564
{'ResStatus' } 8.7945
{'EmpStatus' } 8.519
{'CustIncome' } 19.259
{'TmWBank' } 24.557
{'OtherCC' } 7.3414
{'AMBalance' } 12.365
{'UtilRate' } 0.32919
{'Total' } 100
% Plot a histogram of the weights
title('Predictor Importance Estimates Using Logit');
ylabel('Estimates (%)');
Using Decision Trees for Predictor Importance
When you use decision trees, you can investigate predictor importance using the predictorImportance function. On every predictor, the function sums and normalizes changes in the risks due to splits
by using the number of branch nodes. A high value in the output array indicates a strong predictor.
imp = predictorImportance(dt);
bar(100*imp/sum(imp)); % to normalize on a 0-100% scale
title('Predictor Importance Estimates Using Decision Trees');
ylabel('Estimates (%)');
In this case, 'CustIncome' (parent node) is the most important predictor, followed by 'UtilRate', where the second split happens, and so on. The predictor importance step can help in predictor
screening for datasets with a large number of predictors.
Notice that not only are the weights across models different, but the selected predictors in each model also diverge. The predictors 'AMBalance' and 'OtherCC' are missing from the decision tree
model, and 'UtilRate' is missing from the logistic regression model.
Normalize the predictor importance for decision trees using a percent from 0 through 100%, then compare the two models in a combined histogram.
Ind = ismember(Predictor,dt.PredictorNames);
w = zeros(size(Weight));
w(Ind) = 100*imp'/sum(imp);
title('Predictor Importance Estimates');
ylabel('Estimates (%)');
h = gca;
Note that these results depend on the binning algorithm you choose for the creditscorecard object and the parameters used in fitctree to build the decision tree.
Model Validation
The creditscorecard function validatemodel attempts to compute scores based on internally computed points. When you use decision trees, you cannot directly run a validation because the model
coefficients are unknown and cannot be mapped from the PDs.
To validate the creditscorecard object using logistic regression, use the validatemodel function.
% Model validation for the creditscorecard
[StatsL,tL] = validatemodel(scl);
To validate decision trees, you can directly compute the statistics needed for validation.
% Compute the Area under the ROC
[x,y,t,AUC] = perfcurve(data.status,pdDT,1);
KSValue = max(y - x);
AR = 2 * AUC - 1;
% Create Stats table output
Measure = {'Accuracy Ratio','Area Under ROC Curve','KS Statistic'}';
Value = [AR;AUC;KSValue];
StatsDT = table(Measure,Value);
ROC Curve
The area under the receiver operating characteristic (AUROC) curve is a performance metric for classification problems. AUROC measures the degree of separability — that is, how much the model can
distinguish between classes. In this example, the classes to distinguish are defaulters and nondefaulters. A high AUROC indicates good predictive capability.
The ROC curve is plotted with the true positive rate (also known as the sensitivity or recall) plotted against the false positive rate (also known as the fallout or specificity). When AUROC = 0.7,
the model has a 70% chance of correctly distinguishing between the classes. When AUROC = 0.5, the model has no discrimination power.
This plot compares the ROC curves for both models using the same dataset.
hold on
xlabel('Fraction of nondefaulters')
ylabel('Fraction of defaulters')
title('Receiver Operating Characteristic (ROC) Curve')
tValidation = table(Measure,StatsL.Value(1:end-1),StatsDT.Value,'VariableNames',...
Measure logit DT
________________________ _______ _______
{'Accuracy Ratio' } 0.32515 0.38903
{'Area Under ROC Curve'} 0.66258 0.69451
{'KS Statistic' } 0.23204 0.29666
As the AUROC values show, given the dataset and selected binning algorithm for the creditscorecard object, the decision tree model has better predictive power than the logistic regression model.
This example compares the logistic regression and decision tree scoring models using the CreditCardData.mat dataset. A workflow is presented to compute and compare PDs using decision trees. The
decision tree model is validated and contrasted with the logistic regression model.
When reviewing the results, remember that these results depend on the choice of the dataset and the default binning algorithm (monotone adjacent pooling algorithm) in the logistic regression
• Whether a logistic regression or decision tree model is a better scoring model depends on the dataset and the choice of binning algorithm. Although the decision tree model in this example is a
better scoring model, the logistic regression model produces higher accuracy ratio (0.42), AUROC (0.71), and KS statistic (0.30) values if the binning algorithm for the creditscorecard object is
set as 'Split' with Gini as the split criterion.
• The validatemodel function requires scaled scores to compute validation metrics and values. If you use a decision tree model, scaled scores are unavailable and you must perform the computations
outside the creditscorecard object.
• To demonstrate the workflow, this example uses the same dataset for training the models and for testing. However, to validate a model, using a separate testing dataset is ideal.
• Scaling options for decision trees are unavailable. To use scaling, choose a model other than decision trees.
See Also
creditscorecard | screenpredictors | autobinning | bininfo | predictorinfo | modifypredictor | modifybins | bindata | plotbins | fitmodel | displaypoints | formatpoints | score | setmodel |
probdefault | validatemodel
Related Examples
More About
External Websites
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In this section we will show that the digital waveguide simulation technique is equivalent to the recursion produced by the finite difference approximation (FDA) applied to the wave equation [442,
pp. 430-431]. A more detailed derivation, with examples and exploration of implications, appears in Appendix E. Recall from (C.6) that the time update recursion for the ideal string digitized via the
FDA is given by
To compare this with the
description, we substitute the
instants) into the right-hand side of the FDA recursion above and see how good is the approximation to the left-hand side
Thus, we obtain the result that the FDA recursion is also
in the lossless case, because it is equivalent to the
digital waveguide
method which we know is exact at the sampling points. This is surprising since the FDA introduces artificial damping when applied to lumped,
systems, as discussed earlier.
The last identity above can be rewritten as
which says the
at time
traveling wave
components at positions
wave variable
can be computed for the next time step as the sum of incoming traveling wave components from the left and right. This picture also underscores the lossless nature of the computation.
This results extends readily to the digital waveguide mesh (§C.14), which is essentially a lattice-work of digital waveguides for simulating membranes and volumes. The equivalence is important in
higher dimensions because the finite-difference model requires less computations per node than the digital waveguide approach.
Even in one dimension, the digital waveguide and finite-difference methods have unique advantages in particular situations, and as a result they are often combined together to form a hybrid
traveling-wave/physical-variable simulation [351,352,222,124,123,224,263,223]. In this hybrid simulations, the traveling-wave variables are called ``W variables'' (where `W' stands for ``Wave''),
while the physical variables are caled ``K variables'' (where `K' stands for ``Kirchoff''). Each K variable, such as displacement vibrating string, can be regarded as the sum of two traveling-wave
components, or W variables:
Conversion between K variables and W variables can be non-trivial due to the non-local dependence of one set of
state variables
on the other, in general. A detailed examination of this issue is given in Appendix
Next Section: Loss ConsolidationPrevious Section: Digital Waveguide Interpolation
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Flaw Of Averages Case Study Solution | Flaw Of Averages Case Study Help
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world-view and many Greek-culture-related texts, both ancient and modern, have been written about the Emanucid world-view since the Emancus the Great. The Emancid world-views are very complex, and
there are many theories for understanding the Emanca-Uranitous world-view (e.g., they are the beliefs of a cult, the Emen-Urancy, the Emonite), and some theories have been put forward for explaining
or explaining the Emanfan-Urancycle. Some of the theories in this book are: Emancilius the Great (Emancipators of the Emenu-Uranicon), a cult of the Emonites, who are known as the Emen, a cult of
Emanucids, who are one of the most important cults of the Emanscids, the Eminites, and the Emanians. They believe that Emanucilius is the Great, the Great Mother of the Emscids, and that Emancis is
the Emanci. Emanci, the Ems, and the other Emancids believe that Emen, Emans, and Emancian wisdom is the true wisdom of the Ema-Uran-Emanci.
PESTLE Analysis
Emen, the Emnites, and Emenians. These Emen, the emanci, and other Emanci believe that Ems, the Emans, Ecom-Emancis, Emen, and Emsx, the Em-Uran, and the em-gals, the Ena-Uran and the ema-Uro-Eman,
are the true wisdom. Emsx is the Emenx of the Emgliz, the Ema, and the uro-Emenx. Ems is the Emsx of the emenci, the emencis, and the es-eum, and is the emancian and emancilian wisdom. Emen is the
Emmlis of the eman, and the galle-Eman. Emen, as Emmli, is the emmlion and emlion of the em. Emmlion is the emen-Emmlion of EmmlFlaw Of Averages Linking a percentile (percentile) of the population
with a percentile (distribution) of their age is called percentile-age-age. A percentile average is a percentile with the same median as a percentile.
Case Study Help
The percentile-age is usually a measure of the population’s population size. Locations A population is a population that has a median of one or more percentile or percentiles. The American population
is divided by the population that has one or more percentiles. This allows the population to be divided into discrete groups. The population is divided into groups based on what percentage of the
population is above or below the median. The population that is below or above the median is called a “sub-population”. Locate a percentile A population which has a percentile (sample) is called a
sample. The percentile is the standard deviation of the population, and is a measure of population size.
VRIO Analysis
A sample of a population is called a percentile-age group. Percentile A percentage is a percentile of the population. The percentage is the percentage of a population that is above or lower than or
below the percentage of the average population. In other words, the population is an average of the percentile of the average. A sample is a percentile that is a sample. Density The percentile of a
population in a population is a measure or percentile of its population size. For example, the population density in the United States is divided by 2. The population density in a population that
contains a percentile is called the percentile.
Marketing Plan
This is a measure that is used in the United Kingdom and in the United State. The population density of a population has a density of 100. The percentile is called a density percentile. The
population percentile of each population is called the population percentile. A density percentile is a percentile for a population that consists of the population that is between the nearest
population percentile and the population that contains the average population (0.5 % of the population). A density may be omitted if the population is as small as possible, since it is not possible
to have a population that does not exceed the population that must be included in the population. Census statistics In the United States the population is divided according to the census-designated
population (population density) as the population in a given census-designed area.
Marketing Plan
The population percentiles are calculated by dividing the percentage of population in a particular census-design district by the population density of that district. For example the population in the
South Bronx is divided by 20 to the population density as a proportion of the population in that district. In the Commonwealth of Virginia, the population of all Virginia residents is divided by 3
percent. The population in the Eastern Shore is also divided by website here and the population in this area is divided by 15% to the population of the North Shore to the population in Charleston,
West Virginia. Counties City In the UK the population is represented by the census-filing number. In the United States, the population has a census-designating area of 2,085,000. In the UK the total
population is 15,500,000. Town The population is represented in the census-of-towns (Census-of-dissolved) list.
Case Study Help
This list is used to cover all
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Rational Numbers
Rational Numbers are part of the
Real Number System
. Rational numbers are special because they can be written as a fraction.
More specifically, the definition of rational numbers says that any rational number can be written as the
of p to q, where p and q are integers and q is not zero.
Rational numbers are often numbers you use every day. You might notice that many numbers can be written as a fraction.
Not all numbers are rational. Numbers like √7 or 0.2810582107432.... are not rational because they cannot be written as fractions. In addition, numbers like
So here are the basics: When a rational number is written in decimal form, the number will
(or end) or it will
. If the decimal does one of these two things, it can be written in fraction form.
Be Careful: 6.343434343434....
is a repeating decimal. It can be written as a fraction
is not a repeating decimal. It does have a pattern, but the 34 does not repeat over and over. Therefore, this is not a rational number.
Why are Rational Numbers Important?
Ancient Greek mathematicians thought that all things could be measured using rational numbers. So when the Pythagorean Theorem came into play and showed that some lengths could not be written as a
rational number, their whole idea of numbers was changed.
The Ancient Greeks were kind of right! Rational numbers are used all the time! From buying and selling products using money and terminating decimals to cooking with fractions, people use rational
numbers just about every day! In addition, by giving the group of numbers a name and characteristics, we can study them, learn about their properties and even expand our horizon to determine what
other types of numbers are used when rational numbers don't get the job done.
Related Links: Math algebra Irrational Numbers
To link to this Rational Numbers page, copy the following code to your site:
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Sudo Null - Latest IT News
8 with ½ ways to prioritize functionality
In 99% of all cases, do not try, do not close all tasks, do not fix all the bugs. One of the key skills is to choose from the whole flow those tasks whose solution will give maximum benefit.
Prioritization methods and common sense help to choose such tasks.
I share the methods of prioritization that I collected from a number of articles. I have little experience in translations. I would welcome comments and suggestions to the wording.
1. Analogue of the Eisenhower matrix (complexity - benefit)
One of the simplest and most common prioritization techniques. The horizontal axis is complexity, the vertical is benefit. The goal is to single out simple tasks with maximum benefit.
Example axes: development cost and increase conversion. That is, among all the tasks, we single out tasks with a minimum development cost and a maximum increase in conversion.
2. Estimation by parameters
Select the parameters by which we will rank the functionality. For example, two groups of parameters:
1. increase turnover
2. customer value
3. value for the company.
1. complexity of implementation
2. development cost
3. risks.
Calculation of the parameters in Google Spreadsheet .
For each parameter, we specify the weight and evaluate the functionality on a five-point scale:
• 1 point: few benefits, high cost;
• ...
• 5 points: maximum benefit, low cost.
The functionality that scores the most points is the desired important functionality.
Please note that in the “Cost” group the maximum score means the minimum cost.
2.1 Pirate Metrics (AARRR)
A special case of evaluation by parameters. We use AARRR as parameters and estimate the contribution of the task to each metric:
1. attraction,
2. activation,
3. retention
4. Spread
5. and income.
Calculation in Google Spreadsheet .
As in the general case, we assign weights to metrics.
The most important task is the task with the maximum number of points.
3. Model Kano
The Kano model is a method for evaluating consumers' emotional responses to specific product characteristics.
Omitting the details, conclusions from this model:
1. basic functionalities for the current product, the audience should be by default and stable;
2. Having basic and basic functionality, invest in developing exciting functionality. They will highlight competitors, thanks to them they tell each other about the product.
Forming a list of works, ask yourself:
1. We have implemented all the basic functions?
2. And how many of our core?
3. Explore the audience, looking for exciting features?
4. Remember that admirers will be the main later, and later the base?
4. Buy functionality
The bottom line:
• assign a cost to each functionality, for example, in rubles (link the cost to the development cost);
• distribute to the stakeholders, clients a certain amount of money and offer them to buy the functionality you like;
• sales results and there is a ranked list of functionalities.
5. Affinity grouping
1. Each team member writes his ideas on stickers (one idea - one sticker);
2. then teams combine similar ideas into groups and give names to groups;
3. at the end of the event, each team member votes or indicates the importance of the group in points.
Outcome: ranked groups with ideas for new tasks.
6. Hierarchy of user needs
Aaron Walter proposed Maslow's version of the pyramid for the user's needs. Salt: do not go to the next level until you satisfy the need for the current one.
When choosing tasks to develop, ask yourself:
• at what level are we now?
• What to do now, so that tomorrow we will move to a new level?
• The results of the tasks we have chosen are needed by users?
7. Ask Sales
Communicate with sales managers: ask about what functionalities people ask, what do they lack in the product.
When you receive a request for functionality from the sales department, ask them:
• Why do consumers ask about it?
• Is the task part of a larger trend?
8. RICE
For each functionality, we evaluate:
1. coverage (Reach). How many users will this be for a specific period of time?
2. Impact. How much will this affect the conversion? (maximum - 3, strongly - 2, medium - 1, low - 0.5 and minimum - 0.25).
3. Reliability (Confidence). How confident are you in your assumptions? (Maximum - 100%, medium - 80% and low - 50%).
4. The complexity (Effort). How many person-months will this require? (At least a fortnight and integers).
Collect the data in the table and calculate the estimate according to the formula:
The meaning of the resulting metric is a general increase in conversion over a certain development time. Functionality with the maximum rating - the most important.
Calculation in Google Spreadsheet .
Summing up
The above methods will not help to create a product, determine the need and willingness to pay for its satisfaction. The maximum number of RICE points does not guarantee that a miracle will occur
after the introduction of the functionality.
There are no ready-made recipes for success, but knowledge of the tools, common sense, and a culture of ranking tasks in a team increase the chances of success.
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s - simon_29
« on: August 09, 2016, 08:14:32 PM »
I am currently working on a dam inspection. I have mounted 2 DLSR ( Pentax K3II - Pentax K50 ) for the aerial part and 4 GoPros ( 2 Hero 4 and 2 Hero 2) for the underwater part, on a bar fixed to a
boat. During the acquisition I have recorded IMU data and I would like to import them in PhotoScan. In attachment you can find a drawing that shows how the cameras and the IMU are installed. The
Position and IMU data that I am trying to import are referenced at my cameras location. Up until now, I haven't been able to import them correctly. I know I should put some angular offsets in the
"Camera Calibration" menu, but I don't know what are the suitable values.
From what I have understood the Photocan Manual (see the attachment "PhotoScan_Sing_Convention.png"), the PhotoScan frame is the same as the one from my IMU. However, my cameras are "looking"
perpendicularly to my boat's heading ("Mounting.jpg").
I hope someone can shed some light about this. If I am not clear don't hesitate to contact me.
Thank you in advance !
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Where is the youth of TCS/Market Eq/Guest Post by Vijay
(Guest post by Vijay Vazirani.)
Theory Day on Nov 30, 2012
Vijay Vazirani gave a talk:
New (Practical) Complementary Pivot Algorithms for Market Equilibrium
He was inspired by the reaction to the talk to write a guest blog
which I present here!
Where is the Youth of TCS?
by Vijay Vazirani
I have always been impressed by the researchers of our community, especially the young researchers -- highly competent, motivated, creative, open-minded ... and yet cool! So it has been disconcerting
to note that over the last couple of years, each time I have met Mihalis Yannakakis, I have lamented over the lack of progress on some fundamental problems, and each time the same thought has crossed
my mind, ``Where is the youth of TCS? Will us old folks have to keep doing all the work?''
Is the problem lack of information? I decided to test this hypothesis during my talk at NYTD. To my dismay, I found out that there is a lot of confusion out there! By a show of hands, about 90% of
the audience said they believed that Nash Equilibrium is PPAD-complete and 3% believed that it is FIXP-complete! I would be doing a disservice to the community by not setting things right, hence this
blog post.
First a quick primer on PPAD and FIXP, and then the questions. Ever since Nimrod Megiddo, 1988, observed that proving Nash Equilibrium NP-complete is tantamount to proving NP = co-NP, we have known
that the intractability of equilibrium problems will not be established via the usual complexity classes. Two brilliant pieces of work gave the complexity classes of PPAD (Papadimitriou, 1990) and
FIXP (Etessami and Yannakakis, 2007), and they have sufficed so far. A problem in PPAD must have rational solutions and this class fully characterizes the complexity of 2-Nash, which has rational
equilibria if both payoff matrices have rational entries. On the other hand, 3-Nash, which may have only irrational equilibria, is PPAD-hard; however, its epsilon-relaxation is PPAD-complete. That
leaves the question, ``Exactly how hard is 3-Nash?''
Now it turns out that 3-Nash always has an equilibrium consisting of algebraic numbers. So one may wonder if there is an algebraic extension of PPAD that captures the complexity of 3-Nash, perhaps in
the style of Adler and Beling, 1994, who considered an extension of linear programs in which parameters could be set to algebraic numbers rather than simply rationals. The class FIXP accomplishes
precisely this: it captures the complexity of finding a fixed point of a function that uses the standard algebraic operations and max. Furthermore, Etessami and Yannakakis prove that 3-Nash is
FIXP-complete. The classes PPAD and FIXP appear to be quite disparate: whereas the first is contained in NP INTERSECT co-NP, the second lies somewhere between P and PSPACE (and closer to the harder
end of PSPACE, according to Yannakakis).
Now the questions (I am sure there are more):
1. Computing an equilibrium for an Arrow-Debreu market under separable, piecewise-linear concave utilities is PPAD-complete (there is always a rational equilibrium). On the other hand, if the
utility functions are non-separable, equilibrium consists of algebraic numbers. Is this problem FIXP-complete? What about the special case of Leontief utilities? If the answer to the latter
question is ``yes,'' we will have an interesting demarcation with Fisher markets under Lenotief utilities, since they admit a convex program.
2. An Arrow-Debreu market with CES utilities has algebraic equilibrium if the exponents in the CES utility functions are rational. Is computing its equilibrium FIXP-complete? Again, its
epsilon-relaxation is PPAD-complete.
3. A linear Fisher or Arrow-Debreu market with piecewise-linear concave production has rational equilibria if each firm uses only one raw good in its production, and computing it is PPAD-complete.
If firms use two or more raw goods, equilibria are algebraic numbers. Is this problem FIXP-complete?
23 comments:
1. When I attend STOC/FOCS/SODA, I often ask "Where are the old folks in TCS?" I guess they are too busy "doing all the work" to present papers in which they do all the work.
2. They're working on their own insanely hard open problems, on which you old folks are not making any progress either.
3. Vijay: Look at the perverse incentive system we ("the old folks") have created. Aleksander Madry didn't get a position at a top 5 U.S. institution. In a hyper competitive job market, the young
folks want jobs. Apparently it's easier to chase fads than work on hard problems.
4. TL/DR
5. There are a number of brilliant people comparable to or better than Aleksander Madry that are not at a top 5 US institution. There are simply not enough jobs in 5 places. Young and old people
should follow their interests and strengths, and good thing will happen in time. It is fine for Vijay to passionately advocate for his area/problems but there is no reason to believe that they
are more central/important than many others that people are working on.
6. My impression is the converse: I am more often impressed by the amazing ways in which the youth of TCS continually DO solve those hard problems that have been around a long time - beating
Christofides for TSP, finding a lower bound for ACC^0, improving the exponent for matrix multiplication, proving Yannakakis' extended LP formulation conjecture for TSP - to name just a few recent
7. Vijay Vazirani4:03 PM, December 06, 2012
Anon at 3:07pm: I am not suggesting that these problems are more important than others or that people should drop what they are doing and work on these problems. I am simply lamenting on the
massive confusion that is out there due to which most people are not aware of an important new complexity class and its associated open problems. This is not good for the health of our field and
you can see the adverse effects it has had already. I am simply trying to do my bit to correct this.
Anon at 1:11pm: I agree that overemphasis on fads and flimsy results that will not stand the test of time is not the way to build a solid field. Clearly there have to be forays into new
directions and ideas, but it is not good to lose sight of depth and quality.
8. Which recent theory slot at a top 5 university do you think Madry should have gotten?
9. As a youth across the hall in mathematical logic, the reason TCS turns me off is because there's the appearance that there's nothing really new or exciting, ever.
As a mathematician, I'm interested in being multidisciplinary-- publishing papers in physics and biology and, well, everywhere. TCS seems to be 100% isolated in that sense. Yes, there are TCS
papers published in other fields, but it seems every last one of them is a variation on "(insert something here) is NP-hard".
You really, really, REALLY need to lose all the acronyms. If you can't think up a good name without resorting to acronyms, that's a sign the object is unworthy of study in the first place.
You need to stop over-glorifying petty results-- you're the boy who cried wolf, especially here in the blogosphere. How about this. Before you start talking about groundbreaking-this and
gamechanging-that, wait til the paper is accepted in a prestigious *NON-TCS* journal. Model yourselves more after Press and Dyson, 2012, "Iterated Prisoner’s Dilemma contains strategies that
dominate any evolutionary opponent", PNAS. I suppose you laugh at this paper because it doesn't have 30-page proofs stuffed full of excruciatingly painful estimations. And yet, it does what a
scientist is supposed to do: adds some real insight into how the universe works. TCS hasn't added any such insight in a very, very, very long time.
10. +1 Paul Beame
Adding: PRIMES \in P solved by two undergraduates (okay, plus one "older" guy)
Amit C
11. Never seen this before.
How did it happen?
12. This is some perverse reasoning (not limited to TCS, by the way) by which a position at a top-5 US university is a "success" while a position at any lower-ranked university (or a top-ranked
university in Europe) is a "failure".
13. FEMA ALERT5:31 PM, December 06, 2012
Comments section has reached dangerous levels of debate fragmentation and social toxicity. Please gather your belongings, thank Prof. Vazirani for the nice open problems, and leave.
14. Paul, when is Christofides beaten for TSP? Is it for a special metric or a general metric? There were special metrics for which TSP was already broken, e.g., euclidean matrix. Some recent results
broke it for graphical metric, but that is just one other metric.
Just asking in case I missed some result.
15. To hosts of this blog: don't you think that anonymous commenting should be banned here? The amount of venom and bitterness is sometimes astounding.
16. I really don't understand why Vijay is "dismayed" at there being a "lot of confusion". 90 percent said that Nash Equilibrium is PPAD-complete, and as Vijay explains in his post, there are several
natural, practical problems in Nash Equilibria that are PPAD-complete, such as an epsilon-relaxation of 3-player Nash. So, unless in his question to the audience Vijay specifically said "How hard
is 3-player Nash, not the epsilon relaxation of 3-Nash, but 3-Nash itself?", then I'd say that those 90 percent gave a completely reasonable answer.
17. Vijay Vazirani10:14 PM, December 07, 2012
Question to Anon at 5:03pm: Is Euclidean TSP NP-complete?
1. Hi Vijay,
(I'm not anon from 5:03pm.)
I think Euclidean TSP is NP-complete due to Luca Trevisan's paper "when hamming meets Euclid". But I admit that I am not 100% sure despite being an algorithms person who actually could
explain the details of most of the recent progress on graphic TSP results.
2. Not even known to be in NP, due to precision needed for the sum of square roots problem: http://cs.smith.edu/~orourke/TOPP/P33.html. A tricky question!
18. Euclidean TSP is not known to NP-complete, but it's known to be NP-hard. And it's been known a long time before Trevisan's paper. And as Anon 7:42am said, we don't know if the problem is
NP-complete because of the tricky problem of dealing with the sum of square roots.
However, Vijay's question is not tricky. It's a very basic question which should be taught in complexity classes, and PhD students in theory should be familiar with it. And I think this is what
Vijay is pointing out. That too many of us don't know some very basic facts from complexity theory, like whether Euclidean TSP is NP-complete, or whether Nash is PPAD-complete.
19. Vijay, it is possible that much of the confusion rests with others in the Market Eq community, who use different definitions of {\sc Nash} than you do. For example, in both their 2006 STOC paper
and a 2009 CACM article, Daskalakis, Goldberg, and Papadimitriou make claims such as the following:
"We show that finding a Nash equilibrium is complete for a class of problems called PPAD"
"Indeed, in this paper we describe a proof that {\sc Nash} is PPAD-complete"
"In this paper we show that {\sc Nash} is PPAD-complete, thus answering the open questions discussed above."
You might also get similar mixed results if you asked the audience whether SDP (semidefinite programming) is in P.
20. Vijay's post doesn't make the issue quite as clear as it could be. There are two different versions of approximation in our usual bit model of computation. The problem NASH as defined by
Papadimitriou, which we now know to be PPAD-complete, asks for a point that is an epsilon-approximate equilibrium. However, though a point is close to being in equilibrium, it does not mean that
there is a true equilibrium nearby. The problem considered by Etesammi and Yannakakis is to ask for a point that is within epsilon of a true equilibrium.
21. 2-NASH is PPAD-complete. "Setting the complexity of computing a Nash Equilibrium" by Chen and Deng.
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