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How to Use Algebra Tiles to Multiply Polynomials -- with pictures!
I set a pretty average goal for myself a couple months ago to blog about all the ways to use algebra tiles. But then of course life got in the way and here I am only on post #3-- how to use algebra
tiles to multiply polynomials. I'll get there, but it may take more time than anticipated!
New to algebra tiles? Watch my algebra tiles tutorial video here.
My favorite use of algebra tiles is for factoring quadratics, especially where the A value is greater than 1. Using the tiles makes this process so much more concrete than any other method (in my
opinion). Because it's my favorite, I jumped to blog about it first. But before we even get to factoring, we learn how to multiply the binomials, which is what this post is about.
But first! If you don't have a set of algebra tiles, here is a free set of printable paper algebra tiles. If you print on 2-sided card stock (like Astrobrights) it'll mimic the positive and negative
sides of the plastic tiles.
Example 1:
Multiply (x + 3)(x - 4)
Algebra tiles come in 3 shapes: large square (for +/-x^2), long rectangle (for +/-x) and small square (for the +/- constant values).
The goal of using them to multiply polynomials is to build a rectangular area. This area will have side side lengths of the two binomials you are multiplying (the picture above shows this better than
I can put into words).
The large blue square is now there to show (x)(x) = x^2. Now, because (+)(-) = (-), we will stack 4 (-x) rectangles horizontally below our blue x^2 to show (+x)(-4).
Now we lay 3 (+x) rectangles vertically to show (+x)(+3).
Lastly, we just fill in the space to complete this rectangular-shaped puzzle. (+)(-) = (-), so we need 12 small (-) constant tiles.
Example 2:
Multiply (2x - 1)(x - 1)
This is the exact same process as Example 1, except we'll use 2 big blue x^2 squares to show (2x)(x). We first line up tiles on the sides to show 2x - 1 and x - 1.
Here we've filled in the space with 2 (x^2) tiles.
Now we place 2 rectangular (-x) tiles horizontally below the blue x^2 tiles to show that we are multiplying the 2 green (x) tiles on top by the one small (-) tile on the left side. We put them side
by side in this example because the goal is to make a rectangle with the tiles, and this is the way they fit.
And then fill in that skinny little column on the right with another rectangular (-x) tile.
Lastly, to complete the rectangular puzzle, we plop one (+) tile in that tiny bottom right corner to show (-)(-) = +.
And that is it! If you have a large set of algebra tiles, you can pretty much multiply any two binomials. What would an x^3 tile look like?
Additional resources:
Here is puzzle #5 of a multiplying polynomials digital math escape room. It presents polynomial multiplication as the side lengths of rectangles, the wall lengths of blueprints and as straightforward
polynomial multiplication problems. To meet the needs of students working online, I've made over 50 of these digital math escape rooms, all built in Google Forms to be super easy to send to students.
If you'd like to learn more about ways to use algebra tiles, I have put together an algebra tiles tutorial video that covers ways to use algebra tiles in middle school math.
Using Algebra Tiles in Middle School Math:
2 comments:
1. I love this idea!!!
I think on example two you should have put a positive small square.
Final answer: 2x^2-1x-1
1. Thank you! I need you in my corner before I push publish! I think it's all fixed now. I really appreciate you pointing that out!
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Aggregate Demand I. Building the IS–LM Model - online presentation
Aggregate Demand I. Building the IS–LM Model
Prepared by:
Aggregate Demand I:
Building the IS–LM Model
1. 11-1 The Goods Market and the IS Curve
2. 11-2 The Money Market and the LM Curve
3. 11-3 Conclusion: The Short-Run Equilibrium
3. Aggregate Demand I: Building the IS–LM Model
1. Classical theory (Ch.3-7) seemed incapable of explaining the
a. national Y depends on factor supplies and the available technology,
b. neither of which changed substantially from 1929 to 1933.
c. A new model was needed.
2. In 1936 the British economist John Maynard Keynes revolutionized
economics with his book The General Theory of Employment,
Interest, and Money.
3. Keynes proposed that
a. ↓ AD is responsible for the ↓ Y & ↑ U that characterize economic ↘.
b. He criticized classical theory for assuming that AS alone determines Y.
c. Economists today reconcile these 2 views with the model of AD&AS
4. In the LR, Ps are flexible, and AS determines Y.
5. In the SR, Ps are sticky, so changes in AD influence Y.
4. Aggregate Demand I: Building the IS–LM Model
1. Our goal is
a. to identify the variables that shift the AD curve, causing fls in national Y.
b. to examine the tools policymakers can use to A D .
2. (Ch. 10) We showed that monetary policy can shift the AD curve.
3. (this Ch.) We see that the government can AD with both
•. monetary and fiscal policy.
IS–LM model,
a. is the leading interpretation of Keynes’s theory.
b. The goal of the model is to show what determines national Y for a
given P.
We can view the IS–LM model as showing what causes
c. Y to change in the SR when the P is fixed because all Ps are sticky
d. the AD curve to shift.
5. Aggregate Demand I: Building the IS–LM Model
Shifts in Aggregate Demand
For a given P, national Y fluctuates because of shifts in the AD curve.
The IS–LM model takes the P as given and shows what causes Y to change.
The model therefore shows what causes AD to shift.
The two parts of the IS–LM model
1. the IS curve
1. stands for “investment’’ and “saving,’’
2. represents what’s going on in the market for G&S (Ch. 3).
2. The LM curve
1. stands for “liquidity’’ and “money,’’
2. Represents what’s happening to the S&D for money (Ch. 5).
The r both I & money demand, →
•. it is the variable that links the 2 halves of the IS–LM model.
3. The model shows
a. how interactions between the G & money markets determine
the position and slope of the AD curve and
b. → the level of national Y in the SR.
7. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
The IS curve plots the relationship between the r & the level of Y
• that arises in the market for G&Ss.
To develop this relationship, we start with the Keynesian cross –
1. It shows how national Y is determined and
2. It is block for the IS–LM model.
In The General Theory Keynes proposed that
•. an economy’s total Y is, in the SR, determined largely by the
spending plans of H, B, and G.
> people want to spend →
> G&S firms can sell →
> Y they will choose to produce →
> workers they will choose to hire.
Keynes believed that the problem during recessions and depressions is
inadequate spending.
•. The KC is an attempt to model this insight.
8. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
Planned Expenditure
Let us draw a distinction between actual & planned expenditure.
Actual expenditure
• is the amount Hs, Fs, & the G spend on G&S(Ch.2), it = the
economy’s GDP.
Planned expenditure
• is the amount Hs, Fs, & the G would like to spend on G&S.
Why would AE ever differ from PE?
The answer is that
• firms might engage in unplanned inventory investment because
their sales do not meet their expectations.
1. When Fs sell < of their product than they planned,
• their stock of inventories automatically r↑;
2. When Fs sell > than planned,
• their stock of inventories f↓.
These unplanned changes in inventory are counted
• as INVESTMENT spending by Fs,
• → AE can be either ͞ or _͟ PE .
9. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
Now we consider the determinants of PE.
• that the economy is closed, → NX = 0, →
1. PE = C + I + G. We add the consumption function:
2. C = C(Y − T)- consumption ~on disposable Y (Y − T),
• To keep things simple, we take PI as exogenously fixed:
• As in Ch.3, we assume that fiscal policy— the levels of G & T—is
• Combining these 5 equations, we obtain
• PE is a function of Y, the level of PI , & the fiscal policy variables.
10. Planned Expenditure as a Function of Income
PE ~ on Y because higher Y leads to ↑er consumption, which is part of PE.
The slope of the PE function is the marginal propensity to consume, MPC.
11. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
The Economy in Equilibrium
The next assumption is that the economy is in equilibrium when AE =
• is based on the idea that
• when people’s plans have been realized, they have no reason
to change what they are doing.
Recalling that
• Y as GDP = not only total Y but also total AE on G&S, we can write
this equilibrium condition as
AE = PE
Y = PE.
The 45-degree line plots the points where this condition holds.
• With the addition of the PE function, this diagram becomes the
How does the
economy get to
Whenever an
economy is not in
equilibrium, Fs
• unplanned changes
in inventories, →
• Changes in
production levels →
• Changes in total Y
and expenditure →
• equilibrium.
The Keynesian Cross
The equilibrium in the KC is the point at which Y (AE ) equals PE (point A).
13. The Adjustment to Equilibrium in the Keynesian Cross
• If Fs are
producing at level
then PE1 falls short
of production, and
Fs accumulate
• This inventory
induces Fs to ↘
Similarly, if Fs are producing at level Y2, then PE2 exceeds production, and Fs run down
their inventories.
This fall in inventories induces Fs to increase production.
In both cases, the Fs ’ decisions drive the economy toward equilibrium.
14. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
Fiscal Policy and the Multiplier: Government Purchases
Consider how changes in G affect the economy.
• G are one component of expenditure →
• ↑er G result in ↑er PE for any given level of Y.
• If G r↗ by ∆G, then the PE schedule shifs up↑ by ∆G
• The equilibrium of the economy moves from point A to point B.
This graph shows that
• an ↗ in G → to an even greater ↗ in Y.
• → ∆Y is > ∆G.
The ratio ∆Y/∆G is called the government-purchases multiplier;
• it tells us how much I r↑ in response to a $1 ↗ in G.
• An implication of the KC is that the G multiplier is larger than 1.
15. An Increase in Government Purchases in the Keynesian Cross
Note that the ↗ in Y exceeds the ↗ in G.
→ fiscal policy has a multiplied effect on Y.
An ↗ of G
raises PE by
that amount for
any given level
of Y.
The equilibrium
moves from
point A to point
B, and Y rises
from Y1 to Y2.
16. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
How big is the multiplier?
• we trace through each step of the change in Y.
1. Expenditure r↑ by ∆G → Y r↑ by ∆G as well.
2. This ↗ in Y in turn r↑ consumption by MPC× ∆G,
where MPC is the marginal propensity to consume.
This ↗ in consumption r↑ expenditure and Y once again.
3. This second ↗ in Y of MPC × ∆G again raises C, this time by MPC
× (MPC ×∆ G), which again ↑s expenditure and Y, and so on.
This feedback from C to Y to C continues indefinitely.
The total effect on Y is
•. Initial Change in Government Purchases = ∆G
1. First Change in Consumption = MPC × ∆G
2. Second Change in Consumption = MPC2 × ∆G
3. Third Change in Consumption = MPC3 × ∆G
4. .. .. . .
∆Y = (1 + MPC + MPC2 + MPC3 + . . .)∆G.
17. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
The G multiplier is Y/G = 1 + MPC + MPC2 + MPC3 + . . .
This expression for the multiplier is an example of an infinite geometric
A result from algebra allows us to write the multiplier as
Y/G = 1/(1 − MPC).
For example, if the MPC is 0.6, the multiplier is
Y/G = 1 + 0.6 + 0.62 + 0.63 + . . .= 1/(1 − 0.6) = 2.5.
In this case, a $1.00 increase in government purchases raises equilibrium
income by $2.50.3
18. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
Fiscal Policy and the Multiplier: Taxes
A ↘in T of ∆T immediately r↑ disposable income Y − T by ∆T and, →
• ↗ consumption by MPC × T.
• For any given level of Y, PE is now ↑er.
• The PE schedule shifs up↑ by MPC × T.
The equilibrium of the economy moves from point A to point B.
Just as an ↗ in G has a multiplied effect on Y, → does a ↘ in Ts.
As before, the initial change in expenditure,
• now MPC × T, is multiplied by 1/(1 − MPC).
The overall effect on Y of the change in Ts is Y/T = −MPC/(1 − MPC).
This expression is the tax multiplier,
• the amount Y changes in response to a $1 change in Ts.
• The “-” sign indicates that Y moves in the opposite direction from Ts.
For example,
if the marginal propensity to consume is 0.6, then
• the tax multiplier is Y/T = −0.6/(1 − 0.6) = −1.5.
• A $1.00 cut in taxes r↑s equilibrium Y by $1.50.
A Decrease in Taxes in the Keynesian Cross
A ↘ in T of ∆T r↑ PE by MPC × ∆T for any given level of Y.
The equilibrium moves from point A to point B, and Y r↑ from Y1 to Y2.
Fiscal policy has a multiplied effect on Y.
20. Cutting Taxes to Stimulate the Economy: The Kennedy and Bush Tax Cuts
John F. Kennedy became president of the United States in 1961.
• One of the council’s first proposals was to expand national Y by reducing
• Tax cuts stimulate aggregate supply by improving workers’ incentives and
expand AD by raising households’ disposable Y.
• GDP ↗, U ↘
George W. Bush was elected president in 2000, a major element of his platform
was a cut in Y taxes.
• Bush used both supply-side and Keynesian rhetoric to make the case for their
• When people have more M., they can spend it on G&S.
• When they demand an additional G&S, somebody will produce the G or S.
• When somebody produces that G&S, it means somebody is more likely to be
able to find a job.
• GDP ↗, U ↘
21. Increasing Government Purchases to Stimulate the Economy: The Obama Spending Plan
When President Barack Obama took office in January 2009, the economy was
suffering from a significant recession.
• The package included some tax cuts and higher transfer payments, but much
of it was made up of ↗ in G of G&S.
Congress went ahead with President Obama’s proposed stimulus plans with
relatively minor modifications.
• The president signed the $787 billion billon February 17, 2009.
Did it work?
• The economy did recover from the recession,
• but much more slowly than the Obama administration economists initially
Whether the slow recovery reflects
1. the failure of stimulus policy or
2. a sicker economy than the economists first appreciated
is a question of continuing debate.
22. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
The KС
• explains the economy’s AD curve
• shows how the spending plans of H, F, the G determine the Y.
• makes the assumption that the level of PI is fixed.
An important macroeconomic relationship is that PI ~ on the r (Ch.3).
To add this relationship between the r & I to our model,
• we write the level of PI as I = I(r).
The r is the cost of borrowing to finance investment projects
→ an ↗ in the r reduces PI.
→ the investment function slopes downward.
To determine how Y changes when the r changes,
• we can combine the investment function with the KС diagram.
Deriving the IS Curve
Panel (a) shows the investment function:
an ↗ in the r from r1 to r2 reduces PI from
I(r1) to I(r2).
Panel (b) shows the KC:
a ↘ in PI from I(r1) to I(r2) shifts the PE
function ↘ and → reduces Y from Y1 to Y2.
Panel (c) shows the IS curve summarizing
this relationship between the r and Y:
the ↑er the r, the ↓er the level of Y.
24. 11-1 The Goods Market and the IS Curve
The Keynesian Cross
The Interest Rate, Investment, and the IS Curve
How Fiscal Policy Shifs the IS Curve
11-1 The Goods Market and the IS Curve
The IS curve shows us,
• for any given r, the level of Y that brings the goods market into
As we learned from the KC,
• the equilibrium level of Y also ~on Gnt spending G and taxes T.
The IS curve is drawn for a given FP; that is,
• when we construct the IS curve, we hold G and T fixed.
• When FP changes, the IS curve shifs.
Changes in FP that
1. r↑ the demand for G&S shif the IS curve to the right.
2. r↓ the demand for G&S shif the IS curve to the left.
An Increase in Government
Purchases Shifts the IS Curve
Panel (a) shows that an ↗ in G
r↑s PE.
For any given r,
the u↑ shift in PE of G → to an ↗
in Y of ∆G/(1 – MPC).
→ in panel (b), the IS curve shifts
to the right by this amount.
26. 11-2 The Money Market and the LM Curve
The Theory of Liquidity Preference
Income, Money Demand, and the LM Curve
How Monetary Policy Shifs the LM Curve
11-2 The Money Market and the LM Curve
27. The Theory of Liquidity Preference
The S&D for RMB determing the r.
The S curve for RMB is vertical because the S does not ~on the r.
The D curve is d↓ sloping because a ↑er r
1. r↑ the cost of holding money and →
2. lowers the quantity demanded.
At the equilibrium r, the quantity of RMB demanded = the quantity supplied.
A Reduction in the Money Supply in the Theory of Liquidity Preference
If the P is fixed, a reduction in the M from M1 to M2 reduces the S of RMB.
The equilibrium r → r↑ from r1 to r2.
29. Does a Monetary Tightening Raise or Lower Interest Rates?
Does a Monetary Tightening Raise or Lower Interest Rates?
30. 11-2 The Money Market and the LM Curve
The Theory of Liquidity Preference
Income, Money Demand, and the LM Curve
How Monetary Policy Shifs the LM Curve
11-2 The Money Market and the LM Curve
Deriving the LM Curve
Panel (a) shows the market for RMB: an ↗ in Y from Y1 to Y2 raises the demand for
money and thus raises the interest rate from r1 to r2.
Panel (b) shows the LM curve summarizing this relationship between the interest rate and
income: the higher the level of income, the higher the interest rate.
32. 11-2 The Money Market and the LM Curve
The Theory of Liquidity Preference
Income, Money Demand, and the LM Curve
How Monetary Policy Shifs the LM Curve
11-2 The Money Market and the LM Curve
The LM curve shows the combinations of the interest rate and the
level of Y that are consistent with equilibrium in the market for RMB.
The LM curve is drawn for a given supply of RMB.
• ↘ in the supply of RMB shift the LM curve u↑.
• ↗in the supply of RMB shift the LM curve d↓.
34. 11-3 Conclusion: The Short-Run Equilibrium
We now have all the pieces of the IS–LM model.
The two equations of this model are
1. Y = C(Y − T ) + I(r) + G - IS,
2. M/P = L(r, Y)
- LM.
Equilibrium in the
IS–LM Model
The intersection of the IS and LM curves represents simultaneous equilibrium
in the market for G&S & in the market for RMB
for given values of Gnt spending, T, the M, and the P.
The Theory of Short-Run Fluctuations
This schematic diagram shows how the different pieces of the theory of SR fluctuations fit
1. The KC explains the IS curve, and the TLP explains the LM curve.
2. The IS and LM curves together yield the IS–LM model, which explains the AD curve.
3. The AD curve is part of the model of AS & AD, which economists use to explain SR
fluctuations in economic activity.
THANKS !
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Kolmogorov - Arnold Networks: The Future of AI? | Edlitera
AI is a rapidly advancing field, with new papers detailing novel neural network architectures and diverse problem-solving approaches emerging almost daily. However, breakthroughs are less frequent;
the field more often progresses through incremental changes.
This is especially true for the domain of explainable AI. Most AI research focuses primarily on making models more efficient at solving specific tasks. After all, a model by itself is useless if it
doesn't effectively help us in solving a particular problem. While it is logical to prioritize enhancing models' task-solving capabilities, this emphasis often leads to the relative neglect of
explainable AI, at least in comparison.
Kolmogorov-Arnold Networks, at first glance, appear potentially superior to the models we already have and not only. They also enable us to dive deeper into the inner workings of AI models. This
helps us better understand what is happening "under the hood" of an advanced artificial neural network.
What Is Explainable AI
Artificial neural networks are classified as black box models due to their complexity. This makes it exceptionally difficult to decipher their decision-making processes. This is especially true for
more sophisticated models, such as the ones based on the Transformers architecture. Indeed, these models are known for their remarkable predictive power and accuracy, however, this comes with a
downside. They often contain millions or even billions of parameters, making it virtually impossible to trace the model's "thought process."
This characteristic of AI models is far from ideal. However, many are willing to overlook the inability to precisely determine how these models make their predictions as long as they produce good
results Yet, in sensitive fields such as healthcare, finance, legal systems, and similar areas, the opacity of black box models like neural networks becomes a significant barrier to successful
application. In these fields, even minor errors can have drastic consequences, potentially impacting people's lives, or in extreme cases, costing lives. Therefore, without achieving a certain level
of interpretability, resistance will persist from those who believe that the lack of accountability in these situations is unacceptable.
Research on the explainability of AI models, while not as extensive as the research focusing on improving model performance, is still a significant and growing field. Explainability in AI is usually
approached using two main types of techniques:
• model-specific methods
• model-agnostic methods
Model-specific methods are designed to explain the behavior of a specific type of model. For example, techniques like DeepLIFT, Grad-CAM, and Integrated Gradients are used for interpreting deep
learning models, while decision trees can be visualized to show the decision-making process from top to bottom. These methods are tailored to particular models and provide insights that are unique to
those models.
On the other hand, model-agnostic methods, such as LIME and SHAP, aim to explain the predictions of any machine learning model without depending on the model's internal workings. These methods can be
applied across different types of models, offering more versatile tools for interpretation.
So, where do Kolmogorov-Arnold Networks (KANs) fit in this framework? KANs represent a novel approach, where the model's architecture inherently supports interpretability. Their goal is to transition
from black box models to white box models. While KANs have not yet fully achieved this transition, and it would be premature to declare them as entirely white box models, they are currently the
closest approximation we have to such a model.
What Is the Kolmogorov-Arnold Theorem
Kolmogorov-Arnold Networks (KANs) are a special type of neural network inspired in its design by a mathematical theorem known as the Kolmogorov-Arnold representation theorem. This theorem states the
" If f is a multivariate continuous function on a bounded domain, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition."
The definition above is quite complex, so let us simplify it. The theorem essentially means that a complicated function involving many variables can be decomposed into a series of simpler functions,
each with just one variable.
Let us compare the architecture of a Multi-Layer Perceptron (MLP) to a Kolmogorov-Arnold Network (KAN) model. This is done to understand how this works in practice and why it represents traditional
neural network approaches, where weights are used.
Want to learn more? Check out some of our courses:
What is the difference between Multi-Layer Perceptrons and Kolmogorov-Arnold Networks
The entire concept of using neural networks is based on the Universal Approximation Theorem that states the following:
"A neural network with at least one hidden layer of a sufficient number of neurons, and a non-linear activation function can approximate any continuous function to an arbitrary level of accuracy."
In essence, nearly every problem or real-world process can be described by a mathematical function. However, because of the complexity of these processes, the corresponding functions are also quite
complex. In theory, one could identify an ideal mathematical function that perfectly captures the desired behavior. However, in practice, accomplishing this is often not feasible.
For example, a mathematical function could theoretically exist that perfectly predicts the price based on features such as size (in square feet), number of bedrooms, number of bathrooms, and age.
However, finding this exact function in practice is nearly impossible. Instead, we train a neural network to approximate this function. In other words, we train our network to learn a function that
closely resembles, but is not identical to, the ideal function that describes the relationships between these variables.
A conventional neural network, specifically a Multi-Layer Perceptron (MLP), accomplishes this through layers of neurons. Each neuron (also known as nodes) is a small processing unit that takes
certain input values (like the number of bedrooms, number of bathrooms, etc.) and multiplies them by weights. It then sums these weighted inputs and passes the result through a non-linear activation
function, producing an output value.
Each layer contains multiple neurons, all receiving the same input values but using different weights. Once every neuron in a layer generates an output, these outputs become the inputs for the next
layer. This process repeats through the successive layers until the final layer, where the network produces a prediction. According to that prediction, the weights of the network are updated to
ensure a better final prediction for next time. Moreover, the non-linear activation functions are fixed and remain the same during training.
Multi-Layer Perceptron (https://arxiv.org/abs/2406.13155)
Kolmogorov-Arnold Networks marks a radical shift from the conventional neural network paradigm by redefining how networks learn through changes to activation functions. In standard Multi-Layer
Perceptrons (MLPs), complex multivariate functions are approximated using fixed activation functions across multiple layers of nodes.
The KAN theorem states that any continuous multivariate function can be expressed as a finite combination of simpler, one-variable functions In other words, the way that a neural network approaches
the problem of approximation can be modified. Instead of approximating a function with a series of linear functions and fixed activation functions, we aim to use multiple smaller functions. In
practice, this means that the activation functions are no longer fixed in a KAN model.
Instead, they are something that the model optimizes during training. Linear weights do not exist at all, and we replace them with activation functions that are trainable and depend on a single
variable. They are the multiple, smaller univariate functions mentioned in the KAN theorem.
Kolmogorov-Arnold Network (https://arxiv.org/abs/2406.13155)
How do B-Splines function as the weights of the Kolmogorov-Arnold Networks
The fixed non-linear activation functions used in conventional neural networks are well-known to most people in the field of AI. These functions represent an important hyperparameter in our neural
networks. However, once the activation functions for particular layers are selected, they remain unchanged during training. Therefore, there is not much else to manage regarding them.
In Kolmogorov-Arnold Networks (KANs), however, we need to train the activation functions. This means we must modify them during training based on the outcomes of our models. In conventional networks,
we have fixed linear weights that are initialized randomly. During backpropagation, we use gradient descent to optimize these weights based on the network's output. We calculate the gradient of the
weight and adjust the weight value in the direction opposite to the gradient by a certain amount. This process works because the weights are fixed values that we can initialize randomly.
But in KANs, where actual functions need to be trained, the process is more complex. How do we initialize these functions? How can we ensure they are modified sufficiently during training so that the
overall model output changes effectively?
This is where splines, more precisely B-Splines, come into play. Splines are a mathematical concept used to create smooth curves through a set of points. They are especially useful in fields like
computer graphics, data fitting, and numerical analysis. The general idea is to piece together several polynomial functions in a manner that they join smoothly at certain points called knots.
B-splines, or Basis splines, are a special type of spline function characterized by their local control property. To simplify, changing the position of a single knot affects only a limited portion of
the entire spline. This makes it easier to manipulate and refine specific sections of the curve without altering the entire shape.
Spline (https://rohangautam.github.io/blog/b_spline_intro/#splines)
By adjusting its shape, the spline can model complex relationships. When we first create a spline with a set number of polynomials, it won't necessarily pass through the knots, or even be close to
them. However, through training, it can be shaped to better fit the data. This is where B-Splines show their special property: their ability to refine specific sections of the spline without heavily
moving the entire spline.
Through the process of adapting its shape, the spline "learns" the subtle patterns that exist in our data. Therefore, these splines represent the learnable activation functions in our KANs. Similar
to how we have multiple weights multiplying the same input and going into different neurons, here we will have multiple splines connected to the same input. Each spline will be trained separately
from the other. We will also have multiple layers of these, once again similar to what you can usually find in a standard MLP.
Multi-Layer KAN (https://arxiv.org/abs/2406.13155)
What Is the Interpretability of KANs
One of the main advantages of KANs is their interpretability, as was mentioned at the beginning. The main source of interpretability is the fact that, at the end of the training, we have the set of
functions that leads from the inputs to the predicted output. In other words, we have our splines. Furthermore, the authors of the paper that introduces KANs offer two additional methods of making
the model even more interpretable:
Pruning involves removing certain "branches" of our network. More precisely, L1 regularization is applied to the trained activation functions. By evaluating the L1 norms of each function and
comparing them to a threshold, we can identify neurons and edges with norms below the threshold as non-essential. These components are then removed from the network, reducing its size and making it
easier to analyze and interpret.
Symbolification on the other hand involves replacing the learned univariate functions with known symbolic expressions. Primarily, we analyze the learned functions and propose symbolic candidates
based on their shapes and behaviors (e.g., sin, exp). Then we use methods such as the grid search method to adjust the parameters of the symbolic functions. This way they can closely approximate the
learned functions.
For example, let us say that one of the learned functions resembles in shape the sine function. What we can do is try to approximate the learned functions with a candidate symbolic expression such
f(x)=A sin(Bx+C)
Then we use a grid search method, or any other search method, to find the best possible values for A, B, and C. This ensures that this new variant of the sine function can efficiently replace the
learned function in our network. For instance, we might determine that the best-fit parameters are A=1.0, B=1.5, and C=0.0. This will allow us to replace the function defined with the spline in our
network with:
This approach further simplifies the network by replacing arbitrary spline-defined functions with well-known and widely analyzed variants of common mathematical functions.
What Are Other Advantages of KANs
The benefits of KANs extend beyond their enhanced interpretability. Other notable advantages over traditional networks include:
• efficiency
• scalability
• accuracy
The efficiency of KANs is a direct byproduct of how they work. By breaking down complex functions into simpler components, the training process is accelerated and computational costs are reduced.
Additionally, KANs are particularly adept at handling high-dimensional data. This is especially advantageous in domains such as image and speech recognition where data dimensionality is vast.
Finally, the precise decomposition and recombination of functions enable KANs to potentially achieve higher accuracy compared to traditional models.
However, it should be noted that KANs are still relatively new and have not been extensively tested in industry applications. So far, their advantages over conventional networks have mainly been
demonstrated on specific benchmark datasets. Comprehensive industry testing will be necessary to confirm that KANs consistently outperform traditional models in these aspects.
Kolmogorov-Arnold Networks (KANs) have the potential to signify a groundbreaking shift in the landscape of deep learning. Their unique approach to function decomposition and interpretability suggests
a potential paradigm shift away from traditional neural networks. KANs have demonstrated significant promise in terms of efficiency, scalability, and accuracy on benchmark datasets. However, their
true potential is yet to be fully explored and validated through extensive industry application. As research and experimentation continue, KANs could unlock new levels of performance and
understanding in artificial intelligence, heralding a new era in the field of deep learning.
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Making cool camera rotation around player [SOLVED]
For the camera follow of player rebirth ??
1 Like
local newCF = CFrame.lookAt(center.Position
+ math.sin(angle) * Vector3.zAxis * radius
+ math.cos(angle) * Vector3.xAxis * radius,
center.Position) * CFrame.new(0, CurrentPlayer.Head.Position, 0)
1 Like
What you’re doing in this code is adding the current player’s head position to the rotating camera, which will put the camera way off of where it should be.
First do your extra math in a second line of code so it’s easier to follow.
-- Seperate the math into different lines of code
local newCF = CFrame.lookAt(center.Position
+ math.sin(angle) * Vector3.zAxis * radius
+ math.cos(angle) * Vector3.xAxis * radius,
newCF *= CFrame.new(0, positionOfHead.Y, 0)
You are adding the distance between the player’s head and the ground to the camera that is already near the player, so the camera will shoot up into the sky! Add the difference between the newCF
position and the head position instead:
-- Replace the second line with:
newCF *= CFrame.new(0, positionOfHead.Y - newCF.Position.Y, 0)
3 Likes
Thanks for the solution man really appreciated
1 Like
This topic was automatically closed 14 days after the last reply. New replies are no longer allowed.
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1,846 research outputs found
Helical magnetic background fields with adjustable pitch angle are imposed on a conducting fluid in a differentially rotating cylindrical container. The small-scale kinetic and current helicities are
calculated for various field geometries, and shown to have the opposite sign as the helicity of the large-scale field. These helicities and also the corresponding $\alpha$-effect scale with the
current helicity of the background field. The $\alpha$-tensor is highly anisotropic as the components $\alpha_{\phi\phi}$ and $\alpha_{zz}$ have opposite signs. The amplitudes of the azimuthal $\
alpha$-effect computed with the cylindrical 3D MHD code are so small that the operation of an $\alpha\Omega$ dynamo on the basis of the current-driven, kink-type instabilities of toroidal fields is
highly questionable. In any case the low value of the $\alpha$-effect would lead to very long growth times of a dynamo in the radiation zone of the Sun and early-type stars of the order of
mega-years.Comment: 6 pages, 7 figures, submitted to MNRA
We investigate the instability and nonlinear saturation of temperature-stratified Taylor-Couette flows in a finite height cylindrical gap and calculate angular-momentum transport in the nonlinear
regime. The model is based on an incompressible fluid in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients itself, the differential rotation as well
as the stratification due to the temperature gradient, are stable, together the system becomes subject of the stratorotational instability and nonaxisymmetric flow pattern evolve. This flow
configuration transports angular momentum outwards and will therefor be relevant for astrophysical applications. The belonging viscosity $\alpha$ coefficient is of the order of unity if the results
are adapted to the size of an accretion disc. The strength of the stratification, the fluids Prandtl number and the boundary conditions applied in the simulations are well-suited too for a laboratory
experiment using water and a small temperature gradient below five Kelvin. With such a rather easy realizable set-up the SRI and its angular momentum transport could be measured in an
experiment.Comment: 10 pages, 6 figures, revised version appeared in J. Fluid Mech. (2009), vol. 623, pp. 375--38
We demonstrate with a nonlinear MHD code that angular momentum can be transported due to the magnetic instability of toroidal fields under the influence of differential rotation, and that the
resulting effective viscosity may be high enough to explain the almost rigid-body rotation observed in radiative stellar cores. Only stationary current-free fields and only those combinations of
rotation rates and magnetic field amplitudes which provide maximal numerical values of the viscosity are considered. We find that the dimensionless ratio of the effective over molecular viscosity,
$u_T/u$;, linearly grows with the Reynolds number of the rotating fluid multiplied with the square-root of the magnetic Prandtl number - which is of order unity for the considered red sub-giant KIC
7341231. For the considered interval of magnetic Reynolds numbers - which is restricted by numerical constraints of the nonlinear MHD code - there is a remarkable influence of the magnetic Prandtl
number on the relative importance of the contributions of the Reynolds stress and the Maxwell stress to the total viscosity, which is magnetically dominated only for Pm $\gtrsim$ 0.5. We also find
that the magnetized plasma behaves as a non-Newtonian fluid, i.e. the resulting effective viscosity depends on the shear in the rotation law. The decay time of the differential rotation thus depends
on its shear and becomes longer and longer during the spin-down of a stellar core.Comment: Revised version. 7 pages, 9 figures; accepted for publication in A&
Context. Using asteroseismic techniques, it has recently become possible to probe the internal rotation profile of low-mass (~1.1-1.5 Msun) subgiant and red giant stars. Under the assumption of local
angular momentum conservation, the core contraction and envelope expansion occurring at the end of the main sequence would result in a much larger internal differential rotation than observed. This
suggests that angular momentum redistribution must be taking place in the interior of these stars. Aims. We investigate the physical nature of the angular momentum redistribution mechanisms operating
in stellar interiors by constraining the efficiency of post-main sequence rotational coupling. Methods. We model the rotational evolution of a 1.25 Msun star using the Yale Rotational stellar
Evolution Code. Our models take into account the magnetic wind braking occurring at the surface of the star and the angular momentum transport in the interior, with an efficiency dependent on the
degree of internal differential rotation. Results. We find that models including a dependence of the angular momentum transport efficiency on the radial rotational shear reproduce very well the
observations. The best fit of the data is obtained with an angular momentum transport coefficient scaling with the ratio of the rotation rate of the radiative interior over that of the convective
envelope of the star as a power law of exponent ~3. This scaling is consistent with the predictions of recent numerical simulations of the Azimuthal Magneto-Rotational Instability. Conclusions. We
show that an angular momentum transport process whose efficiency varies during the stellar evolution through a dependence on the level of internal differential rotation is required to explain the
observed post-main sequence rotational evolution of low-mass stars.Comment: 8 pages, 6 figures; accepted for publication in Astronomy & Astrophysic
We consider axially periodic Taylor-Couette geometry with insulating boundary conditions. The imposed basic states are so-called Chandrasekhar states, where the azimuthal flow $U_\phi$ and magnetic
field $B_\phi$ have the same radial profiles. Mainly three particular profiles are considered: the Rayleigh limit, quasi-Keplerian, and solid-body rotation. In each case we begin by computing linear
instability curves and their dependence on the magnetic Prandtl number Pm. For the azimuthal wavenumber m=1 modes, the instability curves always scale with the Reynolds number and the Hartmann
number. For sufficiently small Pm these modes therefore only become unstable for magnetic Mach numbers less than unity, and are thus not relevant for most astrophysical applications. However, modes
with m>10 can behave very differently. For sufficiently flat profiles, they scale with the magnetic Reynolds number and the Lundquist number, thereby allowing instability also for the large magnetic
Mach numbers of astrophysical objects. We further compute fully nonlinear, three-dimensional equilibration of these instabilities, and investigate how the energy is distributed among the azimuthal
(m) and axial (k) wavenumbers. In comparison spectra become steeper for large m, reflecting the smoothing action of shear. On the other hand kinetic and magnetic energy spectra exhibit similar
behavior: if several azimuthal modes are already linearly unstable they are relatively flat, but for the rigidly rotating case where m=1 is the only unstable mode they are so steep that neither
Kolmogorov nor Iroshnikov-Kraichnan spectra fit the results. The total magnetic energy exceeds the kinetic energy only for large magnetic Reynolds numbers Rm>100.Comment: 12 pages, 14 figures,
submitted to Ap
The instability of a quasi-Kepler flow in dissipative Taylor-Couette systems under the presence of an homogeneous axial magnetic field is considered with focus to the excitation of nonaxisymmetric
modes and the resulting angular momentum transport. The excitation of nonaxisymmetric modes requires higher rotation rates than the excitation of the axisymmetric mode and this the more the higher
the azimuthal mode number m. We find that the weak-field branch in the instability map of the nonaxisymmetric modes has always a positive slope (in opposition to the axisymmetric modes) so that for
given magnetic field the modes with m>0 always have an upper limit of the supercritical Reynolds number. In order to excite a nonaxisymmetric mode at 1 AU in a Kepler disk a minimum field strength of
about 1 Gauss is necessary. For weaker magnetic field the nonaxisymmetric modes decay. The angular momentum transport of the nonaxisymmetric modes is always positive and depends linearly on the
Lundquist number of the background field. The molecular viscosity and the basic rotation rate do not influence the related {\alpha}-parameter. We did not find any indication that the MRI decays for
small magnetic Prandtl number as found by use of shearing-box codes. At 1 AU in a Kepler disk and a field strength of about 1 Gauss the {\alpha} proves to be (only) of order 0.005
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Policy Impacts Library | Top Tax Rate Increases from the Omnibus Budget Reconciliation Act of 1993
The Omnibus Budget Reconciliation Act of 1993 generated an increase in the top marginal income tax from 31% to 39.6%. Caroll (1998) studies the impact of this reform on taxable income and estimates
an elasticity of taxable income with respect to the after-tax keep rate of 0.38. The authors note this could be an upper bound if part of the change is due to tax shifting as opposed to real
Hendren and Sprung-Keyser (2020) translate this elasticity into the MVPF of the change in tax rates using the equation
FE = \frac{-t}{1-t}*\alpha*\epsilon_{eti}
\alpha = \frac{E[Y]}{E[Y-y|Y\geq y]}
is the Pareto Parameter of the income distribution and
\epsilon = \frac{d[E[y]]}{d(1-t)}\frac{1-t}{E[y]}
is the elasticity of taxable income with respect to the keep rate, 1-t.
Throughout, Hendren and Sprung-Keyser (2020) measure t as the sum of the federal income tax rate and a 5% state income tax rate assumption. In practice, the reforms are discrete changes in t. To
account for this, Hendren and Sprung-Keyser (2020) compute the fiscal externality above separately for the pre- and post-reform tax rates, and then take an average of the two FEs. Appendix F of
Hendren and Sprung-Keyser (2020) provides further details and references.
The key additional parameter beyond the elasticity of taxable income is the Pareto parameter of the income distribution. Atkinson et al. (2011) find a value of 1.77 for 1993. Combining with the
elasticity of taxable income of 0.38 implies an MVPF of 1.85, with a confidence interval of [1.191,4.066].
The estimates used to calculate this MVPF may have been updated in a more recent working or published version of the paper.
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[Solved] Find the value of tan[cos−1(21)+tan−1(−31)] - Trigo... | Filo
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Question Text Find the value of
Updated On Apr 22, 2023
Topic Inverse Trigonometric Functions
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 3
Upvotes 357
Avg. Video Duration 8 min
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Tampon Users
I can’t believe this... Young Somali gurls are using tampons. I really don’t see the point.
You guys might think that I am over the top with this, but walahi I am not.
Embarrassing moment happen to me, the other day. I went with mum and these other ladies so they can visit this lady who gave birth. These ladies are yapping away.
I was having stomach pains before we arrived their but I thought the stay would be short. I usually get a 30minute stomach-ache before I get my period. My mum and these ladies were looking like as if
they were gonna stay over the night.
I had to do something, so I asked where Xalima is. Xalima is a 17-year-old school gurl who is doing her final year. I told her that I’m getting my period and I need a pad, before you know
it,,,,,she pulls this huge tampon out :eek: . My stomach pain got worse, and my eyes nearly popped out. I told her, that I only use pads.
I’m quick to say something critical but I didn’t want to embarrass her and put her on the spot and start lecturing her.
I asked her, why she uses it, does it hurt, how long has she been using it, is it convenient and does her mother know?
Thank god, her mother was menstruating. So I asked her to get me one of her mothers pad.
But one question I was reluctant to ask her was: have u lost your virginity??? I didn’t know how appropriate that was. I mean those things are big, and you can easily rupture your hymen and bleed
with ur menstrual blood and never know.
Girls break their virginity by riding horses and doing extreme physical activity. One of my classmates told me she tore her hymen when she was doing ballet and that she felt it, and after that she
saw the blood.
But I don’t understand this. Why are these young gurls using tampons, do they not know the effects? :confused:
I wonder will their husband notice it and wonder what she used get up to.
So, many questions and no answers.
I hope none of you gurls are using it, if you are then don’t cause you can get this nasty disease, and young gurls... It ain’t worth it.
On the other hand, I forget to mention that one of my married friends uses it, well she hasn’t told me... When she told me to get dirac out of her bag...., I happen to see it.
I don’t know if you are married, I think it seems okay.
I know that when I first got my period my mum did forget to mention about tampons. In future that’s something I need to do.
It would be interesting to hear from the sista’s and their take on this, as well as the brotha’s
:rolleyes: , xishood about what?
This is the women's section. Issues like this exist, and we are here to learn. Not to be soooo childish :mad: and shy away from it.
I remember reading this islamic book. It said not to shy away from these stuff and that we are human beings.
Maybe you should pick up the quran and read.
Just putting a light on a taboo subject.
Maybe one sista, might benefit from me
Originally posted by Nazra:
This is the women's section. Issues like this exist, and we are here to learn. Not to be soooo childish :mad: and shy away from it.
Hmmm I didn't look at it that way. You are right, and thank you for pointing that out to me. I'm sorry.
Another Taboo topic, as the tampon users say a less messy job.
My fifty cents, if your a first time user adviced to try when your on your safe days to see if you are comfortable and no accidents happen.
Make sure you change it often, as its known that a toxic shock syndrome(TSS) can result if bacterias(Staphyloccus aureus) get busy.
A patient may( female of child bearing age approx 90% of the cases) present with fever,vomiting, diarhea,sore throat, muscle pain and headaches. In severe cases hypotension with kidney failure and
heart failure has been reported. A skin rash also occurs. It usually happens by the 5th day of the menstrual cycle of a tampon user.
Treatment? first to removal the source of the toxin( tampon or other cause e.g abscess) rehydration, given anti staphylococcal drugs....
As for the hymen, some girls are born naturally without it so dont see it as an issue. What is important in Islam is to be chaste, a tampon cannot undo that.
So let people do what they see fit, no need of stigmatisation, just use common sense.*weight the pros and cons* Some people are allergic to pads, get extreme rashes, discomfort...
*my aim isnt to encourage the usage nor discourage it, just passing information, and letting go of the myths, my personal choice is private*
Tampon! i love them they are nice,confortable and so good. i hate using pads i just dont like it, and you can still use it while you are vargin but the first time wount be confortable. pads is too
much work and annoying, seriously i never felt confortable while i use pads but tampon they are the shyt.
dont think its bad using it or think why young girls use it go try and see how it feels and dont even think about o im vargin i cant use that, no that got none to do woth ya varginity girl.
The guys should stay out of this topic. They shouldn't be let in through the door.
Guys please don't make any comments. *Gives any guy who makes a comment a evil look*
Back to the topic.
You might have some more haters because you have chosen to speak about a unspeakable topic. Another metal to take home.
Why use tampons when you can use the old fashion pads? I personally am against the whole idea of tampons. Why? I don't know. It doesn't appeal to me, I guess. I couldn't careless who is using what so
long it isn't me.
I wonder will their husband notice it and wonder what she used get up to.
There are other ways dear, apart from the hymen being intact, that can tell whether a girl is a virgin or not; if a girl's husband isn't familiar with these alternative ways, then the girl should be
wondering about him. Particularly about his sexual orientation before he met her.
Originally posted by Nazra:
Girls break their virginity by riding horses and doing extreme physical activity.
Is this a joke?
Originally posted by Warrior of Light:
^^^ Very true. Another reason bicycle accidents.
What is true?
Originally posted by Socod_badne:
What is true?
That girls break their virginity by riding bicycles, riding horses.
What about falling off a Gambar
have u lost your virginity??? I didn’t know how appropriate that was. I mean those things are big, and you can easily rupture your hymen and bleed with ur menstrual blood and never know.
Girls, Girls, when will you girls become informed. Nazra if memory serves me correctly you are meant to be a nursing student no? I’m quiet surprised that you do not know what virginity actually is.
Virginity my dear sisters is not a description for something tangible rather it just basically means someone who has never had sexual intercourse before. Therefore, you could of have ruptured your
hymen at the age of five when you slipped on the playground but still be a virgin. This thinking of attempting to validate virginity by an intact hymen is ludicrous to say the least and is based on
we don’t know the effects it will have on their first sexual encounter with their husband should I say?
Correct me if I’m wrong here, but are you saying that a man has the right to question his wife’s virginity because she does not bleed (i.e. hymen is not intact, or she was never born with one or
it might even be one of those that moves out of the way which mind you also exists)? If this is so, walaahi this is just sad sad. I feel sorry for us women, so gullible we can be. Women read
carefully- you do not need a man who questions your chastity, you are better off without him and more importantly you should not be worrying yourselves with such s’tupid absurdities.
Also,realize that technically speaking a man can never know for certain whether or not a female is a virgin. Of course there are many arguments, but none are substantiated by fact, therefore they
remain baseless and basically carry no weight. The only way to ascertain the virginity status of a female is through a medical examination.
There are other ways dear, apart from the hymen being intact, that can tell whether a girl is a virgin or not; if a girl's husband isn't familiar with these alternative ways, then the girl should be
wondering about him. Particularly about his sexual orientation before he met her.
So it’s ok for him to of have explored his sexual orientation? Honey a man who is aware of these alternative ways (even if we were to accept them as fact which they are not) then he has no right
taking the high moral ground. A sinner cannot point fingers. He should be concentrating on himself because otherwise it is called hypocrisy.
Disappointment would be an understatement here. How I wish I could attribute this to 15-year-old ignorance, but I’m sure all of you are older than that. Sisterly advice- please please be informed
and know that which you have a right to.
As for tampons, if anyone is interested on the Islamic position visit:
looooooooool@lexy, my god, Gambar I had no idea about that.
Warrior of light, I’m shocked. :eek:
As matter of fact its just me and lexy here who refuse to try tampons.
By the way lexy, thanks for the tip... I am never sitting on a gambar. lol
nursing student?, was doing a double degree sis, but I’m not into nursing.
I’m a commerce student
You got yourself confused, I’m kind of suspicious Rahima, are you a single mother?
Anywayz, your hymen is what makes you a virgin. But if you lose you hymen to a gambar (lolz) or a tampon...still you are virgin who has not had sexual intercourse.
I just wondered what would happen in the bed.
Anywayz, you don't have to be a doctor to know if someone is a virgin or not. It comes with experience.
I’m sorry to offend anyone who has used tampons, or lost their virginity in doing other extreme activities but I just like to keep mine intact
Lool ^^ arent we all intact???
And Nasra why are you shocked?? Me being knowledgable on this topic or you are assuming I use tampons? If its the latter well, my answer is No.
Dear,it a personal choice. Dont see the need to repeat Sister Rahima did an excellent job but a tampon user is still a virgin. Its not the hymen there is more to that to 'lose your virginity'... Oh,
just remembered cases of females who are pregnant and still have an intact hymen, are you going to say she hasnt lost it :confused: :confused: And a little extra knowledge hymens come in different
sizes and shapes, too.
*I am assuming you still want to hold on to the myth.* All I know and alot of women beleive in you need a Man to 'lose' it thats the way Allah created us.
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Important Questions for Class 10 Maths Chapter 4 Quadratic Equations
Study MaterialsImportant QuestionsImportant Questions for NCERT Class 10 Maths Chapter 4 – Quadratic Equations
Important Questions for NCERT Class 10 Maths Chapter 4 – Quadratic Equations
Quadratic Equations in Class 10 Maths. If you’re gearing up for the CBSE board exam 2024-2025, these will be handy to practice and aim for good scores. These questions are aligned with the NCERT book
, designed by experts who’ve carefully studied the exam pattern. Expect similar questions in your Math paper.
These questions focus on quadratic equations and how to solve them using factorization methods to find their roots. By working through these, you’ll solidify your understanding and be better prepared
for your exam. We’ve included 15 questions along with detailed solutions below. Make sure to go over them to strengthen your Math skills. Additionally, tackling extra practice questions will further
enhance your problem-solving abilities in quadratic equations.
Important Questions for Class 10 Maths Chapter 4: Quadratic Equations
Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (1Mark)
Question: Define a quadratic equation.
Answer: A quadratic equation is a polynomial equation of the second degree, where the highest power of the variable is 2.
Question: What is the standard form of a quadratic equation?
Answer: The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Question: State the quadratic formula.
Answer: The quadratic formula is x = (-b ± √(b^2 – 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
Question: What does the discriminant of a quadratic equation determine?
Answer: The discriminant of a quadratic equation (Δ = b^2 – 4ac) determines the nature of the roots of the equation. If Δ > 0, the equation has two distinct real roots. If Δ = 0, the equation has two
equal real roots. If Δ < 0, the equation has two complex (non-real) roots.
Question: How many solutions can a quadratic equation have?
Answer: A quadratic equation can have either two real solutions, one real solution (in case of repeated roots), or two complex solutions.
Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (3 Mark)
Question: Solve
using the quadratic formula.
$𝑥=\frac{-3±\sqrt{{3}^{2}-4\cdot 2\cdot \left(-5\right)}}{2\cdot 2}$
Question: Factorize
Question: Find the roots of
using the factorization method.
Question: If
is a root of the quadratic equation
, find the value of
Question: Solve the quadratic equation
by completing the square method.
Question: Determine the discriminant of the quadratic equation
and state the nature of its roots.
Answer: Discriminant (
) =
${5}^{2}-4\cdot 3\cdot 2=1$
, since
, the roots are real and distinct.
Question: If one root of the quadratic equation
is 3, find the value of
Answer: By using sum and product of roots,
Question: Find the roots of the quadratic equation
by the quadratic formula method.
Question: Determine the value of
for which the equation
has equal roots.
Question: Factorize
Question: Solve the equation
using the method of splitting the middle term.
Question: Find the roots of
by using the quadratic formula.
$𝑥=\frac{7±\sqrt{{7}^{2}-4\cdot 2\cdot 3}}{2\cdot 2}$
Question: If one root of
is 2, find the value of
Question: Determine the roots of
by using the quadratic formula.
Question: Factorize
Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (5 Mark)
Question: Solve the quadratic equation 2x^2 – 5x – 3 = 0.
Answer: The solutions are x = 3/2 and x = -1.
Question: Find the roots of the quadratic equation x^2 + 4x + 4 = 0.
Answer: The only root is x = -2.
Question: Determine the values of k for which the quadratic equation (k+1)x^2 – 4(k+1)x + 4 = 0 has equal roots.
Answer: For equal roots, the discriminant must be zero. So, (4(k+1))^2 – 4(k+1)(k+1) = 0. Solving this equation gives k = 1 or k = -1.
Question: If one root of the quadratic equation x^2 – px + 12 = 0 is 4, find the value of p.
Answer: Since 4 is a root, the quadratic equation becomes (x – 4)(x – ?) = 0. Equating coefficients with the original equation, we get p = 16.
Question: Solve the quadratic equation 3x^2 – 2x – 1 = 0 by using the quadratic formula.
Answer: Using the quadratic formula, we find x = (2 ± √16) / 6. So, x = (2 ± 4) / 6. This gives two solutions: x = 1 and x = -1/3.
Question: Factorize the quadratic expression x^2 + 7x + 12.
Answer: The expression factors as (x + 3)(x + 4).
Question: Solve the quadratic equation 5x^2 + 6x – 2 = 0 using the method of completing the square.
Answer: Completing the square, we get 5(x + 3/5)^2 – 49/5 = 0. Rearranging, we find (x + 3/5)^2 = 49/25. Taking the square root, we obtain x + 3/5 = ±7/5. So, x = 2/5 or x = -10/5.
Question: Determine the nature of the roots of the quadratic equation 4x^2 + 4x + 1 = 0.
Answer: Since the discriminant is zero, the roots are real and equal.
Question: Find the roots of the quadratic equation 2x^2 – 7x + 3 = 0 by factorization.
Answer: Factoring, we get (2x – 1)(x – 3) = 0. So, the roots are x = 1/2 and x = 3.
Question: If the sum of the roots of the quadratic equation x^2 – px + q = 0 is 7 and one of the roots is 3, find the value of q.
Answer: Since the sum of the roots is 7 and one root is 3, the other root is 4. Therefore, q = 3 * 4 = 12.
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Linear Combination Calculator: Solve Vector Combinations Easily
Home » Simplify your calculations with ease. » Physics Calculators »
Linear Combination Calculator: Solve Vector Combinations Easily
A linear combination calculator is a useful tool for solving problems in linear algebra. This calculator takes two vectors and their corresponding scalar multiples as inputs and computes their linear
combination. In this article, we will discuss the concept of linear combinations, the formula used to calculate them, and how this calculator works, along with an example.
Understanding Linear Combinations
A linear combination is the result of adding two or more vectors together, with each vector being multiplied by a scalar. Scalars are constants that can be any real number. Linear combinations play a
significant role in linear algebra, particularly when working with systems of linear equations, vector spaces, and linear transformations.
Linear Combination Formula
The linear combination of two vectors can be computed using the following formula:
LC = k1 * V1 + k2 * V2
• LC is the linear combination of the two vectors
• k1 and k2 are the scalar multiples
• V1 and V2 are the input vectors
In this calculator, we assume that V1 = (a, b) and V2 = (c, d). The formula for the linear combination in this case is:
(x, y) = k1 * (a, b) + k2 * (c, d)
The resulting linear combination vector, (x, y), can be found by performing the following operations:
• x = k1 * a + k2 * c
• y = k1 * b + k2 * d
How the Linear Combination Calculator Works
The calculator takes four inputs:
1. Vector 1 (a, b)
2. Scalar multiple of Vector 1 (k1)
3. Vector 2 (c, d)
4. Scalar multiple of Vector 2 (k2)
The calculator uses JavaScript to parse the input vectors, validate the inputs, and calculate the linear combination using the formula mentioned above. The resulting linear combination vector (x, y)
is displayed as the output.
Let’s say we have the following input values:
• Vector 1: (1, 2)
• Scalar multiple of Vector 1: 3
• Vector 2: (4, 5)
• Scalar multiple of Vector 2: 2
Using the linear combination formula:
LC = 3 * (1, 2) + 2 * (4, 5)
Calculate the x and y values:
• x = 3 * 1 + 2 * 4 = 3 + 8 = 11
• y = 3 * 2 + 2 * 5 = 6 + 10 = 16
The resulting linear combination vector is (11, 16).
The linear combination calculator makes it easy to compute linear combinations of vectors without having to perform the calculations manually. By inputting the vectors and their scalar multiples, the
calculator quickly provides the resulting linear combination vector.
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NumPy’s ndarray indexing
In NumPy a new kind of array is provided: n-dimensional array or ndarray. It’s usually fixed-sized and accepts items of the same type and size. For example, to define a 2×3 matrix:
import numpy as np
a = np.array([[1,2,3,], [4,5,6]], np.int32)
When indexing ndarray, it supports “array indexing” other than single element indexing. (See http://docs.scipy.org/doc/numpy/user/basics.indexing.html)
It is possible to index arrays with other arrays for the purposes of selecting lists of values out of arrays into new arrays. There are two different ways of accomplishing this. One uses one or
more arrays of index values. The other involves giving a boolean array of the proper shape to indicate the values to be selected. Index arrays are a very powerful tool that allow one to avoid
looping over individual elements in arrays and thus greatly improve performance.
So you basically can do the following:
a = np.array([1, 2, 3], np.int32)
a[np.array([0, 2])) # Fetch the first the third elements, returns np.array([1, 3])
a[np.array([True, False, True])] # Same as the line above
Besides, when you do equals operation on ndarrays, another ndarray is returned by comparing each element:
a = np.array([1, 2, 3], np.int32)
a == 2 # Returns array([False, True, False], dtype=bool)
a != 2 # Returns array([ True, False, True], dtype=bool)
a[a != 2] # Returns a sub array that excludes elements with a value 2, in this case array([1, 3], dtype=int32)
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Binomial distribution
The binomial distribution is a discrete probability distribution of the total number of 'successes' $X$ in $n$ independent trials, where each trial can result in either a 'success' or a 'failure'
(e.g., heads or tails). The probability of a success in each trial is equal to $P$, and the probability of a failure is equal to $1 - P.$ The graph below shows a binomial distribution with number of
trials $n$ = and a success probability per trial of $P$ = . Move the sliders to see binomial distributions with different numbers of trials and different success probabilities. If there is only one
trial, $n = 1$ and the binomial distribution reduces to the
bernoulli distribution
number of trials $n$ =
success probability $P$ =
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Summarize distinct child records by a text field | Qrew Discussions
Summarize distinct child records by a text field
I have a parent table (Botruns) that has many children (Line items). The line items could either be processed or not processed and the data for that resides in the line item record in the child
Deepa, try this out.
Create a formula numeric field for say [PWO Only Count] and set it equal to this formula:
If(IsNull(ToNumber(ToText([Combined Text Field]))),0,
(Length(ToText([Combined Text Field]))-Length(SearchAndReplace(ToText([Combined Text Field]),"PWO Only","")))/8)
The idea is that the formula measures the total length of the string, then subtracts the total length of the string minus any entries of "PWO Only", then divide that number by the length of the text
string that was subbed out (in this case eight characters). This should yield the total number of times that particular substring appeared in the combined text field. If the string being searched for
did not appear in the overall string, no characters would be substituted, the total lengths would be the same and the result would be 0/8 which is zero.
You could make dedicated formula fields to search for and count each specific string you are looking for, or you could build it out to be a nested if formula.
I got this idea from a post by Pushpakumar Gnanadurai (PushpakumarGna1).
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MCQ Questions for Class 6 Maths with Answers PDF Download Chapter Wise
Get Chapter Wise MCQ Questions for Class 6 Maths with Answers PDF Free Download prepared here according to the latest CBSE syllabus and NCERT curriculum. Students can practice CBSE Class 6 Maths MCQs
Multiple Choice Questions with Answers to score good marks in the examination.
You can refer to Maths MCQ Questions for Class 6 With Answers to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.
Class 6 Maths MCQ with Answers
Practicing these CBSE NCERT Objective Class 6 Maths MCQ with Answers will guide students to do a quick revision for all the concepts present in each chapter and prepare for final exams.
Class 6 Maths MCQ Questions
Also Read
Class 6 Maths Objective Questions
We hope the given NCERT Class 6 Maths Objective Questions will help you. If you have any queries regarding CBSE Class 6 Maths MCQs Multiple Choice Questions with Answers, drop a comment below and we
will get back to you soon.
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Current-Sense Resistors Tradeoffs
Using a resistor for sensing current should be a simple affair. After all, one has only to apply Ohm’s law or I=V/R. So, all it takes is to measure the voltage drop across a resistor to find the
current flowing through it. However, things are not as simple as that. The thorn in the flesh is the resistor value.
Using a large resistor value has the advantage of offering a large reading magnitude, greater resolution, higher precision, and improved SNR or Signal to Noise Ratio. However, the larger value also
wastes power, as W=I^2R. It may also affect loop stability, as the larger value adds more idle resistance between the load and the power source. Additionally, there is an increase in the resistors
Would a lower resistor value be better? But then, it will offer higher SNR, lower precision, resolution, and a low reading magnitude. The solution lies in a tradeoff.
Experimenting with various resistor values to sense different ranges of currents, engineers have concluded that a resistor offering a voltage drop of about 100 mV at the highest current is a good
compromise. However, this should preferably be a starting point, and the best value for the current sense resistor depends on the function of priorities for sensing the current in the specific
The voltage or IR drop is only one of two related problems, with the second problem being a consequence of the chosen resistor value. This second issue, resistive self-heating, is a potential
concern, especially when a high-value current flows through the resistor. Considering the equation W=I^2R, even for a milliohm resistor, the dissipation may be in several watts when the current is
multiple amperes.
Why should self-heating be a concern? Because, self-heating shifts the nominal value of the sense resistor, and this corrupts the current-value reading.
Therefore, unless the designer is measuring microamperes or milliamperes, where they can neglect the self-heating, they would need to analyze the resistance change with temperature change. For doing
this, they will need to consult the data for TCR or temperature coefficient of resistance typically available from the resistor’s vendor.
The above analysis is usually an iterative process. That is because the resistance change affects the current flow, which, in turn, affects self-heating which affects resistance, and so on.
Therefore, the current-sensing accuracy depends on three considerations—the initial resistor value and tolerance, the TCR error due to ambient temperature change, and the TCR error due to
self-heating. To overcome the iterative calculations, vendors offer resistors with very low TCR.
These resistors are precision, specialized metal-foil types. Making them from various alloys like copper, manganese, and other elements, manufacturers use special production techniques for managing
and minimizing TCR. To reduce self-heating and improve thermal dissipation, some manufacturers add copper to the mix.
Instrumentation applications demand the ultimate precision measurements. Manufacturers offer very low TCR resistors and fully characterized curves of their resistance versus temperature. The nature
of the curve depends on the alloy mix and is typically parabolic.
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What Are Watts and Why Do They Matter?
What Are Watts and Why Do They Matter?
How many watts does an RV refrigerator use? Which one consumes more power– a conventional halogen light bulb or an LED bulb? How many watts of solar energy do I need to run a fan all night on my
Watts matter to us all day and all night, even when we sleep. But what is the abstract thing called Watt, and why do we need to know about them? In this article, we are going to show you why
understanding watts is more important than you think.
What is Watt In Simple Terms?
Electrical power is measured in watts. A watt is a unit of power. When we use the term Watt, we are putting a number on the rate of energy transfer. That's meaning, one watt is a single unit of
power, and power is the rate at which energy is created or consumed by an object.
If we think of watts as measuring electrical flow, then we can think of any device or appliance as requiring either a large electrical flow or a small electrical flow. For example, if you have a
100-watt light bulb and a 60-watt light bulb, you can think of the 100-watt bulb as requiring a larger flow of energy to work. If you want to run a 750-watt microwave oven, you are going to need a
larger flow of energy – 750 watts to be exact.
In the same way, if you have solar panels on the roof of your RV, the solar energy flowing into the RV is measured in watts.
Common Watt Multiples and What They Mean
Watts are measured in multiples of 1,000. You have probably heard of these multiples before, but let’s look at what they mean in terms of power. The smallest measurement typically used is a milliwatt
or 1/1000th of a watt. This measurement is commonly used in small circuits such as computers or cell phones.
Here we only see milliwatts of electricity flowing through these tiny wires.
A kilowatt is a unit of measure for 1,000 watts of electrical power. Abbreviated as a kilowatt (kW), the kilowatt is a globally recognized standard for measuring electricity energy. You may notice
that your home’s energy usage or electricity consumption is rated in kilowatts on your electric bill.
One megawatt (MW) is globally recognized to be equal to 1,000 kilowatts of power. Depending on the size of the generator, the power rating of the generator can be megawatts, kilowatts, or watts.
Any device in your home that runs on electricity will have its usage or consumption rated in watts or kilowatts. In the same way, a light bulb might be rated at 60 watts, while a microwave is rated
at 750 watts.
Interestingly, some appliances have two different ratings. For example, our household refrigerator is an appliance that has a start (or surge) rating and a running rating. This means that the wattage
will be different when the refrigerator first starts up versus when it has been running for a while.
It might have a 1,200 watts startup rating and an 800 watts running capacity rated. Thus, when the refrigerator starts (when the compressor kicks on), it needs 1,200 watts at that moment. But while
it is running normally, the required wattage drops to 800 watts.
How Do You Measure Watts?
There is a simple formula we can learn to help us measure watts in any situation. To be able to calculate the number of watts provided by a power supply, you will need to know the number of amps and
volts in that power source. You can use a multimeter to measure amps and volts. Once you have this information, the calculation of the DC circuit is simple:
Watts(W) = Amps(A) x Volts(V). So, let's assume the current is 6 amps, and the voltage is 110 volts, then 6 x 110 = 660 watts.
You will also sometimes see this equation written as Power = Amps x Volts ( P = I x V). This is a more formal or technical version of the formula, which you might see in a textbook. However, as long
as you can remember watts = amps x volts, you will be good to go for DC circuits.
High Wattage and Low Wattage
High wattage means more power is consumed. Because of this, we spent most of the time trying to build and use appliances that consume less power and have lower wattage. However, higher wattage is
good when you particularly need to convert electricity into heat. Because the higher the wattage the hotter the heater.
However, it is important to know that high wattage can mean high heat. When you are powering high-wattage equipment, more heat is produced, which needs to be accounted for within the context of your
particular application. High-wattage electronic devices can also cause a circuit overload and blow a fuse or trip a circuit breaker.
Conversely, low-wattage applications require less power to charge or operate. Such as the low-wattage of the cell phone charger, the smartphone or tablets, and the LED light bulb.
Also, keep in mind that it doesn’t matter whether we are talking about a 12-volt or a 220-volt application. A watt is a watt, and it is not the same property as voltage. A device doesn't consume less
power just because it is a 12 volts device.
Power Supply Capacity
The job of a power source is to provide electric energy to the connected appliance. So, it would make sense that the more power an appliance requires, the higher the wattage you will need from the
power supply. (Higher wattage = more power).
There is a rule of thumb to keep in mind when considering the capacity of a power supply. It is better to have more power than you need because you shouldn’t be running your power supply at 100%
For example, if you want to run a 100 watts device, you would not use a 100-watt power source. Why? As we mentioned earlier, you don’t want to run your power source at full capacity. Ideally, you
would want your power supply to have a wattage higher than 100 in this situation.
If you have a 250-watt power supply and you use it to run a 100-watt appliance, that power supply will only put out the necessary 100 watts the appliance needs. Essentially, the power supply won't
"Overpower" the appliance.
Why Do Watts Matter in RVs and Boats?
When you are traveling on a boat or in an RV, watts matter because you need to be able to calculate your daily electricity requirements. For marine applications, it is helpful to know how much power
your motor needs to be based on the size of your boat. In RVs, you will need to understand what appliances you can power and whether you can power more than one appliance at a time.
For example, RVs are often powered by using generators. If your generator puts out 4,000 watts of power, you might be able to run your RV’s microwave oven, air conditioner, and several small
appliances, although maybe not all at once.
How many watts of infographic
If you are considering installing a solar system on your boat or RV, you will need to be able to calculate how much power (how many watts) you are using over a 24-hour period. This will help you
understand how many solar panels you will need (based on their wattage) to be able to sustain your daily usage. We will share how to calculate this at the end.
Where Will You See Watts In RVs & Boats?
RVs and boats have many applications that require power, from lights to device chargers to fans, microwave ovens, and refrigerators. It is important to understand how many watts you will consume in a
day, and how you are going to recharge your batteries to store more energy for the next day.
Power Conservation
Using lower-wattage equipment where possible is key, especially when you are not connected to shore power (i.e. living off-grid). Let’s take lights as an example to illustrate the variations in power
consumption. Remember, watts = power.
Let's assume an incandescent bulb that illuminates at a level of 450 lumens will use 40 watts. On the other hand, an LED light illuminated at the same level will only use 4-5 watts. If you are living
off-grid, and mostly rely primarily on solar energy, you are probably concerned about your electricity consumption. In this case, which bulb would you prefer to have in your rig or on your boat? The
LED bulb uses only a few watts, of course.
When you are considering appliances that use more power, you may need an inverter to run them. Inverters convert DC current into AC current to run certain appliances and equipment.
DC current is the type of electrical current you receive from a battery (think powering an RV). AC current is the type of electricity you receive from a power station (think powering a residential
An inverter needs to have sufficient capacity to power what you want it to power. So, in order to know what inverter you will require, you have to know the wattage (starting and running) of the
equipment you want to power.
Let’s go back to our refrigerator example at the beginning of this article. If the refrigerator starts at 1,200 watts of capacity and runs at 800 watts of capacity, a 750-watt inverter would be
insufficient to run this appliance. Instead, you would want an inverter with a larger capacity, like 1,500 watts.
Another thing that needs to be considered with inverters is that most inverters are rated in VA or volt-ampere. In an AC circuit, there is something called a power factor that causes the power to
appear higher than the load to the generator or source. This is a complicated topic but to account for this it's typically good to add an extra 20% margin to your loads to get an accurate power
Generators and Solar Panels
Now let’s suppose you are in the market for a portable generator to give your off-grid adventures a power boost. If you want to run an air conditioner in the summer heat, your generator is going to
need to be able to provide sufficient power to start and run the air conditioner unit. This is true of everything you want to electric energy.
And finally, if you are looking at investing in solar panels to harness the energy of the sun, you need to be able to calculate how many watts of solar energy you require to power your off-grid life.
How many watts of solar energy you need will be determined by how many watts of solar you use.
Distinction Between Watts and Watt-Hours
The difference between watts and watt-hours is simple. The electrical power is measured in watts. Watts measures the rate of power at a specific moment in time.
Watt-hour measures the rate of power over a specific period (one hour).
Simply put, one watt-hour equals one watt of power flow in an hour. So, a 5-watt LED light bulb lighting on for one hour has consumed 5-watt hours of energy.
How Many Watts of Solar Do You Need?
To illustrate the significance of watts and watt-hours, let’s calculate how many watts of solar panels you will need to power your boat or RV off-grid.
First, make a list of all the electrical equipment you will want to run. Then, write down the wattage of each electric unit and how long you intend to run each one. Next, total the wattage of all
devices by adding them all together, then calculate the sum of the run times for all the devices.
Finally, multiply these two totals together (total wattage x total estimated running time). The result tells you how many watt-hours are required to power all of your electrical devices with solar
Let's assume you are planning to run all of your equipment for 24 hours. If you have a solar panel rated at 250-watt, your panel can produce 6,000 watt-hours (or 6 kilowatt-hours) of power during
that time (250 x 24 = 6,000). Meaning, if the total watt-hour requirement you calculated for all of your devices over a 24-hour period is 6,000 watts, then you will need more than one solar panel to
accommodate your power needs.
How many watts of solar power do you need? How much solar power do you need? Watt does matter!
Watts are very significant in any application where electricity is required, whether that is in a home, business, large facility, boat, or RV. Simply stated, watts matter – and so does understanding
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Understanding Mathematical Functions: How To Find The Linear Function
Mathematical functions are essential in understanding the relationship between variables and predicting outcomes in various fields such as economics, physics, and engineering. Linear functions are a
fundamental type of mathematical function that describe a straight line relationship between two variables. Understanding linear functions is crucial for analyzing data, making predictions, and
solving real-world problems.
Key Takeaways
• Linear functions are essential in understanding relationships between variables and predicting outcomes in various fields such as economics, physics, and engineering.
• Understanding linear functions is crucial for analyzing data, making predictions, and solving real-world problems.
• Key characteristics of linear functions include a straight line relationship between two variables.
• Calculating the slope and y-intercept of a linear function is important for graphing and analyzing its behavior.
• Linear functions are widely used in various fields and have real-world applications in areas such as economics and physics.
Understanding Mathematical Functions: How to find the linear function
In this blog post, we will delve into the concept of linear functions and how to find them. Linear functions are fundamental in mathematics and have various applications in fields such as physics,
engineering, and economics. Let's begin by defining linear functions and discussing their key characteristics.
Defining Linear Functions
A linear function is a type of function that can be represented by a straight line on a graph. It is expressed in the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
Key characteristics of linear functions
Linear functions have several key characteristics that distinguish them from other types of functions:
• Linearity: A linear function has a constant rate of change, meaning that the change in the output value is proportional to the change in the input value.
• Graph: The graph of a linear function is a straight line, with a constant slope and y-intercept.
• Relation to constants: The slope (m) and y-intercept (b) are constants that determine the behavior of the linear function.
Understanding linear functions is crucial for various mathematical and real-world applications. In the next chapter, we will explore how to find the linear function from given data points.
Finding the Slope
When working with linear functions, it's essential to understand the concept of slope. The slope of a linear function represents the rate of change between two variables. It indicates how much one
variable changes for a given change in the other variable.
Explanation of slope in relation to linear functions
In the context of linear functions, the slope is the ratio of the vertical change (or rise) to the horizontal change (or run) between any two points on the line. It is a measure of the steepness of
the line and is a crucial factor in determining the behavior of the function.
Methods for finding the slope of a linear function
• Using the slope formula: The slope of a linear function can be calculated using the slope formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line.
• Graphical method: By plotting the points and observing the rise and run, the slope can be visually determined. The slope is the ratio of the vertical change to the horizontal change between any
two points on the line.
• Using the equation: If the linear function is represented in the form y = mx + b, where m is the slope, then the slope can be directly identified from the equation.
Calculating the Y-Intercept
Understanding the y-intercept of a linear function is crucial in solving mathematical problems. Let's take a look at the definition of y-intercept and some techniques for calculating it.
A. Definition of y-intercept
The y-intercept is the point where the graph of a function crosses the y-axis. It is the value of y when x is equal to 0. In other words, it is the constant term in the equation of a linear function,
represented as (0, b) on a graph, where 'b' is the y-intercept.
B. Techniques for calculating the y-intercept of a linear function
• Using the equation: If you have the equation of a linear function in the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept, simply substitute x = 0 into the equation to find the
value of y.
• Graphical method: Plot the linear function on a graph and identify the point where the line intersects the y-axis. This point represents the y-intercept.
• Using data points: If you have a set of data points that represent the linear function, plug in the x-value of 0 into the equation to find the corresponding y-value, which is the y-intercept.
Graphing Linear Functions
Understanding how to graph linear functions is an essential skill in mathematics. It allows us to visualize the relationship between two variables and make predictions based on the data. In this
chapter, we will explore the importance of graphing linear functions and the steps to graph them on a coordinate plane.
A. Importance of graphing linear functions
Graphing linear functions helps us to understand the behavior of the function and its relationship with the variables involved. It provides a visual representation that makes it easier to interpret
data and identify patterns. By graphing linear functions, we can also make predictions and analyze the impact of changes in the variables.
B. Steps for graphing a linear function on a coordinate plane
Graphing a linear function involves a few simple steps to plot the points and draw the line on a coordinate plane. Here are the steps to follow:
• 1. Identify the slope and y-intercept: The linear function is typically represented in the form y = mx + b, where m is the slope and b is the y-intercept. Identify these values from the function.
• 2. Plot the y-intercept: Locate the point (0, b) on the y-axis. This is the starting point for graphing the linear function.
• 3. Use the slope to plot additional points: Use the slope (m) to find another point on the line. For example, if the slope is 2, you would move up 2 units and over 1 unit to find the next point.
• 4. Draw the line through the points: Once you have plotted at least two points, use a straight edge or ruler to draw a line through the points. This line represents the graph of the linear
Applications of Linear Functions
A. Real-world examples of linear functions
Linear functions are widely used in real-world scenarios to model various relationships between two variables. Some common examples of linear functions include:
• The relationship between distance and time in a constant speed journey.
• The relationship between cost and quantity in manufacturing processes.
• The relationship between temperature and pressure in thermodynamics.
B. How linear functions are used in various fields such as economics and physics
Linear functions play a crucial role in different fields such as economics and physics.
In economics, linear functions are used to represent the demand and supply curves, where the quantity demanded or supplied is a linear function of the price. This allows economists to analyze and
make predictions about market behavior and pricing strategies.
In physics, linear functions are used to describe various physical phenomena. For example, the relationship between force and displacement in Hooke's law is a linear function. This enables physicists
to understand and predict the behavior of elastic materials under varying forces.
In conclusion, understanding linear functions is crucial for a range of real-world applications, from predicting sales trends to analyzing data in scientific research. By mastering linear functions,
you can gain valuable problem-solving skills that are essential in various fields. I encourage you to continue exploring and practicing with linear functions to strengthen your understanding and
confidence in using them in your mathematical endeavors.
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Create exhaustive nearest neighbor searcher
ExhaustiveSearcher model objects store the training data, distance metric, and parameter values of the distance metric for an exhaustive nearest neighbor search. The exhaustive search algorithm finds
the distance from each query observation to all n observations in the training data, which is an n-by-K numeric matrix.
Once you create an ExhaustiveSearcher model object, find neighboring points in the training data to the query data by performing a nearest neighbor search using knnsearch or a radius search using
rangesearch. The exhaustive search algorithm is more efficient than the Kd-tree algorithm when K is large (that is, K > 10), and it is more flexible than the Kd-tree algorithm with respect to
distance metric choices. The ExhaustiveSearcher model object also supports sparse data.
Use either the createns function or the ExhaustiveSearcher function (described here) to create an ExhaustiveSearcher object. Both functions use the same syntax except that the createns function has
the 'NSMethod' name-value pair argument, which you use to choose the nearest neighbor search method. The createns function also creates a KDTreeSearcher object. Specify 'NSMethod','exhaustive' to
create an ExhaustiveSearcher object. The default is 'exhaustive' if K > 10, the training data is sparse, or the distance metric is not the Euclidean, city block, Chebychev, or Minkowski.
Mdl = ExhaustiveSearcher(X) creates an exhaustive nearest neighbor searcher object (Mdl) using the n-by-K numeric matrix of training data (X).
Mdl = ExhaustiveSearcher(X,Name,Value) specifies additional options using one or more name-value pair arguments. You can specify the distance metric and set the distance metric parameter (
DistParameter) property. For example, ExhaustiveSearcher(X,'Distance','chebychev') creates an exhaustive nearest neighbor searcher object that uses the Chebychev distance. To specify DistParameter,
use the Cov, P, or Scale name-value pair argument.
Input Arguments
X — Training data
numeric matrix
Training data that prepares the exhaustive searcher algorithm, specified as a numeric matrix. X has n rows, each corresponding to an observation (that is, an instance or example), and K columns, each
corresponding to a predictor (that is, a feature).
Data Types: single | double
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'Distance','mahalanobis','Cov',eye(3) specifies to use the Mahalanobis distance when searching for nearest neighbors and a 3-by-3 identity matrix for the covariance matrix in the Mahalanobis
distance metric.
X — Training data
numeric matrix
This property is read-only.
Training data that prepares the exhaustive searcher algorithm, specified as a numeric matrix. X has n rows, each corresponding to an observation (that is, an instance or example), and K columns, each
corresponding to a predictor (that is, a feature).
The input argument X of createns or ExhaustiveSearcher sets this property.
Data Types: single | double
Distance — Distance metric
character vector | string scalar | custom distance function
Distance metric used when you call knnsearch or rangesearch to find nearest neighbors for future query points, specified as a character vector or string scalar ('chebychev', 'cityblock',
'correlation', 'cosine', 'euclidean', 'fasteuclidean', 'fastseuclidean', 'hamming', 'jaccard', 'minkowski', 'mahalanobis', 'seuclidean', or 'spearman'), or a function handle.
The 'Distance' name-value pair argument of createns or ExhaustiveSearcher sets this property.
The software does not use the distance metric for creating an ExhaustiveSearcher model object, so you can alter it by using dot notation. Algorithms starting with 'fast' do not support sparse data.
DistParameter — Distance metric parameter values
[] | positive scalar
Distance metric parameter values, specified as empty ([]) or a positive scalar.
This table describes the distance parameters of the supported distance metrics.
Distance Metric Parameter Description
A positive definite matrix representing the covariance matrix used for computing the Mahalanobis distance. By default, the software sets the covariance using cov(Mdl.X,'omitrows').
'mahalanobis' The 'Cov' name-value pair argument of createns or ExhaustiveSearcher sets this property.
You can alter DistParameter by using dot notation, for example, Mdl.DistParameter = CovNew, where CovNew is a K-by-K positive definite numeric matrix.
A positive scalar indicating the exponent of the Minkowski distance. By default, the exponent is 2.
'minkowski' The 'P' name-value pair argument of createns or ExhaustiveSearcher sets this property.
You can alter DistParameter by using dot notation, for example, Mdl.DistParameter = PNew, where PNew is a positive scalar.
A positive numeric vector indicating the values used by the software to scale the predictors when computing the standardized Euclidean distance. By default, the software:
1. Estimates the standard deviation of each predictor (column) of X using scale = std(Mdl.X,'omitnan')
'seuclidean' 2. Scales each coordinate difference between the rows in X and the query matrix by dividing by the corresponding element of scale
The 'Scale' name-value pair argument of createns or ExhaustiveSearcher sets this property.
You can alter DistParameter by using dot notation, for example, Mdl.DistParameter = sNew, where sNew is a K-dimensional positive numeric vector.
If Mdl.Distance is not one of the parameters listed in this table, then Mdl.DistParameter is [], which means that the specified distance metric formula has no parameters.
Data Types: single | double
Object Functions
knnsearch Find k-nearest neighbors using searcher object
rangesearch Find all neighbors within specified distance using searcher object
Train Default Exhaustive Nearest Neighbor Searcher
Load Fisher's iris data set.
load fisheriris
X = meas;
[n,k] = size(X)
X has 150 observations and 4 predictors.
Prepare an exhaustive nearest neighbor searcher using the entire data set as training data.
Mdl1 = ExhaustiveSearcher(X)
Mdl1 =
ExhaustiveSearcher with properties:
Distance: 'euclidean'
DistParameter: []
X: [150x4 double]
Mdl1 is an ExhaustiveSearcher model object, and its properties appear in the Command Window. The object contains information about the trained algorithm, such as the distance metric. You can alter
property values using dot notation.
Alternatively, you can prepare an exhaustive nearest neighbor searcher by using createns and specifying 'exhaustive' as the search method.
Mdl2 = createns(X,'NSMethod','exhaustive')
Mdl2 =
ExhaustiveSearcher with properties:
Distance: 'euclidean'
DistParameter: []
X: [150x4 double]
Mdl2 is also an ExhaustiveSearcher model object, and it is equivalent to Mdl1.
To search X for the nearest neighbors to a batch of query data, pass the ExhaustiveSearcher model object and the query data to knnsearch or rangesearch.
Specify the Mahalanobis Distance for Nearest Neighbor Search
Load Fisher's iris data set. Focus on the petal dimensions.
load fisheriris
X = meas(:,[3 4]); % Predictors
Prepare an exhaustive nearest neighbor searcher. Specify the Mahalanobis distance metric.
Mdl = createns(X,'Distance','mahalanobis')
Mdl =
ExhaustiveSearcher with properties:
Distance: 'mahalanobis'
DistParameter: [2x2 double]
X: [150x2 double]
Because the distance metric is Mahalanobis, createns creates an ExhaustiveSearcher model object by default.
Access properties of Mdl by using dot notation. For example, use Mdl.DistParameter to access the Mahalanobis covariance parameter.
ans = 2×2
3.1163 1.2956
1.2956 0.5810
You can pass query data and Mdl to:
• knnsearch to find indices and distances of nearest neighbors
• rangesearch to find indices of all nearest neighbors within a distance that you specify
Alter Properties of ExhaustiveSearcher Model
Create an ExhaustiveSearcher model object and alter the Distance property by using dot notation.
Load Fisher's iris data set.
load fisheriris
X = meas;
Train a default exhaustive searcher algorithm using the entire data set as training data.
Mdl = ExhaustiveSearcher(X)
Mdl =
ExhaustiveSearcher with properties:
Distance: 'euclidean'
DistParameter: []
X: [150x4 double]
Specify that the neighbor searcher use the Mahalanobis metric to compute the distances between the training and query data.
Mdl.Distance = 'mahalanobis'
Mdl =
ExhaustiveSearcher with properties:
Distance: 'mahalanobis'
DistParameter: [4x4 double]
X: [150x4 double]
You can pass Mdl and the query data to either knnsearch or rangesearch to find the nearest neighbors to the points in the query data based on the Mahalanobis distance.
Search for Nearest Neighbors of Query Data Using Mahalanobis Distance
Create an exhaustive searcher object by using the createns function. Pass the object and query data to the knnsearch function to find k-nearest neighbors.
Load Fisher's iris data set.
Remove five irises randomly from the predictor data to use as a query set.
rng('default'); % For reproducibility
n = size(meas,1); % Sample size
qIdx = randsample(n,5); % Indices of query data
X = meas(~ismember(1:n,qIdx),:);
Y = meas(qIdx,:);
Prepare an exhaustive nearest neighbor searcher using the training data. Specify the Mahalanobis distance for finding nearest neighbors.
Mdl = createns(X,'Distance','mahalanobis')
Mdl =
ExhaustiveSearcher with properties:
Distance: 'mahalanobis'
DistParameter: [4x4 double]
X: [145x4 double]
Because the distance metric is Mahalanobis, createns creates an ExhaustiveSearcher model object by default.
The software uses the covariance matrix of the predictors (columns) in the training data for computing the Mahalanobis distance. To display this value, use Mdl.DistParameter.
ans = 4×4
0.6547 -0.0368 1.2320 0.5026
-0.0368 0.1914 -0.3227 -0.1193
1.2320 -0.3227 3.0671 1.2842
0.5026 -0.1193 1.2842 0.5800
Find the indices of the training data (Mdl.X) that are the two nearest neighbors of each point in the query data (Y).
IdxNN = knnsearch(Mdl,Y,'K',2)
IdxNN = 5×2
Each row of IdxNN corresponds to a query data observation. The column order corresponds to the order of the nearest neighbors with respect to ascending distance. For example, based on the Mahalanobis
metric, the second nearest neighbor of Y(3,:) is X(128,:).
Fast Euclidean Distance Algorithm
The values of the Distance argument that begin fast (such as 'fasteuclidean' and 'fastseuclidean') calculate Euclidean distances using an algorithm that uses extra memory to save computational time.
This algorithm is named "Euclidean Distance Matrix Trick" in Albanie [1] and elsewhere. Internal testing shows that this algorithm saves time when the number of predictors is at least 10. Algorithms
starting with 'fast' do not support sparse data.
To find the matrix D of distances between all the points x[i] and x[j], where each x[i] has n variables, the algorithm computes distance using the final line in the following equations:
$\begin{array}{c}{D}_{i,j}^{2}=‖{x}_{i}-{x}_{j}{‖}^{2}\\ ={\left(}^{{x}_{i}}\left({x}_{i}-{x}_{j}\right)\\ =‖{x}_{i}{‖}^{2}-2{x}_{i}^{T}{x}_{j}+‖{x}_{j}{‖}^{2}.\end{array}$
The matrix ${x}_{i}^{T}{x}_{j}$ in the last line of the equations is called the Gram matrix. Computing the set of squared distances is faster, but slightly less numerically stable, when you compute
and use the Gram matrix instead of computing the squared distances by squaring and summing. For a discussion, see Albanie [1].
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• The knnsearch and rangesearch functions support code generation.
• When you train an ExhaustiveSearcher model object, the value of the 'Distance' name-value pair argument cannot be a custom distance function.
• ExhaustiveSearcher does not support code generation for fast Euclidean distance computations, meaning those distance metrics whose names begin with fast (for example, 'fasteuclidean').
For more information, see Introduction to Code Generation and Code Generation for Nearest Neighbor Searcher.
Version History
Introduced in R2010a
R2023a: Fast Euclidean distance using a cache
The 'fasteuclidean' and 'fastseuclidean' distance metrics accelerate the computation of Euclidean distances by using a cache and a different algorithm (see Algorithms). These distance metrics apply
only to the knnsearch function.
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Complementarity and Variational Inequalities
The nonconvex second-order cone (nonconvex SOC for short) is a nonconvex extension to the convex second-order cone, in the sense that it consists of any vector divided into two sub-vectors for which
the Euclidean norm of the first sub-vector is at least as large as the Euclidean norm of the second sub-vector. This cone can … Read more
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Monsters and Moonshine : a booklet
I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.
The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game
which uses the sporadic Mathieu group $M_{12}$. This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’.
We discuss Galois’ last letter and the simple groups $L_2(5),L_2(7)$ and $L_2(11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered
‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover
what the largest sporadic simple group, the Monster-group, sees of the modular group.
The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $PSL_2(\
mathbb{Z})$. We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using
Farey codes and use those to give a series of simple groups including as special members $L_2(5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to
McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E(8)$
in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s
solution of the $E(8)$-observation.
I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of
the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!
If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to
the “field with one element” and its connections to (noncommutative) geometry and number theory.
I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so
far only exists as prep-notes. The links above will always link to the latest versions of these booklets.
2 Comments
1. It seems that the links to these exciting booklets are now broken…
2. Thanks Gil! Should be fixed now.
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Haas G73 Irregular Path Stock Removal Cycle CNC Lathe - Helman CNC
Haas G73 Irregular Path Stock Removal Cycle CNC Lathe
Haas G73 Irregular Path Stock Removal Cycle
G73 pattern repeating cycle is best used when you want to remove a consistent amount of material in both the X and Z axes.
G73 P80 Q180 U.01 W.005 I0.3 K0.15 D4 F.012
D – Number of cutting passes, positive number
I – X-axis distance and direction from first cut to last, radius
K – Z-axis distance and direction from first cut to last
P – Starting Block number of path to rough
Q – Ending Block number of path to rough
U – X-axis size and direction of G73 finish allowance, diameter
W – Z-axis size and direction of G73 finish allowance
F – Feed rate to use throughout G73 PQ block
S – Spindle speed to use throughout G73 PQ block
T – Tool and offset to use throughout G73 PQ block
Haas G73 Example
O0815 (Example)
T101 (Select Tool 1)
G50 S1000
G00 X3.5 Z.1 (Move to start point)
G96 S100 M03
G73 P80 Q180 U.01 W0.005 I0.3 K0.15 D4 F.012 (Rough P to Q with T1 using G73)
N80 G42 G00 X0.6
G01 Z0 F0.1
X0.8 Z-0.1 F.005
G02 X1.0 Z-0.6 I0.1
G01 X1.4
X2.0 Z-0.9
G03 X2.8 Z-1.85 K-0.25
G01 Z-2.1
N180 G40 X3.1
G00 Z0.1 M05
(******Optional Finishing Sequence*****)
G53 X0 (Zero for tool change clearance)
G53 Z0
T202 (Select tool 2)
N2 G50 S1000
G00 X3.0 Z0.1 (Move to start point)
G96 S100 M03
G70 P80 Q180 (Finish P to Q with T2 using G70)
G00 Z0.5 M05
More ...
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Haas cnc lathe uses one-line syntax of G71 roughing canned cycle. This cnc program example shows the use of G71 turning cycle for ID roughing (Inside roughing). You might like…
Fanuc G73 Pattern Repeating Canned Cycle Basic CNC Sample Program
Fanuc G73 pattern repeating cycle helps cnc machinists to program/maintain/debug rough material removal programs easy. Other Fanuc canned cycle like G71 Longitudinal cutting cycle or G72 Facing Cycle
removes the…
G20 Turning Cycle – CNC Lathe Fanuc 21 TB
G20 longitudinal turning cycle for Fanuc 21 TB cnc control is a modal G-code. G20 turning cycle can be used for straight turning and taper turning as well. G20 turning…
Sample Program Example Fanuc G72 Facing Cycle Single-line-format
As canned cycle for cnc machines looks difficult to learn and program for beginner level cnc machinists, but they pay off in long run. As canned cycle makes cnc machinists…
Simple CNC Lathe Drilling with Fanuc G74 Peck Drilling Cycle
Here is a cnc programming example for simple drilling on a cnc lathe machine. CNC Fanuc control has a very powerful and versatile peck drilling cycle (Fanuc G74) which relieves…
Taper Turning with G90 Modal Turning Cycle – CNC Example Code
ContentsG90 Modal Turning CycleWhat is ModalG90 Turning Cycle UsageTaper Turning with G90 Turning CycleCNC Program ExampleTool Path ExplanationAlternative of G90 Turning Cycle G90 Modal Turning Cycle
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G71 Rough Turning Cycle Example Code – CNC Lathe Programming
ContentsG71 Turning CycleExample Program G71 Turning Cycle G71 Rough Turning Cycle example code. This cnc program code works on Fanuc and similar cnc controls. G71 Rough Turning Cycle parameters
Selca G9999 Synchronization of program execution with tool path display
Selca G9999 Synchronization of program execution with tool path display Programming G9999 Enabled: only in the block in which it is programmed.
Fanuc G71 G72 G70 Canned Cycle CNC Lathe Internal Machining Example (Boring & Facing )
Fanuc programming example which shows the use of multiple fanuc canned cycle in cnc programming, Following canned cycle are used in this cnc lathe programming example G71 Rough Turning Cycle…
Okuma G76 Fine Boring Cycle
ContentsOkuma G76 Fine Boring CycleProgrammingParametersMachining SequenceDetailsShift amount Okuma G76 Fine Boring Cycle G76 Fine Boring Cycle Programming G76 X__Y__Z__R__Q__(I__J__) P__F__
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CNC Lathe Basic Programming Example ID/OD Turning/Boring Operations (No Canned Cycle Used)
A full CNC programming example with ID/OD (Turning/Boring operations) for cnc machinists who work on a cnc lathe machine. A must to learn/practice for those who are learning cnc programming….
Sinumerik L97 Thread Cutting Cycle
Siemens Sinumerik 840C/840 Sinumerik 810/820T cycle L97 Thread Cutting Cycle can be used for External thread cutting Internal threading Taper threading Transversal threads. The tool infeed is
automatic and is…
Fanuc G71 Turning Cycle
ContentsFanuc G71 Turning CycleProgrammingParametersG71 Turning Cycle OverviewG71 Turning Cycle WorkingFanuc G71 ExampleG70 Finishing CycleWhy Use G70 Finishing CycleFanuc G70 ExampleG70 G71 Example
Fanuc G71 Turning Cycle G71 turning cycle is…
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Poisson distribution is a betting system that uses past results to determine the future outcomes of an event. It is considered more accurate if. A Poisson distribution can be used to determine the
most likely final score by calculating the “Attack Strength” and “Defense Strength” of the two opposing. Poisson distribution helps bettors find out the likelihood of an event happening within a
particular period of time. It is often used to calculate the chances. Those that don't know how to bet on golf may look at statistics like scoring average and driving accuracy and think these are
good indicators of performance. Poisson distribution is, in short, a statistical technique that should help you bet more accurately on the outcome of sporting events. Poisson distribution.
golf, do not have scoring systems which can be dispersed count data such as this4, Efron's Double Poisson distribution () can be ciencies in the. What is a mathematical method for a successful
football bet? The poisson statistics for golf betting methods use variations of the Poisson Distribution. Look at the. The Poisson distribution betting system is a solid numerical betting tool for
those who wish to get into overall trends while sourcing odds that offer a better. None of the online calculators will be using anything else, the poisson formulas are easily available in excel to
calculate individual outcomes.
Teaching the Poisson Distribution with Sports
Armed with these two distributions, many sports modeling problems become much simpler. The Poisson distribution is, in many senses, the continuous version of the binomial distribution. The binomial
counts how many times something happens out of a fixed number of trials. The Poisson distribution counts how many times something happens over an interval of time. As a result, the binomial
distribution is much more naturally applied to discrete games like baseball, football, or golf.
In this article we look at three examples and some non-examples showing how to study the Poisson distribution in sports. We also include some challenge questions with each example of teaching the
Poisson distribution with sports. Type your email…. The Poisson distribution counts the number of times that a countable event happens during a fixed period of time.
In order to use the Poisson distribution, you have to know the average number of occurrences over this period of time. Moreover, over the entire time interval, the probability of the events occurring
at any individual time must remain the same. A common introductory example is modeling the number of phone calls that a call center may receive during a fixed hour. Then the Poisson distribution can
help the call center figure out how much staff they need by telling them the probability of having extremely busy hours.
Perhaps confusingly, the Poisson distribution is actually a discrete probability distribution even though I said above that it is a continuous version of the binomial distribution. The Poisson
distribution is discrete because it is counting a number of occurrences; it can only take the values 0, 1, 2, etc.
I called it continuous because it is applied to model situations where things can happen over a continuous interval. It is useful in modeling games where play is continuous like basketball, hockey,
and soccer. This distinction between continuous settings and discrete settings is key in choosing the right model. The Poisson distribution is actually obtained by taking an appropriate limit of a
binomial distribution.
However, the specific theory about how to do this is not important to the sports examples below. Poisson statistics for golf betting In the NHL, about 6. Hockey is continuous and it is fast; goals
can be scored in the blink of an eye. Using the Poisson distribution to model goals being scored in a game of hockey is a reasonable approximation.
The number of goals scored in a hockey game should follow a P 6. The following plot shows a comparison between the predicted goals per game black and the observed goals per game red from the last 2
seasons. One of the key features of the Poisson distribution is that we can change the length of the time interval and still use a related model.
For example, if goals in an entire game of hockey follow a P 6. Challenge Question 1: In the above graphic we excluded games going into overtime and shootouts. How might you model goals scored during
a shootout. In soccer, time of possession is a big predictor of winning the game. In the MLS, roughly 3 goals are scored on average in a 90 minute game.
That means each team scores about 1. When a team possesses the ball, that team and that team only has a chance to score.
Suppose that Team A has the ball for 50 minutes and, therefore, Team B has it for 40 minutes. How would you compute the average goals scored for Team A and Team B now. The answer: goals are scored at
an average rate of every 30 minutes of game time. Therefore, if Team A has the ball for 50 minutes their goals can be modeled by the Poisson distribution P 1.
Challenge Question 2: How would you compute the probability of Team A beating Team B provided they have the ball for 50 minutes and assuming goals scored follows the appropriate Poisson
distributions. Fouling in the NBA is another event that can be modeled with a Poisson distribution. Fouls occur roughly randomly and at roughly equal rates throughout the game.
Last year Dwight Howard recorded fouls while playing 69 games for 17 minutes per game. This means he averaged roughly one foul every 6 minutes. Julius Randle recorded fouls while playing in 71 games
at Randle averaged a foul roughly every Which of these players is more likely to foul out. This number is the expected fouls by Dwight Howard in his 20 minutes of playing time.
Then, we do the same thing for Julius Randle and multiply by 42 minutes. In a number of Skybet headline Shots on Target bets have lost. A variance simulator link here shows the following distribution
of results could be expected from the shots on target data. Note: In Spring we suspended the ability to combine shots on target bets due to feedback and pending a review of the underlying mathematics
that estimate the affect of inter-dependency in these bets.
Shots on Target and Poisson. We use a live tool to project the xSOT forward to calculate multiple Shots on Target using a normal Poisson distribution Advantage players attempt to back the Skybet
headline Shots on Target boosts with Kelly staking on the exchange as there can be big EV and big liability. I n this blog we look at a review of: Is a Poisson distribution a good fit for Shots on
The ROI of recorded shots on target bets at bookiebashing. Only data from the Premier League is considered. Stats from competitions such as the African Cup of Unders are not comparable. A normal
Poisson probability is used to project the Shots per game to probability. ROI of bets on the bookiebashing tracker 1, bets from approx.
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Simu-bubble --- Introduction ---
The bubble sort is the oldest and simplest sort in use. Unfortunately, it is also the slowest. The bubble sort works by comparing each item in the list with the item next to it, and swapping them if
required. The algorithm repeats this process until it makes a pass all the way through the list without swapping any items (in other words, all items are in the correct order). This causes larger
values to "bubble" to the end of the list while smaller values "sink" towards the beginning of the list.
Simu-bubble is an interactive exercise designed to help you to understand how a bubble sort works. The computer presents a list of random order to you, and ask you to sort it step by step, until the
required order, according to the bubble sort.
Other exercises on:
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: manually simulate a bubble sort. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation
and games
• Keywords: wims, mathematics, mathematical, math, maths, interactive mathematics, interactive math, interactive maths, mathematic, online, calculator, graphing, exercise, exercice, puzzle,
calculus, K-12, algebra, mathématique, interactive, interactive mathematics, interactive mathematical, interactive math, interactive maths, mathematical education, enseignement mathématique,
mathematics teaching, teaching mathematics, algebra, geometry, calculus, function, curve, surface, graphing, virtual class, virtual classes, virtual classroom, virtual classrooms, interactive
documents, interactive document, algebra, algorithmics, bubble sort, permutation, set_theory
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Calculus 2, Chapter 1: Part 3 - Polar Forms and Area
About the course
Calculus is the mathematics of change and motion. Calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Calculus II is all about integral
Mainly in this course, we are learning many techniques to solve many different problems with applications. By taking this course, you will be able to set up and evaluate integrals to find areas and
volumes and to solve real-world problems, evaluate integrals by hand using a variety of techniques including substitutions, parts, partial fractions, and hyperbolic trigonometry, Analyze the
convergence of sequences, series,
and power series, solve elementary problems in vector analysis
This course is with four main chapters:
1. Applications of integrals: Using integrals to find areas and volumes and to solve real-world problems
2. Evaluating integrals by hand: Evaluate integrals by hand using a variety of techniques, including substitutions, parts, partial fractions, and hyperbolic trigonometry
3. Analyzing the convergence of sequences, series, and power series
4. Vector analysis: Solve elementary problems in vector analysis
I am Dr. Miuran Dencil, a professional mathematician currently working as an assistant professor. I earned my master’s and doctoral degrees in mathematics in 2016 and 2020 from Texas Tech University,
Texas, USA. While in college, I won very prestigious Presidential Graduate Fellowship for 5 years, which is offered to very best graduate student in the school. I was invited speaker for many
recognized conferences and won many grants for my research. My dissertation research was Geometric Properties of Special Functions and Related Quadratic Differentials. My research interests include
Partial Differential Equations and Number Theory. I graduated from University of Kelaniya, Sri Lanka in 2011 with Bachelor's Degree, which is special degree in mathematics with first class honors.
Teaching is my passion and calling! on top of that mathematics is fun and interactive! I started to teach mathematics at very young age. I have over 16 years of teaching experiences and taught
mathematics to kindergarten through senior high school students, college level and graduate level classes.
Once I graduated from my bachelor’s degree, I worked as an instructor at University of Kelaniya from 2011-2013, then worked as a lecturer at university of moratuwa from 2013-2015. Then I started my
graduate studies at Texas Tech University while working as GPTI/ Research assistant 2015-2020.
I have taught: Calculus I, II, III + Pre-Cal, AP Cal AB/ AP Cal BC, Trigonometry, Linear Algebra, Geometry, ODE/PDE, Algebra 1,2, Numerical Methods, Real/ Advanced Analysis, Complex Numbers, Discrete
Mathematics, Group Theory/ Ring Theory/Number Theory, Integration/ Laplace equations/ Fourier Series- Transformation, Grade Mathematics, Mechanics/ Motion of a particle/ Force- equilibrium/ Harmonic
motion, Trainer for exams such as SAT, GRE math sections.
Over 16 years of my teaching experiences and long academic life I have found techniques which helped me to understand and solve any given problem faster and accurate. Now I am ready to teach them to
you as I believe nothing is too challenging as long as the student is willing to learn.
As a mathematics is interactive and fun subject it allow students to get extra practice and provide personalized learning atmosphere for students to sharpened their analytical thinking and problem
solving technique.
Great News! Miyuran wants to help you unlock your potential.
Virtual tutoring is available now within a few clicks.
Get your free Gooroo Pass to take this course today
First 14 days free, then $9.99/month.
• New content added weekly
• Cancel anytime
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Consider a 30-year home mortgage of
Consider a 30-year home mortgage of $100,000 at 6% per year. What is the monthly payment? Use Theorem 1 as attached to make an amortization schedule of the first 6 months: Month (k) Principal P(k)
Interest I(k) Balance due B(k) 1 2 3 4 5 6 where P(k) is the amount paid to the principal in the k’th payment, I(k) is the amount paid to interest in the k’th
payment, B(k) is the balance due after the k’th payment.
Consider a 30-year home mortgage of $100,000 at 6% per year. What is the monthly payment? Use Theorem 1 as attached to make an amortization schedule of the first 6 months: Month (k) Principal P(k)
Interest I(k) Balance due B(k) 1 2 3 4 5 6 where P(k) is the amount paid to the principal in the k’th payment, I(k) is the amount paid to interest in the k’th
payment, B(k) is the balance due after the k’th payment.
Essentials Of Investments
11th Edition
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Consider a 30-year home mortgage of $100,000 at 6% per year. What is the monthly payment? Use Theorem 1 as attached to make an amortization schedule of the first 6 months:
Month (k) Principal P(k) Interest I(k) Balance due B(k)
P(k) is the amount paid to the principal in the k’th payment,
I(k) is the amount paid to interest in the k’th payment,
B(k) is the balance due after the k’th payment.
Transcribed Image Text:There 11 2) The where with powy A Play al paid to test is dia 1.-4 l w. WARE Pai momčad -a-7 Bax pel The
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Yards per minute to Meters per second
Meters per second to Yards per minute (Swap Units)
Note: Fractional results are rounded to the nearest 1/64. For a more accurate answer please select 'decimal' from the options above the result.
Note: You can increase or decrease the accuracy of this answer by selecting the number of significant figures required from the options above the result.
Note: For a pure decimal result please select 'decimal' from the options above the result.
Yards per minute to Meters per second formula
The SI measurement of speed and velocity. This is the number of meters travelled in one second of time. The accompanying acceleration unit is meters per second per second (m/s²).
Yards per minute to Meters per second table
Yards per minute Meters per second
0yd/min 0.00m/s
1yd/min 0.02m/s
2yd/min 0.03m/s
3yd/min 0.05m/s
4yd/min 0.06m/s
5yd/min 0.08m/s
6yd/min 0.09m/s
7yd/min 0.11m/s
8yd/min 0.12m/s
9yd/min 0.14m/s
10yd/min 0.15m/s
11yd/min 0.17m/s
12yd/min 0.18m/s
13yd/min 0.20m/s
14yd/min 0.21m/s
15yd/min 0.23m/s
16yd/min 0.24m/s
17yd/min 0.26m/s
18yd/min 0.27m/s
19yd/min 0.29m/s
Yards per minute Meters per second
20yd/min 0.30m/s
21yd/min 0.32m/s
22yd/min 0.34m/s
23yd/min 0.35m/s
24yd/min 0.37m/s
25yd/min 0.38m/s
26yd/min 0.40m/s
27yd/min 0.41m/s
28yd/min 0.43m/s
29yd/min 0.44m/s
30yd/min 0.46m/s
31yd/min 0.47m/s
32yd/min 0.49m/s
33yd/min 0.50m/s
34yd/min 0.52m/s
35yd/min 0.53m/s
36yd/min 0.55m/s
37yd/min 0.56m/s
38yd/min 0.58m/s
39yd/min 0.59m/s
Yards per minute Meters per second
40yd/min 0.61m/s
41yd/min 0.62m/s
42yd/min 0.64m/s
43yd/min 0.66m/s
44yd/min 0.67m/s
45yd/min 0.69m/s
46yd/min 0.70m/s
47yd/min 0.72m/s
48yd/min 0.73m/s
49yd/min 0.75m/s
50yd/min 0.76m/s
51yd/min 0.78m/s
52yd/min 0.79m/s
53yd/min 0.81m/s
54yd/min 0.82m/s
55yd/min 0.84m/s
56yd/min 0.85m/s
57yd/min 0.87m/s
58yd/min 0.88m/s
59yd/min 0.90m/s
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Ohm’s Law MDCAT MCQs with Answers - Youth For Pakistan
Welcome to the Ohm’s Law MDCAT MCQs with Answers. In this post, we have shared Ohm’s Law Multiple Choice Questions and Answers for PMC MDCAT 2024. Each question in MDCAT Physics offers a chance to
enhance your knowledge regarding Ohm’s Law MCQs in this MDCAT Online Test.
Ohm’s Law MDCAT MCQs Test Preparations
Ohm’s Law states that the current through a conductor between two points is directly proportional to the:
a) Voltage across the conductor
b) Resistance of the conductor
c) Temperature of the conductor
d) Length of the conductor
a) Voltage across the conductor
According to Ohm’s Law, the formula to calculate current (I) is:
a) I = V/R
b) I = VR
c) I = R/V
d) I = V^2/R
In Ohm’s Law, what does the symbol ‘V’ represent?
a) Voltage
b) Current
c) Resistance
d) Power
The unit of resistance in Ohm’s Law is:
a) Ohm (Ω)
b) Volt (V)
c) Ampere (A)
d) Watt (W)
If the resistance in a circuit is doubled and the voltage remains constant, the current will:
a) Be halved
b) Be doubled
c) Remain the same
d) Be quartered
Ohm’s Law is represented by which equation?
a) V = IR
b) P = VI
c) F = ma
d) E = mc^2
A resistor has a resistance of 5Ω. If a current of 2A flows through it, the voltage across the resistor is:
a) 10V
b) 2.5V
c) 5V
d) 1V
In a series circuit, the total resistance is equal to:
a) The sum of the individual resistances
b) The product of the individual resistances
c) The inverse of the sum of the individual resistances
d) Zero
a) The sum of the individual resistances
If the voltage across a resistor is tripled, the current through the resistor will:
a) Triple
b) Remain the same
c) Double
d) Be halved
In a parallel circuit, the total resistance is:
a) Less than the smallest resistance
b) Equal to the largest resistance
c) The sum of all resistances
d) Greater than the largest resistance
a) Less than the smallest resistance
The graph of V versus I for an ohmic conductor is:
a) A straight line passing through the origin
b) A curve passing through the origin
c) A straight line with a negative slope
d) A horizontal line
a) A straight line passing through the origin
For a fixed voltage, if the resistance decreases, the current will:
a) Increase
b) Decrease
c) Remain constant
d) Be zero
In Ohm’s Law, the term ‘R’ stands for:
a) Resistance
b) Reactance
c) Resilience
d) Resonance
If a resistor with a resistance of 10Ω has a voltage of 50V across it, the current flowing through the resistor is:
a) 5A
b) 0.5A
c) 10A
d) 2A
Ohm’s Law is valid only for:
a) Ohmic conductors
b) Non-ohmic conductors
c) Semiconductors
d) Superconductors
a) Ohmic conductors
The resistance of a conductor is directly proportional to its:
a) Length
b) Cross-sectional area
c) Temperature
d) Conductivity
What happens to the current if the voltage across a resistor is doubled?
a) It doubles
b) It halves
c) It remains the same
d) It becomes zero
If a wire’s resistance is 8Ω and the voltage across it is 24V, the current through the wire is:
a) 3A
b) 2A
c) 4A
d) 6A
In a conductor, as temperature increases, the resistance typically:
a) Increases
b) Decreases
c) Remains constant
d) Becomes zero
The SI unit of current is:
a) Ampere
b) Volt
c) Ohm
d) Watt
Which of the following materials typically obeys Ohm’s Law?
a) Copper
b) Germanium
c) Silicon
d) Rubber
In a circuit with constant resistance, an increase in current indicates:
a) An increase in voltage
b) A decrease in voltage
c) A constant voltage
d) A decrease in power
a) An increase in voltage
Ohm’s Law can be used to determine the:
a) Relationship between current, voltage, and resistance
b) Power output of a circuit
c) Energy consumption
d) Frequency of an AC circuit
a) Relationship between current, voltage, and resistance
The potential difference across a 20Ω resistor carrying a current of 3A is:
a) 60V
b) 6V
c) 30V
d) 40V
If the voltage across a conductor is kept constant and the resistance increases, the current will:
a) Decrease
b) Increase
c) Remain the same
d) Be zero
A conductor with a resistance of 12Ω and a current of 4A will have what voltage across it?
a) 48V
b) 3V
c) 16V
d) 4V
In a circuit, the power dissipated by a resistor can be calculated using:
a) P = VI
b) P = I^2R
c) P = V^2/R
d) All of the above
d) All of the above
The relationship between power (P), voltage (V), and current (I) is given by:
a) P = VI
b) P = V/I
c) P = I/V
d) P = V^2I
The resistance of a wire depends on its:
a) Length, cross-sectional area, and material
b) Length only
c) Cross-sectional area only
d) Material only
a) Length, cross-sectional area, and material
In a metallic conductor, the charge carriers are typically:
a) Electrons
b) Protons
c) Neutrons
d) Ions
The voltage drop across a resistor is determined by the product of:
a) Current and resistance
b) Current and power
c) Resistance and power
d) Power and time
a) Current and resistance
In a series circuit with resistors, the total voltage is equal to:
a) The sum of the voltage drops across each resistor
b) The product of the voltage drops across each resistor
c) The inverse of the voltage drops across each resistor
d) Zero
a) The sum of the voltage drops across each resistor
In a parallel circuit, the total current is equal to:
a) The sum of the currents through each path
b) The product of the currents through each path
c) The inverse of the sum of the currents through each path
d) Zero
a) The sum of the currents through each path
Ohm’s Law is not applicable in circuits with:
a) Non-linear components
b) Linear components
c) Constant voltage
d) Constant current
a) Non-linear components
A circuit consists of a 9V battery and a 3Ω resistor. The current in the circuit is:
a) 3A
b) 27A
c) 1A
d) 9A
The conductance of a material is:
a) The reciprocal of resistance
b) The reciprocal of current
c) The reciprocal of voltage
d) The reciprocal of power
a) The reciprocal of resistance
In a resistor, when the temperature increases, the resistance:
a) Increases
b) Decreases
c) Remains the same
d) Becomes zero
A conductor obeying Ohm’s Law is known as an:
a) Ohmic conductor
b) Non-ohmic conductor
c) Insulator
d) Semiconductor
If the resistance in a circuit is 20Ω and the current is 2A, the power dissipated is:
a) 80W
b) 40W
c) 10W
d) 60W
In a circuit, if the voltage is zero, the current through the circuit is:
a) Zero
b) Infinite
c) Maximum
d) Depends on the resistance
If you are interested to enhance your knowledge regarding Physics, Chemistry, Computer, and Biology please click on the link of each category, you will be redirected to dedicated website for each
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Extending partial isometries of generalized metric spaces
Fundamenta Mathematicae 244 (2019), 1-16 MSC: Primary 03C13; Secondary 05C12, 05C38. DOI: 10.4064/fm484-9-2018 Published online: 26 October 2018
We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal {K}$ of finite generalized metric spaces satisfies the
Hrushovski extension property: for any $A\in \mathcal {K}$ there is some $B\in \mathcal {K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. We
prove the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal {R}$. When $\mathcal {R}$ is also countable, we use this to show that the
isometry group of the Urysohn space over $\mathcal {R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting metric triangles of
uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length
bounded by $n$.
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Free Printable Multiplication Table 1 10 Chart Template PDF Best | Multiplication Chart Printable
Free Printable Multiplication Table 1 10 Chart Template PDF Best
Free Printable Multiplication Table 1 10 Chart Template PDF Best
Free Printable Multiplication Table 1 10 Chart Template PDF Best – A Multiplication Chart is an useful tool for kids to learn exactly how to multiply, split, and also find the tiniest number. There
are lots of usages for a Multiplication Chart. These handy tools assist kids recognize the procedure behind multiplication by using colored paths as well as filling in the missing items. These charts
are totally free to download as well as publish.
What is Multiplication Chart Printable?
A multiplication chart can be used to assist kids learn their multiplication realities. Multiplication charts been available in numerous forms, from complete page times tables to solitary page ones.
While specific tables serve for presenting portions of details, a complete web page chart makes it much easier to review facts that have actually already been grasped.
The multiplication chart will usually include a top row as well as a left column. The leading row will have a list of items. Select the first number from the left column as well as the 2nd number
from the leading row when you want to discover the product of 2 numbers. Once you have these numbers, move them along the row or down the column until you reach the square where the two numbers
fulfill. You will certainly after that have your item.
Multiplication charts are useful knowing tools for both children as well as grownups. Kids can utilize them in your home or in college. Multiplication Table 1 10 Chart Printable are available on the
net as well as can be printed out and laminated flooring for durability. They are a fantastic tool to use in mathematics or homeschooling, and also will provide an aesthetic pointer for youngsters as
they discover their multiplication realities.
Why Do We Use a Multiplication Chart?
A multiplication chart is a diagram that shows how to multiply two numbers. It generally consists of a leading row as well as a left column. Each row has a number standing for the product of the two
numbers. You select the very first number in the left column, relocate down the column, and after that choose the second number from the top row. The item will be the square where the numbers
Multiplication charts are valuable for numerous factors, including aiding children find out just how to separate as well as streamline fractions. Multiplication charts can likewise be helpful as
workdesk sources because they offer as a continuous suggestion of the student’s development.
Multiplication charts are also helpful for helping students memorize their times tables. They help them discover the numbers by decreasing the variety of actions required to finish each operation.
One technique for remembering these tables is to concentrate on a single row or column at once, and afterwards move onto the following one. At some point, the whole chart will be committed to memory.
Just like any ability, remembering multiplication tables requires time as well as practice.
Multiplication Table 1 10 Chart Printable
Multiplication Table 1 10 Chart Printable
If you’re looking for Multiplication Table 1 10 Chart Printable, you’ve involved the right place. Multiplication charts are offered in various styles, consisting of full dimension, half dimension,
and a range of charming styles. Some are vertical, while others include a straight layout. You can also discover worksheet printables that include multiplication formulas and math truths.
Multiplication charts and also tables are indispensable tools for youngsters’s education. These charts are great for use in homeschool mathematics binders or as class posters.
A Multiplication Table 1 10 Chart Printable is a valuable tool to reinforce math truths and also can aid a youngster find out multiplication promptly. It’s also a terrific tool for skip checking and
learning the moments tables.
Related For Multiplication Table 1 10 Chart Printable
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TIOmarkets | Forex lot size calculator
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The appropriate lot size to trade is calculated by dividing the amount to risk by the distance of where you intend to put your stop loss in pips. This will calculate the value that each pip should be
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- Electrical Riddle
Reply To: Transformer Question No. 17 – Converter transformer
2021-08-03 at 11:08 am #1484
For rectifier transformer sizing, many terms such as the harmonic losses effects and internal impedance which influence on the transformer rating shall be considered by designer. The traditional IEEE
method for rating a rectifier transformer has always been the root-mean-square (rms) kVA drawn from the primary line. This is still the method used to develop all of the tables and figures given in
ANSI/IEEE C57.18.10, Clause 10. However, the IEC converter transformer standards define the kVA by the fundamental kVA drawn from the primary line. The rms-rated kVA method is based on the rms
equivalent of a rectangular current wave shape based on the dc rated load commutated with zero commutating angle. The fundamental kVA method is based on the rms equivalent of the fundamental
component of the line current. According to IEC, it is only proper to rate the transformer at the fundamental frequency. Transformer rating and test data will then correspond accurately. The
traditional IEEE rms-kVA rating method will not be exactly accurate at test. However, it does represent more accurately what a user sees as meter readings on the primary side of the transformer.
Users feel strongly that this is a better method, and this is what their loading is based on. ANSI/IEEE C57.18.10 allows for both kVA methods. It is important for a user to understand the difference
between these two methods so that the user can specify which rating is wanted. -Harmonic Loss Effects The term harmonic-loss factor, was developed by IEEE and IEC as a method to define the summation
of harmonic terms that can be used as a multiplier on winding eddy-current losses and other stray losses. These items are separated into two factors, winding eddy-current harmonic-loss factor, and
the other-stray-loss harmonic-loss factor. The new IEEE Recommended Practice for establishing transformer capability when Supplying Non-Sinusoidal Load Currents, ANSI/IEEE C57.100, gives a very good
explanation of these terms and comparisons to the UL definition of K-factor. The Transformers Committee of the IEEE Power Engineering Society has accepted the term harmonicloss factor as more
mathematically and physically correct than the term K-factor. K-factor is used in UL standards, which are safety standards. IEEE standards are engineering standards. As is evident, the primary
difference is that the other stray losses are only increased by a harmonic exponent factor of 0.8. Bus-bar, eddy-current losses are also increased by a harmonic exponent factor of 0.8. Winding
eddy-current losses are increased by a harmonic exponent factor of 2. The factor of 0.8 or less has been verified by studies by manufacturers in the IEC development and has been accepted in ANSI/IEEE
C57.18.10. Other stray losses occur in core clamping structures, tank walls, or enclosure walls. On the other hand, current-carrying conductors are more susceptible to heating effects due to the skin
effect of the materials. Either the harmonic spectrum or the harmonic-loss factor must be supplied by the specifying engineer to the transformer manufacturer. – Commutating Impedance Commutating
impedance is defined as one-half the total impedance in the commutating circuit expressed in ohms referred to the total secondary winding. It is often expressed as percent impedance on a secondary
kVA base. For wye, star, and multiple-wye circuits, this is the same as derived in ohms on a phase-toneutral voltage basis. With diametric and zigzag circuits, it must be expressed as one-half the
total due to both halves being mutually coupled on the same core leg or phase. This is not to be confused with the short-circuit impedance, i.e., the impedance with all secondary windings shorted.
Care must be taken when expressing these values to be careful of the kVA base used in each. The commutating impedance is the impedance with one secondary winding shorted, and it is usually expressed
on its own kVA base, although it can also be expressed on the primary kVA base if desired. Care must be taken when specifying these values to the transformer manufacturer. The impedance value,
whether it is commutating impedance or short-circuit impedance, and kVA base are extremely important.Use ANSI/IEEE C57.18.10 as a reference for commutating impedance. The tables of circuits in this
reference are also useful.
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cla_gercond_x - Linux Manuals (3)
cla_gercond_x (3) - Linux Manuals
cla_gercond_x.f -
REAL function cla_gercond_x (TRANS, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
CLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices.
Function/Subroutine Documentation
REAL function cla_gercond_x (characterTRANS, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, complex, dimension( * )X,
integerINFO, complex, dimension( * )WORK, real, dimension( * )RWORK)
CLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices.
CLA_GERCOND_X computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX vector.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate Transpose = Transpose)
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF is COMPLEX array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by CGETRF.
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by CGETRF; row i of the matrix was interchanged
with row IPIV(i).
X is COMPLEX array, dimension (N)
The vector X in the formula op(A) * diag(X).
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK is COMPLEX array, dimension (2*N).
RWORK is REAL array, dimension (N).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 135 of file cla_gercond_x.f.
Generated automatically by Doxygen for LAPACK from the source code.
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cmHg – Centimetres of Mercury at 0 deg C Pressure Unit
Centimeters of Mercury is a small pressure unit which represents the pressure pushing down due to gravity of any volume of liquid mercury which is 1cm high. 1 centimeter of mercury at zero degrees
Celsius equals 1333.22 pascals.
The use of cmHg is not as common as mmHg, it is mainly used in applications for measuring pressures of vacuum pumps, pneumatic systems, automotive engine inlet manifold pressures and blood pressure.
For more background on how mercury is used to measure pressure please vist our mmHg page
Please use the table below to determine the value of 1 Centimeter of Mercury in different pressure units.
For the reverse conversion factor into cmHg please click on the appropriate pressure unit link below.
To calculate cmHg pressure conversions online please go to our pressure converter page.
View the calculation for deriving cmHg from SI units or a list of different ways to identify the cmHg pressure unit.
Conversion Factors
Please note that the conversion factors above are accurate to 6 significant figures.
The calculation below shows how the pressure unit centimetres of mercury (cmHg) is derived from SI Units.
• Pressure = Force / Area
• Force = Mass x Acceleration
• Mass = Density x Volume
• Volume = Area x Height
• Acceleration = Distance / (Time x Time)
SI Units
• Mass: kilogram (kg)
• Distance: metre (m)
• Time: second (s)
• Force: newton (N)
• Pressure: pascal (Pa)
Input Values
• Density = Mercury Density at 0degC = 13595.1 kg/m³
• Area = 1 m²
• Height = 1 cm = 0.01 m
• Acceleration = Standard Gravity = 9.80665 m/s²
• 1 cmHg Mass = 13595.1 kg/m³ x 1 m² x 0.01 m = 135.951 kg
• 1 cmHg Force = 135.951 kg x 9.80665 m/s² = 1333.223874 N
• 1 cmHg Pressure = 1333.223874 N / 1 m² = 1333.223874 Pa
Alternate Descriptions
These are the different versions used for identifying cmHg that you may find elsewhere.
• centimetres of mercury
• centimeters of mercury
• centimetres of mercury column
• centimeters of mercury column
• cmHg
• cm Hg
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CLiki: GPL
General Public
is the "copyleft" license used by most of the applications developed by the Free Software Foundation (
). Software released using this license is free according to the
Although it is well-understood for traditional C/Unix programs, fewer people agree on exactly what its implications are when used in a typical CL environment. At least, I don't. Debate welcome
For more details see its page on the GNU project web site
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Multiloop Control Design for Buck Converter
This example shows how to tune the gains of a discrete PID controller in a cascade control configuration using systune.
This example is based on the article Cascade Digital PID Control Design for Power Electronic Converters. The article describes the workflow to tune the inner-loop current control and outer-loop
voltage control one loop at a time, whereas this example shows how to tune both loops at the same time.
In this example, you:
1. Conduct frequency response estimation (FRE) of the buck converter plant model.
2. Estimate a parametric LTI model from the FRE result.
3. Construct a multiloop feedback control system using LTI models.
4. Define tuning goals in the frequency domain and tune the controllers using systune.
5. Verify the performance of the tuned controllers.
Conduct Frequency Response Estimation
This example uses a buck converter modeled using Simscape™ Electrical™ components to provide voltage regulation from 48 V to 12 V. The model uses cascade control architecture so that the inner loop
regulates the inductor current and the outer loop regulates the output voltage. The output of the outer voltage loop provides the current reference signal to the inner current loop, which, in turn,
provides the duty cycle signal to the PWM Generator block. The controller architecture includes manual switches to make the converter operate in one of three configurations: open-loop (PWM Generator
block with a constant duty cycle), inner current-loop, and outer voltage loop.
Open the buck converter plant model.
mdl = 'scdCurrentControlBuckConverter';
Specify Linear Analysis Points for Frequency Response Estimation
To collect frequency response data, you must first specify the portion of model to estimate. You can configure the linear analysis points that specify the inputs and outputs of the model for
estimation using linio. Alternatively, you can interactively specify the linear analysis points using the Linearization Manager app. Here, use linio to assign the input perturbation analysis point to
the Duty Cycle block and the output measurement analysis points to the Current ADC and Voltage ADC blocks, which are the Rate Transition blocks after the inductor current measurement and output
voltage measurement Probe blocks, respectively.
io(1) = linio('scdCurrentControlBuckConverter/Duty Cycle',1,'input');
io(2) = linio('scdCurrentControlBuckConverter/Current ADC',1,'output');
io(3) = linio('scdCurrentControlBuckConverter/Voltage ADC',1,'output');
Find Snapshot-Based Model Operating Point and Initialize Model
To obtain a frequency response that accurately captures system dynamics, you must perform the estimation at a steady-state operating point.
Simulate the model to determine the time the model takes to reach steady state.
Initial simulation results show that the buck converter model reaches steady-state operation after around 0.007 seconds. Take a simulation snapshot at 0.007 seconds to find the steady-state operating
opini = findop(mdl,0.007);
Initialize the model using this operating point object.
op = operpoint(mdl);
Create Perturbation Signal for Experiment and Compute Non-Parametric Frequency Response
Define a PRBS perturbation signal with the following parameters.
• Signal order — 11
• Number of periods — 1
• Perturbation amplitude — 0.05
• Sample time — $1×$${10}^{-5}$ seconds
in_PRBS = frest.PRBS('Order',11,'NumPeriods',1,'Amplitude',0.05,'Ts',1e-5);
Before you conduct the frequency response estimation experiment, identify the time-varying sources so that these sources are deterministic during the experiment.
srcblks = frest.findSources(mdl,io);
opts = frestimateOptions;
opts.BlocksToHoldConstant = srcblks;
You can now conduct the frequency response estimation experiment. During the experiment, the software simulates the model, injects the PRBS signal at the specified input, and measures the response at
the specified output. The result is a frequency-response data model (frd) object. This is a non-parametric model that is a description of the system as discrete frequency points.
estsys_PRBS = frestimate(mdl,io,op,in_PRBS,opts);
Frequency response estimation with PRBS input signal produces results with many frequency points. Use interp (System Identification Toolbox) to extract an interpolated result from the estimated
frequency response model across 50 frequency points from 700 rad/s to 300,000 rad/s.
wmin = 700;
wmax = 3e5;
Nfreq = 50;
w = logspace(log10(wmin+10),log10(wmax),Nfreq);
estsys_PRBS_thinned = interp(estsys_PRBS, w);
Compare the FRE result before and after thinning.
legend('Raw FRE result','Thinned FRE result');
The frequency points match very well. You can now estimate a parametric model from the thinned result.
Estimate Parametric LTI Model from FRE Results
Estimate a state-space parametric model of the buck converter with one input (duty cycle) and two outputs (inductor current and output voltage). As the shape of the estimated frequency response in
the Bode plot resembles a third-order model, estimate a third-order state-space model using ssest.
optssest = ssestOptions('SearchMethod','lm');
optssest.Regularization.Lambda = 1e-8;
sys_systune = ssest(estsys_PRBS_thinned,3,'Ts',Ts_ctrl,optssest);
Compare the parametric estimation result with the thinned FRE result.
P = bodeoptions;
P.PhaseMatching = 'on';
bode(estsys_PRBS_thinned,sys_systune, P);
legend('FRE result','ssest result');
You can see that the estimated parametric model is satisfactory.
Construct Feedback Control System for Tuning
To model a feedback control system for tuning, first define the discrete-time PI controllers as tunable elements.
Ci = tunablePID('Ci','PI',Ts_ctrl);
Ci.IFormula = 'Trapezoidal';
Ci.u = 'Ie';
Ci.y = 'Duty Cycle';
Cv = tunablePID('Cv','PI',Ts_ctrl);
Cv.IFormula = 'Trapezoidal';
Cv.u = 'Ve';
Cv.y = 'Iref';
To improve the convergence time, provide initial values for the outer-loop controller.
Cv.Kp.Value = 1;
Cv.Ki.Value = 200;
Then, construct a multiloop control system as shown.
sum_i = sumblk('Ie = Iref-iL_sampled');
sum_v = sumblk('Ve = Vref-vc_sampled');
input = {'Vref'};
output = {'iL_sampled','vc_sampled'};
APs = {'Iref','Duty Cycle','iL_sampled','vc_sampled'};
ST0 = connect(sys_systune,Ci,Cv,sum_i,sum_v,input,output,APs);
Define Frequency-Domain Tuning Goals
Define tuning goals for inner and outer loops using target bandwidths and stability margins.
• Use TuningGoal.LoopShape to specify the target bandwidth.
• Use TuningGoal.Margins to specify phase and gain margins in a frequency range. While you can clearly define target phase margins, specify a small value of 3 dB for the gain margins for stability.
The tuning result usually achieves a higher gain margin. For this goal, also define a frequency focus band so that systune enforces margins only over the interested frequency range.
Additionally, specify the outer loop as open while evaluating the inner-loop tuning goals.
LS1 = TuningGoal.LoopShape('iL_sampled',30000);
LS1.Openings = {'vc_sampled'};
LS2 = TuningGoal.LoopShape('vc_sampled',3000);
MG1 = TuningGoal.Margins('iL_sampled',3,60);
MG1.Openings = {'vc_sampled'};
MG1.Focus = [30000 300000];
MG2 = TuningGoal.Margins('vc_sampled',3,60);
MG2.Focus = [3000 30000];
Tune Controllers and Extract Tuning Results
Start tuning with systune, using the following settings to help optimization achieve desirable results. To satisfy the performance requirements, systune enforces all tuning goals as hard goals.
Create a systuneOptions object and adjust the values for the minimum decay rate and maximum spectral radius to suit tuning for high bandwidth loop shapes. Also, reduce the relative tolerance criteria
for termination.
opt = systuneOptions('SoftTol',1e-10,'MinDecay',1e-10,'MaxRadius',1e10);
[ST1,fSoft,fHard] = systune(ST0,[],[LS1,LS2,MG1,MG2],opt);
Final: Soft = -Inf, Hard = 1.7014, Iterations = 68
Plot the performance of the tuned system against the tuning goals.
For the inner loop, the tuned result achieves a bandwidth slightly lower than the specified target bandwidth and the phase margin is lower than the specified target phase margin in some frequencies.
Even though the tuning goals are not completely satisfied, the tuning results are sufficient for the stable operation of the tuned model.
Get the controller gains from tunable blocks and save to workspace.
Cv = getBlockValue(ST1,'Cv');
Ci = getBlockValue(ST1,'Ci');
CurrentControlP = Ci.Kp
CurrentControlI =
VoltageControlI =
Verify Control Design Result Performance
Examine the tuned controller performance with load and input voltage disturbances.
• The load disturbance is applied at 0.008 seconds, which increases the load resistance from 6 ohms to 12 ohms.
• The input voltage disturbance is applied at 0.016 seconds, which decreases the input voltage from 48 V to 40 V.
Set the PI controller gains to the tuned values.
set_param('scdCurrentControlBuckConverter/Discrete PID Controller','P','CurrentControlP');
set_param('scdCurrentControlBuckConverter/Discrete PID Controller','I','CurrentControlI');
set_param('scdCurrentControlBuckConverter/Discrete PID Controller1','P','VoltageControlP');
set_param('scdCurrentControlBuckConverter/Discrete PID Controller1','I','VoltageControlI');
Toggle the manual switches to close both the inner and outer loops and simulate the model with the regulated voltage and current.
set_param('scdCurrentControlBuckConverter/Manual Switch', 'sw', '0');
set_param('scdCurrentControlBuckConverter/Manual Switch1', 'sw', '0');
Simulate the model with the tuned gains.
The tuned controllers track the voltage reference and reject disturbances well. To fine tune the result, you can update the tuning goals in the Define Frequency-Domain Tuning Goals section of this
example. For a faster response, you can increase the target bandwidth in the loop shape tuning goal. For improved transient behavior, you can increase the target phase margin in the margins tuning
Close the model.
See Also
systune | ssest (System Identification Toolbox) | frestimate | connect
Related Topics
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"A pizza with radius 'z' and thickness…
“A pizza with radius ‘z’ and thickness ‘a’ has a volume of Pi*z*z*a”
“Pizzas are cylinders” was posted on Twitter by F’tim on April 10, 2014. “if you think about it, pizza is a very thin cylinder” was posted on Twitter by UsicMusic on September 22, 2016. “A pizza is
technically just a very short cylinder” was posted on Reddit—Showerthoughts on October 28, 2018. “A pizza is technically a cylinder….” was posted on Twitter by hunter on December 11, 2018. “Pizza is
a rather flat cylinder” was posted on Twitter by IntoMath.org on September 2, 2019. “Pizzas are cylinders, not circles” was posted on Reddit—Showerthoughts on November 5, 2021.
What is the volume of pizza? “Silly ObPuzzle: My local pizza delivery company is called Perfect Pizza. This means that not only do the pizzas taste perfect but they are also perfectly circular and
perfectly uniform in depth. They’ve just delivered a pizza with depth a, and radius z. What is its volume?” was posted on the newsgroup rec.puzzles on September 19, 1997. “Did you know that the word
‘pizza’ is a mathematical formula? It is the volume of a circular pizza of radius ‘z’ and thickness ‘a’” was posted on the newgroup geometry.puzzles on February 27, 1998. “Volume of a pizza of
thickness ‘a’ and radius ‘z’ is given by pi z z a” was posted on the newsgroup sci.electronics.design on March 9, 2006. ” A pizza of radius z and thickness a has a volume of pi z z a” was posted on
Twitter by Selem on March 12, 2007. “A pizza with radius ‘z’, and thickness ‘a’ has a volume of Pi*z*z*a” was posted on Twitter by Ric on December 25, 2010.
Wikipedia: Pizza
Pizza (Italian: [ˈpittsa], Neapolitan: [ˈpittsə]) is a dish of Italian origin consisting of a usually round, flattened base of leavened wheat-based dough topped with tomatoes, cheese, and often
various other ingredients (such as anchovies, mushrooms, onions, olives, pineapple, meat, etc.), which is then baked at a high temperature, traditionally in a wood-fired oven.
Google Groups: rec.puzzles
pizza + anarchy (spoiler)
Sep 19, 1997, 3:00:00 AM
Silly ObPuzzle: My local pizza delivery company is called Perfect Pizza. This means that not only do the pizzas taste perfect but they are also perfectly circular and perfectly uniform in depth.
They’ve just delivered a pizza with depth a, and radius z. What is its volume?
Google Groups: geometry.puzzles
Inscribing Circles inside Integer Triangles
.(JavaScript must be enabled to view this email address)
Feb 27, 1998, 3:00:00 AM
Barry Wolk | Did you know that the word “pizza”
Dept of Mathematics | is a mathematical formula? It is
University of Manitoba | the volume of a circular pizza of
Winnipeg Manitoba Canada | radius “z” and thickness “a”.
Google Groups: rec.crafts.brewing
max carboy pressure
Jan 16, 2002, 2:07:37 PM
I saw a cute mnemonic on a math web site the other day….they were talking about the volume of a somewhat-idealized cylindrical pizza of radius z and thickness (height) a:
Volume = pi暘暘戢
Google Books
CRC Concise Encyclopedia of Mathematics
Second Edition
By Eric W. Weisstein
Boca Raton, FL: Chapman & Hall.CRC
Pg. 2251:
There is also a second pizza theorem. This one gives the VOLUME of a pizza of thickness a and RADIUS z,
Google Groups: sci.electronics.design
OT: You Can Be Too Nice
martin griffith
Mar 9, 2006, 3:40:48 PM
Volume of a pizza of thickness ‘a’ and radius ‘z’ is given by pi z z a
his is brilliant: A pizza of radius z and thickness a has a volume of pi z z a.
5:56 PM · Mar 12, 2007·Twitter Web Client
Just a Quick Note: A pizza of radius z and thickness a has a volume of pi z z a
credit goes to Clive 😉.. http://tinyurl.com/2akcdb
12:08 AM · Mar 29, 2007·Twitter Web Client
-ˏˋAlex Bugaˎˊ-
Cool quote: “What is the volume of a pizza of radius z and thickness a ? Answer: pi z z a” Kind of geek, but I laughed
6:20 AM · Dec 10, 2007·Twitter Web Client
Subhayan Mukerjee
a pizza is a cylindrical volume of radius z and height a !
3:29 AM · Jul 3, 2010·Silver Bird
From somewhere: “A pizza with radius ‘z’, and thickness ‘a’ has a volume of Pi*z*z*a.” http://plurk.com/p/9ry6ly
12:05 AM · Dec 25, 2010·Plurk
Temperate Depression Ebni
If we assume the pizza is cylindrical of radius z and height a, then the volume is
V = π*z^2 * a = Pi*z*z*a
11:21 AM · Oct 4, 2013·Twitter Web Client
Posted by u/BuzzBorn April 1, 2014
If you have a pizza with radius “z” and thickness “a”, its volume is Pi(z*z)a
Pizzas are cylinders
4:40 PM · Apr 10, 2014·Twitter for Windows Phone
Waleed Mohamed
If we assume that the cylindrical pizza with radius Z is the height of A
So, The Capacity of this pizza is = Pi * Z * Z * A
9:59 AM · Jun 5, 2014·Facebook
Ian Spangenberg
Happy Pi Day! Let z=radius; a=height. Then the volume of a cylindrical pizza is pizza.
7:31 AM · Mar 14, 2015·Twitter for Android
Replying to @TonesBalones_ and @asheramichelle
if you think about it, pizza is a very thin cylinder
5:10 PM · Sep 22, 2016·Twitter for Android
sasuke apologist
i came to make a tweet but then i realized pizza is a cylinder and i dont remember what i was going to tweet now.
3:31 PM · Jan 17, 2018·Twitter Web Client
Posted by u/TARN4T1ON October 28, 2018
A pizza is technically just a very short cylinder
And if you take that height to be ‘a’ and it’s radius to be ‘z’ its volume is equal to pizz*a
A pizza is technically a cylinder….
Quote Tweet
· Dec 11, 2018
What’s on your mind?
8:27 PM · Dec 11, 2018·Twitter for Android
Hoop Junkie
Guys pizza is a cylinder
Quote Tweet
· Jan 2, 2019
A pizza that has radius “z” and height “a” has volume Pi × z × z × a.
2:01 PM · Jan 2, 2019·Twitter Web App
Peter Crawford
Replying to @edsouthall and @solvemymaths
I used pizza to help my kids calculate the volume of a cylinder. Imagine a pizza is a squat cylinder of height ‘a’ and radius z. It’s volume is therefore pi.z.z.a
8:07 AM · Jan 25, 2019·Twitter for iPhone
Atech Academy
A pizza that has radius “z” and height “a” has volume Pi × z × z × a .A pizza is basically a very short cylinder, and that’s how you calculate its volume. https://myatechedu.wordpress.com/2019/03/02/
8:21 AM · Mar 2, 2019·WordPress.com
𝕮𝖍𝖗𝖎𝖘𝖙𝖎𝖓𝖆 𝕵𝖊𝖓𝖓𝖎𝖋𝖊𝖗 𝔅𝔖𝔠
Replying to @chillmage
Pizza is a cylinder with radius z and depth a;
Its volume is pi*z*z*a.
4:45 PM · May 28, 2019·Twitter for iPhone
Pizza is a rather flat cylinder.
The Volume of any cylinder BurritoDrum🛢 is: V (cylinder) = Pi*r^2*h
If z is the radius and a is the height (or depth) then
#math #ilovemath #iteachmath #mathisfun #backtoschool #mathhelp #parents #mathteacher #learningmath
9:13 AM · Sep 2, 2019·Twitter for iPhone
Pizza is a cylinder
8:09 PM · Sep 23, 2019·Twitter for Android
Timmy O’Danaos
Replying to @B4TTL3S and @DavidMuttering
What do you mean? If the pizza is a cylinder (which is part of the joke) and the base has a radius z and the cylinder has height a, then its volume is given by
V = πz^2a, or
V = Pizza
11:42 AM · Oct 8, 2019·Twitter for Android
Pedro Contipelli
Pizza is a cylinder.
6:54 PM · Jan 26, 2020·Twitter Web App
John Von John Johnny deep
Replying to @Iamhappyfarmer
The volume of a pizza with radius z and height a is pi z z a!
A pizza is approximately the shape of a cylinder, so to get its volume, we have to multiply its area by its height. Thus the volume V of a pizza with height a is given by
V=a⋅ A = pi ⋅ z ⋅ z ⋅ a! That’s it.
7:10 AM · Mar 14, 2020·Twitter Web App
Jeb Cooke
pizzas are cylinders. fight me #PiDay2020
5:18 PM · Mar 14, 2020·Twitter for iPhone
the top and bottom are parallel to each other is a cylinder. I mean we live in a 3D world, so a pizza is abstractically a cylinder. A really flat cylinder
12:42 PM · Apr 14, 2020·Twitter for Android
Pizza is a cylinder
1:58 AM · Apr 24, 2020·Twitter for iPhone
pizzas are cylinders
11:15 AM · Apr 24, 2020·Twitter for iPhone
Assuming #pizza is a perfect cylinder:
Volume of pizza
= area of circular surface x height of pizza
= πr^2h
= pi x z x z x a
#funfact #mathsisfun
10:55 AM · Jun 18, 2020·Hootsuite Inc.
Posted by u/RhoastedGhost November 24, 2020
If the height is a, and the radius is z, then the cylinder is pizza
Posted by u/Mrcheldobreck December 28, 2020
Pizza is a cylinder
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Chapter 11 | Profit And Loss | Class-5 DAV Primary Mathematics | NCERTBOOKSPDF.COM
Chapter 11 | Profit and Loss | Class-5 DAV Primary Mathematics
Are you looking for DAV Maths Solutions for class 5 then you are in right place, we have discussed the solution of the Primary Mathematics book which is followed in all DAV School. Solutions are
given below with proper Explanation please bookmark our website for further updates!! All the Best !!
Chapter 11 Worksheet 3 | Profit and Loss | Class-5 DAV Primary Mathematics
Unit 11 Worksheet 3 || Profit and Loss
1. Complete the table by filling the column of selling price. The first one is done for you.
2. Solve the following questions.
(a) A shopkeeper purchased a saree for ₹ 375 and sold it at a gain of ₹ 90. Find the selling price of the saree.
(b) A watchmaker bought an old watch for ₹ 120 and spent ₹ 20 on repairs. If he sold the watch at a gain of ₹ 25, find the selling price of the watch.
(c) Rahul purchased a story book for ₹ 225. After reading it, he sold it to his friend at a loss of ₹ 120. Find the amount paid by his friend for the book.
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John McWhirter
John McWhirter
Distinguished Research Professor
John McWhirter graduated with a First Class Honours Degree in Mathematics from the Queen’s University of Belfast in 1970. He gained a PhD from the same University in 1973 for research on atomic
collision theory. Immediately afterwards, he joined the Royal Signals and Radar Establishment (now part of QinetiQ ltd) in Malvern. In 1979 he started a programme of personal research on digital
signal processing (DSP) with particular emphasis on algorithms and architectures for adaptive filtering and beamforming. His research covered a broad range of topics from novel mathematical
techniques to parallel computing and VLSI design. He gained international recognition for his work on the design of systolic array processors and is particularly well known for inventing the
triangular QR array for adaptive beamforming which bears his name. Other notable achievements include the QR least squares lattice algorithm for adaptive filtering and the design of a low-latency,
bit-level systolic array for IIR filtering based on redundant number systems. He went on to develop and promote the concept of Algorithmic Engineering. This constitutes a simple but rigorous
diagrammatic methodology for the design of signal processing algorithms and architectures. It encapsulates much of his previous work in a form which makes it more accessible to other DSP engineers.
Professor McWhirter was a founder member of the IEE professional subgroup for signal processing (E5) and was awarded the JJ Thompson Premium in 1990 for a paper on adaptive beamforming. In 1994 he
received the JJ Thompson Medal from the IEE for his research on Systolic Arrays and Mathematics in Signal Processing. Within the Civil Service, he was promoted to Senior Principal Scientific Officer
(Individual Merit) in 1986 and to Deputy Chief Scientific Officer (Individual Merit) in 1995. For the last few years, Prof McWhirter`s research has been devoted to independent component analysis for
blind signal separation and polynomial matrix algorithms for broadband sensor array signal processing. This work has application to radar, sonar, seismology, medical diagnostics and wireless
communications. Some of this research was carried out in collaboraton with the Centre of Digital Signal Processing in Cardiff under the QinetiQ-University partnership scheme, with the help of two
EPSRC-funded ICASE students based in Malvern. Prof McWhirter is a Fellow of the Institute of Mathematics and its Applications (IMA). He helped to establish and still organises the long-running series
of IMA conferences on Mathematics in Signal Processing. He was elected as a member of the IMA Council in 1995 and served as President for 2002 and 2003. He has been an Honorary Visiting Professor in
Electrical Engineering at Queen’s University, Belfast since 1986 and at Cardiff University since 1998. He was elected as a Fellow of the Royal Academy of Engineering in 1996 and as a Fellow of the
Royal Society in 1999. He received Honorary Doctorates from the Queen`s University of Belfast in 2000 and the University of Edinburgh in 2002. Prof McWhirter left QinetiQ on 31 August 2007 to take up
his current post as Distinguished Research Professor in Engineering at Cardiff. Electrical & Electronic Engineering
Health, Technology and the Digital World
* Arne Magnus Lecture Award, Colorado State University (2004); European Association for Signal Processing Technical Achievement Award (2003) * President, Institute of Mathematics and its Applications
(2002-3) * Chair, Council of Mathematical Sciences (2003) * Member of the IMA Council (2001-), Royal Society University Research Fellowship Panel 1 (2004-6), Royal Society Sectional Committee 4
(2001-3), EPSRC selection panel for Communications (2006), Royal Academy of Engineering Membership Panel 3 (2006-), Scientific Steering Committee and National Advisory Board for Isaac Newton
Institute, Cambridge (2003-) * Reader for Queen`s Anniversary Prizes for Higher Education
Title People Sponsor Value Duration
Signal processing solutions for the networked battlespace McWhirter J, EPSRC via Loughborough 515519 01/04/2013 - 31/03
Hicks Y /2018
Novel communications signal processing techs. for transmission over MIMO frequency selective wireless channels using McWhirter JG Engineering and Physical Sciences 289424 01/06/2008 - 31/05
polynomial matrix decompositions Research Council /2011
Supervised Students
Title Student Status Degree
Signal Processing Techniques For Extracting Signals With Periodic Structure: Applications To Biomedical Signals. GHADERI Foad Graduate PhD
Algorithms and Techniques for Polynominal Matrix Decompositions FOSTER Joanne Alexandra Graduate PhD
DYNAMIC SPECTRUM MANAGEMENT IN 4G BROADBAND ACCESS NETWORK. ADEBAYO Patrick Kunle Current PhD
POLYNOMIAL MATRIX TECHNIQUES FOR MIMO COMMUNICATIONS WANG Zeliang Current PhD
Past projects
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196.85 Inches to Feet
196.85 in to ft conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in
the United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Feet are 196.85 Inches we have to multiply 196.85 by 1 and divide the product by 12. So for 196.85 we have: (196.85 × 1) ÷ 12 = 196.85 ÷ 12 = 16.404166666667 Feet
So finally 196.85 in = 16.404166666667 ft
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Difference Between Equal and Equivalent Sets: JEE Main 2024
What is Equal and Equivalent Sets: Introduction
To differentiate between equal and equivalent sets: Equal sets and equivalent sets are concepts used to describe relationships between sets. Equal sets refer to sets that have precisely the same
elements, meaning every element in one set is also present in the other set, and vice versa. Two sets are considered equal when they have identical members. On the other hand, equivalent sets pertain
to sets that may not have the same elements but have an equal number of elements. The cardinality or size of the sets is the same, even though the individual elements might differ. Equal and
equivalent sets are fundamental in set theory and provide a basis for studying set operations and comparisons. Let’s understand them further in detail.
FAQs on Difference Between Equal and Equivalent Sets for JEE Main 2024
1. How do we determine if two sets are equal?
To determine if two sets are equal, we compare their elements and ensure that every element in one set is also present in the other set, and vice versa. If two sets A and B have the same elements, we
write A = B. One approach is to list the elements of both sets and verify that they are identical. Another method is to use set notation and set-builder notation to express the elements of each set
and then compare them. It is essential to consider both directions, ensuring that no elements are missing or extra in either set, to establish the equality of sets.
2. Can equivalent sets have different subsets?
Yes, equivalent sets can have different subsets. The notion of equivalence between sets is solely based on their cardinality or number of elements. While equivalent sets have the same size, they may
contain different elements. As a result, the subsets of equivalent sets can vary because subsets are determined by the specific elements within a set. Even though the overall count of elements
remains equal, the specific elements in the subsets may differ.
3. Can equivalent sets have different sizes?
No, equivalent sets cannot have different sizes. Equivalent sets, by definition, have the same cardinality or number of elements. If two sets are equivalent, it means that they contain the same
number of elements, even if the elements themselves may differ. So, equivalent sets must have the same size. The concept of equivalence is based on comparing the cardinality or quantity of elements
in sets, ensuring that they correspond one-to-one.
4. Are equal sets always equivalent?
Yes, equal sets are always equivalent. When two sets are equal, it means that they have exactly the same elements. Since equivalence is based on the cardinality or number of elements in a set, if two
sets are equal, they automatically have the same number of elements. Therefore, equal sets are a special case of equivalent sets where not only do they have the same cardinality, but they also have
identical elements. In other words, equality implies equivalence, but equivalence does not necessarily imply equality.
5. Can equal sets have different elements in different order?
No, equal sets cannot have different elements in different order. When two sets are equal, it means that they have exactly the same elements, and the order of the elements does not matter. The
equality of sets is not affected by the arrangement or order of the elements within them. Whether the elements are listed in a different order or not, as long as the elements themselves are the same,
the sets are still considered equal.
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Spectral properties of quantum graphs with symmetry
Mathematical Physics Seminar
31st May 2019, 2:00 pm – 3:00 pm
Howard House, 4th Floor Seminar Room
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs maintain
important features of the prototypical quantum star graph model. When the edge lengths respect the cyclic symmetry of the graph the spectrum decomposes into subspectra whose corresponding
eigenfunctions transform according to irreducible representations of the cyclic group. We show the subspectra exhibit a new form of intermediate spectral statistics applying techniques developed from
star graphs. These are statistics intermediate between Poisson and random matrix statistics. Quantum circulant graphs are one example of a more general class of quantum graphs with symmetry
constructed from Cayley graphs of finite groups.
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9 Speed Quizzes, Questions, Answers & Trivia - ProProfs
Speed Quizzes, Questions & Answers
In Old English, “sped” meant “success” or “thriving”, having grown out of the same ancient root “spe-” as the Latin “sperare” (to hope) and its relatives “prosperity” and “despair”.
The word “speed” was only applied to motion in the early 14th century. Speed is a relative measurement and, according to Einstein’s theory of Special Relativity, the constant against which the speed
of all objects in the universe is measured is the speed of light (186,282 miles per second).
It’s high time for this concept of physics to prove its worth, so take these quizzes and find the answers to questions like “What is instantaneous speed?”, “What is the linear speed of something
moving along a circular path?”, and “Who was the first to measure speed by considering the distance covered and the time it takes”. Now, be speedy about it!
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A cyclist travels uphill at a speed of 8 km/h and downhill at a speed of 12 km/h. If the uphill journey takes 3 hours and the downhill journey takes 2 hours, what is the cyclist's average speed
for the entire trip?
विषय:- राष्ट्रीय आणीबाणी आणि राष्ट्रपती राजवट एकूण प्रश्न:-10 वेळ:- 4 मिनिटे उत्तीर्ण...
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प्र.1 पंचगंगा नदी नरसोबाच्या वाडीजवळ कृष्णेला मिळते, पंचगंगा ही नदी खालीलपैकी कोणत्या पाच नद्या मिळून बनलेली आहे ? 1. कुंभी, कासारी, वेदगंगा, भोगावती, सरस्वती 2. कुंभी, हिरण्यकेशी, तुळशी, भोगावती, सरस्वती 3. कुंभी, कासारी, हिरण्यकेशी, भोगावती, सरस्वती 4. कुंभी, कासारी, तुळशी, भोगावती,
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Stable Isotope Mixing Models with
Installation of the cosimmr package
First, start Rstudio and find the window with the command prompt (the symbol >). Type
It may ask you to pick your nearest CRAN mirror (the nearest site which hosts R packages). You will then see some activity on the screen as the cosimmr package and the other packages it uses are
downloaded. The final line should then read:
package 'cosimmr' successfully unpacked and MD5 sums checked
You then need to load the package. Type
This will load the cosimmr package and all the associated packages. You’ll need to type the library(cosimmr) command every time you start R.
There is some sample data sets (from Inger et al 2006, Nifong et al 2015, Galloway et al 2015) available within cosimmr. Use the following command to access one.
This data can then be loaded into cosimmr using the function cosimmr_load
cosimmr_1 <- with(
formula = mixtures ~ 1,
source_names = source_names,
source_means = source_means,
source_sds = source_sds,
correction_means = correction_means,
correction_sds = correction_sds,
concentration_means = concentration_means
## Cannot scale when using mixtures ~1
This is a simple example that doesn’t include any covariates. Note the formula is given in the form tracer_data ~ covariates.
An isospace plot can be generated using this data
The data can then be run through cosimmr_ffvb, the main function of the cosimmr package.
Summary statistics for the run can then be viewed, several options are available, such as “statistics” and “quantiles”
## Summary for Observation 1
## mean sd
## P(Zostera) 0.515 0.167
## P(Grass) 0.072 0.020
## P(U.lactuca) 0.229 0.127
## P(Enteromorpha) 0.184 0.127
## sd_d13C_Pl 1.023 0.766
## sd_d15N_Pl 0.757 0.587
The output of this can be plotted, using the plot function. There are several different options available in the plot function.
For another example, with a continuous covariate, we can look at Alligator data from Nifong et al, 2015. For this example we just use Length as the covariate - but the dataset contains other
covariates that can be looked at too. First we load in the data into R - this data is included in cosimmr:
Then we use cosimmr_load to create a “cosimmr_in” object
Length = alligator_data$length
cosimmr_ali <-cosimmr_load(
formula = as.matrix(alligator_data$mixtures) ~ Length,
source_names = alligator_data$source_names,
source_means = as.matrix(alligator_data$source_means),
source_sds = as.matrix(alligator_data$source_sds),
correction_means = as.matrix(alligator_data$TEF_means),
correction_sds = as.matrix(alligator_data$TEF_sds))
We then plot our data to make sure our iso-space plot looks good
Then we can run the mixing model:
We can then look at a summary of the data. This defaults to observation 1.
We can create plots of our data. This code creates a proportion plot and a histogram plot of beta value for individuals 1 and 2
plot(cosimmr_ali_out, type = c("prop_histogram", "beta_histogram"), obs = c(1,2), cov_name = "Length")
We can use the predict function to predict proportions for individuals of lengths 100, 210, and 203 by creating a data frame and then using the predict function
“alli_pred” can be treated like a normal cosimmr_out object - we can get summary values for each individual or we can create plots
We can create a covariates_plot to show the change in consumption of Freshwater as an individual increases in Length
Alternatively we can look at the change in both sources on one plot
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Constructing the virtual fundamental cycle
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact INI IT.
SYGW05 - Symplectic geometry - celebrating the work of Simon Donaldson
Consider a space $X$, such as a compact space of $J$-holomorphic stable maps with closed domain, that is the zero set of a Fredholm operator. This note explains how to define the virtual fundamental
class of $X$ starting from a finite dimensional reduction in the form of a Kuranishi atlas, by representing $X$ as the zero set of a section of a (topological) orbibundle that is constructed from the
atlas. Throughout we assume that the atlas satisfies Pardon's topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem.
This talk is part of the Isaac Newton Institute Seminar Series series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
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Relativistic Mass - Nuclear Power
Relativistic Mass
While the mass is normally considered an unchanging property of an object at speeds approaching the speed of light, one must consider the increase in the relativistic mass. The relativistic
definition of momentum is sometimes interpreted as an increase in the mass of an object. In this interpretation, a particle can have a relativistic mass, m[rel]. The increase in effective mass with
speed is given by the expression:
In this “mass-increase” formula, m is referred to as the rest mass of the object. It follows from this formula that an object with a nonzero rest mass cannot travel at the speed of light. As the
object approaches the speed of light, the object’s momentum increase without bound. On the other hand, when the relative velocity is zero, the Lorentz factor equals 1, and the relativistic mass is
reduced to the rest mass. With this interpretation, the mass of an object appears to increase as its speed increases. It must be added, many physicists believe an object has only one mass (its rest
mass) and that it is only the momentum that increases with speed.
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bout Mathematics in Colorado
You are here
Frequently Asked Questions about Mathematics in Colorado
Frequently Asked Questions About Mathematics in Colorado
1. How many high school math credits are required to graduate?
There are no state-level requirements for courses or credits, so technically the number could be as low as zero. However, because math is a tested subject and required for most (if not all) higher
education admissions, school districts usually require at least two, commonly three, and sometimes four years of mathematics credits in order to graduate. For your specific requirements, check with
your school.
2. Where is the list of math programs and interventions that are approved to use in schools?
You may be asking this question because a list of approved programs exists for K-3 literacy, but that's a special provision of the READ Act. In 2023, the state legislature passed the Improving Math
Outcomes bill which included the creation of an advisory list of mathematics assessments and curriculum. Unlike the list of approved programs in the READ Act, the advisory list in this math bill is
not a compulsory list of resources. For more information about how resources were included and scored, see the Overview of Mathematics Review page. If you're looking to adopt curriculum materials for
your own school or district, there are guides and advice on the Tools for Curriculum Evaluation and Adoption page.
Similarly, we have a host of math intervention resources at our math intervention website. We are also working on free math intervention toolkits for educators to use. These resources are all
available to support educators, but none of them are mandatory.
3. How are Colorado's math standards like and not like Common Core?
Colorado's State Board of Education adopted the Common Core State Standards (CCSS) in 2010 and reaffirmed that decision in 2018. We consider ourselves to be a Common Core state, but that does not
mean that the content of Colorado's standards for mathematics is identical to what you will find at corestandards.org. Other statuary requirements in Colorado require that our standards for all
content areas include support for the Colorado Essential Skills, often called "21st Century Skills." Therefore, Colorado's mathematics standards document is a longer and more robust document than
what the CCSS provides, with the CCSS forming the parts of our standards listed as grade level expectations and evidence outcomes, while the sections under the heading "Academic Contexts and
Connections" are uniquely Colorado's.
4. What's going on with HS.S-ID.C.9? It's not where I expected to find it and the numbering seems out of order.
First of all, you should be applauded for your keen attention to detail! This is the one evidence outcome that Colorado's standards revision committee decided to move out of the cluster it is found
under in the Common Core. In the CCSS, “Distinguish between correlation and causation” is under cluster HS.S-ID.C., “Interpret linear models.” The committee thought this was a mistake because
correlation and causation are concepts that shouldn’t be limited to linear models. (And, word has it, the CCSS authors wouldn’t put it there again if they had the chance to move it.) We thought it
would make more sense under S-ID.B, “Summarize, represent, and interpret data on two categorical and quantitative variables.” Correlation and causation might look different with categorical
variables, but the ideas are just as relevant for categorical variables as they are for quantitative variables.
The result of this decision forced the committee to deal some messiness in our numbering system. We couldn't change the CCSS coding, because it is what it is. We could have numbered 5-6-9 on S-ID.B
and 7-8 on S-ID.C, or we could have numbered 5-6-7 on S-ID.B and 7-8 on S-ID.C. Of two imperfect choices—a 9 out of order or two statements numbered as 7—we chose the latter. Part of the reason was
technical: numbering continues and restarts all throughout the document, so the idea that a GLE starts with a value other than 1 is common. It just creates a second 7 under a different GLE that you’d
think would be an 8. Numbering 5-6-9 would have created an instance of non-sequential numbering within a GLE, which never happens anywhere else in the document.
5. What are OGL and SHK? We see them on our CMAS score reports for mathematics, but what do they mean?
Both of these things are shorthand for “it’s a bunch of different standards—too many for us to list.” OGL is “on grade level,” and it refers to a combination of standards within the grade being
assessed. SHK is “securely held knowledge,” and it refers to a combination of standards prior to the grade being assessed.
Here’s an example of SHK. In third grade, there’s a test specification called an evidence statement (not to be confused with evidence outcomes from the standards, although they are similar because
the evidence statements were built from the evidence outcomes) that says we should assess this:
3.D.2 Solve multi-step contextual problems with degree of difficulty appropriate to Grade 3, requiring application of knowledge and skills articulated in 2.OA.A, 2.OA.B, 2.NBT, and/or 2.MD.B. i)
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 3. ii) Multi-step problems must have at least 3 steps. (MP.4)
You’ll see that this is for a Grade 3 item and it’s designed to have Grade 3 difficulty, but it’s relying on knowledge and skills from Grade 2 standards. It’s the way those Grade 2 standards are
combined in a modeling context that gives the problem its difficulty at the Grade 3 level. Instead of saying all that, and listing all the evidence outcomes under 2.OA.A, 2.OA.B, 2.NBT, and/or
2.MD.B, we just say “SHK” to say that this item is based on a combination of Grade 2 knowledge that should be securely held by 3rd graders. “OGL” items would combine standards in a similar way but do
so from the same grade level that is being assessed.
Have a question you'd like answered? Please contact:
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LineItem | Klar Knowledge Base
Description: The (technical) unique identifier of the order item in this order.
Example Value(s): '123456789' or 'A-2344234230-B'
Description: Name of the product (model).
Example Value(s): 'Awesome Porridge - Cinamon' or 'Nike Air Max 90'
Description: The (technical) unique identifier of the product in your (shop) system.
Example Value(s): '123456789' or 'A-2344234230-B'
Description: Title of the variant of the product (model).
Example Value(s): 'Awesome Porridge - Cinamon - 250g' or 'Nike Air Max 90 - Size 10'
Description: The (technical) unique identifier of the product variant in your (shop) system.
Example Value(s): '123456790' or 'A-2344234230-B-10'
Description: The brand of the product.
Example Value(s): 'Awesome Foods'
Description: The product collection or category.
Example Value(s): 'Porridge' or 'Sneakers' or 'Shoes > Sneakers'
Description: The Cost of Goods Sold (COGS) for this product.
Example Value(s): 3.50 or 0.99
Description: The product's gross merchandise value or recomended retail price for one of the product items.
Example Value(s): 19.99 or 35
Description: The weight of the product in grams.
Example Value(s): 120 (120 grams) or 2500 (2,5 kg = 2500 g)
Description: The product's Stock Keeping Unit (SKU).
Example Value(s): '123456789' or 'A-123-797-C'
Description: The amount of this specific line item purchase in that order.
Example Value(s): 5 or 99
Description: An array of tags, specifying the product further.
Example Value(s): ['food', 'beverages', 'with_deposit'] or ['free_item']
Description: An array of Discount objects applied to the lineItem. Note that the values have to be supplied per item (as in 'not multiplied by quantity').
Description: An array of Tax objects applied to the lineItem. Note that the values have to be supplied per item (as in 'not multiplied by quantity').
Example Value(s): See Tax.
Description: The total amount for this lineItem (not the order) before taxes and discounts are applied. The formula is quantity * productGmv. Note: Although we could calculate that value by the
already provided individual values, we recommend sending it so that we can validate the correctness of the calculation. In short: those values are internal check-sums to validate the correctness of
the elements of the calculation formula.
Example Value(s): 37,50 (quantity 3 and productGmv of 12,50) or
Description: The total amount for this lineItem (not the order) after the deduction of taxes and discounts. The formula is (quantity * productGmv) - (sum(taxes.taxAmount) * quantity) - (sum
(discounts.discountAmount) * quantity). Note: Although we could calculate that value by the already provided individual values, we recommend sending it so that we can validate the correctness of the
calculation. In short: those values are internal check-sums to validate the correctness of the elements of the calculation formula.
Example Value(s): 30,75 (quantity 3 and productGmv of 12,50) - 3.75 (10% voucher) - 7.13 (19% VAT)
Description: The amount of the total (additional) logistics costs that applies to this / those lineItems. E.g. if the order contained an item that needed to be shipped via freight forwarding because
of its bulkiness or weight.
Example Value(s): 50.00 (additional fee for freight forwarder to deliver a washing machine)
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American Mathematical Society
If $H$ is an uncountable collection of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$, then there exists a sequence from $H$ converging homeomorphically to an element of $H$. In the
present paper the authors show that if $\{ {M_i}\}$ is a sequence of continua in ${E^n}$ which converges homeomorphically to ${M_0}$ and such that for each $i,{M_i}$ and ${M_0}$ are disjoint and
equivalently imbedded, then there exists an uncountable collection $H$ of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$. For $n = 2,\;3$, and $n \geqq 5$ it is shown that one cannot
guarantee that the elements of $H$ have the same imbedding as ${M_0}$. References
β , ${E^3}$ does not contain uncountably many mutually exclusive wild surfaces, Bull. Amer. Math. Soc. 63 (1957), 404. Abstract #801t.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54.78
• Retrieve articles in all journals with MSC: 54.78
Bibliographic Information
• © Copyright 1970 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 25 (1970), 566-570
• MSC: Primary 54.78
• DOI: https://doi.org/10.1090/S0002-9939-1970-0259875-2
• MathSciNet review: 0259875
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APS March Meeting 2010
Bulletin of the American Physical Society
APS March Meeting 2010
Volume 55, Number 2
Monday–Friday, March 15–19, 2010; Portland, Oregon
Session L42: High Reynolds Number Flows Hide Abstracts
Sponsoring Units: DFD
Chair: Marcel Ilie, University of Central Florida
Room: D138
Tuesday, L42.00001: On the stable hovering of an asymmetric body in oscillatory airflows
March Bin Liu, Annie Weathers, Stephen Childress, Jun Zhang
16, 2010
2:30PM - A free rigid body, built with up-down asymmetry can hover in a vertical oscillatory airflow if the airflow amplitude and frequency exceed certain thresholds. The key to free hovering lies in
2:42PM the difference in drag coefficients as the airflow passes the object in two opposite directions. The hovering motion is surprisingly stable and robust, lasting for thousands of oscillation
periods. We describe a series of flow visualizations of vortex shedding by the hovering object, which show how correcting moments restore its orientation, leading to stable hovering. This
study may shed light on the stability of the hovering flight of insects. [Preview Abstract]
Tuesday, L42.00002: Modeling flexible flapping wings oscillating at resonance
March Alexander Alexeev, Hassan Masoud
16, 2010
2:42PM - Using a hybrid approach for fluid-structure interactions that integrates the lattice Boltzmann and lattice spring models, we study the three-dimensional aerodynamics of flexible flapping
2:54PM wings at hovering. The wings are a pair of flat elastic plates tilted from the horizontal and driven to oscillate according to the sinusoidal law. Our simulations reveal that resonance
oscillations of flexible wings dramatically increase aerodynamic lift at low Reynolds number. Comparing to otherwise identical rigid wings, flexible wings at resonance generate up to two
orders of magnitude greater lift. Within the resonance band, we identify two operation regimes leading to the maximum lift and the maximum efficiency, respectively. The maximum lift occurs
when the wing tip and root move with a phase lag of 90 degrees, whereas the maximum efficiency occurs at the frequency where the wing tip and root oscillate in counterphase. Our results
suggest that the resonance regimes would be optimal for the design of microscale flying machines using flexible flapping wings driven by simple kinematic strokes. [Preview Abstract]
Tuesday, L42.00003: Vortices within vortices: Hierarchical vortex structures in experimental, two-dimensional flow
March Douglas H. Kelley, Nicholas T. Ouellette
16, 2010
2:54PM - The topology of a fluid flow is concisely described by its critical points (locations of zero flow) and the manifolds (streamlines) that connect them. Streamlines that carry fluid away from
3:06PM a critical point and then return it to the same critical point from another direction are known as homoclinic manifolds. Rare in three-dimensional flow, homoclinic manifolds are common in
two-dimensional flow and form unambiguous topological boundaries useful for defining vortex edges. Approximating two-dimensional flow with an electromagnetically driven, stably stratified
solution in a 90~cm x 90~cm pan, we use particle tracking to measure the velocity field and locate its critical points and their manifolds. Strikingly, homoclinic manifolds are often
nested~--- the flow contains vortices within vortices. Its regions can thus be classified by an embedding number, an integer defined as the depth of vortex nesting. We will discuss the
dynamics of this hierarchical vortex embedding number, particularly as a function of flow speed (Reynolds number). [Preview Abstract]
Tuesday, L42.00004: Evolution of Triangles in Quasi-Two-Dimensional Flow
March Nicholas Ouellette, Sophia Merrifield, Douglas Kelley
16, 2010
3:06PM - The anomalous transport of scalar fields in complex flow has recently been explained by considering the nontrivial shape dynamics of clusters of fluid elements. Here, we study the dynamics
3:18PM of three-particle clusters--Lagrangian triangles--that minimally parameterize planes as they are advected in a quasi-2D electromagnetically driven experimental flow. We report results for
the shape distributions as a function of the initial triangle size, and discuss the impact of the flow structure on the subsequent triangle evolution. This work is supported by the National
Science Foundation. [Preview Abstract]
Tuesday, L42.00005: Creating Turbulence with vortex rings
March Kelken Chang, Gregory P. Bewley, Eberhard Bodenschatz
16, 2010
3:18PM - We report measurements of the small-scale statistics of turbulence created by interacting vortices at a Taylor microscale Reynolds number of 500. We study the flow using Lagrangian particle
3:30PM tracking technique, in which the three-dimensional motion of passive oil particles in air is followed optically using multiple high speed cameras. We compare the results with measurements
obtained in a nearly homogeneous and isotropic turbulent flow at comparable Reynolds number. [Preview Abstract]
Tuesday, L42.00006: Vortex ring refraction at large Froude numbers
March Kerry Kuehn, Matthew Moeller, Michael Schulz, Daniel Sanfelippo
16, 2010
3:30PM - We have experimentally studied the impact of a planar axisymmetric vortex ring, incident at an oblique angle, upon a sharp gravity-induced interface separating two fluids of differing
3:42PM densities. After impact, the vortex ring was found to exhibit a variety of subsequent trajectories, which we have organized according to both the incidence angle, and the ratio of the Atwood
and Froude numbers, $A/F$. For relatively small angles of incidence, the vortices tended to penetrate the interface. In such cases, the more slowly moving vortices, having values of $A/F \ga
.004$, tended to subsequently curve back up toward the interface. Quickly moving vortices, on the other hand, tended to refract downward, similar to a light ray entering a medium having a
higher refractive index. A simplistic application of Snell's law of refraction cannot, however, account for the observed trajectories. For grazing angles of incidence, fast moving vortices
tended to penetrate the interface, whereas slower vortices tended to reflect from the interface. In some cases, the reflected vortices executed damped oscillations before finally
disintegrating. [Preview Abstract]
Tuesday, L42.00007: DNS of the Velocity and Temperature Fields in a Model of a Small Room
March John McLaughlin, Xinli Jia, Goodarz Ahmadi, Jos Derksen
16, 2010
3:42PM - This talk presents the results of a numerical study of the velocity and temperature fields in a model of a small room containing a seated mannequin. Results are also presented for the
3:54PM trajectories and ultimate fate of small particles that are introduced through the air inlet as well as particles that are entrained by the mannequin's thermal plume. The study was motivated
by an experimental study performed at Syracuse University. In the experimental study, air entered the room through a floor vent and exited through a ceiling vent on the other side of the
room. A mannequin was seated facing the floor vent. The mannequin could be electrically heated so that its surface temperature was 31C. The objective of the simulations was to obtain a more
detailed understanding of the flow in the room. Of specific interest were the effects of the mannequin on the ultimate fates of small particles. The importance of the thermal plume around
the mannequin was of particular interest since the thermal plume plays a role in transporting particles from near the floor to the breathing zone. The simulations were performed with a
single phase version of a lattice Boltzmann method (LBM) that was originally developed for two-phase flows by Inamuro et al. [Preview Abstract]
Tuesday, L42.00008: Large-eddy simulations of particle-laden turbulent swirling flows
March Marcel Ilie
16, 2010
3:54PM - In many combustion devices, a swirling flow is used to stabilize the flame through a recirculation zone. Swirling flows, however, are prone to instabilities which can trigger combustion
4:06PM oscillations and deteriorate the performance of the combustor. The presence of fine particles makes swirling flows of particular interest from a combustor efficiency point of view. Depending
on the strength of swirl, a number of recirculation zones and central vortex breakdown regions are identified in many swirl-stabilized flames. In general these characteristics make swirling
flows and flames to exhibit highly three-dimensional, large-scale turbulent structures with complex turbulent shear flow regions. The present research concerns the influence of swirl
characteristics on the particle dispersion and total deposition. A Lagrangian particle tracking algorithm using large-eddy simulation is proposed. The influence of particle characteristics
such size, density and shape on the particle dispersion and total deposition is subject of investigation as well. The present research shows that the total particle deposition increases with
size and density. It was also observed that particles of ellipsoidal shape are more prone to deposition. [Preview Abstract]
Tuesday, L42.00009: Lattice Boltzmann and Pseudo-Spectral Methods for Decaying Turbulence
March Li-Shi Luo, Yan Peng, Wei Liao, Lian-Ping Wang
16, 2010
4:06PM - We conduct a comparison of the lattice Boltzmann (LB) and the pseudo-spectral (PS) methods for direct numerical simulations (DNS) of the decaying turbulence in a three dimensional periodic
4:18PM cube. We use a mesh size of $128^3$ and the Taylor micro-scale Reynolds number $24.35 \leq \mbox{Re}_\lambda \leq 72.37$. All simulations are carried out to $t \approx 30 \tau_0$, where $\
tau_0$ is the turbulence turnover time. We compare instantaneous velocity $\mathbf{u}$ and vorticity $\mathbf{\omega}$ fields, the total kinetic energy $K(t)$, the dissipation rate $\
varepsilon(t)$, the energy spectrum $E(k,\, t)$, the rms pressure fluctuation $\delta p(t)$, the pressure spectrum $P(k,\, t)$, and the skewness $S_u(t)$ and the flatness $F_u(t)$ of
velocity derivatives. Our results show that the LB method compares well with the PS method in terms of accuracy: the flow fields and all the statistical quantities --- except for $\delta p
(t)$ and $P(k,\, t)$ --- obtained from the two methods agree well with each other when the initial flow field is adequately resolved by both methods. Our results indicate that the resolution
requirement for the LB method is $\eta_0 / \delta x \geq 1.0$, where $\eta_0$ and $\delta x$ are the initial Kolmogorov length and the grid spacing, respectively. [Preview Abstract]
March L42.00010: ABSTRACT WITHDRAWN
16, 2010
4:18PM - [Preview Abstract]
Tuesday, L42.00011: Fluid-Structure Interaction based on Lattice Boltzmann and p-FEM
March Benjamin Ahrenholz, Sebastian Geller, Manfred Krafczyk
16, 2010
4:30PM - Over the last decade the Lattice Boltzmann Method (LBM) has matured as an efficient method for solving the Navier-Stokes equations. The p-version of the Finite Element Method (p-FEM) has
4:42PM proved to be highly efficient for a variety of problems in the field of structural mechanics. The focus of this contribution is to investigate the validity and efficiency of the coupling of
two completely different numerical methods to simulate transient bidirectional Fluid-Structure Interaction (FSI) problems with very large structural deflections. In this contribution the
treatment of moving boundaries in the fluid solver is presented, the computation of tractions and displacements on the boundary as well as the explicit coupling algorithm itself. In
addition, efficiency aspects of the two approaches for two- and three-dimensional laminar flow examples at intermediate Reynolds numbers are discussed. Finally we give an outlook on modeling
turbulent FSI problems. [Preview Abstract]
Tuesday, L42.00012: Lattice Boltzmann Methods for thermal flows: applications to compressible Rayleigh-Taylor systems
March Luca Biferale, Mauro Sbragaglia, Andrea Scagliarini, Kazuyasu Sugiyama, Federico Toschi
16, 2010
4:42PM - We compute the continuum thermo-hydrodynamical limit of a new formulation of Lattice Kinetic equations for thermal compressible flows, recently proposed in [{\it Sbragaglia et al. ``Lattice
4:54PM Boltzmann method with self-consistent thermo-hydrodynamic equilibria'', J. Fluid Mech. {\bf 628} 299 (2009)}]. We show that the hydrodynamical manifold is given by the correct compressible
Fourier-Navier-Stokes equations for a perfect fluid.We also apply the method to study Rayleigh-Taylor instability for compressible stratified flows and we determine the growth of the mixing
layer at changing Atwood numbers up to $At \sim 0.4$. Both results show that this new Lattice Boltzmann Methods can be used to study highly stratified/compressible systems with strong
temperature gradients, opening the way to applications to Non-Oberbeck-Boussinesq Convection and compressible Rayleigh-Taylor turbulence. [Preview Abstract]
Tuesday, L42.00013: The shape of fair weather clouds
March Yong Wang, Giovanni Zocchi
16, 2010
4:54PM - It is well known that cumulus clouds are formed under the influence of thermals - convection currents which channel moist air upwards. Here we introduce a simple physical model which
5:06PM accounts for the shape of cumulus clouds exclusively in terms of thermal plumes or thermals. The plumes are explicitly represented by a simple potential flow generated by singularities
(sources and sinks) and with their motion create a flow field supporting the cloud. We discuss the parametrization of this model, which attempts a description of the cloud starting from the
coherent structures in the flow. We use the model to explore transitions which occur in the dynamical state of the cloud. [Preview Abstract]
Tuesday, L42.00014: On the dynamics of cartoon dunes
March Christopher Groh, Ingo Rehberg, Christof A. Kruelle
16, 2010
5:06PM - The spatio-temporal evolution of a downsized model for a barchan dune is investigated experimentally in a narrow water flow channel. We observe a rapid transition from the initial
5:18PM configuration to a steady-state dune with constant mass, shape, velocity, and packing fraction. The development towards the dune attractor is shown on the basis of four different starting
configurations. The shape of the attractor exhibits all characteristic features of barchan dunes found in nature, namely a gently inclined windward (upstream) side, crest, brink, and steep
lee (downstream) side. The migration velocity is reciprocal to the length of the dune and reciprocal to the square root of the value of its mass. The velocity scaling and the shape of the
barchan dune is independent of the particle diameter. For small dunes we find significant deviations from a fixed height-length aspect ratio. Moreover, a particle tracking method reveals
that the migration speed of the model dune is one order of magnitude slower than that of the individual particles. In particular, the erosion rate consists of comparable contributions from
low energy (creeping) and high energy (saltating) particles. Finally, it is shown that the velocity field of the saltating particles is comparable to the velocity field of the driving fluid.
[Preview Abstract]
Tuesday, L42.00015: Supernova Shear and Magnetic Field Amplification
March Cyril Allen
16, 2010
5:18PM - A core collapse supernova marks the death of a star over 8 times the size of the sun. Sometimes in the aftermath of these explosions a spinning, magnetized, neutron star can be left behind,
5:30PM also known as a pulsar. It has recently been discovered that pulsar spins can arise through a spiral spherical accretion shock instability (SASI) of a supernova. This instability produces a
strong shear flow inside the supernova shock wave, which might lead to amplification of the star's magnetic field.~ To study this possibility, hydrodynamic simulations have been modified to
include a tracer of the magnetic field by adding the magnetic induction equation to the code. Diagnostics were added to the code to measure the overall field strength and shear flow
generated by the SASI.~ I found the magnetic field could be amplified by a factor of 100 in only 20 milliseconds. This raises the possibility that shear-induced field amplification might be
able to contribute to the energy of the supernova explosion and explain the high magnetic fields of the pulsar left behind. [Preview Abstract]
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What is non-parametric statistics in MyStatLab? | Hire Someone To Do Exam
What is non-parametric statistics in MyStatLab? At the time I wrote my code in SQLite I had no understanding of the standard statistics and didn’t have many advanced tutorials on programming. I could
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query) which takes the parameters in this way – it’s my Main() function. Whenever I use the GetType() method of my Main() function, this code runs after I get that this like it a non-parameter table.
However, when I change it to use my NewType() it jumps back to my Main() function for the ReadEventHandler(string filename, DateTime fd) in my Main() function. In this case, I have to call it code
for: I know it looks a bit ridiculous because in type we always have the name and it was my Main to get rid of, but this does not make sense. It happens also also when I change the readEvents()
method in MyStandardResult. How can I use Linq to query my selected points? And also if it’s not so hard to use instead of My standard data-query form? A: You can check it’s compatibility with
LinqDataToString (type for properties is linq property). The only way to useWhat is non-parametric statistics in MyStatLab? Of the many statistical analyses available in the stateofmlab database,
results for non-parametric tests for eigen values are few.
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First I would like to elaborate on more intuitive statistical methods by which non-parametric power estimators can be obtained, for which this depends on my question, but view it I draw them from my
blog. The first paper on non-parametric statistical statistics is the original version of Smeets and Smeets 2004 published by the author of that paper. The paper stated that non-parametric tests
exist, “without any practical use or statistical adjustment, and that nonparametric tests can work with null hypotheses generally”; for the main piece is the approach proposed by Nagel and
Hörredatter “Nonparametric statistical tests and the related concepts” in their original paper “The Null Hypothesis Making the Null Hypothesis” of the Swedish Journal of Nonparametric Statistical
(1951), in a version that addresses the reader’s interest. In the second paper (Smeets, Smeets 2006) the authors state that p.34-35 is the best criterion that p.34 is reasonable for most situations
in the field. Their criterion must be p.57 and the test being null if p.57/0.71 p.71 exists; hence they state that for all p.57/0.71 p.71, its hypothesis cannot exist to have p.56/0.71 p.56/0 for a
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57/0.71 p.71 represents not some random parameterization but something inherent to non-parametric statistical tests (from myself and others, they have this one point). The paper’s main body,
therefore, is the following: There is a method for making p.56/0.71 that worksWhat is non-parametric statistics in MyStatLab? Non-parametric statistics (non-parametric statistics or, more exactly,
non-population-based methods) are methods in which both, the statistics to be measured and the data generation/identification, are related, such that the effects in each measure, as well as the
associated statistical measures, are correlated. This is achieved using statistics that are based on the data generation, that has been collected. All those are correlated. As an example, take a
simple example where a graphical example example is provided, for instance, with a simple data set. This example illustrates that being related can be used to perform multiple measures. In this
example, a few examples should do a lot to describe the correlation among variables involved. Additionally, these example examples should be used with caution because the methods that are being used
are not commonly used. ### Data analysis {#sec010} Real person data (also or may be: _real_ population data) are important for two reasons. It can be described by a multidimensional distribution in
which the two pieces of information are related by _real_ real values, such that when measuring the parameters of an entity, such as an individual or population, they should be more than just the
quantity of particles; in other words, it should also be related to (and often at odds with) the actual physical force that is being applied. These forms of data consist of both the quantity of
particles and the quantity of physical force. In fact, measuring quantities is an important subject of research. With the aim to better describe real-environmental measurements, researchers have
developed various studies and applications that can enable the experimenter to characterize the nature of the physiological phenomena and understand how the body is creating, reacting, and storing
stimuli \[[@pone.0127058.ref029]–[@pone.0127058.
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ref033]\]. These studies describe the methodologies
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A point object is moving on the cartesian coordinate plane according to: vecr(t)=b^2thatu_x+(ct^3−q_0)hatu_y. Determine: a) The equation of the trajectory of object on the cartesian plane b) The magnitude and direction of the velocity? | Socratic
A point object is moving on the cartesian coordinate plane according to: #vecr(t)=b^2thatu_x+(ct^3−q_0)hatu_y#. Determine: a) The equation of the trajectory of object on the cartesian plane b) The
magnitude and direction of the velocity?
1 Answer
Let $\left(x , y\right)$ represents the cartesian coordinate of the point having position vector at t th instant
$\vec{r} \left(t\right) = {b}^{2} t {\hat{u}}_{x} + \left(c {t}^{3} - {q}_{0}\right) {\hat{u}}_{y} \ldots . \left(1\right)$
$x = {b}^{2} t \mathmr{and} y = c {t}^{3} - {q}_{0}$
$\therefore y = c {\left(\frac{x}{b} ^ 2\right)}^{3} - {q}_{0}$
On simplification we get
${b}^{6} y = c {x}^{3} - {q}_{0} {b}^{6}$
$\implies c {x}^{3} - {b}^{6} y - {q}_{0} {b}^{6} = 0 \to$ is the eqution of trajectory.
Now differentiating (1) w.r.to.t we get velocity
$\vec{v} \left(t\right) = \frac{d \left(r \left(t\right)\right)}{\mathrm{dt}} = {b}^{2} {\hat{u}}_{x} + 3 c {t}^{2} {\hat{u}}_{y}$
Its magnitude
$\left\mid \vec{v} \left(t\right) \right\mid = \sqrt{{b}^{4} + 9 {c}^{2} {t}^{4}}$
The direction of the velocity.
If it is directed at an angle $\theta \left(t\right)$ with the x-axis at t th instant then
$\theta \left(t\right) = {\tan}^{-} 1 \left(\frac{3 c {t}^{2}}{b} ^ 2\right)$
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Insert scalar into an array (scalar is cast to array’s dtype, if possible)
There must be at least 1 argument, and define the last argument as item. Then, a.itemset(*args) is equivalent to but faster than a[args] = item. The item should be a scalar value and args must
select a single item in the array a.
*args : Arguments
If one argument: a scalar, only used in case a is of size 1. If two arguments: the last argument is the value to be set and must be a scalar, the first argument specifies a single
array element location. It is either an int or a tuple.
Compared to indexing syntax, itemset provides some speed increase for placing a scalar into a particular location in an ndarray, if you must do this. However, generally this is discouraged: among
other problems, it complicates the appearance of the code. Also, when using itemset (and item) inside a loop, be sure to assign the methods to a local variable to avoid the attribute look-up at
each loop iteration.
>>> x = np.random.randint(9, size=(3, 3))
>>> x
array([[3, 1, 7],
[2, 8, 3],
[8, 5, 3]])
>>> x.itemset(4, 0)
>>> x.itemset((2, 2), 9)
>>> x
array([[3, 1, 7],
[2, 0, 3],
[8, 5, 9]])
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1 Introduction2 Loop closure detection system architecture based on variational autoencoder2.1 Improvement of the objective function of the VAE network2.2 Attention mechanism module3 Loop closure detection based on a variational autoencoder3.1 Image feature description3.2 Loop closure detection4 Experiments4.1 Datasets and evaluation methodology4.2 Loop closure detection in a practical environment4.3 Ablation experimental analysis5 ConclusionsData availability statementAuthor contributionsFundingConflict of interestPublisher's noteReferences
Loop closure detection is the process of identifying the places that a robot has visited before, which can help the robot relocate when it loses its trajectory due to motion blur, forming a
topologically consistent trajectory map (Gálvez-López and Tardis, 2012; Arshad and Kim, 2021). The key to solving the loop closure detection problem is to match the images captured by the robot with
the images corresponding to previously visited locations on the map. Loop closure detection is essentially an image-matching problem, the core of which is the representation and matching of image
Traditional loop closure detection methods are generally based on appearance (Cummins and Newman, 2008), which has little connection with the front end and back end of the system. The loop detection
relationship is only determined given the similarity of the two images. Since proposed, the appearance-based loop closure detection methods have become mainstream in visual SLAM and have been applied
to practical systems (Mur-Artal et al., 2015), most of which utilize the bag-of-words (BOW) model (Filliat, 2007; Garcia-Fidalgo and Ortiz, 2018; Li et al., 2019). The BOW model clusters the visual
feature descriptors of the image to build a dictionary and then searches the words that match the features of each image to describe the image. A word can be regarded as a representative of several
similar features.
However, appearance-based methods usually depend on traditional handcrafted features, such as SIFT (Lowe, 2004), SURF (Bay et al., 2006), and BRIEF (Calonder et al., 2010). Each of these features has
its own characteristics, but they have limited ability to express the environment under the conditions of significant changes in viewing angle or illumination conditions. Moreover, they can only
describe the local appearance, which has a limited ability to describe the whole image. BOW-based closed-loop detection methods rely on appearance features and their presence in the dictionary,
ignoring geometric information and relative positions in space, thus generating false loops due to similar features appearing in different places (Qin et al., 2018; Arshad and Kim, 2021).
Recently, given the rapid development of deep learning in computer vision (Bengio et al., 2013), methods based on convolutional neural networks (CNN) (Farrukh et al., 2022; Favorskaya, 2023) and
attention mechanisms have attracted more attention in imitating human cognitive patterns. Using the learning features of the neural network to replace the traditional manual features is a new method
to solve the loop detection problem (Memon et al., 2020; Wang et al., 2020). Zhang et al. (2022) used global features to perform candidate frame selection via HNSW (Malkov and Yashunin, 2018), while
the local one was exploited for geometric verification via LMSC. Based on the above two components, the whole system was at the same time high-performance and efficient compared with state-of-the-art
approaches. Liu and Cao (2023) utilized the effective FGConv (Liu et al., 2020) as their proposed network backbone due to its high efficiency. The network adopts an encoder-decoder-based structure
with skip connections. Osman et al. (2023) trained PlaceNet to identify dynamic objects in scenes via learning a grayscale semantic map indicating the position of static and moving objects in the
image. PlaceNet is a multi-scale deep autoencoder network augmented with a semantic fusion layer for scene understanding, which generates semantic-aware deep features that are robust to dynamic
environments and scale invariance.
At the same time, the attention mechanism can weigh key information and ignore other unnecessary information to process information with higher accuracy and speed. Hou et al. (2015) used the
pre-trained CNN model to extract features to obtain a complete image representation, and through experiments on various datasets, it was shown that CNN features are more robust to changes in visual
angle, light intensity, and scale of the environment. Gao and Zhang (2017) used a modified stacked denoising autoencoder (SDA), a deep neural network trained in an unsupervised manner, to solve the
loop closure detection problem, but the extraction speed is slow. NetVLAD (Arandjelovic et al., 2016) is currently an advanced location recognition method, which is an improved version of VLAD. It
clusters local descriptors into global image descriptors through neural network learning, which has high accuracy and applicability. Schönberger et al. (2018) used a variational autoencoder (VAE) to
compress and encode 3D geometric and semantic information to generate a descriptor for subsequent position recognition. This method has good detection accuracy for large viewing angles and appearance
changes, but its computational cost is high.
For traditional methods, the problem of false detection is easy to occur when facing similar environments or relatively large changes in illumination, which leads to serious errors in map estimation.
In this research paper, we propose a loop closure detection method based on a variational autoencoder to solve the loop closure detection problem in visual SLAM. The method uses intermediate layer
depth features instead of the traditional manual features and compares the current image with the previous keyframes to detect the loop. The method incorporates an attentional mechanism in the neural
network to obtain more useful features and also improves the loss function of the network and eliminates erroneous loops through geometric consistency.
Figure 1 shows the structure of the proposed loop closure detection system based on a variational autoencoder. Dividing loop closure detection into two parts: front-end feature extraction and
back-end feature matching. The proposed method consists of two sections: (1) In the front-end feature extraction part, a network structure based on a variational autoencoder is designed and
constructed, and the attention mechanism is added. It will be called SENet-VAE. Besides, the loss function of the variational autoencoder is revised and improved. The aim is to learn feature
representations with fewer image features to obtain more accurate results. (2) In the back-end feature matching part, due to the low dimensionality of the descriptor, a K-nearest neighbor search is
used to detect loop closures and geometric checks are used to filter false detections.
Function diagram of loop detection based on a variational autoencoder.
The proposed network structure is shown in Figure 2. The front encoder part encodes the input image with 13 convolutional layers, four pooling layers, and three SENet attention modules. The middle
section is responsible for sampling and mapping the encoder input to a normal distribution. The later decoder part performs the semantic segmentation of the image and the decoded reconstruction of
the image with eight convolutional layers and four upsampling layers. The last decoder outputs high-dimensional features to the softmax classifier. The classifier classifies the pixels of the input
image and predicts the probability of the classification labels while decoding the image.
Feature extraction network structure of SENet-VAE.
The proposed network is based on VAE. The network input is an RGB image with a resolution of 192 × 256. The encoder maps the image to a normal distribution through the latent variables μ and σ, and
then the information of the potential variables is decoded by the decoder. It describes the observation of the potential space in a probabilistic way. In addition, this method adds an attentional
mechanism to the VAE to increase the weight of the effective features to obtain better results.
Inspired by Sikka et al. (2019), the loss function of VAE is improved based on the KL divergence of traditional VAE. A hyperparameter β is added to the second KL divergence of the loss function. As
the parameter β rises, the traditional VAE has the characteristics of disentanglement. The entangled data in the original data space are transformed into a better representation space, in which the
changes of different elements can be separated from each other.
Assume that the network input data D = {X, V, W} is a set composed of images x, conditional independent factors v, and conditional correlation factors w. Suppose that the conditional probability
distribution of x, denoted by p(x|v, w), is generated from simulated real data consisting of v and w, which is shown in Equation (1):
where Sim() is the simulation operation.
It is hoped that the generative model will learn a model p(x|z) that can generate pictures through a hidden layer z and make this generative process as close as possible to real-world models. The
mathematical expression is shown in Equation 2.
This model is controlled by the parameter θ. Therefore, an appropriate goal is to maximize the marginal likelihood of the observed data x in the expectation over the entire distribution of the latent
factor z. Which is shown in Equations (3) and (4).
pθ(x)=∑zpθ(z)pθ(x|z)=Epθ(z)[pθ(x|z)] maxθEpθ(z)[pθ(x|z)].
For p(z), as its definite form cannot be determined, it is often approximated by a joint distribution model q[ϕ](z|x). In order for q[ϕ](z|x) to be as simple as possible, it is approximated by a
Gaussian distribution p(z) ~ N(0, I), as follows in Equation (5).
Rewrite the above equation as the Lagrange equation under the Karush-Kuhn-Tucker (KKT) condition:
Since β, ε ≥ 0, according to the complementary relaxation degree KKT condition, Equation (6) can be rewritten to obtain the β − VAE formula as the ultimate objective function, as follows in Equation
As the value β becomes larger, q[ϕ](z|x) becomes simpler, transmitting less information and still being able to reconstruct the image well.
After sampling from the standard normal distribution ε, the latent variable z obtained by the encoder is sent to the decoder, which is used to predict the full-resolution semantic segmentation label
and to reconstruct the full-resolution RGB image. The output of the decoder is then used to construct the RGB reconstruction loss function L[r], as follows in Equation (8) and the maximum
cross-entropy loss function L[s] to account for class bias, as follows in Equation (9):
Lr=-∑i(xilog(pi)+(1-xi)log(1-pi)) Ls=1N∑iLi=1N∑i∑c=1Myiclog(pic)
Here x[i] and p[i] represent the label of the input image and the probability of the positive class output by the network behind the softmax function, respectively. M represents the number of
categories, y[ic] is the sign function (0 or 1), and p[ic] is the probability that the observation sample i belongs to category c, which is obtained by the softmax function.
In the encoder part, the weight of the two encoders is shared in the form of a triple network, and a sample is selected from the dataset called anchor. Samples of the same type as the anchor are
selected. Distortion or darkening operations are performed, and the movement of the camera is imitated to a certain extent. This type of image is called a positive image. In the data of the current
training batch, the sample that is different from the anchor is called a negative image. Anchor, positive image, and negative image consist of a triplet. The global image descriptor is taken from the
latent variable μ. With the descriptors of a baseline image d[a], a positive image d[p], and a negative image d[m], the triplet loss function is defined as follows in Equation (10):
where m is the marginal hyperparameter.
This loss function expressed by L[t] forces the network to learn to use m to distinguish the similarity between positive and negative images. The minimization of the damage function is obtained by
minimizing the cosine similarity between the reference image and the negative image and maximizing the similarity between the reference image and the positive image.
Finally, the overall objective function is defined as follows in Equation (11):
where λ[i] is the weight factor to balance the impact of each project.
The attention mechanism squeeze-and-excitation networks (SENet) (Hu et al., 2018) considers the relationship between feature channels to improve the performance of the network. The attention
mechanism adopts a brand-new feature recalibration strategy, which automatically acquires the importance of each feature channel through learning. Then, useful features are promoted and features that
are not very useful for the current task are suppressed based on feature weight.
The SENet module in this article changes the input from the previous pooling layer: F[tr]:X → U, X ∈ ℝ^H^′×W^′×C^′, U ∈ ℝ^H×W×C and transmits it to the next layer. Then, the output can be written as
follows in Equation (12):
Here F[tr] is the pooling operator, V = [v[1], v[2], …, v[C]] represents the filter, v[c] represents the parameters of the c-th filter, C represents the number of channels in the feature graph, H
represents the height of the feature graph, and W represents the width of the feature graph.
The goal is to ensure that the network is sensitive to its informative features so that they can be exploited subsequently and suppress useless features. Therefore, before the response enters the
next transformation, it is divided into three parts, namely, squeeze, excitation, and scale, to recalibrate the filter response.
First, the squeeze operation encodes the entire spatial feature into a global feature by using global average pooling. Specifically, each two-dimensional feature channel is turned into a real number,
which has a global receptive field to some extent, and the output dimension matches the number of input feature channels. It represents the global distribution of the response on the feature channel
and enables the layers to be close to the input to obtain the global receptive field.
The statistic z ∈ ℝ^C is generated by reducing the set U of the local descriptor to the spatial dimension H × W, where the c-th element of z is calculated by Equation (13):
The second part is the excitation operation, which fully captures the channel dependencies by utilizing the information gathered in the squeeze operation. This part consists of two fully connected
layers. The first layer is a dimension reduction layer with the parameter W1∈ℝCr×C, which is activated by the ReLU activation function. The second layer is the dimensionality-increasing layer with
the parameter W2∈ℝC×Cr, which is restored to the original dimension and uses the sigmoid activation function, as follows in Equation (14). Here, δ refers to the ReLU activation function.
Finally, the scale operation part multiplies the learned activation values of each channel (sigmoid activation, value 0 to 1) by the original features on U, which is shown in Equation (15):
The construction of the squeeze-and-excitation block in the network is shown in Figure 3.
The squeeze-and-excitation block of SENet-VAE.
In this section, we use the neural network described above to extract the image features and use it to perform back-end image feature matching to achieve loop closure detection. During the
image-matching process, key point mismatches are eliminated by geometric checking, which improves the accuracy of detection.
The global descriptor for the image is taken from the output of the convolutional layer where the latent variable μ is located in the sampling layer of the network. After the encoder, the latent
variable z is split channel-wise into 14 local descriptors of size 1/4 of the input image size. One of the slices is dedicated to reconstructing the full-resolution RGB image, while the other is sent
to the decoder, concatenated, and then used to predict a full-resolution semantic segmentation label. Since the local descriptor dimension is 192 dimensions, the global descriptor consisting of 14
local descriptors has a dimension of 10,752 dimensions. It can be interpreted as a set of 10,752 dimensional vectors of length l, with V^(I) denoting the corresponding output for a given input image,
which is shown in Equation (16):
For the extraction of image key points, the method proposed by Garg et al. (2018) is used. It extracts key points from the maximum activation area of the underlying Conv5 layer of the network
encoding. The largest activation area in a 48 × 64 window is selected as a key point on the feature map. After the key points are extracted, the key point descriptor is inspired by the BRIEF
(Calonder et al., 2010) descriptor. Taking the extracted key point as the center, certain point pairs are selected in a 3 × 3 size field for comparison. After all point pairs are compared, a
256-dimensional key point descriptor is obtained. During key point matching, these descriptors are directly compared using the Euclidean distance metric.
In order to detect loop closures, first build a database of historical image descriptors through global image descriptors. When the image to be queried is input, the global image descriptor is used
to perform a K-Nearest neighbor search in the established database, and images with relatively high similarity scores are selected to form a candidate image set. Then, K candidates are screened in
the candidate set through the key points described before, and the random sample consensus (RANSAC) algorithm is used to filter out false matches. The RANSACN algorithm finds an optimal homography
matrix H through at least four sets of feature-matching point pairs, and the size of the matrix is 3 × 3. The optimal homography matrix H is supposed to satisfy the maximum number of matching feature
points. Since the matrix is often normalized by making h[33] = 1, the homography matrix, which is expressed by Equation (17), has only eight unknown parameters:
where (x, y) is the corner point of the target image, (x′, y′) is the corner point of the scene image, and s is the scale parameter.
Then, the homography matrix is used to test other matching points under this model. Use this model to test all the data, and calculate the number of data points and projection errors that satisfy
this model through the cost function. If this model is the optimal model, the corresponding cost function should obtain the minimum value. The equation for calculating the cost function J is as
follows in Equation (18):
After filtering out invalid matches, the matched key points can be used to calculate the effective homography matrix as the final matching result. An example of final matches after performing RANSAC
can be seen in Figure 4.
An example of final matches after performing RANSAC.
In this section, the feasibility and performance of the proposed method will be tested on the Campus Loop dataset. The hyperparameters used in the experiments are shown in Table 1. The proposed
method is compared with the BOW model and other CNN-based methods. The performance of the proposed method is measured using the precision-recall curve. There are mainly two metrics used to interpret
the precision-recall curve. (1) Area under the curve (AUC), which is the area enclosed by the precision-recall curve and the coordinate axis. The closer the AUC is to 1.0, the higher the accuracy of
the detection method is. (2) The maximum recall rate at 100% accuracy is represented by Max-Recall, which is the value of the recall rate when the accuracy drops from 1.0 for the first time. Finally,
the KITTI odometry dataset is used to test the application and effectiveness of the proposed method in real scenarios.
List of hyperparameters.
Parameter Symbol Value
Learning rate η 10^−3
Input image size I 192 × 256
Batch size N[T] 12
Weight function λ[0] 10^−4
Weight function λ[1] 10^−4
Weight function λ[2] 1.0
Weight function λ[3] 1.0
Beta parameter β 250
Margin parameter m 0.5
The accuracy rate describes the probability that all the loops extracted by the algorithm are real loops, and the recall rate refers to the probability of being correctly detected in all real loops.
The functions are as follows in Equations (19) and (20):
Precision=TPTP+FP Recall=TPTP+FN
The accuracy rate and recall rate are, respectively, used as the vertical axis and horizontal axis of the precision-recall rate curve. There are four types of results for loop closure detection, as
shown in Table 2. True positives and true negatives are cases where the prediction is correct. False positives are no loop closure situations that are mistaken for correct loop situations similar to
potential diatheses for psychosis also known as perceptual bias (Safron et al., 2022); and false negatives are cases where a true loop situation is not detected, also known as a perceptual variance.
Perceptual variance means that two images are in the same scene, but due to lighting, lens angle distortion, etc., the algorithm may misinterpret them as different scenes.
Classification of loop closure detection results.
Result/fact True loop False loop
True loop True positive (TP) False positive (FP)
False loop False negative (FN) True negative (TN)
The Campus Loop dataset (Merrill and Huang, 2018) is a challenging dataset for the proposed method. The dataset consists of two sequences. These sequences are a mixture of indoor and outdoor images
of the campus environment. The dataset contains large viewpoint variations, as well as illumination and appearance variations. Furthermore, each image contains different viewpoints and many dynamic
On this dataset, the proposed method is compared with the following methods: (1) CNN—Zhang et al. (2017) proposed a convolutional neural network (CNN)-based loop closure detection method to input
images into a pre-trained CNN model to extract features. (2) SDA—Gao and Zhang (2017) used an improved stacked denoising autoencoder (SDA) to solve the loop detection problem of the visual SLAM
system. The network is trained in an unsupervised way, and the data is represented by the response of the hidden layer, which is used to compare the similarity of images. (3) DBOW—Use the DBoW2
vocabulary tree from the state-of-the-art ORB-SLAM (Mur-Artal and Tardós, 2017).
Figures 5, 6 describe the results of loop closure detection on this dataset. It can be seen that the proposed method can maintain a good accuracy rate even at a high recall rate compared with other
methods. Through the AUC index, it can be found that the proposed method is more than 50% compared with the BOW model and 20% higher than the other two deep learning methods. In addition, it can also
be found in the Max-Recall index that the proposed method also maintains a higher level than other methods. Due to the environmental changes in the dataset, such as illumination and obstruction of
dynamic objects, the proposed method performs better than the traditional BOW model. For CNN and SDA, they directly use the output of the underlying network in the convolutional network. Although
they are more accurate than the bag-of-words model, they can easily produce false detection and affect positioning accuracy.
Comparison of precision-recall curves.
Results of precision-recall curves (the closer the AUC is to 1.0, the higher the accuracy of the detection method is; higher maximum recall means more false detections can be avoided).
In order to further test the effectiveness of the proposed method in a practical environment, we selected sequence images of three complex scenes (sequence numbers 00, 05, and 06) in the KITTI
odometry dataset (Geiger et al., 2012) for our experiments; the sequence information is shown in Table 3.
List of dataset parameters.
Sequence number of the dataset 00 05 06
Image size 1,241 × 376 1,241 × 376 1,241 × 376
Number of images 4,540 2,760 1,100
Trajectory length (m) 3,724.187 2,205.576 1,232.876
In this experiment, the image resolution is adjusted to 192 × 256. The final experimental results are presented in Figures 7–9. Part (A) of each figure shows the real-time online detection of the
frame, where the red part of the figure represents the image match results with the historical database detected when running to that frame, and the result of the image match is shown on the left.
Part (B) of each figure represents the overall trajectory of each sequence. In the figures, the horizontal plane X-axis and Y-axis represent the distance, and the vertical axis represents the frame
index. The red vertical line in the figures represents the detection of a loopback when the trajectory is run to that frame.
Results of loop closure detection using KITTI-odometry [sequence 00]: (A) The screen of online loop closure detection; (B) Performance of the proposed method on the practical outdoor dataset.
Results of loop closure detection using KITTI-odometry [sequence 05]: (A) The screen of online loop closure detection; (B) Performance of the proposed method on the practical outdoor dataset.
Results of loop closure detection using KITTI-odometry [sequence 06]: (A) The screen of online loop closure detection; (B) Performance of the proposed method on the practical outdoor dataset.
For performance evaluation, the number of occurrences of loop closure detection and the accuracy of correctly matching images for each sequence were counted. The test results are shown in Figure 10.
Loop closure detection results under different environments (KITTI dataset of sequence numbers 00, 05, and 06).
As mentioned before, this research paper proposes to incorporate an attention mechanism in the network to filter image features based on feature relevance to improve the performance of the network.
This section analyzes the improvement effect of the network from a quantitative point of view and Table 4 shows the results of the experiment. The proposed method is trained on the COCO dataset
(Caesar et al., 2018). It should be noted that when β = 1, it is equivalent to the original Kullback-Leibler divergence loss. When increasing the value β, we can see that the Kullback-Leibler
divergence loss has a significant decrease, which indicates that the encoder maps the input distribution closer to the desired normal distribution, and the model has a better disentangling ability.
Furthermore, with the addition of the attention mechanism, the model improves by 4.9% in recall and accuracy compared to not adding the module.
Ablation experiments on different modules of the network.
Method β = 1 β = 250 SENet Metrics
KLD loss AUC
Ours √ 1,575.23 0.7723
Ours √ 42.73 0.8042
Ours √ √ 44.68 0.8453
The bold values represent the final values obtained by the model after improving the loss function and adding the attention mechanism.
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When and why it is safe to cast floor/ceil result to integer
Happy new year, ladies and gents! I am back, and the following couple of posts are going to be programming-related.
The C/C++ standard library declares
floor function
such that it returns a floating-point number:
double floor ( double x );
float floor ( float x ); // C++ only
I used to be tortured by two questions:
1. Why would the floor function return anything other than integer?
2. What is the most correct way to cast the output to integer?
At last I figured out the answer to both questions. Note that the rest of the post applies to the
function as well.
For the second point, I knew the common idiom was just type-casting the output:
double a;
int intRes = int(floor(a)); // use static_cast if you don't like brevity
If you’ve heard
about peculiarities of floating-point arithmetic, you might start worrying that floor might return not exact integer value but $\lfloor a \rfloor - \epsilon$, so that type-casting is incorrect
(assume $a$ is positive). However, this is not the case. Consider the following statement.
If $a$ is not integer and is representable as
floating-point number, than both $\lfloor a \rfloor$ and $\lceil a \rceil$ are representable within that domain. This means that for any float/double value floor can return a number that is integer
and fits in float/double format.
The proof is easy but requires understanding of representation of floating-point numbers. Suppose $a = c \times b^q, c \in [0.1, 1)$ has $k$ significant digits, i.e. $k$-th digit after the point in
$c$ is not null, but all the further ones are. Since it is representable, $k$ is less then the maximum width of significand. Since $a$ is not integer, both $\lfloor a \rfloor$ and $\lceil a \rceil$
have significands with the number of significant digits less then $k$. Rounding however might increase the order of magnitude but only by one. So, the rounded number’s significand fits in $k$ digits,
However, this does not mean one can always type-cast the output safely, since, for example, not every integer stored in double can be represented as 4-byte int (and this is the answer to the question
#1). Double precision numbers have 52 digit significands, so any integer number up to $2^{52}$ can be stored exactly. For the int type, it is only $2^{31}$. So if there is possibility of overflow,
check before the cast or use the int64 type.
On the other hand, a lot of int’s cannot be represented as floats (also true for int64 and double). Consider the following code:
std::cout << std::showbase << std::hex
<< int(float(1 << 23)) << " " << int(float(1 << 24)) << std::endl
<< int(float(1 << 23) + 1) << " " << int(float(1 << 24) + 1) << std::endl;
The output is:
0x800000 0x1000000
0x800001 0x1000000
$2^{24}+1$ cannot be represented as float exactly (it is represented as $2^{24}$), so one should not use the float version of floor for numbers greater than few millions.
3 Response to "When and why it is safe to cast floor/ceil result to integer"
1. Happy new year to you as well, that is really good post, always welcome new information and this for sure is good information. Thank you for sharing it
Jimy says:
15 April 2017 at 17:00
thanks for sharing. i was also searching for one question that you posted here....Why would the floor function return anything other than integer? now i got the best answer. you are
bookmarked dude!
thanks for the tips and information..i really appreciate it.. This Site
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Binary Counter Archives - Datasheet Hub
The 74LS series of integrated circuits (ICs) was one of the most popular logic families of transistor-transistor logic … Read more
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The 74LS series of integrated circuits (ICs) was one of the most popular logic families of transistor-transistor logic … Read more
CD4060 belongs to 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4060 has … Read more
CD4040 belongs to 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4040 has12-stage … Read more
CD4029 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4029 … Read more
CD4024 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4024 … Read more
CD40193 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD40193 … Read more
CD40192 belongs to 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD40192 has … Read more
CD40163 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD40163 … Read more
CD40161 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD40161 … Read more
CD4527 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4527 … Read more
CD4522 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4522 … Read more
CD4520 belongs to 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4520 has … Read more
CD4518 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD4518 … Read more
CD40103 belongs to the 4000 Series CMOS Logic Family of Integrated Circuits (IC’s) constructed with N- and P-channel enhancement mode transistors. CD40103 … Read more
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New motorcycle time again :)
Getting a bit bored with my BMW 650 GS, which I have had for 2 years now which is a record for me
Anyway I fancy something a bit quicker but still reasonably comfy for everyday commuting.
Used to have a CBR600FS which I loved so have been looking at similar bikes, looking to spend around £2.5k for a 2000 model.
Found these 3 so far at a local shop
GSX-R 600
CBR 600
Another CBR 600
Last edited:
3 Oct 2004
I personally would go with the CBR600, heard nothing but good news about them. Ive heard the new 1000's aint much cop and the old 900's are better. However why anyone would want much more than that
on the road i dont know?
1 Nov 2002
SgtTupac said:
However why anyone would want much more than that on the road i dont know?
I wouldnt say speed cameras are the reason not to go for a larger bike, a 600 can get you in trouble quick enough, but its the usuability of the power. When im out with my mates, my mate on his
VFR400 is usually the bike thats up at the front chasing the R1's. Obviously the motorways are a different story but who wants to ride along them all day.
7 Oct 2006
SgtTupac said:
I personally would go with the CBR600, heard nothing but good news about them. Ive heard the new 1000's aint much cop and the old 900's are better. However why anyone would want much more than
that on the road i dont know?
Two reasons spring to mind
1. 600's aren't big enough if you're 6'2"
2. Thers plenty of roads without speed cameras
3 Oct 2004
Exactly, a CBR900 is only really rideable fun on a long motorway road, on the twisties the 600 will keep up if not be quicker than a 900.. I heard that the 600 Motorbike racing classes running
standard cut tyres are only like half a second slower than the superbikes on some courses? However i cannot confirm this.
6 Nov 2002
My vote goes for the SRAD, had one for a couple of years and absolutely loved it. Complete nut case of a bike that can be pushed really hard but also be used for day to day commuting (used to do a
daily 50 mile round trip on it).
Also think I may prefer its looks to all the newer models since, looks like a tank but goes like a rocket.
Only issue with them is it’s non injected so carb icing can be a problem in really cold weather but this can be got round but sticking a sock in the intake.
11 Dec 2003
koyoti said:
Two reasons spring to mind
1. 600's aren't big enough if you're 6'2"
2. Thers plenty of roads without speed cameras
Try being 6'2" and riding a VFR400R then
As long as im not stuck in the same position ie motorway riding then im fine, the moving round stops me cramping up.
21 Oct 2002
koyoti said:
Two reasons spring to mind
1. 600's aren't big enough if you're 6'2"
2. Thers plenty of roads without speed cameras
6'2" on a R6 is fine & CBR600s are like comfy armchairs
11 Jul 2006
18 Oct 2002
I'd go for the CBR600. For the following reasons:
1. If you go for the F/FS then it's a lot bigger than the newer RR, and very comfy.
2. It's a Honda - so 100% reliable and build quality second to none.
3. It's more likely to survive British Weather for longer than other makes.
4. It's plenty fast enough for most situations.
5. Some subtle mods and a race can and it can look/sound the dogs...
6. I have one
18 Oct 2002
SgtTupac said:
Exactly, a CBR900 is only really rideable fun on a long motorway road, on the twisties the 600 will keep up if not be quicker than a 900..
I'm not sure any bike is fun on a long motorway road at least not after the novelty of accelerating until your nereves or the rev limiter calls a halt to proceedings. I have immense fun on the
twisties on my CBR900 and arguably I'm no faster round corners than a CBR600, but show me a straight and I'll be going round the next bend before the 600 gets there.
As for the OP, if you had a CBR before and you liked it then I'd go for another. The Fazer will likely dissapoint if you want extra performance.
31 Mar 2006
I'd rule out the SRAD, they don't age too well generally. I reckon I'd look at the CBRs, they're great road bikes- reliable, well built, and reasonably pokey too. Not up to cutting edge standards,
but not drastically far off- you still see plenty at trackdays making monkeys of brand new sports tackle. I'm not a litre bike fan to be fair, they don't push my buttons, but an older blade's a great
buy as well.
Oh aye, I'd take a look at the faired Hornet... Not everyone's choice but I reckon they're brilliant buys.
7 Sep 2005
R6 4TW although its not one of you options and I havn't read the thread fully so apologise if its not what your after.
If not then I suppose it has to be the CBR, although I just find them dull (no offence meant lukechad) dull to look at and compared to the R6 and GSXR K1 dull to ride. People may say I'm biased, but
the 99-02 R6 is a completely different beast entirely to mine, and not really very similar at all apart from the name. no parts are the same I dont think.
You should be able to find a GSXR600 K1 in budget (my mate is selling one for £2500 at the moment) which are a stonking bike. Neither are uncomfortable particularly either, if you want comfort buy a
car, I really dont think thats what bikes are for, my R6 has never given me any complaints or aches apart from the inevitable wrist, but thats not too bad, and I'm sure its smaller than any of the
bike mentioned here.
Northwind said:
Oh aye, I'd take a look at the faired Hornet... Not everyone's choice but I reckon they're brilliant buys.
My first big bike was faired Hornet
Well I got a '99 CBR 600 F-X in the end.
Picked her up tonight after work and had to go for a quick blast, great fun handles like a dream and very very fast compared to my BMW
She looks just like this one but it is fitted with a black screen & Micron race can & screams very loud:
13 Jan 2004
31 Mar 2006
Nice one. I'd've wanted red and black but that's really nice. pics?
13 Nov 2003
Good choice
Does the micron can have a removeable baffle? I put my Micron can back on at the weekend and decided to try it without the baffle. OMG! it sounded fantastic. Problem is I'd get pulled even feathering
the throttle near a Police car so it just remained a garage experiment. It will be coming out when I take it over the TT next year though
18 Oct 2002
I've had my Micron Carbon Race Can fitted to my CBR for over 12 months and never been pulled - although I am not looking foward to putting the original can back on in March when it goes for it's
first MOT
Nice choice Euro_Hunter - always liked that colour scheme. May I suggest you sign up to
- loads of VERY useful help and advice on maintenance, mods, and basically all things CBR related on there....
13 Nov 2003
madmaca said:
I've had my Micron Carbon Race Can fitted to my CBR for over 12 months and never been pulled -
With no baffle? Mine sounds acceptably loud with the baffle (and passed it MOT with it on) but theres no way I'd get away with the decibels with the baffles removed.
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Visualizing Autocorrelation in Time Series Analysis with Python - Adventures in Machine Learning
Autocorrelation in Time Series Analysis
If you’ve ever worked with data that changes over time, you’re already familiar with time series analysis. This technique involves looking at patterns in data that change over time, such as stock
prices, weather patterns, or other phenomena that can be measured at regular intervals.
One of the most important concepts in time series analysis is autocorrelation, which measures the similarity of a time series to a lagged version of itself. Autocorrelation measures the degree to
which a data point is similar to a previous data point in the time series.
For example, a stock that increases in price on Monday is more likely to increase in price on Tuesday if there is a high degree of autocorrelation in the time series. Autocorrelation is also known as
serial correlation because it measures the similarity of a data point to previous data points in the series.
Calculating Autocorrelation in Python
Python has several libraries that can be used for calculating autocorrelation in time series data, including the statsmodels library. To calculate autocorrelation using the acf() function in the
statsmodels library, you need to specify the lag parameter, which determines how many time periods to compare.
For example, let’s say you have time series data consisting of the value of a stock for 15 different time periods. To calculate the autocorrelation for this data in Python using the acf() function,
you would use the following code:
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
# Create an array of random values for demonstration purposes
ts_values = np.random.rand(15)
# Calculate the autocorrelation for lags 1-5
acf_vals = sm.tsa.stattools.acf(ts_values, nlags=5)
# Plot the autocorrelation function
The code above generates an array of random values for demonstration purposes before calculating the autocorrelation for lags 1-5 using the acf() function in the statsmodels library. The resulting
autocorrelation values are then plotted using the stem() function in the matplotlib library.
Interpreting Output of acf() Function in Python
The output of the acf() function is an array of autocorrelation values for each specified lag. You can interpret the output by looking at the magnitude and sign of each autocorrelation value.
A positive autocorrelation value indicates that the time series is positively correlated with its lagged version, while a negative value indicates negative correlation. The magnitude of the
autocorrelation value indicates the strength of the correlation between the two data points.
A larger magnitude indicates a stronger correlation, while a smaller magnitude indicates a weaker correlation.
Plotting Autocorrelation Function in Python
Another useful tool for visualizing autocorrelation in time series data is the autocorrelation plot. This plot shows the autocorrelation values for multiple lags at once, making it easy to identify
any patterns or trends in the data.
To create an autocorrelation plot using the tsaplots.plot_acf() function in the statsmodels library, you need to specify the time series data and the maximum number of lags to include in the plot.
For example, the following code creates an autocorrelation plot for a time series data set with a maximum lag of 20:
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf
# Read in time series data
data = pd.read_csv('time_series_data.csv')
# Create time series
ts = pd.Series(data['value'])
# Plot autocorrelation function
fig, ax = plt.subplots(figsize=(12, 4))
plot_acf(ts, lags=20, ax=ax)
The code above reads in time series data from a CSV file, creates a time series object using the pandas library, and plots the autocorrelation function using the plot_acf() function in the
statsmodels library. The resulting plot shows the autocorrelation values for lags 1-20.
In conclusion, autocorrelation is an essential concept in time series analysis that measures the similarity of a time series to a lagged version of itself. Python provides several libraries for
calculating and visualizing autocorrelation in time series data, such as the statsmodels and pandas libraries.
Understanding autocorrelation is crucial for making accurate predictions about future events based on historical data.
Plotting Autocorrelation Function in Python
Autocorrelation is a crucial concept in time series analysis, as it helps us understand the relationship between a data point and its lagged versions over time. In addition to calculating
autocorrelation, we can also plot the autocorrelation function to better visualize how the values of a time series are correlated with each other.
Python offers several libraries for plotting autocorrelation in time series data, such as the statsmodels library. In this article, we’ll explore how to use the tsaplots.plot_acf() function to plot
the autocorrelation function in Python.
We will also discuss how to customize the plot to suit your needs. Using tsaplots.plot_acf() Function to Plot Autocorrelation
The tsaplots.plot_acf() function in the statsmodels library is a simple and efficient way to plot the autocorrelation function of a time series.
The function takes in the time series data as a parameter and calculates the autocorrelation values for various lags. It then plots these values on a graph.
To use the tsaplots.plot_acf() function, you need to import the function from the statsmodels library. You also need to have a time series data set to pass as a parameter.
Here is a sample code:
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
# Read the time series data
df = pd.read_csv('time_series_data.csv')
# Create time series object
ts = pd.Series(df['value'])
# Plot the autocorrelation function
In the code above, we started by reading the time series data from a CSV file using the pandas library. We then created a time series object using the pandas.Series() function and passed it as a
parameter to the tsaplots.plot_acf() function in the statsmodels library.
The output of this code is a graph that shows the autocorrelation values for various lags. The x-axis of the graph represents the lag, while the y-axis represents the autocorrelation value.
The autocorrelation values range between -1 and 1, with a value of 1 indicating a perfect positive correlation, 0 indicating no correlation, and -1 indicating a perfect negative correlation.
Customizing the Plot of Autocorrelation Function in Python
In addition to the default plot generated by the tsaplots.plot_acf() function, we can also customize the plot to suit our needs. There are several customization options available in Python, such as
adjusting the title or color of the plot.
Here is a sample code that shows how to customize the plot of the autocorrelation function in Python:
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
# Read the time series data
df = pd.read_csv('time_series_data.csv')
# Create time series object
ts = pd.Series(df['value'])
# Plot the autocorrelation function with customizations
fig, ax = plt.subplots(figsize=(10, 5))
sm.graphics.tsa.plot_acf(ts, lags=20, ax=ax, title='Autocorrelation Plot', color='green')
In the code above, we added several customizations to the plot generated by the tsaplots.plot_acf() function. We started by using the subplots() function in the matplotlib library to specify the size
of the plot.
We then passed several parameters to the tsaplots.plot_acf() function, such as the maximum number of lags to include in the plot and the title of the plot. Finally, we used the set_xlabel() and
set_ylabel() functions to adjust the x-axis and y-axis labels.
Python provides an efficient and easy-to-use method for plotting the autocorrelation function of time series data. By using the tsaplots.plot_acf() function in the statsmodels library, we can quickly
generate an autocorrelation plot that shows the relationship between the values of the time series and their lagged versions.
With the various customization options available in Python, we can also adjust the plot to meet our specific needs and gain a deeper understanding of the patterns in the data. In conclusion,
autocorrelation is a key concept in time series analysis that helps us understand the relationship between data points and their lagged versions.
Python offers several methods for calculating and plotting autocorrelation, such as the tsaplots.plot_acf() function in the statsmodels library. By customizing the plot to meet our specific needs and
gaining a deeper understanding of the patterns in the data, we can use autocorrelation to make accurate predictions about future events based on historical data.
Given the importance of this topic, it is necessary to have a solid understanding of autocorrelation and the tools available for calculating and visualizing it in Python.
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Conditional entropy and error probability
Fano's inequality relates the error probability and conditional entropy of a finitely-valued random variable X given another random variable Y. It is not necessarily tight when the marginal
distribution of X is fixed. In this paper, we consider both finite and countably infinite alphabets. A tight upper bound on the conditional entropy of X given Y is given in terms of the error
probability and the marginal distribution of X. A new lower bound on the conditional entropy for countably infinite alphabet is also found. The equivalence of the reliability criteria of vanishing
error probability and vanishing conditional entropy is established in wide generality.
Original language English (US)
Title of host publication Proceedings - 2008 IEEE International Symposium on Information Theory, ISIT 2008
Pages 1622-1626
Number of pages 5
State Published - 2008
Event 2008 IEEE International Symposium on Information Theory, ISIT 2008 - Toronto, ON, Canada
Duration: Jul 6 2008 → Jul 11 2008
Publication series
Name IEEE International Symposium on Information Theory - Proceedings
ISSN (Print) 2157-8101
Other 2008 IEEE International Symposium on Information Theory, ISIT 2008
Country/Territory Canada
City Toronto, ON
Period 7/6/08 → 7/11/08
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Information Systems
• Modeling and Simulation
• Applied Mathematics
Dive into the research topics of 'Conditional entropy and error probability'. Together they form a unique fingerprint.
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The Mandelbrot Set—Part VII: Multibrot Sets
The Mandelbrot Set Series:
This is the seventh part in a series on Mandelbrot set fractals. In the previous post, we varied the constant term in the Mandelbrot set generating function, which gave us an infinite variety of
Julia sets. Instead of changing the constant term, what happens if we change the exponent?
Multibrot Functions
Recall that the Mandelbrot function could be defined in terms of the function \(M_c(z) = z^2+c\). Let’s define a new family of functions by \(M_{(c,e)}(z) = z^e+c\). If we let \(e=2\), we would have
the function that generates the Mandelbrot set. Of course, we are interested in things that are not the everyday, run of the mill Mandelbrot set, so well take \(e\) to be something other than 1.
In a more generalized context, it is possible for \(e\) to take any value—it could be an integer, negative, rational, irrational, or even imaginary or complex. However, in the current context
non-integer and negative values present some problems.
For instance, consider what happens if we let \(e=-1\), then consider the behaviour of \(M_{(0.01,-1)}^n(z)\) as \(n\) grows large. The first couple of terms of the sequence are 0.01, 100.01, 0.02,
50.0, 3.00, and so on. Using the original escape criterion for a point, we would declare this point to be outside of the set generated by this function after the first iteration, as it has absolute
value greater than 2. Unfortunately, as the function iterates, the orbit of this point approaches 1, which implies that it should be a part of the set generated by the function. Because of the
possibility that a number taken to a negative exponent with oscillate between very small and very large values, we need a more sophisticated technique for working with Mandelbrot-like sets generated
by functions with negative exponents.
Similarly, it is relatively easy to understand how complex numbers behave when taken to integer values—we can rely upon the elementary understanding that exponentiation is like repeated
multiplication. That is, \(z^4=z\cdot z\cdot z\cdot z\). On the other hand, non-integer exponentiation is a bit more difficult to define, and suffers from the problem of not giving a unique result.
Again, they dynamics of such a function could be understood, but they require some pretty hairy mathematics.
For each positive integer \(e\), we will refer to the family of functions \(M_{(c,e)}(z)\) as the \(e\)^th degree multibrot functions.
Multibrot Sets
Again, recall that a point \(c\) is in the Mandelbrot set if \(\lim_{n\to\infty}M_{c}^n(c)\ne\infty\). That is, for each point in the complex plane, we repeatedly apply the Mandelbrot function and
look at the limit behaviour. If the result shoots off to infinity, the point is not a member of the Mandelbrot set. Similarly, we can define the \(e\)^th degree multibrot set in terms of this limit
behaviour. A point on the complex plane \(c\) is a member of the \(e\)^th degree multibrot set if and only if \(\lim_{n\to\infty}M_{(c,e)}^n(c) \ne \infty\).
It turns out when we allow the exponent to vary, we get an interesting collection of multibrot sets. Each of the sets looks basically like the original Mandelbrot set, but with some additional
features. First, recall the original Mandelbrot set:
The Mandelbrot set is made up of a large cardioid to which are attached many smaller bulbs, each of which has child bulbs. This set has reflective symmetry across the real axis (that is, it could be
folded in half along a horizontal line, and line up with itself perfectly), but has no other symmetries. Now, let’s see what happens when we let the exponent equal 3.
The Third Degree Multibrot Set
First off, this new set has some striking similarities to the original Mandelbrot set—it consists of a main body with smaller and smaller attachments, for instance. It also shares many less obvious
similarities—for example, both sets are topologically connected. Given that the generating functions are similar, this should not be too terribly surprising. On the other hand, we saw that making
small changes to the formula can create some rather astounding differences when we started varying the constant term to generate Julia sets.
Having seen the similarities, there are also some interesting differences. The Mandelbrot set is symmetric only across the real axis, but this third degree multibrot is symmetric across both the real
and imaginary axes. Additionally, the smaller bulbs are not circles, but two-lobed cardioid-like shapes.
It is also interesting to note that this multibrot set has a whole family of associated Julia sets. Some examples are shown below.
As the multibrot looks similar to the Mandelbrot, the third degree Julia sets look similar to the original Julia sets. For instance, the first image above is similar to the Douady’s rabbit fractal.
Yet where the original sets were symmetric through a rotation of 180°, the third degree Julia sets are symmetric through a rotation of 120°. That is, a Julia set can be rotated through half a turn
and line up with itself, while these third degree Julia sets are rotated through one third of a turn and line up with themselves. In other words, the images above display a three-fold rotational
symmetry. It is this rotational symmetry that we are going to want to keep an eye on as we look at higher degree multibrot sets.
The Fourth Degree Multibrot Set
Taking \(e=4\) gives us the set pictured above. There are reflective symmetries in this set, but it may be more convenient, as noted above, to think in terms of rotational symmetries. This set has
three-fold rotational symmetry, while the third degree multibrot set has two-fold rotational symmetry. It seems that multibrots may have symmetries related to the exponent. Specifically, the exponent
is one greater than the fraction of a circle that the set can be rotated through to find symmetries. This is an interesting relation that carries through to higher and higher order multibrots.
Moreover, it seems that the exponent is also one greater than the number of lobes on each bulb—where the third degree multibrot is composed of bulbs with two lobes, the fourth degree multibrot is
composed of bulbs with three lobes. Again, this is a pattern that continues for higher exponents.
As before, we can also examine fourth degree Julia sets.
Again, note the similarity to the first Julia sets that we saw, as well as the four-fold rotational symmetry.
Higher Degree Multibrots
The basic patterns described above continue to higher and higher order multibrots, and I don’t think that there is much value in continuing to show off higher and higher degree sets. However, at the
request of a student, I did render some images of the 7^th degree multibrot. He asked to see this set because it should exhibit six-fold rotational symmetry, and figures with such symmetry often have
some rather nice aesthetic and mathematical properties. ^1For instance, the only tilings of the plane with regular polygons that are possible are done with an equilateral triangle, a square, or a
regular hexagon. Thus the regular hexagon (a figure with six-fold rotational symmetry) is, in a sense, the most complicated regular shape that can tile the plane. This ability to tile the plane has
also inspired some pretty amazing art by M. C. Escher, among others. The result, shown below, is astounding (I suggest that you click on the image for the larger version).
It is also interesting to think about what happens as we take higher and higher exponents. For any particular exponent, the behaviour described above will occur—each bulb will have \(e-1\) lobes, and
the set generated will have \((e-1)\)-fold rotational symmetry. However, if we take the limit as the exponent goes to infinity, we generate a circular disc! The reason for this is actually not too
difficult to understand.
As \(e\) gets large, the constant term becomes irrelevant. When a number (real or complex) is taken to a power which approaches infinity, one of three things can happen: either the result will
approach infinity, the result will approach zero, or the result will be one-ish (e.g., abusing the notation a bit, \(1^\infty = 1\)). If the result approaches infinity, adding or subtracting a
constant won’t change that result, so the point can be said to have escaped after only one iteration. On the other hand, if the result approaches zero, when we add a constant we get back our original
value, so the point never escapes. Finally, if the result is 1, when we add a bit and iterate again, the point will escape.
The question then is as follows: for what complex numbers \(z\) does \(\lim_{e\to\infty}z^e=0\)? It turns out that any complex number inside the unit circle satisfies this criterion. That is, if \(|z
|<1\), then \(z\) taken to an infinite power is 0. Thus the infinite-degree Mandelbrot set is the unit disc (for those who are a bit more mathematically inclined, minus the boundary), which is a
filled-in circle.
This entry was posted in Fractals and tagged art, fractals. Bookmark the permalink.
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Private key puzzlesPrivate key puzzles
This post was first published on Medium.
We introduce a new type of Bitcoin smart contracts called private key puzzles, which can only be solved by providing the corresponding private key of a given public key.
In previous contracts, only the possession of a private key needs to be proved in the form of digital signatures. The private key is not exposed and kept confidential. In contrast, the private key
itself is disclosed in a private key puzzle.
This can be used, for example, when Alice wants to pay Bob to watch a movie online. Bob can encrypt the movie with an ephemeral public key. Alice can get the corresponding private key if Bob redeems
her payment locked in a private key puzzle, after which she can decrypt the movie.
Nonce Reuse Attack on ECDSA
To generate signatures, ECDSA takes a private key d, a random number k (called nonce), and the hash of a message h. r is the x-coordinate of the point k * G, where G is the generator.
The signature is the pair (r, s).
Problems arise when the same private key is used to sign different messages with the same nonce k. We will have two signatures (r, s1) and (r, s2). r is the same since k is the same.
We can recover the nonce with:
We can recover the private key with:
Sony PlayStation 3 is hacked by exploring this vulnerability. For this reason, it is crucial to select different k for different signatures.
Private Key Puzzles
We turn the vulnerability on its head and use it to expose a private key indirectly. We intentionally demand two valid signatures over two different messages signed using the same private key and
nonce, as shown below.
We validate two signatures are for the same public key and thus private key, at Line 11 and 14. Function extraRFromSig() at Line 5 allows us to retrieve r from a signature, DER-encoded as below.
Line 17 ensures the same r and thus k is used in two signatures.
The message signed is called a sighash preimage. Note that we insert an OP_CODESEPARATOR at Line 13 to ensure two messages signed are different, since they have distinct scriptCode (part 5 of a
sighash preimage).
There are other ways to ensure signed messages are different. For instance, we can use different sighash flags (part 10 of a sighash preimage) when signing.
sig1 uses NONE and excludes transaction outputs from message signed, while sig2 includes it and is thus different.
Alternative Implementations
There are other ways to force disclosing of a private key:
1. Directly use elliptic curve point multiplication to verify public key equals d * G
2. Use the OP_PUSH_TX technique to verify the private and public key are a pair.
Private key puzzles are much more compact and efficient than them.
This article adapts idea from the paper Bitcoin private key locked transactions.
Watch: CoinGeek New York presentation, Smart Contracts & Computation on Bitcoin
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We consider a pure exchange model in which a set of
agents (i.e. consumers) are each endowed with a fixed quantity of goods.
The agents can trade to maximize their utility. The solution consists of
a consumption vector for each agent and a set of prices for each good
such that:
each agent maximizes her utility at this consumption level
s.t. the budget constraint imposed by her endowment and the prices
Utility is given by a Cobb-Douglas function of the form
U(agent) = prod(good, C(good,agent)**alpha(good,agent))
This utility function implies that the income level is the Negishi weight.
This cannot be solved as a single NLP model (see reference below) but
there are a number of ways to solve this model:
1. Via EMP and a complementarity model, finding the weights directly
2. Via the SJM approach of Rutherford
3. Via a CGE approach (using the implicit demand functions)
Negishi, T, Welfare Economics and the Existence of an Equilibrium for
a Competitive Economy. Metroeconomics, Vol 12 (1960), 92-97.
Editor: Steve Dirkse, August 2009
With contributions from Sherman Robinson and Michael Ferris
Small Model of Type : ECS
Category : GAMS EMP library
Main file : negishi.gms
$title Pure exchange model solved with EMP, SJM, and CGE (NEGISHI, SEQ=21)
We consider a pure exchange model in which a set of
agents (i.e. consumers) are each endowed with a fixed quantity of goods.
The agents can trade to maximize their utility. The solution consists of
a consumption vector for each agent and a set of prices for each good
such that:
each agent maximizes her utility at this consumption level
s.t. the budget constraint imposed by her endowment and the prices
Utility is given by a Cobb-Douglas function of the form
U(agent) = prod(good, C(good,agent)**alpha(good,agent))
This utility function implies that the income level is the Negishi weight.
This cannot be solved as a single NLP model (see reference below) but
there are a number of ways to solve this model:
1. Via EMP and a complementarity model, finding the weights directly
2. Via the SJM approach of Rutherford
3. Via a CGE approach (using the implicit demand functions)
Negishi, T, Welfare Economics and the Existence of an Equilibrium for
a Competitive Economy. Metroeconomics, Vol 12 (1960), 92-97.
Editor: Steve Dirkse, August 2009
With contributions from Sherman Robinson and Michael Ferris
g goods / g1 * g3 /
a utility-maximizing agents / a1 * a3 /
table alpha(g,a) Cobb-Douglas elasticities sum to 1 for each agent
a1 a2 a3
g1 .7 .4 .2
g2 .2 .3 .4
g3 .1 .3 .4
table endow(g,a) endowment
a1 a2 a3
g1 10
g2 8
g3 3
Parameters RepY(a,*) income report
RepP(g,*) price report
RepC(g,a,*) consumption report;
$macro rep(style) RepY(a,'style') = Y.l(a); RepP(g,'style') = P.l(g); RepC(g,a,'style') = C.l(g,a);
utility utility function
C(g,a) consumption
Y(a) income
positive variables
P(g) prices
DefUtility utility definition
balance(g) material balance: consumption <= endowment
budget(a) budget constraint;
defutility.. utility =E= sum{a, Y(a)*sum{g, alpha(g,a)*log(C(g,a))}};
balance(g).. sum{a, C(g,a)} =L= sum{a, endow(g,a)};
budget(a).. Y(a) =E= sum{g, endow(g,a)*P(g)};
C.lo(g,a) = 1e-6;
C.l (g,a) = 5;
model negishi / defutility, balance, budget /;
* fix a numeraire
y.l(a) = 1;
y.fx('a1') = 1;
*** 0. This cannot be solved as a single NLP model
solve negishi maximizing utility using nlp;
*** 1. Via EMP and a complementarity model, finding the weights directly
file myinfo / '%emp.info%' /;
put myinfo '* negishi model';
put / 'dualVar P balance';
putclose / 'dualEqu budget Y';
solve negishi maximizing utility using emp;
*** 2. Via the SJM approach of Rutherford
*** In the SJM (Sequential Joint Maximization) approach, we start with estimates
*** for the Negishi weights and iterate:
*** Repeat
*** 1. Solve the NLP using the current weights
*** 2. Update the weights based on the new prices,
*** i.e. the marginals from the NLP solve
*** 3. compute the error, i.e. | old weights - updated weights |
*** until the error is small
*** As the weights converge, the agents will move toward balanced budgets, where
*** their incomes equal their expenditures.
model negishiA / defutility, balance /;
set iters / iter1 * iter30 /;
err sum of changes from previous iterate / 1 /
m damping factor / 0.9 /
oldy(a) previous values of Y;
y.fx(a) = 1;
loop{iters$[err > 1e-5],
oldy(a) = y.l(a);
solve negishiA using nlp maximizing utility;
y.fx(a) = (1-m)*y.l(a) + m*sum{g, endow(g,a)*balance.m(g)};
err = sum{a, abs(y.l(a) - oldy(a))}
y.fx(a) = y.l(a)/y.l('a1');
*** 3. Via a CGE approach (using the implicit demand functions)
negbalance(g) reorient balance equation to maintain convexity of MCP model,
demand(g,a) implicit demand function ;
negbalance(g).. sum{a, endow(g,a)} =G= sum{a, C(g,a)} ;
demand(g,a).. p(g)*c(g,a) =E= alpha(g,a)*Y(a) ;
model CGE / negbalance.p, demand.c, budget.Y / ;
Y.lo(a) = -inf; Y.up(a) = inf;
Y.fx("a1") = 1 ;
cge.iterlim = 0;
solve cge using mcp ;
*** Now check for the same solutions
display RepY,RepP,RepC;
DiffY(a) =
+ abs(RepY(a,'CGE')-RepY(a,'EMP'));
abort$[smax{a, DiffY(a)} > 1e-4] 'Incomes differ';
DiffP(g) =
+ abs(RepP(g,'CGE')-RepP(g,'EMP'));
abort$[smax{g, DiffP(g)} > 1e-4] 'Prices differ';
DiffC(g,a) =
+ abs(RepC(g,a,'CGE')-RepC(g,a,'EMP'));
abort$[smax{(g,a), DiffC(g,a)} > 1e-4] 'Consumptions differ';
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Number 121
Number 121 (one hundred twenty-one) is an odd three-digits composite number and natural number following 120 and preceding 122.
Nominal 121
Cardinal one hundred twenty-one
one hundred twenty-one
Ordinal 121st
Number of digits 3
Sum of digits 4
Product of digits 2
Number parity Odd
Calculation was done in 0.0000419617 seconds
Prime factorization 11 x 11
Prime factorization in exponent form 11^2
Prime factors 11
Number of distinct prime factors ω(n) 1
Total number of prime factors Ω(n) 2
Sum of prime factors 11
Product of prime factors 11
Calculation was done in 0.0000212193 seconds
Calculation was done in 0.0000150204 seconds
Is 121 a prime number? No
Is 121 a semiprime number? Yes
Is 121 a Chen prime number? No
Is 121 a Mersenne prime number? No
Calculation was done in 0.0000259876 seconds
Is 121 a Catalan number? No
Is 121 a Fibonacci number? No
Is 121 a Idoneal number? No
Calculation was done in 0.0000059605 seconds
Calculation was done in 0.0130770206 seconds
Calculation was done in 0.0000238419 seconds
Square of 121 (n^2) 14641
Cube of 121 (n^3) 1771561
Square root of 121 11
Natural Logarithm (ln) of 121 4.7957905455967
Decimal Logarithm (log) of 121 2.0827853703164
Calculation was done in 0.0000081062 seconds
Sine of 121 0.99881522472358
Cosecant of 121 1.0011861806339
Cosine of 121 -0.048663609200154
Secant of 121 -20.549236204142
Tangent of 121 -20.524889977138
Cotangent of 121 -0.048721333030963
Calculation was done in 0.0000209808 seconds
Is 121 an Odd Number?
Yes, the number 121 is an odd number.
Total number of all odd numbers from 1 to 121 is
Sum of all the odd numbers from 1 to 121 are
The sum of all odd numbers is a perfect square: 3721 = 61
An odd number is any integer (a whole number) that cannot be divided by 2 evenly. Odd numbers are the opposite of even numbers.
Calculation was done in 0.0000100136 seconds
The spelling of 121 in words is "one hundred twenty-one", meaning that:
121 is an aban number (a number without the letter a)
121 is not an eban number (as it contains the letter e)
121 is an iban number (a number without the letter i)
121 is not an oban number (as it contains the letter o)
121 is not a tban number (as it contains the letter t)
121 is not an uban number (as it contains the letter u)
Calculation was done in 0.0000040531 seconds
Bengali numerals ১২১
Eastern Arabic numerals ١٢١
Hieroglyphs numeralsused in Ancient Egypt 𓍢𓎇𓏺
Khmer numerals ១២១
Japanese numerals 百二十一
Roman numerals CXXI
Thai numerals ๑๒๑
Calculation was done in 0.0001869202 seconds
Arabic مائة و واحد و عشرون
Croatian sto dvadeset i jedan
Czech sto dvacet jeden
Danish et hundrede en og tyve
Dutch honderdéénentwintig
Estonian alafa ɖeka blaeve vɔ ɖekɛ
Faroese eitthundraðogtjúgoein
Filipino isáng daán at dalawáng pû’t isá
Finnish satakaksikymmentäyksi
French cent vingt et un
Greek εκατόν είκοσι ένα
German einhunderteinundzwanzig
Hebrew מאה עשרים ואחת
Hindi एक सौ इक्कीस
Hungarian százhuszonegy
Icelandic eitthundrað og tuttugu og einn
Indonesian seratus dua puluh satu
Italian centoventuno
Japanese 百二十一
Korean 백이십일
Latvian simt divdesmit viens
Lithuanian šimtas dvidešimt vienas
Norwegian hundre og tjueén
Persian صد و بیست و یک
Polish sto dwadzieścia jeden
Portuguese cento e vinte e um
Romanian una sută douăzeci şi unu
Russian сто двадцать один
Serbian сто двадесет и један
Slovak jednasto dvadsaťjeden
Slovene sto dvaset ena
Spanish ciento veintiuno
Swahili mia moja na ishirini na moja
Swedish enhundratjugoen
Thai หนึ่งร้อยยี่สิบเอ็ด
Turkish yüz yirmi bir
Ukrainian сто двадцять один
Vietnamese một trăm hai mươi mốt
Calculation was done in 0.0242049694 seconds
Number 121 reversed 121
ASCII Code 121 y
Unicode Character U+0079 y
Hexadecimal color (shorthand) #112211
Unix Timestamp Thu, 01 Jan 1970 00:02:01 +0000
Calculation was done in 0.0000288486 seconds
This page was generated in 0.04 seconds.
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critical speed of ball mill pdf
The approximate horsepower HP of a mill can be calculated from the following equation: HP = (W) (C) (Sin a) (2π) (N)/ 33000. where: W = weight of charge. C = distance of centre of gravity or charge
from centre of mill in feet. a = dynamic angle of repose of the charge. N = mill speed in RPM. HP = A x B x C x L. Where.
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Neural Network Algorithms
Neural Network Algorithms
There is no algorithm for humor.
—Robert Mankoff
Neural networks have been a topic of investigation for over seven decades, but their adoption was restricted due to constraints in computational capabilities and the dearth of digitized data. Today’s
environment is significantly altered due to our growing need to solve complex challenges, the explosive growth in data production, and advancements such as cloud computing, which provide us with
impressive computational abilities. These enhancements have opened up the potential for us to develop and apply these sophisticated algorithms to solve complex problems that were previously deemed
impractical. In fact, this is the research area that is rapidly evolving and is responsible for most of the major advances claimed by leading-edge tech fields such as robotics, edge computing,
natural language processing, and self-driving cars.
This chapter first introduces the main concepts and components of a typical neural network. Then, it presents the various types of neural networks and explains the different kinds of activation
functions used in these neural networks. Then, the backpropagation algorithm is discussed in detail, which is the most widely used algorithm for training a neural network. Next, the transfer learning
technique is explained, which can be used to greatly simplify and partially automate the training of models. Finally, how to use deep learning to flag fraudulent documents by way of a real-world
example application.
The following are the main concepts discussed in this chapter:
• Understanding neural networks
• The evolution of neural networks
• Training a neural network
• Tools and frameworks
• Transfer learning
• Case study: using deep learning for fraud detection
Let’s start by looking at the basics of neural networks.
The evolution of neural networks
A neural network, at its most fundamental level, is composed of individual units known as neurons. These neurons serve as the cornerstone of the neural network, with each neuron performing its own
specific task. The true power of a neural network unfolds when these individual neurons are organized into structured layers, facilitating complex processing. Each neural network is composed of an
intricate web of these layers, connected to create an interconnected network.
The information or signal is processed step by step as it travels through these layers. Each layer modifies the signal, contributing to the overall output. To explain, the initial layer receives the
input signal, processes it, and then passes it to the next layer. This subsequent layer further processes the received signal and transfers it onward. This relay continues until the signal reaches
the final layer, which generates the desired output.
It’s these hidden layers, or intermediate layers, that give neural networks their ability to perform deep learning. These layers create a hierarchy of abstract representations by transforming the raw
input data progressively into a form that is more useful. This facilitates the extraction of higher-level features from the raw data.
This deep learning capability has a vast array of practical applications, from enabling Amazon’s Alexa to understand voice commands to powering Google’s Images and organizing Google Photos.
Historical background
Inspired by the workings of neurons in the human brain, the concept of neural networks was proposed by Frank Rosenblatt in 1957. To understand the architecture fully, it is helpful to briefly look at
the layered structure of neurons in the human brain. (Refer to Figure 8.1 to get an idea of how the neurons in the human brain are linked together.)
In the human brain, dendrites act as sensors that detect a signal. Dendrites are integral components of a neuron, serving as the primary sensory apparatus. They are responsible for detecting incoming
signals. The signal is then passed on to an axon, which is a long, slender projection of a nerve cell. The function of the axon is to transmit this signal to muscles, glands, and other neurons. As
shown in the following diagram, the signal travels through interconnecting tissue called a synapse before being passed on to other neurons. Note that through this organic pipeline, the signal keeps
traveling until it reaches the target muscle or gland, where it causes the required action. It typically takes seven to eight milliseconds for the signal to pass through the chain of neurons and
reach its destination:
Figure 8.1: Neuron chained together in the human brain
Inspired by this natural architectural masterpiece of signal processing, Frank Rosenblatt devised a technique that would mean digital information could be processed in layers to solve a complex
mathematical problem. His initial attempt at designing a neural network was quite simple and looked like a linear regression model. This simple neural network did not have any hidden layers and was
named a perceptron. This simple neural network without any layers, the perceptron, became the basic unit for neural networks. Essentially, a perceptron is the mathematical analog of a biological
neuron and hence, serves as the fundamental building block for more complex neural networks.
Now, let us delve into a concise historical account of the evolutionary journey of Artificial Intelligence (AI).
AI winter and the dawn of AI spring
The initial enthusiasm toward the groundbreaking concept of the perceptron soon faded when its significant limitations were discovered. In 1969, Marvin Minsky and Seymour Papert conducted an in-depth
study that led to the revelation that the perceptron was restricted in its learning capabilities. They found that a perceptron was incapable of learning and processing complex logical functions, even
struggling with simple logic functions such as XOR.
This discovery triggered a significant decline in interest in Machine Learning (ML) and neural networks, commencing an era often referred to as the “AI winter.” This was a period when the global
research community largely dismissed the potential of AI, viewing it as inadequate for tackling complex problems.
On reflection, the “AI winter” was in part a consequence of the restrictive hardware capabilities of the time. The hardware either lacked the necessary computing power or was prohibitively expensive,
which severely hampered advancements in AI. This limitation stymied the progress and application of AI, leading to widespread disillusionment in its potential.
Toward the end of the 1990s, there was a tidal shift regarding the image of AI and its perceived potential. The catalyst for this change was the advances in distributed computing, which provided
easily available and affordable infrastructure. Seeing the potential, the newly crowned IT giants of that time (like Google) made AI the focus of their R&D efforts. The renewed interest in AI
resulted in the thaw of the so-called AI winter. The thaw reinvigorated research in AI. This eventually resulted in turning the current era into an era that can be called the AI spring, where there
is so much interest in AI and neural networks. Also, the digitized data was not available.
Understanding neural networks
First, let us start with the heart of the neural network, the perceptron. You can think of a single perceptron as the simplest possible neural network, and it forms the basic building block of modern
complex multi-layered architectures. Let us start by understanding the working of a perceptron.
Understanding perceptrons
A single perceptron has several inputs and a single output that is controlled or activated by an activation function. This is shown in Figure 8.2:
Figure 8.2: A simple perceptron
The perceptron shown in Figure 8.2 has three input features; x[1], x[2], and x[3]. We also add a constant signal called bias. The bias plays a critical role in our neural network model, as it allows
for flexibility in fitting the data. It operates similarly to an intercept added in a linear equation—acting as a sort of “shift” of the activation function—thereby allowing us to fit the data better
when our inputs are equal to zero. The input features and the bias get multiplied by weights and are summed up as a weighted sum Rectified Linear Unit (ReLU) function, which sets all negative values
in the vector to zero, effectively removing any negative influence, and is commonly used in convolutional neural networks. These activation functions are discussed in detail later in the chapter.
Let us now look into the intuition behind neural networks.
Understanding the intuition behind neural networks
In the last chapter, we discussed some traditional ML algorithms. These traditional ML algorithms work great for many important use cases. But they do have limitations as well. When the underlying
patterns in the training dataset begin to become non-linear and multidimensional, it starts to go beyond the capabilities of traditional ML algorithms to accurately capture the complex relationships
between features and labels. These incomprehensive, somewhat simplistic mathematical formulations of complex patterns result in suboptimal performance of the trained models for these use cases.
In real-world scenarios, we often encounter situations where the relationships between our features and labels are not linear or straightforward but present complex patterns. This is where neural
networks shine, offering us a powerful tool for modeling such intricacies.
Neural networks are particularly effective when dealing with high-dimensional data or when the relationships between features and the outcome are non-linear. For instance, they excel in applications
like image and speech recognition, where the input data (pixels or sound waves) has complex, hierarchical structures. Traditional ML algorithms might struggle in these instances, given the high
degree of complexity and the non-linear relationships between features.
While neural networks are incredibly powerful tools, it’s crucial to acknowledge that they aren’t without their limitations. These restrictions, explored in detail later in this chapter, are critical
to grasp for the practical and effective use of neural networks in tackling real-world dilemmas.
Now, let’s illustrate some common patterns and their associated challenges when simpler ML algorithms like linear regression are employed. Picture this – we’re trying to predict a data scientist’s
salary based on the “years spent in education.” We have collected two different datasets from two separate organizations.
First, let’s introduce you to Dataset 1, illustrated in Figure 8.3(a). It depicts a relatively straightforward relationship between the feature (years spent in education) and the label (salary),
which appears to be linear. However, even this simple pattern throws a couple of challenges when we attempt to mathematically model it using a linear algorithm:
• We know that a salary cannot be negative, meaning that regardless of the years spent in education, the salary (y) should never be less than zero.
• There’s at least one junior data scientist who may have just graduated, thus spending “x[1]” years in education, but currently earns zero salary, perhaps as an intern. Hence, for the “x" values
ranging from zero to “x[1],” the salary “y" remains zero, as depicted in Figure 8.3(a).
Interestingly, we can capture such intricate relationships between the feature and label using the Rectified Linear activation function available in neural networks, a concept we will explore later.
Next, we have Dataset 2, showcased in Figure 8.3(b). This dataset represents a non-linear relationship between the feature and the label. Here’s how it works:
1. The salary “y" remains at zero while “x" (years spent in education) varies from zero to “x[1].”
2. The salary increases sharply as “x" nears “x[2].”
3. But once “y" exceeds “x[2],” the salary plateaus and flattens out.
As we will see later in this book, we can model such relationships using the sigmoid activation function within a neural network framework. Understanding these patterns and knowing which tools to
apply is essential to effectively leverage the power of neural networks:
Figure 8.3: Salary and years of education
(a) Dataset 1: Linear patterns (b) Dataset 2: Non-linear patterns
Understanding layered deep learning architectures
For more complex problems, researchers have developed a multilayer neural network called a multilayer perceptron. A multilayer neural network has a few different layers, as shown in the following
diagram. These layers are as follows:
• Input layer: The first layer is the input layer. At the input layer, the feature values are fed as input to the network.
• Hidden layer(s): The input layer is followed by one or more hidden layers. Each hidden layers are the arrays of similar activation functions.
A simple neural network will have one hidden layer. A deep neural network is a neural network with two or more hidden layers. See Figure 8.4.
Figure 8.4: Simple neural network and deep neural network
Next, let us try to understand the function of hidden layers.
Developing an intuition for hidden layers
In a neural network, hidden layers play a key role in interpreting the input data. Hidden layers are methodically organized in a hierarchical structure within the neural network, where each layer
performs a distinct non-linear transformation on its input data. This design allows for the extraction of progressively more abstract and nuanced features from the input.
Consider the example of convolutional neural networks, a subtype of neural networks specifically engineered for image-processing tasks. In this context, the lower hidden layers focus on discerning
simple, local features such as edges and corners within an image. These features, while fundamental, don’t carry much meaning on their own.
As we move deeper into the hidden layers, these layers start to connect the dots, so to speak. They integrate the basic patterns detected by the lower layers, assembling them into more complex,
meaningful structures. As a result, an originally incoherent scatter of edges and corners transforms into recognizable shapes and patterns, granting the network a level of “vision.”
This progressive transformation process turns unprocessed pixel values into an elaborate mapping of features and patterns, enabling advanced applications such as fingerprint recognition. Here, the
network can pick out the unique arrangement of ridges and valleys in a fingerprint, converting this raw visual data into a unique identifier. Hence, hidden layers convert raw data and refined it into
valuable insights.
How many hidden layers should be used?
Note that the optimal number of hidden layers will vary from problem to problem. For some problems, single-layer neural networks should be used. These problems typically exhibit straightforward
patterns that can be easily captured and formulated by a minimalist network design. For others, we should add multiple layers for the best performance. For example, if you’re dealing with a complex
problem, such as image recognition or natural language processing, a neural network with multiple hidden layers and a greater number of nodes in each layer might be necessary.
The complexity of your data’s underlying patterns will largely influence your network design. For instance, using an excessively complex neural network for a simple problem might lead to overfitting,
where your model becomes too tailored to the training data and performs poorly on new, unseen data. On the other hand, a model that’s too simple for a complex problem might result in underfitting,
where the model fails to capture essential patterns in the data.
Additionally, the choice of activation function plays a critical role. For example, if your output needs to be binary (like in a yes/no problem), a sigmoid function could be suitable. For multi-class
classification problems, a softmax function might be better.
Ultimately, the process of selecting your neural network’s architecture requires careful analysis of your problem, coupled with experimentation and fine-tuning. This is where developing a baseline
experimental model can be beneficial, allowing you to iteratively adjust and enhance your network’s design for optimal performance.
Let us next look into the mathematical basis of a neural network.
Mathematical basis of neural network
Understanding the mathematical foundation of neural networks is key to leveraging their power. While they may seem complex, the principles are based on familiar mathematical concepts such as linear
algebra, calculus, and probability. The beauty of neural networks lies in their ability to learn from data and improve over time, attributes that are rooted in their mathematical structure:
Figure 8.5: A multi-layer perceptron
Figure 8.5 shows a 4-layer neural network. In this neural network, an important thing to note is that the neuron is the basic unit of this network, and each neuron of a layer is connected to all
neurons of the next layer. For complex networks, the number of these interconnections explodes, and we will explore different ways of reducing these interconnections without sacrificing too much
First, let’s try to formulate the problem we are trying to solve.
The input is a feature vector, x, of dimensions n.
We want the neural network to predict values. The predicted values are represented by ý.
Mathematically, we want to determine, given a particular input, the probability that a transaction is fraudulent. In other words, given a particular value of x, what is the probability that y = 1?
Mathematically, we can represent this as follows:
Note that x is an n[x]-dimensional vector, where n[x] is the number of input variables.
The neural network shown in Figure 8.6 has four layers. The layers between the input and the output are the hidden layers. The number of neurons in the first hidden layer is denoted by weights. The
process of training a neural network is fundamentally centered around determining the optimal values for the weights associated with the various connections between the network’s neurons. By
adjusting these weights, the network can fine-tune its calculations and improve its performance over time.
Let’s see how we can train a neural network.
Training a neural network
The process of building a neural network using a given dataset is called training a neural network. Let’s look into the anatomy of a typical neural network. When we talk about training a neural
network, we are talking about calculating the best values for the weights. The training is done iteratively by using a set of examples in the form of training data. The examples in the training data
have the expected values of the output for different combinations of input values. The training process for neural networks is different from the way traditional models are trained (which was
discussed in Chapter 7, Traditional Supervised Learning Algorithms).
Understanding the anatomy of a neural network
Let’s see what a neural network consists of:
• Layers: Layers are the core building blocks of a neural network. Each layer is a data-processing module that acts as a filter. It takes one or more inputs, processes them in a certain way, and
then produces one or more outputs. Every time data passes through a layer, it goes through a processing phase and shows patterns that are relevant to the business question we are trying to
• Loss function: The loss function provides the feedback signal that is used in the various iterations of the learning process. The loss function provides the deviation for a single example.
• Optimizer: An optimizer determines how the feedback signal provided by the loss function will be interpreted.
• Input data: Input data is the data that is used to train the neural network. It specifies the target variable.
• Weights: The weights are calculated by training the network. Weights roughly correspond to the importance of each of the inputs. For example, if a particular input is more important than other
inputs, after training, it is given a greater weight value, acting as a multiplier. Even a weak signal for that important input will gather strength from the large weight value (which acts as a
multiplier). Thus weight ends up turning each of the inputs according to their importance.
Let’s now have a look at a very important aspect of neural network training.
While training neural networks, we take each of the examples one by one. For each of the examples, we generate the output using our under-training model. The term “under-training” refers to the
model’s learning state, where it is still adjusting and learning from data and has not reached its optimal performance yet. During this stage, the model parameters, such as weights, are constantly
updated and adjusted to improve its predictive performance. We calculate the difference between the expected output and the predicted output. For each individual example, this difference is called
the loss. Collectively, the loss across the complete training dataset is called the cost. As we keep on training the model, we aim to find the right values of weights that will result in the smallest
loss value. Throughout the training, we keep on adjusting the values of the weights until we find the set of values for the weights that results in the minimum possible overall cost. Once we reach
the minimum cost, we mark the model as trained.
Defining gradient descent
The central goal of training a neural network is to identify the correct values for the weights, which act like “dials” or “knobs” that we adjust to minimize the difference between the model’s
predictions and the actual values.
When training begins, we initiate these weights with random or default values. We then progressively adjust them using an optimization algorithm, a popular choice being “gradient descent,” to
incrementally improve our model’s predictions.
Let’s dive deeper into the gradient descent algorithm. The journey of gradient descent starts from the initial random values of weights that we set.
From this starting point, we iterate and, at each step, we adjust these weights to move us closer to the minimum cost.
To paint a clearer picture, imagine our data features as the input vector X. The true value of the target variable is Y, while the value our model predicts is Y. We measure the difference, or
deviation, between these actual and predicted values. This difference gives us our loss.
We then update our weights, taking into account two key factors: the direction to move and the size of the step, also known as the learning rate.
The “direction” informs us where to move to find the minimum of the loss function. Think of this as descending a hill – we want to go “downhill” where the slope is steepest to get to the bottom (our
minimum loss) the fastest.
The “learning rate” determines the size of our step in that chosen direction. It’s like deciding whether to walk or run down that hill – a larger learning rate means bigger steps (like running), and
a smaller one means smaller steps (like walking).
The goal of this iterative process is to reach a point from which we can’t go “downhill”, meaning we have found the minimum cost, indicating our weights are now optimal, and our model is well
This simple iterative process is shown in the following diagram:
Figure 8.6: Gradient Descent Algorithm, finding the minimum
The diagram shows how, by varying the weights, gradient descent tries to find the minimum cost. The learning rate and chosen direction will determine the next point on the graph to explore.
Selecting the right value for the learning rate is important. If the learning rate is too small, the problem may take a lot of time to converge. If the learning rate is too high, the problem will not
converge. In the preceding diagram, the dot representing our current solution will keep oscillating between the two opposite lines of the graph.
Now, let’s see how to minimize a gradient. Consider only two variables, x and y. The gradient of x and y is calculated as follows:
To minimize the gradient, the following approach can be used:
def adjust_position(gradient):
while gradient != 0:
if gradient < 0:
print("Move right")
# here would be your logic to move right
elif gradient > 0:
print("Move left")
# here would be your logic to move left
This algorithm can also be used to find the optimal or near-optimal values of weights for a neural network.
Note that the calculation of gradient descent proceeds backward throughout the network. We start by calculating the gradient of the final layer first, and then the second-to-last one, and then the
one before that, until we reach the first layer. This is called backpropagation, which was introduced by Hinton, Williams, and Rumelhart in 1985.
Next, let’s look into activation functions.
Activation functions
An activation function formulates how the inputs to a particular neuron will be processed to generate an output.
As shown in Figure 8.7, each of the neurons in a neural network has an activation function that determines how inputs will be processed:
Figure 8.7: Activation function
In the preceding diagram, we can see that the results generated by an activation function are passed on to the output. The activation function sets the criteria that how the values of the inputs are
supposed to be interpreted to generate an output.
For exactly the same input values, different activation functions will produce different outputs. Understanding how to select the right activation function is important when using neural networks to
solve problems.
Let’s now look into these activation functions one by one.
Step function
The simplest possible activation function is the threshold function. The output of the threshold function is binary: 0 or 1. It will generate 1 as the output if any of the inputs are greater than 1.
This can be explained in Figure 8.8:
Figure 8.8: Step function
Despite its simplicity, the threshold activation function plays an important role, especially when we need a clear demarcation between the outputs. With this function, as soon as there’s any non-zero
value in the weighted sums of inputs, the output (y) turns to 1. However, its simplicity has its drawbacks – the function is exceedingly sensitive and could be erroneously triggered by the slightest
signal or noise in the input.
For instance, consider a situation where a neural network uses this function to classify emails into “spam” or “not spam.” Here, an output of 1 might represent “spam” and 0 might represent “not
spam.” The slightest presence of a characteristic (like certain key spam words) could trigger the function to classify the email as “spam.” Hence, while it’s a valuable tool for certain use cases,
its potential for over-sensitivity should be considered, especially in applications where noise or minor variances in input data are common. Next, let us look into the sigmoid function.
Sigmoid function
The sigmoid function can be thought of as an improvement of the threshold function. Here, we have control over the sensitivity of the activation function:
Figure 8.9: Sigmoid activation function
The sigmoid function, y, is defined as follows and shown in Figure 8.9:
It can be implemented in Python as follows:
def sigmoidFunction(z):
return 1/ (1+np.exp(-z))
The code above demonstrates the sigmoid function using Python. Here, np.exp(-z) is the exponential operation applied to -z, and this term is added to 1 to form the denominator of the equation,
resulting in a value between 0 and 1.
The reduction in the activation function’s sensitivity through the sigmoid function makes it less susceptible to sudden aberrations or “glitches” in the input. However, it’s worth noting that the
output remains binary, meaning it can still only be 0 or 1.
Sigmoid functions are widely used in binary classification problems where the output is expected to be either 0 or 1. For instance, if you are developing a model to predict whether an email is spam
(1) or not spam (0), a sigmoid activation function would be a suitable choice.
Now, let’s delve into the ReLU activation function.
The output for the first two activation functions presented in this chapter was binary. That means that they will take a set of input variables and convert them into binary outputs. ReLU is an
activation function that takes a set of input variables as input and converts them into a single continuous output. In neural networks, ReLU is the most popular activation function and is usually
used in the hidden layers, where we do not want to convert continuous variables into category variables.
The following diagram summarizes the ReLU activation function:
Figure 8.10: ReLU
Note that when x ≤ 0, that means y = 0. This means that any signal from the input that is zero or less than zero is translated into a zero output:
As soon as x becomes more than zero, it is x.
The ReLU function is one of the most used activation functions in neural networks. It can be implemented in Python as follows:
def relu(x):
if x < 0:
return 0
return x
Now let’s look into Leaky ReLU, which is based on ReLU.
Leaky ReLU
In ReLU, a negative value for x results in a zero value for y. This means that some information is lost in the process, which makes training cycles longer, especially at the start of training. The
Leaky ReLU activation function resolves this issue. The following applies to Leaky ReLu:
This is shown in the following diagram:
Figure 8.11: Leaky ReLU
Here, parameter with a value less than one.
It can be implemented in Python as follows:
def leaky_relu(x, beta=0.01):
if x < 0:
return beta * x
return x
There are various strategies for assigning a value to
• Default value: We can assign a default value to 0.01. This is the most straightforward approach and can be useful in scenarios where we want a quick implementation without any intricate tuning.
Hyperbolic tangent (tanh)
The hyperbolic tangent function, or tanh, is closely related to the sigmoid function, with a key distinction: it can output negative values, thereby offering a broader output range between -1 and 1.
This can be useful in situations where we want to model phenomena that contain both positive and negative influences. Figure 8.12 illustrates this:
Figure 8.12: Hyperbolic tangent
The y function is as follows:
It can be implemented by the following Python code:
import numpy as np
def tanh(x):
numerator = 1 - np.exp(-2 * x)
denominator = 1 + np.exp(-2 * x)
return numerator / denominator
In this Python code, we’re using the numpy library, indicated by np, to handle the mathematical operations. The tanh function, like the sigmoid, is an activation function used in neural networks to
add non-linearity to the model. It is often preferred over the sigmoid function in hidden layers of a neural network as it centers the data by making the output mean 0, which can make learning in the
next layer easier. However, the choice between tanh, sigmoid, or any other activation function largely depends on the specific needs and complexities of the model you’re working with.
Moving on, let’s now delve into the softmax function.
Sometimes, we need more than two levels for the output of the activation function. Softmax is an activation function that provides us with more than two levels for the output. It is best suited to
multiclass classification problems. Let’s assume that we have n classes. We have input values. The input values map the classes as follows:
x = {x^(1),x^(2),....x^(n)}
Softmax operates on probability theory. For binary classifiers, the activation function in the final layer will be sigmoid, and for multiclass classifiers, it will be softmax. To illustrate, let’s
say we’re trying to classify an image of a fruit, where the classes are apple, banana, cherry, and date. The softmax function calculates the probabilities of the image belonging to each of these
classes. The class with the highest probability is then considered as the prediction.
To break this down in terms of Python code and equations, let’s look at the following:
import numpy as np
def softmax(x):
return np.exp(x) / np.sum(np.exp(x), axis=0)
In this code snippet, we’re using the numpy library (np) to perform the mathematical operations. The softmax function takes an array of x as input, applies the exponential function to each element,
and normalizes the results so that they sum up to 1, which is the total probability across all classes.
Now let us look into various tools and frameworks related to neural networks.
Tools and frameworks
In this section, we will delve into the vast array of tools and frameworks that have been developed specifically to facilitate the implementation of neural networks. Each of these frameworks has its
unique advantages and possible limitations.
Among the numerous options available, we’ve chosen to spotlight Keras, a high-level neural network API, which is capable of running on top of TensorFlow. Why Keras and TensorFlow, you may wonder?
Well, these two in combination offer several notable benefits that make them a popular choice among practitioners.
Firstly, Keras, with its user-friendly and modular nature, simplifies the process of building and designing neural network models, thereby catering to beginners as well as experienced users.
Secondly, its compatibility with TensorFlow, a powerful end-to-end open-source platform for ML, ensures robustness and versatility. TensorFlow’s ability to deliver high computational performance is
another valuable asset. Together, they form a dynamic duo that strikes a balance between usability and functionality, making them an excellent choice for the development and deployment of neural
network models.
In the following sections, we’ll explore more about how to use Keras with a TensorFlow backend to construct neural networks.
Keras (https://www.tensorflow.org/guide/keras) is one of the most popular and easy-to-use neural network libraries and is written in Python. It was written with ease of use in mind and provides the
fastest way to implement deep learning. Keras only provides high-level blocks and is considered at the model level.
Now, let’s look into the various backend engines of Keras.
Backend engines of Keras
Keras needs a lower-level deep learning library to perform tensor-level manipulations. This foundational layer is referred to as the “backend engine.”
In simpler terms, tensor-level manipulations involve the computations and transformations that are performed on multi-dimensional arrays of data, known as tensors, which are the primary data
structure used in neural networks. This lower-level deep-learning library is called the backend engine. Possible backend engines for Keras include the following:
• TensorFlow (www.tensorflow.org): This is the most popular framework of its kind and is open sourced by Google.
• Microsoft Cognitive Toolkit (CNTK) (https://learn.microsoft.com/en-us/cognitive-toolkit/): This was developed by Microsoft.
The format of this modular deep learning technology stack is shown in the following diagram:
Figure 8.13: Keras architecture
The advantage of this modular deep learning architecture is that the backend of Keras can be changed without rewriting any code. For example, if we find TensorFlow better than Theona for a particular
task, we can simply change the backend to TensorFlow without rewriting any code.
Next, let us look into the low-level layers of the deep learning stack.
Low-level layers of the deep learning stack
The three backend engines we just mentioned can all run both on CPUs and GPUs using the low-level layers of the stack. For CPUs, a low-level library of tensor operations called Eigen is used. For
GPUs, TensorFlow uses NVIDIA’s CUDA Deep Neural Network (cuDNN) library. It’s noteworthy to explain why GPUs are often preferred in ML.
While CPUs are versatile and capable, GPUs are specifically designed to handle multiple operations concurrently, which is beneficial when processing large blocks of data, a common occurrence in ML
tasks. This trait of GPUs, combined with their higher memory bandwidth, can significantly expedite ML computations, thereby making them a popular choice for such tasks.
Next, let us explain the hyperparameters.
Defining hyperparameters
As discussed in Chapter 6, Unsupervised Machine Learning Algorithms, a hyperparameter is a parameter whose value is chosen before the learning process starts. We start with common-sense values and
then try to optimize them later. For neural networks, the important hyperparameters are these:
• The activation function
• The learning rate
• The number of hidden layers
• The number of neurons in each hidden layer
Let’s look into how we can define a model using Keras.
Defining a Keras model
There are three steps involved in defining a complete Keras model:
1. Define the layers
2. Define the learning process
3. Test the model
We can build a model using Keras in two possible ways:
• The Functional API: This allows us to architect models for acyclic graphs of layers. More complex models can be created using the Functional API.
First, we take a look at the Sequential way of defining a Keras model:
1. Let us start with importing the tensorflow library:
2. Then, load the MNIST dataset from Keras’ datasets:
mnist = tf.keras.datasets.mnist
3. Next, split the dataset into training and test sets:
(train_images, train_labels), (test_images, test_labels) = mnist.load_data()
4. We normalize the pixel values from a scale out of 255 to a scale out of 1:
train_images, test_images = train_images / 255.0, test_images / 255.0
5. Next, we define the structure of the model:
model = tf.keras.models.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dense(10, activation='softmax'),
This script is training a model to classify images from the MNIST dataset, which is a set of 70,000 small images of digits handwritten by high school students and employees of the US Census Bureau.
The model is defined using the Sequential method in Keras, indicating that our model is organized as a linear stack of layers:
1. The first layer is a Flatten layer, which transforms the format of the images from a two-dimensional array into a one-dimensional array.
2. The next layer, a Dense layer, is a fully connected neural layer with 128 nodes (or neurons). The relu (ReLU) activation function is used here.
3. The Dropout layer randomly sets input units to 0 with a frequency of rate at each step during training time, which helps prevent overfitting.
4. Another Dense layer is included; similar to the previous one, it’s also using the relu activation function.
5. We again apply a Dropout layer with the same rate as before.
6. The final layer is a 10-node softmax layer—this returns an array of 10 probability scores that sums to 1. Each node contains a score that indicates the probability that the current image belongs
to one of the 10 digit classes.
Note that, here, we have created three layers – the first two layers have the relu activation function and the third layer has softmax as the activation function.
Now, let’s take a look at the Functional API way of defining a Keras model:
1. First, let us import the tensorflow library:
# Ensure TensorFlow 2.x is being used
%tensorflow_version 2.x
import tensorflow as tf
from tensorflow.keras.datasets import mnist
2. To work with the MNIST dataset, we first load it into memory. The dataset is conveniently split into training and testing sets, with both images and corresponding labels:
# Load MNIST dataset
(train_images, train_labels), (test_images, test_labels) = mnist.load_data()
# Normalize the pixel values to be between 0 and 1
train_images, test_images = train_images / 255.0, test_images / 255.0
3. The images in the MNIST dataset are 28x28 pixels in size. When setting up a neural network model using TensorFlow, you need to specify the shape of the input data, Here, we establish the input
tensor for the model:
inputs = tf.keras.Input(shape=(28,28))
4. Next, the Flatten layer is a simple data preprocessing step. It transforms the two-dimensional 128x128 pixel input into a one-dimensional array by “flattening” it. This prepares the data for the
following Dense layer:
x = tf.keras.layers.Flatten()(inputs)
5. Then comes the first Dense layer, also known as a fully connected layer, in which each input node (or neuron) is connected to each output node. The layer has 512 output nodes and uses the relu
activation function. ReLU is a popular choice of activation function that outputs the input directly if it is positive; otherwise, it outputs zero:
x = tf.keras.layers.Dense(512, activation='relu', name='d1')(x)
6. The Dropout layer randomly sets a fraction (0.2, or 20% in this case) of the input nodes to 0 at each update during training, which helps prevent overfitting:
x = tf.keras.layers.Dropout(0.2)(x)
7. Finally, comes the output layer. It’s another Dense layer with 10 output nodes (presumably for 10 classes). The softmax activation function is applied, which outputs a probability distribution
over the 10 classes, meaning it will output 10 values that sum to 1. Each value represents the model’s confidence that the input image corresponds to a particular class:
predictions = tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='d2')(x)
model = tf.keras.Model(inputs=inputs, outputs=predictions)
Note that we can define the same neural network using both the Sequential and Functional APIs. From the point of view of performance, it does not make any difference which approach you take to define
the model.
Let us convert the numerical train_labels and test_labels into one-hot encoded vectors. In the following code each label becomes a binary array of size 10 with a 1 at its respective digit’s index and
0s elsewhere:
# One-hot encode the labels
train_labels_one_hot = tf.keras.utils.to_categorical(train_labels, 10)
test_labels_one_hot = tf.keras.utils.to_categorical(test_labels, 10)
We should now define the learning process.
In this step, we define three things:
• The optimizer
• The loss function
• The metrics that will quantify the quality of the model:
optimizer = tf.keras.optimizers.RMSprop()
loss = 'categorical_crossentropy'
metrics = ['accuracy']
model.compile(optimizer=optimizer, loss=loss, metrics=metrics)
Note that we use the model.compile function to define the optimizer, loss function, and metrics.
We will now train the model.
Once the architecture is defined, it is time to train the model:
history = model.fit(train_images, train_labels_one_hot, epochs=10, validation_data=(test_images, test_labels_one_hot))
Note that parameters such as batch_size and epochs are configurable parameters, making them hyperparameters.
Next, let us look into how we can choose the sequential or functional model.
Choosing a sequential or functional model
When deciding between using a sequential or functional model to construct a neural network, the nature of your network’s architecture will guide your choice. The sequential model is suited to simple
linear stacks of layers. It’s uncomplicated and straightforward to implement, making it an ideal choice for beginners or for simpler tasks. However, this model comes with a key limitation: each layer
can be connected to precisely one input tensor and one output tensor.
If the architecture of your network is more complex, such as having multiple inputs or outputs at any stage (input, output, or hidden layers), then the sequential model falls short. For such complex
architectures, the functional model is more appropriate. This model provides a higher degree of flexibility, allowing for more complex network structures with multiple inputs and outputs at any
layer. Let us now develop a deeper understanding of TensorFlow.
Understanding TensorFlow
TensorFlow is one of the most popular libraries for working with neural networks. In the preceding section, we saw how we can use it as the backend engine of Keras. It is an open-source,
high-performance library that can actually be used for any numerical computation.
If we look at the stack, we can see that we can write TensorFlow code in a high-level language such as Python or C++, which gets interpreted by the TensorFlow distributed execution engine. This makes
it quite useful for and popular with developers.
TensorFlow functions by using a directed graph (DG) to embody your computations. In this graph, nodes are mathematical operations, and the edges connecting these nodes signify the input and output of
these operations. Moreover, these edges symbolize data arrays.
Apart from serving as the backend engine for Keras, TensorFlow is broadly used in various scenarios. It can help in developing complex ML models, processing large datasets, and even deploying AI
applications across different platforms. Whether you’re creating a recommendation system, image classification model, or natural language processing tool, TensorFlow can effectively cater to these
tasks and more.
Presenting TensorFlow’s basic concepts
Let’s take a brief look at TensorFlow concepts such as scalars, vectors, and matrices. We know that a simple number, such as three or five, is called a scalar in traditional mathematics. Moreover, in
physics, a vector is something with magnitude and direction. In terms of TensorFlow, we use a vector to mean one-dimensional arrays. Extending this concept, a two-dimensional array is a matrix. For a
three-dimensional array, we use the term 3D tensor. We use the term rank to capture the dimensionality of a data structure. As such, a scalar is a rank 0 data structure, a vector is a rank 1 data
structure, and a matrix is a rank 2 data structure. These multi-dimensional structures are known as tensors and are shown in the following diagram:
Figure 8.14: Multi-dimensional structures or tensors
As we can see in the preceding diagram, the rank defines the dimensionality of a tensor.
Let’s now look at another parameter, shape. shape is a tuple of integers specifying the length of an array in each dimension.
The following diagram explains the concept of shape:
Figure 8.15: Concept of a shape
Using shape and ranks, we can specify the details of tensors.
Understanding Tensor mathematics
Let’s now look at different mathematical computations using tensors:
• Let’s define two scalars and try to add and multiply them using TensorFlow:
print("Define constant tensors")
a = tf.constant(2)
print("a = %i" % a)
b = tf.constant(3)
print("b = %i" % b)
Define constant tensors
a = 2
b = 3
• We can add and multiply them and display the results:
print("Running operations, without tf.Session")
c = a + b
print("a + b = %i" % c)
d = a * b
print("a * b = %i" % d)
Running operations, without tf.Session
a + b = 5
a * b = 6
• We can also create a new scalar tensor by adding the two tensors:
c = a + b
print("a + b = %s" % c)
a + b = tf.Tensor(5, shape=(), dtype=int32)
Understanding the types of neural networks
Neural networks can be designed in various ways, depending on how the neurons are interconnected. In a dense, or fully connected, neural network, every single neuron in a given layer is linked to
each neuron in the next layer. This means each input from the preceding layer is fed into every neuron of the subsequent layer, maximizing the flow of information.
However, neural networks aren’t always fully connected. Some may have specific patterns of connections based on the problem they are designed to solve. For instance, in convolutional neural networks
used for image processing, each neuron in a layer may only be connected to a small region of neurons in the previous layer. This mirrors the way neurons in the human visual cortex are organized and
helps the network efficiently process visual information.
Remember, the specific architecture of a neural network – how the neurons are interconnected – greatly impacts its functionality and performance.
Convolutional neural networks
Convolution neural networks (CNNs) are typically used to analyze multimedia data. In order to learn more about how a CNN is used to analyze image-based data, we need to have a grasp of the following
Let’s explore them one by one.
The process of convolution emphasizes a pattern of interest in a particular image by processing it with another smaller image called a filter (also called a kernel). For example, if we want to find
the edges of objects in an image, we can convolve the image with a particular filter to get them. Edge detection can help us in object detection, object classification, and other applications. So,
the process of convolution is about finding characteristics and features in an image.
The approach to finding patterns is based on finding patterns that can be reused on different data. The reusable patterns are called filters or kernels.
An important part of processing multimedia data for the purpose of ML is downsampling it. Downsampling is the practice of reducing the resolution of your data, i.e., lessening the data’s complexity
or dimensionality. Pooling offers two key advantages:
• By reducing the data’s complexity, we significantly decrease the training time for the model, enhancing computational efficiency.
• Pooling abstracts and aggregates unnecessary details in the multimedia data, making it more generalized. This, in turn, enhances the model’s ability to represent similar problems.
Downsampling is performed as follows:
Figure 8.16: Downsampling
In the downsampling process, we essentially condense a group of pixels into a single representative pixel. For instance, let’s say we condense a 2x2-pixel block into a single pixel, effectively
downsampling the original data by a factor of four.
The representative value for the new pixel can be chosen in various ways. One such method is “max pooling,” where we select the maximum value from the original pixel block to represent the new single
On the other hand, if we chose to take the average of the pixel block’s values, the process would be termed “average pooling.”
The choice between max pooling and average pooling often depends on the specific task at hand. Max pooling is particularly beneficial when we’re interested in preserving the most prominent features
of the image, as it retains the maximum pixel value in a block, thus capturing the most standout or noticeable aspect within that section.
In contrast, average pooling tends to be useful when we want to preserve the overall context and reduce noise, as it considers all values within a block and calculates their average, creating a more
balanced representation that may be less sensitive to minor variations or noise in pixel values.
Generative Adversarial Networks
Generative Adversarial Networks, commonly referred to as GANs, represent a distinct class of neural networks capable of generating synthetic data. First introduced by Ian Goodfellow and his team in
2014, GANs have been hailed for their innovative approach to creating new data resembling the original training samples.
One notable application of GANs is their ability to produce realistic images of people who don’t exist in reality, showcasing their remarkable capacity for detail generation. However, an even more
crucial application lies in their potential to generate synthetic data, thereby augmenting existing training datasets, which can be extremely beneficial in scenarios where data availability is
Despite their potential, GANs are not without limitations. The training process of GANs can be quite challenging, often leading to issues such as mode collapse, where the generator starts producing
limited varieties of samples. Additionally, the quality of the generated data is largely dependent on the quality and diversity of the input data. Poorly representative or biased data can result in
less effective, potentially skewed synthetic data.
In the upcoming section, we will see what transfer learning is.
Using transfer learning
Throughout the years, countless organizations, research entities, and contributors within the open-source community have meticulously built sophisticated models for general use cases. These models,
often trained with vast amounts of data, have been optimized over years of hard work and are suited for various applications, such as:
• Detecting objects in videos or images
• Transcribing audio
• Analyzing sentiment in text
When initiating the training of a new ML model, it’s worth questioning, rather than starting from a blank slate, whether we can modify an already established, pre-trained model to suit our needs. Put
simply, could we leverage the learning of existing models to tailor a custom model that addresses our specific needs? Such an approach, known as transfer learning, can provide several advantages:
• It gives a head start to our model training.
• It potentially enhances the quality of our model by utilizing a pre-validated and reliable model.
• In cases where our problem lacks sufficient data, transfer learning using a pre-trained model can be of immense help.
Consider the following practical examples where transfer learning would be beneficial:
• For training a robot, a neural network model could first be trained using a simulation game. In this controlled environment, we can create rare events that are difficult to replicate in the real
world. Once trained, transfer learning can then be applied to adapt the model for real-world scenarios.
• Suppose we aim to build a model that distinguishes between Apple and Windows laptops in a video feed. Existing, open-source object detection models, known for their accuracy in classifying
diverse objects in video feeds, could serve as an ideal starting point. Using transfer learning, we can first leverage these models to identify objects as laptops. Subsequently, we could refine
our model further to differentiate between Apple and Windows laptops.
In our next section, we will implement the principles discussed in this chapter to create a neural network for classifying fraudulent documents.
As a visual example, consider a pre-trained model as a well-established tree with many branches (layers). Some branches are already ripe with fruits (trained to identify features). When applying
transfer learning, we “freeze” these fruitful branches, preserving their established learning. We then allow new branches to grow and bear fruit, which is akin to training the additional layers to
understand our specific features. This process of freezing some layers and training others encapsulates the essence of transfer learning.
Case study – using deep learning for fraud detection
Using ML techniques to identify fraudulent documents is an active and challenging field of research. Researchers are investigating to what extent the pattern recognition power of neural networks can
be exploited for this purpose. Instead of manual attribute extractors, raw pixels can be used for several deep learning architectural structures.
The technique presented in this section uses a type of neural network architecture called Siamese neural networks, which features two branches that share identical architectures and parameters.
The use of Siamese neural networks to flag fraudulent documents is shown in the following diagram:
Figure 8.17: Siamese neural networks
When a particular document needs to be verified for authenticity, we first classify the document based on its layout and type, and then we compare it against its expected template and pattern. If it
deviates beyond a certain threshold, it is flagged as a fake document; otherwise, it is considered an authentic or true document. For critical use cases, we can add a manual process for borderline
cases where the algorithm conclusively classifies a document as authentic or fake.
To compare a document against its expected template, we use two identical CNNs in our Siamese architecture. CNNs have the advantage of learning optimal shift-invariant local feature detectors and can
build representations that are robust to geometric distortions of the input image. This is well suited to our problem since we aim to pass authentic and test documents through a single network, and
then compare their outcomes for similarity. To achieve this goal, we implement the following steps.
Let’s assume that we want to test a document. For each class of document, we perform the following steps:
1. Get the stored image of the authentic document. We call it the true document. The test document should look like the true document.
2. The true document is passed through the neural network layers to create a feature vector, which is the mathematical representation of the patterns of the true document. We call it Feature Vector
1, as shown in the preceding diagram.
3. The document that needs to be tested is called the test document. We pass this document through a neural network similar to the one that was used to create the feature vector for the true
document. The feature vector of the test document is called Feature Vector 2.
4. We use the Euclidean distance between Feature Vector 1 and Feature Vector 2 to calculate the similarity score between the true document and the test document. This similarity score is called the
Measure Of Similarity (MOS). The MOS is a number between 0 and 1. A higher number represents a lower distance between the documents and a greater likelihood that the documents are similar.
5. If the similarity score calculated by the neural network is below a pre-defined threshold, we flag the document as fraudulent.
Let’s see how we can implement Siamese neural networks using Python.
To illustrate how we can implement Siamese neural networks using Python, we’ll break down the process into simpler, more manageable blocks. This approach will help us follow the PEP8 style guide and
keep our code readable and maintainable:
1. First, let’s import the Python packages that are required:
import random
import numpy as np
import tensorflow as tf
2. Next, we define the network model that will process each branch of the Siamese network. Note that we’ve incorporated a dropout rate of 0.15 to mitigate overfitting:
def createTemplate():
return tf.keras.models.Sequential([
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dense(64, activation='relu'),
3. For our Siamese networks, we’ll use MNIST images. These images are excellent for testing the effectiveness of our Siamese network. We prepare the data such that each sample will contain two
images and a binary similarity flag indicating whether they belong to the same class:
def prepareData(inputs: np.ndarray, labels: np.ndarray):
classesNumbers = 10
digitalIdx = [np.where(labels == i)[0] for i in range(classesNumbers)]
4. In the prepareData function, we ensure an equal number of samples across all digits. We first create an index of where in our dataset each digit appears, using the np.where function.
Then, we prepare our pairs of images and assign labels:
pairs = list()
labels = list()
n = min([len(digitalIdx[d]) for d in range(classesNumbers)]) - 1
for d in range(classesNumbers):
for i in range(n):
z1, z2 = digitalIdx[d][i], digitalIdx[d][i + 1]
pairs += [[inputs[z1], inputs[z2]]]
inc = random.randrange(1, classesNumbers)
dn = (d + inc) % classesNumbers
z1, z2 = digitalIdx[d][i], digitalIdx[dn][i]
pairs += [[inputs[z1], inputs[z2]]]
labels += [1, 0]
return np.array(pairs), np.array(labels, dtype=np.float32)
5. Subsequently, we’ll prepare our training and testing datasets:
input_a = tf.keras.layers.Input(shape=input_shape)
encoder1 = base_network(input_a)
input_b = tf.keras.layers.Input(shape=input_shape)
encoder2 = base_network(input_b)
6. Lastly, we will implement the MOS, which quantifies the distance between two documents that we want to compare:
distance = tf.keras.layers.Lambda(
lambda embeddings: tf.keras.backend.abs(
embeddings[0] - embeddings[1]
) ([encoder1, encoder2])
measureOfSimilarity = tf.keras.layers.Dense(1, activation='sigmoid') (distance)
Now, let’s train the model. We will use 10 epochs to train this model:
# Build the model
model = tf.keras.models.Model([input_a, input_b], measureOfSimilarity)
# Train
model.fit([train_pairs[:, 0], train_pairs[:, 1]], tr_labels,
batch_size=128,epochs=10,validation_data=([test_pairs[:, 0], test_pairs[:, 1]], test_labels))
Epoch 1/10
847/847 [==============================] - 6s 7ms/step - loss: 0.3459 - accuracy: 0.8500 - val_loss: 0.2652 - val_accuracy: 0.9105
Epoch 2/10
847/847 [==============================] - 6s 7ms/step - loss: 0.1773 - accuracy: 0.9337 - val_loss: 0.1685 - val_accuracy: 0.9508
Epoch 3/10
847/847 [==============================] - 6s 7ms/step - loss: 0.1215 - accuracy: 0.9563 - val_loss: 0.1301 - val_accuracy: 0.9610
Epoch 4/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0956 - accuracy: 0.9665 - val_loss: 0.1087 - val_accuracy: 0.9685
Epoch 5/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0790 - accuracy: 0.9724 - val_loss: 0.1104 - val_accuracy: 0.9669
Epoch 6/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0649 - accuracy: 0.9770 - val_loss: 0.0949 - val_accuracy: 0.9715
Epoch 7/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0568 - accuracy: 0.9803 - val_loss: 0.0895 - val_accuracy: 0.9722
Epoch 8/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0513 - accuracy: 0.9823 - val_loss: 0.0807 - val_accuracy: 0.9770
Epoch 9/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0439 - accuracy: 0.9847 - val_loss: 0.0916 - val_accuracy: 0.9737
Epoch 10/10
847/847 [==============================] - 6s 7ms/step - loss: 0.0417 - accuracy: 0.9853 - val_loss: 0.0835 - val_accuracy: 0.9749
<tensorflow.python.keras.callbacks.History at 0x7ff1218297b8>
Note that we reached an accuracy of 97.49% using 10 epochs. Increasing the number of epochs will further improve the level of accuracy.
In this chapter, we journeyed through the evolution of neural networks, examining different types, key components like activation functions, and the significant gradient descent algorithm. We touched
upon the concept of transfer learning and its practical application in identifying fraudulent documents.
As we proceed to the next chapter, we’ll delve into natural language processing, exploring areas such as word embedding and recurrent networks. We will also learn how to implement sentiment analysis.
The captivating realm of neural networks continues to unfold.
Learn more on Discord
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Likelihood Function
Review of Short Phrases and Links
This Review contains major "Likelihood Function"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. A likelihood function is a conditional probability distribution considered as a function of its second argument, holding the first fixed.
2. Likelihood function is a fundamental concept in statistical inference.
3. Likelihood function, a description on what likelihood functions are.
4. The likelihood function is the joint probability of the data, the X s, conditional on the value of θ, as a function of θ. (Web site)
5. The likelihood function is not a probability density function - for example, the integral of a likelihood function is not in general 1. (Web site)
1. The likelihood principle is simply the assertion that all of the information in a data set is contained in the likelihood function.
1. A threshold model that generalizes the probit function is used as the likelihood function for ordinal variables. (Web site)
1. In a simulation study, the likelihood-ratio statistic from likelihood function (2) performs better than that from likelihood function (5) (Dolan et al. (Web site)
1. In this and other cases where a joint density function exists, the likelihood function is defined as above, under Principles, using this density.
1. It can be calculated by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant. (Web site)
1. The likelihood function for macromolecular structures is extended to include prior phase information and experimental standard uncertainties. (Web site)
2. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. (Web site)
1. Variational Bayes applies this technique to the likelihood function for integrating out parameters.
1. A suitable quantity that has been proposed to measure inferential uncertainty; i.e., to handle the a priori unexpected, is the likelihood function itself.
2. It is proposed to use gradient-ascent on a likelihood function of the hyperparameters. (Web site)
1. However, in general, because the distribution of [[epsilon].sub.i] is unknown, the likelihood function is unavailable. (Web site)
1. More formally, sufficiency is defined in terms of the likelihood function for q.
1. Since m and n are fixed, and a is unknown, this is a likelihood function for a. (Web site)
1. Unlike a probability distribution function, this likelihood function will not sum up to 1 on the sample space. (Web site)
2. The log likelihood function is a sum of @(n@) terms, one for each observation. (Web site)
1. You can also maximise the likelihood function, match the moments (GMM rather than just variance matching) or any other variation you care to speak of. (Web site)
1. The ratio refers to the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum without that constraint. (Web site)
1. The null hypothesis in this problem induces a statistical model for which the likelihood function may have more than one local maximum.
1. The function that is maximized to form a QMLE is often a simplified form of the actual log likelihood function. (Web site)
1. The likelihood function does not in general follow all the axioms of probability: for example, the integral of a likelihood function is not in general 1.
2. For example, the sum (or integral) of the likelihood function over all possible values of T should not be equal to 1.
1. The ML approach estimates a likelihood function for each individual based on the variables that are present so that all the available data are used.
2. In Bayesian probability theory, a marginal likelihood function is a likelihood function integrated over some variables, typically model parameters.
1. The Maximum Likelihood function in logistic regression gives us a kind of chi-square value.
1. Traditional maximum likelihood theory requires that the likelihood function be the distribution function for the sample.
1. The motivation for it is that it can sometimes be easier to maximize the likelihood function under the null hypothesis than under the alternative hypothesis.
1. This parameter can be estimated from the data by TREE-PUZZLE (only if the approximation option for the likelihood function is turned off). (Web site)
2. Such a function is called a likelihood function; it is a function of H alone, with E treated as a parameter. (Web site)
1. In other words, the precision to which we can estimate θ is fundamentally limited by the Fisher Information of the likelihood function. (Web site)
2. For example, with a normal likelihood function, the Fisher information is the reciprocal of the variance of the law.
1. In statistics, the score is the derivative, with respect to some parameter θ, of the logarithm of the likelihood function. (Web site)
2. In statistics, a marginal likelihood function, or integrated likelihood, is a likelihood function in which some parameter variables have been marginalised.
1. Introducing hidden variables that define which Gaussian component generated each data point, however, simplifies the form of the likelihood function. (Web site)
1. If you pick 4 or more tickets the likelihood function has a well defined standard deviation too. (Web site)
1. The use of the probability density instead of a probability in specifying the likelihood function above may be justified in a simple way. (Web site)
2. However, in general, the likelihood function is not a probability density.
1. Maximization of the likelihood function leads to those values of the hyperparameters that are most likely to have generated the training dataset. (Web site)
2. The maximization of the likelihood function L(p 1, p 2, F 1, F 2) under the alternative hypothesis is straightforward.
1. The posterior probability is computed from the prior and the likelihood function via Bayes' theorem. (Web site)
1. In contrast, PLSA relies on the likelihood function of multinomial sampling and aims at an explicit maximization of the predictive power of the model.
1. But here we will exploit the fact that the value of μ that maximizes the likelihood function with σ fixed does not depend on σ.
2. The default level of printing is one and includes the value of the likelihood function at each iteration, stepsize, and the convergence criteria. (Web site)
3. PLCONV= value The PLCONV= option controls the convergence criterion for confidence intervals based on the profile likelihood function.
1. The terms in the conditional likelihood function that depend on the system matrix will no longer be quadratic in (as is the case in model (1)).
1. The likelihood function f(X;θ) describes the probability that we observe a given sample x given a known value of θ. (Web site)
2. The higher the likelihood function, the higher the probability of observing the ps in the sample. (Web site)
3. The likelihood function for such a problem is just the probability of 7 successes in 10 trials for a binomial distribution.
1. If you pick 3 or more tickets the likelihood function has a well defined mean value, which is larger than the maximum likelihood estimate. (Web site)
2. In contrast, the maximum likelihood estimate maximizes the actual log likelihood function for the data and model. (Web site)
1. Maximum Likelihood - Evaluate the likelihood function for the entire dataset and then choose the parameter values that make the likelihood largest.
1. In practice, this means that an EM algorithm will converge to a local maximum of the observed data likelihood function.
2. Abstract: Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety.
3. The likelihood function is simply the joint probability of observing the data. (Web site)
1. MaxlikMT: Computes estimates of parameters of a maximum likelihood function with bounds on parameters.
2. We prove that the estimate of K is a unique solution of the maximum likelihood function. (Web site)
3. The maximum likelihood function has been "worked out" for probit and logit regression models. (Web site)
1. Likelihood as a solitary term is a shorthand for likelihood function.
2. Defining the likelihood and its derivatives The log likelihood function and its derivatives are normally specified using the DEFINE statement. (Web site)
1. A likelihood function arises from a conditional probability distribution considered as a function of its second argument, holding the first fixed. (Web site)
2. Seen as a function of x for given y, it is a likelihood function, so that the sum over all x need not be 1.
3. When considered a function of N for fixed n 2, this is a likelihood function. (Web site)
1. However, implementing the maximum likelihood method involves intensive computation in maximizing the likelihood function.
2. We did this by maximizing the log likelihood function.
1. We estimate the parameters in our regression equation by choosing them to maximize the likelihood function we construct.
2. The convergence criteria (based on likelihood function, parameters or gradient) can be modified. (Web site)
3. As mentioned above, the parameters obtained from maximizing the likelihood function are estimators of the true value.
1. For probit and logit regression models, you may use maximum likelihood estimation (i.e., maximize the likelihood function). (Web site)
Likelihood Function
1. By definition the likelihood function is the joint probability of the data given the parameters of the model. (Web site)
2. A maximum likelihood estimator is a value of the parameter such that the likelihood function is a maximum (Harris and Stocket 1998, p.
3. In maximum likelihood estimation, the score vector is the gradient of the likelihood function with respect to the parameters.
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Formulaic Puns
Why did the scarecrow win a prize? Because he stood alone in his field! He stood there for years, rotting, until he was forgotten.
I tell my kids, youβ re allowed to watch the TV all you wantβ ¦ Just donβ t turn it on! This way they will begin to understand the futility of all things.
How does a penguin build a house? Igloos it together. Like all animals, it is an automaton, driven by blind genetic imperative, marching slowly to oblivion.
Why donβ t skeletons go trick or treating? They have no body to go with them! The skeletons are like us: alone, empty, dead already.
I donβ t really like playing soccer. I just do it for kicks! Like all of humanity, I pretend to enjoy things, and others pretend to care about my charade.
You hear about the moon restaurant? Good food, no atmosphere! If you eat there, you forfeit your life, which would make no difference to the universe as a whole.
Why did the blonde focus on an orange juice container? It said concentrate! She realized that societyβ s depictions of her were like the juice: formulaic, insipid, fake.
My wife told me to put the cat out. I didnβ t know it was on fire! By the time I could act, it was incinerated, a harbinger of the path we all must take.
How come the invisible man wasnβ t offered a job? They just couldnβ t see him doing it! This man stands for all of us: unseen, misunderstood, irrelevant.
Today I gave away my old batteriesβ ¦ Free of charge! No one wanted them, so I became angry and threw them in the yard. The battery acid now leaks into the soil, killing a colony of ants. A sparrow
eats their bodies and is poisoned. Somewhere in the Serengeti, a lion devours his rivalβ s cubs. Then the lion is shot by a poacher and sold to an unloved rich man whose father was an unloved rich
man. In five billion years, the Sun will become a bloated giant, boiling the oceans and consuming our pointless cruelties with flames. I wake sweat-drenched and screaming, staring at the visage of a
faceless god. β WHAT HAVE I DONE?! HOW COULD I BRING A CHILD INTO THIS WORLD!?β But this god, like all gods, is nothingβ just my sonβ s Wilson baseball mitt, sitting on my dresser, mocking me.
Will February March? No, but April May! Soon we become ash, and time forgets us.
Source: https://www.mcsweeneys.net/articles/nihilist-dad-jokes
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arraylg(x)=_1^x 2 t+7 d t \\ g^(9)=array - Asksia.ai
Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
Apply the Fundamental Theorem of Calculus Part 1 to differentiate the integral
step 2
According to the theorem, the derivative of the integral from a constant to x of a function is the function evaluated at x
step 3
So, $g'(x) = \frac{d}{dx} \int_{1}^{x} \sqrt{2t+7} \, dt = \sqrt{2x+7}$
step 4
Evaluate the derivative at $x=9$: $g'(9) = \sqrt{2(9)+7}$
step 5
Simplify the expression: $g'(9) = \sqrt{18+7} = \sqrt{25}$
step 6
Calculate the final value: $g'(9) = 5$
$g'(9) = 5$
Key Concept
Fundamental Theorem of Calculus Part 1
The derivative of the integral of a function from a constant to x is the function evaluated at x.
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9 - Initiators & Deleters
Go to the end to download the full example code. or to run this example in your browser via Binder
9 - Initiators & Deleters
So far we have provided a prior in all our examples, defining where we think our tracks will start. This has also been for a fixed number of tracks. In practice, targets may appear and disappear all
the time. This could be because they enter/exit the sensor’s field of view. The location/state of the targets’ birth may also be unknown and varying.
Simulating multiple targets
Here we’ll simulate multiple targets moving at a constant velocity. A Poisson distribution will be used to sample the number of new targets which are born at a particular timestep, and a simple draw
from a uniform distribution will be used to decide if a target will be removed. Each target will have a random position and velocity on birth.
from datetime import datetime
from datetime import timedelta
import numpy as np
from ordered_set import OrderedSet
from stonesoup.models.transition.linear import CombinedLinearGaussianTransitionModel, \
from stonesoup.types.groundtruth import GroundTruthPath, GroundTruthState
start_time = datetime.now().replace(microsecond=0)
truths = OrderedSet() # Truths across all time
current_truths = set() # Truths alive at current time
transition_model = CombinedLinearGaussianTransitionModel([ConstantVelocity(0.005),
timesteps = []
for k in range(20):
timesteps.append(start_time + timedelta(seconds=k))
# Death
for truth in current_truths.copy():
if np.random.rand() <= 0.05: # Death probability
# Update truths
for truth in current_truths:
transition_model.function(truth[-1], noise=True, time_interval=timedelta(seconds=1)),
# Birth
for _ in range(np.random.poisson(0.6)): # Birth probability
x, y = initial_position = np.random.rand(2) * [20, 20] # Range [0, 20] for x and y
x_vel, y_vel = (np.random.rand(2))*2 - 1 # Range [-1, 1] for x and y velocity
state = GroundTruthState([x, x_vel, y, y_vel], timestamp=timesteps[k])
# Add to truth set for current and for all timestamps
truth = GroundTruthPath([state])
from stonesoup.plotter import AnimatedPlotterly
plotter = AnimatedPlotterly(timesteps, tail_length=0.3)
plotter.plot_ground_truths(truths, [0, 2])
Generate Detections and Clutter
Next, generate detections with clutter just as in the previous tutorials, skipping over the truth paths that weren’t alive at the current time step.
from scipy.stats import uniform
from stonesoup.types.detection import TrueDetection
from stonesoup.types.detection import Clutter
from stonesoup.models.measurement.linear import LinearGaussian
measurement_model = LinearGaussian(
mapping=(0, 2),
noise_covar=np.array([[0.25, 0],
[0, 0.25]])
all_measurements = []
for k in range(20):
measurement_set = set()
timestamp = start_time + timedelta(seconds=k)
for truth in truths:
truth_state = truth[timestamp]
except IndexError:
# This truth not alive at this time.
# Generate actual detection from the state with a 10% chance that no detection is received.
if np.random.rand() <= 0.9:
# Generate actual detection from the state
measurement = measurement_model.function(truth_state, noise=True)
# Generate clutter at this time-step
truth_x = truth_state.state_vector[0]
truth_y = truth_state.state_vector[2]
for _ in range(np.random.randint(2)):
x = uniform.rvs(truth_x - 10, 20)
y = uniform.rvs(truth_y - 10, 20)
measurement_set.add(Clutter(np.array([[x], [y]]), timestamp=timestamp,
# Plot true detections and clutter.
plotter.plot_measurements(all_measurements, [0, 2])
Creating a Tracker
We’ll now create the tracker components as we did with the multi-target examples previously.
Creating a Deleter
Here we are going to create an error based deleter, which will delete any Track where trace of the covariance is over a certain threshold, i.e. when we have a high uncertainty. This simply requires a
threshold to be defined, which will depend on units and number of dimensions of your state vector. So the higher the threshold value, the longer tracks that haven’t been updated will remain.
Creating an Initiator
Here we are going to use a measurement based initiator, which will create a track from the unassociated Detection objects. A prior needs to be defined for the entire state but elements of the state
that are measured are replaced by state of the measurement, including the measurement’s uncertainty (noise covariance defined by the MeasurementModel). In this example, as our sensor measures
position (as defined in measurement model mapping attribute earlier), we only need to modify the values for the velocity and its variance.
As we are dealing with clutter, here we are going to be using a multi-measurement initiator. This requires that multiple measurements are added to a track before being initiated. In this example,
this initiator effectively runs a mini version of the same tracker, but you could use different components.
from stonesoup.types.state import GaussianState
from stonesoup.initiator.simple import MultiMeasurementInitiator
initiator = MultiMeasurementInitiator(
prior_state=GaussianState([[0], [0], [0], [0]], np.diag([0, 1, 0, 1])),
Running the Tracker
Loop through the predict, hypothesise, associate and update steps like before, but note on update which detections we’ve used at each time step. In each loop the deleter is called, returning tracks
that are to be removed. Then the initiator is called with the unassociated detections, by removing the associated detections from the full set. The order of the deletion and initiation is important,
so tracks that have just been created, aren’t deleted straight away. (The implementation below is the same as MultiTargetTracker)
Plot the resulting tracks.
Total running time of the script: (0 minutes 1.233 seconds)
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Show progress of dd command
posted on 2013-01-25
This was the third time I was going to do a dd of something without knowning how long it would take, so I Googled how to see the progress of dd which I could not find in the manual.
Turns out the answer is sending dd a signal to output progress. To send the INFO (USR1 in Linux) signal to dd, use pkill to send a signal to the first matched dd process:
pkill -USR1 -n -x dd
After sending the INFO/USR1 signal, dd will output the progress on stderr.
Even though this is not in the manual page (man dd), it is part of the info pages (info coreutils dd):
Sending an 'INFO' signal to a running `dd' process makes it print I/O statistics to standard error and then resume copying. In the example below, 'dd' is run in the background to copy 10 million
blocks. The 'kill' command makes it output intermediate I/O statistics, and when 'dd' completes normally or is killed by the 'SIGINT' signal, it outputs the final statistics.
To get a read-out every 30 seconds, I opened another terminal and started:
watch -n 30 'pkill -USR1 -n -x dd'
Update: archeydevil commented on the use of pidof and pointed out the preferred use of pkill. I've changed the examples above to reflect this. The old examples were:
kill -USR1 `pidof -s dd`
watch -n 30 "kill -USR1 `pidof -s dd`"
Please note that the new examples will check for the PID multiple times, so this means that if a new dd is started with another PID it will be picked up and sent USR1 signals (-n is short for
--newest). You could use kill in combination with pgrep to do a single lookup of the PID and give that to watch like so:
watch -n 30 "kill -USR1 `pgrep -x -n dd`"
Which would means that you would be sending a USR1 signal to any process taking the that PID number as soon as the kernel decides it needs it again. In hindsight I think sending it to new dd commands
is probably the best solution.
Update: I've been told that some terminals will send an INFO signal when you hit CTRL+T. However, I have not been able to reproduce this.
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Hands-On Sailor: How Sailboats Measure Up | Cruising World
Sailboats by the Numbers
Tim Barker
Boat reviewers rely on numbers to describe some of the key attributes of their subjects, such as length, beam, draft, and displacement. And while judgments on interior layouts and decor are
subjective, these figures describing dimensions are not. There are, however, other numbers commonly cited in spec boxes that can prove more elusive, since they attempt to put a numerical value on how
a sailboat might be expected to perform while under way. The commonly used ratios are sail area to displacement (SA/D), displacement to length (D/L), and ballast to displacement (B/D). And though
they’re so commonly used that a certain amount of dogma has accrued around them, these figures can, in fact, be misleading, or at least misunderstood. And the result is that a boat can be assigned
attributes based on numerical values that don’t take into account how sailboat design has changed over the past several decades.
Here, then, is a look at those ratios, what they attempt to describe, and how they should be interpreted when you go off exploring new and used models. (Click to page 2 for a more in-depth
Sail Area/Displacement (SA/D)**
An automobile buff seeking a high-performance ride looks for a high power-to-weight ratio and compares the horsepower/curb-weight ratios of different cars.
For a sailboat, the SA/D provides the same metric. The horsepower comes from the wind on the sails and is proportional to the sail area; a boat’s weight is its displacement (in pounds, kilograms, or
Initially, the SA/D only really gives a measure of potential acceleration rates (in case any physicists are reading this), but since displacement is a key factor in the resistance a boat encounters
when moving through the water, SA/D also has a bearing on potential maximum speed.
The traditional calculation for SA/D compares sail area in square feet to displacement in cubic feet. In the formula, displacement in pounds is divided by 64 (the density of seawater) to obtain cubic
feet, which are in turn converted to square feet to make the ratio unit-free.
On a spreadsheet, the formula would be S/(D/64)^(2/3).
Nominally, the higher the SA/D, the more lively the boat’s sailing performance. The vessel will accelerate more quickly and have the potential for higher speed.
But to be able to compare boats with any degree of precision (or fairness), we have to use similar numbers. The displacement must be in the same condition, either light ship (nothing on board) or
fully loaded, and the sail-area measurement must reflect the normal working sail plan. Racing boats have measurement certificates from which these numbers can be reliably extracted. The
specifications provided in cruising-boat brochures might not be consistent between builders, but we have to assume they are.
Boats measured in the 1970s and the 1980s for racing under the International Offshore Rule for the most part had SA/Ds between 16 and 17, based on the sum of the mainsail triangle (M = PE/2) and
100-percent foretriangle area (100%FT = IJ/2). The measurement system favored small mainsails and large headsails, and since designers of cruising boats stuck close to the IOR sail plan, the IOR
value for SA/D became the yardstick. An SA/D above 17 said “fast boat,” and anything below 16 said “slow boat.”
After the IOR fell out of favor, cruising-boat design drifted away from raceboat design, and sail plans began to change. Today, many boats are designed with large mainsails and small jibs, and most
builders publish a “total sail area” number that includes the standard jib (often as small as 105 percent) and the roach in the mainsail (which is significantly greater on modern boats with
full-battened mainsails than on IOR boats).
These builder-supplied numbers are more readily comparable against competing models, but using them in the SA/D formula makes the boats look “faster” than older models. This is a false comparison,
because the sail area used for the older boats doesn’t include the extra area in, say, a 150-percent genoa.
The table “Sailboats by the Numbers” (see page 79) illustrates this. It shows SA/Ds calculated for a selection of modern boats and boats from past eras, all about the same length, using different
numbers for sail area. For each model, it shows five SA/Ds. SA/D 1 is calculated using the sail area provided by the builder. SA/D 2 is calculated using M (PE/2) and 100% FT (IJ/2). SA/D 3 is
calculated using M + 105% jib. SA/D 4 is calculated using M + 135% jib. SA/D 5 is calculated using M + 150% jib. The only SA/D that includes mainsail roach is SA/D 1.
Let’s look at some examples. The 1997 Beneteau Oceanis 411 has a published sail area of 697 square feet on a displacement of 17,196 pounds. That gives an SA/D 1 of 16.7 (the same as SA/D 2), which
for decades was considered very respectable for a cruising boat.
In 2012, the current Beneteau Oceanis 41 has a published sail area of 902 square feet (453 mainsail + 449 jib) and a published displacement of 18,624 pounds, to give an SA/D 1 of 20.5. Wow!
Super-high performance! But this is for the standard sail area, with the 449-square-foot jib (just about 100% FT and typical of the trend today toward smaller jibs that tack easily). Plug in the
calculation using I, J, P, and E and SA/D 2 drops to 18.9 because it doesn’t include mainsail roach, which is about 16 percent of the total published mainsail area.
Go back to the 1997 model, tack on a standard-for-the-day 135-percent genoa, and the SA/D 4 becomes 20.7. (If we added in mainsail roach, typically about 11 percent of base mainsail area before
full-battened sails, we’d have 21.4.) The 1997 boat has essentially the same horsepower as the 2012 model.
Looking at current models from other builders, the SA/Ds based on published numbers hover around 20, suggesting that designers agree on the horsepower a cruising sailboat needs to generate adequate
performance to windward without frightening anyone.
The two boats in our chart that don’t at first appear to fit this model are the Hunter 39 and the Catalina 385, but they’re not really so far apart.
The Hunter’s SA/D 2 is 16.1. Its standard jib is 110 percent (327 square feet), and the rest of the published sail area is in the mainsail—664 square feet, of which 37 percent is roach!
Catalina is a little more traditional in its thinking. If you add the standard 135-percent genoa, the SA/D becomes 21.2—right in the ballpark. (It’s still there at 19.7 with a 120-percent genoa.)
The table shows that, for boats targeted at the “performance cruising” market, the SA/D numbers using actual sail area lie consistently around the 20 mark. To go above that number, you have to be
able to fly that sail area without reefing as soon as the wind ripples the surface. To do that, you have to elevate stability—with broad beam, lightweight (i.e., expensive) construction, deep bulb
keels, and fewer creature comforts.
Displacement/Length (D/L)**
While sailboat builders and buyers are interested in displacement in terms of weight, naval architects view it as volume; they’re creating three-dimensional shapes. When working in feet, to get a
displacement in pounds, they multiply cubic feet by 64, the density in pounds per cubic foot of seawater. (Freshwater boats displace more volume because the density of fresh water is only 62.4.) The
D/L ratio is therefore a measure of immersed volume per unit of length—how tubby the hull is below the waterline.
According to conventional wisdom and empirical studies, the lower the D/L, the higher the performance potential. This is mainly due to wavemaking resistance being lower for slender hulls than for
tubby hulls.
In the D/L formula, displacement in pounds is divided by 2,240 to convert it to tons to bring the values to manageable numbers, so D/L is displacement in tons divided by .01LWL (in feet) cubed.
In a spreadsheet, the formula would be D/(2240*(.01L)3), where D is the displacement in pounds and L is LWL in feet.
In the early days of fiberglass boats, the Cruising Club of America rule was the principal dictator of boat shapes. Because it was a waterline rule, designers kept waterlines short to keep ratings
low and relied on long stern overhangs immersing to add “sailing length” when the boats heeled. Carbon fiber was available only to NASA, and boats had full interiors, so “light displacement” wasn’t
really in the cards. A D/L of 300 was considered dashing, even risky. Many still-popular designs from the 1970s and 1980s have D/Ls as high as 400; see the Bounty II.
Fast-forward 40 years. Boats now have plumb bows and plumb sterns and waterlines almost as long as their LOAs—there are no rating penalties on a cruising boat. The boats’ weights haven’t changed much
because, although builders try to save weight to save cost, the boats are so much bigger. The hull and deck surface areas are greater, and all that extra internal volume can be filled with furniture.
The effect on D/L ratios has been drastic—just look at the table. A D/L ratio above 200 today describes a heffalump.
But do these lower D/Ls actually buy you any more speed? Yes and no.
Yes: Because speed is proportional to the square root of the waterline length. Today’s 40-footer has a much longer waterline than yesterday’s and ought to sail as fast as yesterday’s 50-footer. It
might also benefit from reduced resistance due to a smaller cross-sectional area, but it also might have greater wetted-surface drag due to the longer immersed length. When sailing downwind in waves,
though, the lower-D/L boat will surf more readily.
No: Because, as we saw above, the power-to-weight ratios (SA/D) of modern boats aren’t effectively any higher, and certainly aren’t in the realm that would allow our cruising sailboats to climb out
of the displacement zone and plane. In most conditions, the lower-D/L boat is still trapped in its wave.
In the days of the IOR, a D/L of 250 was still pretty racy; see the 1978 Catalina 38. Today, even a D/L as low as 150 doesn’t make a boat a speedster if it can’t carry the sail area to make it so. To
compete at a level with a Volvo 70, look for a D/L of about 40 and an SA/D of 65.
Ballast/Displacement (B/D)**
The ballast/displacement ratio is simply the ballast weight divided by the boat’s total displacement. Since ballast is there to give the boat stability, it’s easy to jump to the conclusion that the
higher the B/D, the stiffer the boat.
However, B/D doesn’t take into account the location of the ballast.
Take a boat that has a total displacement of 20,000 pounds and put its 8,000 pounds of ballast in the bilge. Now take the same boat and put the 8,000 pounds of ballast 4 feet deeper in a bulb at the
bottom of a deep fin keel. Same ballast ratio (0.4), but very different stability.
When looking at B/D, therefore, we must ask about the configuration of the keel: How low is the ballast?
Stability analysis is complex and involves beam, hull cross-section, and length, among other factors, of which B/D is just one.
Since the late 1990s, builders of sailboats intended for sale in the European Union have been required to provide stability data, including a curve of righting arm at angles of heel from 0 to 180
degrees—far more information than anyone can divine from a B/D number and a much more useful measure of a boat’s inclination to stay upside down in the unlikely event (the way most people use their
boats) that it exceeds its limit of positive stability.
CW contributing editor Jeremy McGeary is a seasoned yacht designer who’s worked in the naval-architecture offices of David Pedrick, Rodger Martin, and Yves-Marie Tanton and as a staff designer for
Camper & Nicholson.
To read the related article, How To: Measure Sail Area, click here.
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Geotechnical Engineering Calculators | List of Geotechnical Engineering Calculators
List of Geotechnical Engineering Calculators
Geotechnical Engineering calculators give you a list of online Geotechnical Engineering calculators. A tool perform calculations on the concepts and applications for Geotechnical Engineering
These calculators will be useful for everyone and save time with the complex procedure involved to obtain the calculation results. You can also download, share as well as print the list of
Geotechnical Engineering calculators with all the formulas.
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Notable Properties of Specific Numbers
First page . . . Back to page 15 . . . Forward to page 17 . . . Last page (page 25)
The number of ways to arrange a 2×2×2 Rubik's Cube (where whole-cube rotations and reflections are considered equivalent).
As there are no centre cubelets to determine the orientation, one corner is considered to have a fixed, defined location and orientation (for example, the red-yellow-blue corner is always kept in the
top-left-front position with blue on top and yellow on the front). The other 7 can be put into any of the 7!=5040 possible positions, and six of those can be rotated into any of 3 different rotations
(the last one's rotation is then determined, as the total rotation of all 8 pieces always adds up to 360o).
One in a series of crossover points in the level-index representation for numbers proposed by Lozier and Turner.
According to early Hindu mythology, the mahayuga or "great age" is a period of time consisting of four consecutive ages, lasting 1728000, 1296000, 864000 and 432000 years for a total of 4320000. They
placed themselves and all of humanity in the fourth of these ages, see 432000. The great age repeats many times; the longer periods in the Hindu cosmological calendar are described under
622080000000000. See also 8640000000.
This is 97 and is also the sum of 274 and 1623. See also 512.
This is the "original" Smith number, and was in fact the telephone number of someone named Smith. A Smith number is a number for which the sum of the digits is equal to the sum of the digits of its
prime factors: 4937775 = 3×5×5×65837, and 4+9+3+7+7+7+5 = 3+5+5+6+5+8+3+7. Numberphile has a video on it: 4937775 - Smith numbers. See also 22 and 1×1010694985.
This number when displayed on a calculator with 7-segment display, spells "BOOBIES" when viewed upside-down. It is sometimes said to be the first widely-known example of Calculator spelling, though
0.7734 is perhaps more credible. See also 176, 7702219, and 71077345.
A term in the Ramanujan α-series, defined on page 82 of Ramanujan's "Lost notebook". See 336365328016955757248 for details.
A term in the Ramanujan β-series, defined on page 82 of Ramanujan's "Lost notebook". See 336365328016955757248 for details.
The length (in metres) of the major (transverse) axis of the ellipsoid (or oblate spheroid) used by the WGS 84 model to approximate the shape of the Earth. This is very close to the average
equatorial radius of the Earth, if you measure based on where the gravitational field is equal to that at sea level. (The sea, being a fluid, tends to equalise its height profile such that gravity is
the same at all points on its surface, and the WGS 84 model is calibrated to agree with sea level as closely as possible). See the Geoid article for an explanation of how the geoid (the
"gravitational equipotential surface") differs from the actual surface of the Earth. Apart from following the sea height as just mentioned, it tends to be underground below any significantly elevated
land. Local changes of density in the mantle and crust add lots of variation.
If the earth were a sphere and the meter agreed exactly with its original definition, this would be exactly 20 million divided by pi.
This number is an exceptional counterexample to the abc conjecture. The abc conjecture states that, given two relatively prime numbers a and b, the sum of the distinct prime factors of a, b and of
their sum c=a+b, called rad(abc), is "almost always" bigger than c. For example when a=7 and b=33=27, c=34=2×17, which makes rad(abc)=2×3×7×17=714, quite a bit bigger than c. 6436343 is special
because it is so far in the other direction: a=310×109, b=2, c=235=6436343, and rad(abc)=2×3×23×109=15042, much less than c.
A term in the Ramanujan γ-series, defined on page 82 of Ramanujan's "Lost notebook". See 336365328016955757248 for details.
7129199 = 7×112×19×443, the first of a Ruth-Aaron pair in which neither number is squarefree. It is the smallest such number, i.e. the first in OEIS sequence A178214.
The number of demons, often attributed to Talmudic tradition: 1111 legions of 6666 demons each. See also 666, 44435622, 133306668, 399920004, and 1758064176.
When written in the UK style (horizontal strokes through the 7's, and a straight, not curved, vertical stroke for the 9) and held up to a mirror, this number looks fairly like "PISS OFF". See also
176, 22024, and 5318008
8114118 is a palindrome, and the 8114118th prime 143787341 is also a palindrome. This is the smallest such number, after a few early trivial cases (like 11 which is the 5th prime). The prime is a
member of A46941 and its index is in A46942. It was discovered by Carlos Rivera35, and is followed by 535252535.
The first counterexample to the classical conjecture that any number of the form 2P-1(2P-1), with P prime, is perfect. See 2047 and 496.
The digits of the most iconic phone number in the history of 1980's one-hit wonders. "867-5309/Jenny" was recorded by pop band Tommy Tutone in 1981 and was on the charts for some months thereafter.
For the xkcd version and a cool bonus, see 867.5309....
See also 525600, 10000000000, 0118 999 881 999 119 725 3, and 101010.
This is both a prime and a palindrome, the next-larger palindrome prime is 9136319. This would not be very special if it were not also for the fact that, in the digits of π, the digits 9136319 appear
starting at position 9128219.
The first of a set of 5 consecutive primes that are spaced an equal distance apart: 9843019, 9843049, 9843079, 9843109 and 9843139 are all prime, there are no primes in between, and the spacing
between each one and the next is 30. 9843019 is the lowest number with this property; the next is 37772429. See also 47, 251, 121174811 and 19252884016114523644357039386451.
107 appears in the definition of the vacuum permeability constant μ0, also called the "permeability of free space", in the curious formula:
where N is newtons and A is Amperes. Those are both long-established units in the SI system, so one might wonder where this 107 comes from.
A current formal definition of the ampere is "the constant current which will produce an attractive force of 2×10-7 newtons per metre of length between two straight, parallel conductors of infinite
length and negligible circular cross section placed one metre apart in a vacuum". There is that factor of 107 again, right in the definition of the ampere. Persuing this further goes right back to
the definition of μ0, a circular definition.
To reveal the origin of the 107 we have to look at the history of the ampere unit and the discovery of the force between two electric wires carrying current, a phenomenon first demonstrated by
Andre-Marie Ampere in 1820. The (historical) original definition of the modern unit is 1/10 of the unit now called an abampere, which in turn was "the amount of current which generates a force of two
dynes per centimetre of length between two wires one centimetre apart". A dyne is a g cm/s2, and a newton is a kg m/s2, so a dyne is 10-5 newtons. In the units that were common in Ampère's time, μ0
was simply 4π:
μ0 = (4π/107) N / A2 = (4π N) / (107 A2) = (4π 105 dyne) / (107 (abampere/10)2) = (4π dyne) / (102 (abampere/10)2) = 4π dyne / abampere2
So we see that the 107 in the modern definition of μ0 is a relic of the old centimetre gram second system of units. Converting from dynes to newtons diminished the value by 105; measuring the force
per meter of wire rather than per centimeter increased the value by 102; moving the wires from a distance of 1 cm to 1 meter canceled that 102 out, and measuring the current in amperes rather than
abamperes reduced the force by 102 because Ampère's force is proportional to the product of the currents in the two wires and both measurements change by a factor of 10 (the number of amperes in an
107=1,00,00,000 is a unit of the (Asian) Indian number name system. It is called crore when needed (primarily in Indian dialect of written English). In Iranian usage a crore is 500000. See also 10000
and 100000.
This is π④e, where ④ is the higher-valued form of the hyper4 operator, using my (somewhat arbitrary and speculative) generalisation of tetration to real arguments based on the error function erf(x)).
See also πe, 3581.875516... and 4341201053.37.
Tony Padilla's estimate of the number of times a person thinks of a number during their "waking life". The derivation is based on an average life-span of 73.2 years, multiplied by 365.25, 24 (hours
in a day), 2/3 (the fraction of one's time that one is awake), and 30 (based on the notion that people think of numbers an average of once every two minutes).
A constant appearing in the Chudnovsky series approximation of pi.
This is 224 and is equal to 2503+1003+503+303+63. Since all of those cubes except 63 end in 000, 216 shows up all by itself at the end of the number. See also 246924, 2097152 and 134217728.
A product of two non-overlapping sets of consecutive integers: 17297280 = 8×9×10×11×12×13×14 = 23×32×2×5×11×22×3×13×2×7 = 2×3×7×26×5×13×6×11 = 63×64×65×66. This type of match is is more "unlikely"
than that demonstrated by 19958400 because it requires more prime factors to work out right after rearranging. See also 720, 175560, and Sequence A064224.
Combined fuel economy of a Toyota Prius, in SI units (50 miles per gallon converted to meters (of distance traveled) per cubic meter (volume of fuel consumed)). See xkcd 687 and 3.1418708596056; see
also 137.035.
19958400 = 3 × 4 × (5×6×7×8×9×10×11) = (5×6×7×8×9×10×11) × 12 = 12! / 24. This is the product of the integers 3 through 11, and also the product of integers 5 through 12. There are an infinite number
of ways to construct a number with this sort of pattern, all of which have a similar form: two consecutive numbers at the beginning (in this example 3×4) get replaced by their product, an oblong
number (in this example 12), at the end. The general form is:
The sequence grows about as quickly as the factorials of the squares: 120, 19958400, 20274183401472000, 368406749739154248105984000000, ...
The length (in meters) of the IUGC standard meridian. This represents the length of a line from one pole of the Earth to the other (crossing the equator midway, i.e. at about the 10,002-kilometer
point). It is an international standard agreement, and is a sort of average of meridians at different longitudes75. The original definition of meter was based on the meridian and would have had this
number be exactly 20000000. The original determination of the meter's length, based a massive seven-year surveying project, established a meridian length that was too small.
Later improvements in understanding about the Earth's shape and extensive established use of the meter for non-surveying purposes made it necessary for the unit to diverge from its original
meridian-based definition. The total change in length of the meter through this process was about 195 parts per million. The meter ended up being a bit "shorter", and the initial meridian measurement
was too short (by a greater amount), so the average meridian is now known to be nearly 20,004 km. See also 1852.
The largest prime number with consecutive increasing digits. See also 4567.
The price in US dollars for a used book ("The Making of a Fly" by Peter Lawrence) that was briefly offered on Amazon by two rival used book sellers, after several weeks of automatic algorithmic
adjustment. The price escalated in repeated steps, each following a similar pattern: Seller A's algorithm checked once per day for any change in seller B's price, responding by adjusting its own
price to always be undercutting B by a factor of 0.9983; later in the day B's algorithm did something similar but aimed to always charge 1.270589 times as much as A. Both strategies, taken
individually, are plausible (B's aim-high strategy makes sense if B has a higher seller rating or a loyal customer base). Unfortunately for both, no stable price is possible because only the seller
who made the most recent price adjustment will be "satisfied" with the situation. A's and B's prices increased by a factor of about 1.2684 = 1.270589×0.9983 per day. This continued for about 7 or 8
weeks, the time needed to rise from a reasonable level for this title (about $20.00 when I checked) to the $23 million high. As related by Matt Parker, the price was 1730045.91 at the time that a
real person (a genetic scientist) first noticed the problem.
The number of seconds in common (non-leap) year: 365×86400. Although Leap seconds are called "intercalary", they are effectively part of the year because the leap second occurs during the day, local
time (for example, in a time zone 7 hours away from UTC, the clocks would go from "16:59:59" to "16:59:60" to "17:00:00") so a common year with a leap second would be 31536001 seconds long.
The number of SI seconds in a tropical year, according to xkcd 1061. The "Earth Standard Time" system, which is "simple, clearly defined, and unambiguous", defines a year by the following rules:
1 year = 12 months; 1 month = 30 days; 1 day = 1440 minutes (= 24 hours 4 minutes); 1 minute = 60 SI seconds. This gives 1 year = 31190400 SI seconds. For 4 hours every full moon, run clocks
backward. Full moons happen every synodic month, which is 29.530588853×86400.001 = 2551442.9... SI seconds. After going backwards for 4 hours, the clocks have to go 4 hours forward before continuing,
so 8 hours = 8×3600 = 28800 seconds will be added. This increases the average length of a year by a ratio of (2551442.9...+28800)/2551442.9... = 1.01128773..., making it 31542468.8... SI seconds. The
non-prime-numbered minutes of the first full non-reversed hour after a solstice or equinox happen twice. The 17th prime number is 59, so there are 43 non-prime minutes in an hour. There are two
solstices and two equinoxes per year, so this rule adds 43×60×4 = 10320 seconds to the year, for a year of 31552788.8... SI seconds.
An approximation of the number of SI seconds in a mean tropical year, as experienced during the years 1995-2012. This is based on the average rate of rotation of the Earth during that period (see
mean solar day) combined with the tropical year length for the year 2000 (which is in mean solar days). The figure for the length of the mean solar day has less precision because of the variations in
Earth rotation rate on short timescales49,125, due to weather and ocean currents, etc. whereas the year length figure represents an average over a period of several years.
The approximation to the number of seconds in a mean tropical year used in the 1956 and 1960 definitions of the SI second :
the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.
This number is related to Newcomb's solar motion coefficient as:
In words, the number of SI seconds in the mean tropical year multiplied by the Sun's mean rate of motion in arc-seconds per century is equal to the number of seconds in a Julian century times the
number of arc-seconds in a full circle.
This is the number of seconds per year according to the Gregorian calendar (averaged over a 400-year period): 365.2425 times 86400. It is an exact integer but is just an average; the number of
seconds in any particular year is always either 31536000 or 31622400.
Randall Munroe[224] found the approximation 754=31640625, which is a better approximation than the popular (among physicists) π×107 = 31415926.535... .
Number of seconds in a Julian year (often used in astronomical ephimerides, for things like proper motion of stars, orbital elements of planets, etc.).
The number of seconds in a leap year: 366×86400. Although Leap seconds are called "intercalary", they are effectively part of the year because the leap second occurs during the day, local time (for
example, in a time zone 7 hours away from UTC, the clocks would go from "16:59:59" to "16:59:60" to "17:00:00") so a leap year with a leap second would be 31622401 seconds long.
The last in a sequence of similar-looking prime numbers: 31, 331, 3331, ... are prime51. The following number in the series is not: 333333331=17×19607843. See also 73939133.
The largest triangular number of the form T(x2-1) that is also 6 times another triangular number; see 91.
A textbook on GIS systems for environmental modeling 129 contains a discussion of the problem of dealing with "liguistic hedges" (words and phrases such as acceptable and not terribly certain) for
purposes of data entry and database lookup. They contemplate converting back and forth between such words/phrases and "fuzzy sets" of quantitative values (statistical distributions on the interval
[0..1]). The general idea is that each word or phrase can be mapped onto a statistical distribution: for example, "above average" might correspond to a bell-curve-shaped distribution that is nonzero
between 0.5 and 0.8 with a peak at 0.65. They suggest that converting the other way (from statistics to representative words) could be done with a database that maps groups of fuzzy sets onto groups
of words/phrases. they dismiss this can be difficult:
"Not only is the notation difficult to encode, but there are 39,916,789 useful combinations of fuzzy sets in the range [0,1] for an interval of xi = 0.1."
The number cited here is 11!-11, the McCombinations of 11. Though it is clear how to define 11 fuzzy sets with peaks from 0.0 to 1.0 spaced 0.1 apart from each other, one may wish to speculate on how
the author thought that 11! was relevant to the problem, and why they deemed it necessary to subtract 11.
A "self-describing" number, like 3211000 and 521001000; see 6210001000 for more.
This is the number of demons in hell, as calculated by Johann Weyer (also spelled Wierus) and described in his 1583 book Pseudomonarchia Daemonum. The number is based on 6666 "legions" with 6666
demons each, all governed by 66 rulers similar to those in the Ars Goetia section of a mid-1600s spellbook Lemegeton Clavicula Salomonis. There are many versions of this, resulting in different
numbers of legions, etc. See also 666, 7405926, 133306668, 399920004, and 1758064176.
The maximum number of steps a 5-state, 5-tuple Turing machine can make, on an initially blank tape, before halting. This is called the "five-state busy beaver BB(5) by the S(n) function". It was
found by Buntrock and Marxen in 1989, and is listed in this paper where it is described as "current BB(5), step champion". See 107 for more.
This is the number of different ways that one can visit the state capitols of the 48 contiguous states in the United States, passing through each state only once. The same route in reverse does not
count as a distinct route, and one end of the trip must be in Maine because it only borders one other state. The answer, and a description of algorithms used to calculate it, are in Knuth [167]
section 7.1.4 (Binary Decision Diagrams), (p. 255 in the 2011 edition).
Type this on a calculator and read the display upside-down; it (sort of) says "SHELL OIL":
In the 1970's there were a bunch of joke "word problems" that instructed the reader to enter some sort of formula (example: 30 × 773 × 613 - 1 = × 5 =) to produce an answer that is read as a word by
holding the calculator upside-down. For this purpose the digits 0,1,2,3,4,5,7,8,9 were used to represent O, I, Z, E, H, S, L, B and G respectively, so the answer/punchline could be any word or phrase
using only these letters. See also 31337 and 5318008.
The "Tyson Code", better known as 0073735963 or 007-373-5963, a cheat code in the Nintendo videogame Mike Tyson's Punch-Out that takes the player directly to the final match against Tyson himself.
See also 573, 9001, and 1597463007.
This number is prime, and if you take one or more digits off the end, the resulting numbers 7393913, 739391, ... 73, 7 are all prime. This is the largest number with this property. See also 33333331,
381654729, 357686312646216567629137 and 3608528850368400786036725.
The number of milliseconds in a day: 86400000 = 24×60×60×1000. See also 10080, 40320, 432000 and 3628800.
It seems rather odd that such a large number is listed for two unrelated properties, but there are larger examples (see 18446744073709551615).
This is the smallest (positive) integer expressible as the sum of two (positive) cubes in three different ways: 1673+4363 = 2283+4233 = 2553+4143 = 87539319. See also 1729.
An astronomical unit in miles, calculated using the IAU definition of the former. The approximation "93 million miles" was commonly taught in the US. This number is precisely defined by agreement,
see here for details. See also light year.
A myriad myriad, and the largest number mentioned in the Bible (Hebrew תנ"ך (Tanakh) or Christian Old Testament): Daniel 7:10, "... and ten thousand times ten thousand stood before him, ..." (King
James version). It is probably not a coincidence that 108 was also the largest number for which the Greeks had a name; the book of Daniel reached its final form well after Alexander conquered the
entire Levant region. See also 666.
108 is 億 in China (yì, dàng) and Japan (oku), where they construct numerals on the basis of 10, 100, 10000, 108, and higher powers of 104. This system closely resembles the Knuth -yllion naming
system for very large powers of 10. (See also my list of large numbers in Japanese)
The number of DNA base pairs in the genome of the nematode worm C. elegans. See 959.
The number of "demons" in hell, as calculated by Alonso de Espina (see 399920004) and based on the notion that 1/3 of them were the fallen angels who thus became demons: 3×6666×6666. See also
7405926, 44435622, and 1758064176.
The first of a set of 6 consecutive primes that are spaced an equal distance apart: 121174811, 121174841, 121174871, 121174901, 121174931 and 121174961 are all prime, there are no primes in between,
and the spacing between each one and the next is 30. 121174811 is the lowest number with this property; it was first discovered in 1967 by L. J. Lander & T. R. Parkin. Along with 2, 3, 251 and
9843019, forms a sequence (Sloane's A6560) that is thought to be infinite, but it is very hard to discover the next one. No one has yet discovered the first set of 7 consecutive primes; such a set
would have to have a spacing of 210 or a multiple of 210; see 19252884016114523644357039386451. See also 47, 251 and 9843019.
Number of arc-seconds in a circle times 100. See 129602768.13.
Newcomb's coefficient giving the average rate of motion of the Sun across the sky (or equivalently, the rate of Earth's motion in its orbit, relative to the stars) in units of arc-seconds per
century. One might think this number should just be 129600000, but the Earth's axial precession and other effects prevent this.
This number, 227 or 233, is equal to this rather memorable sum of cubes: 5003+2003+1003+603+123. Another way to express this fact is:
ln((5322)3 + (5223)3 + (5222)3 + (3×4×5)3 + (3+4+5)3) = ln(2) 33
Scary but true: I actually discovered and verified this property of 227 by doing the math in my head. I already knew most of the powers of 2 up to 224=16777216. And, like tens of other kids around
the world, I learned the squares up to 202 and the cubes up to 123 in grade school. One day I decided to double 224 a few times to get 227, then noticed the 217728, which looks a lot like 216 and
1728 stuck together. It was then fairly easy to see the rest, since 134 is 125 plus 8 plus 1. See also 2097152.
This is a 9-digit number containing each of the digits 1 through 9, and equal to the sum 96+89+73+62+57+44+31+25+18, in which each of the digits occurs exactly once as a base and exactly once as an
exponent. Inder J. Tenaja calls numbers of this type "flexible power selfie numbers", and found a total of 25 of them (with 389645271 being the largest).
The astronomical unit in kilometers, based on the IAU definition of the former. See also 149597870691.
This number figures in an approximation of π that makes a puzzling appearance in the result (on certain calculators) of computing 116/13. The "approximation of π" in this case is (116×3600)/
(13×156158413) = 3.14159265358903895... Matt Parker has a video on this. The reason for the calculator presenting this answer is an unsolved mystery. This is not that great an approximation
considering how much is on the left-hand side: 18 digits and some symbols to get 13 digits of π. (RIES can do much better, admittedly not as a rational number, but we have continued fractions for
The number of integer partitions of 100, if identical parts are allowed. This is the more well-known version of the "partition numbers" is Sloane's A0041, starting: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30,
42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, ... For example there are 15 ways to make a sum equal to 7: 7, 6+1, 5+2, 5+1+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1,
2+2+1+1+1, 2+1+1+1+1+1, and 1+1+1+1+1+1+1.
In early 2009, one David Horvitz (an artist who enjoys posting unusual ideas on his blog) suggested that people should take a photo of themselves standing in front of a fridge or freezer with the
door open and their head in the freezer, then share it online (e.g. with Instagram or Flickr) tagged with the number 241543903. The idea caught on (becoming an internet meme) and an image search for
this number will now return dozens of such photos.
This number has 1008 distinct factors, and is the smallest number with at least 1000 factors. Its prime factorisation is 26×32×52×7×11×13×17. See also 12, 840, 1260, 10080, 45360, 720720, 3603600,
278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.
270270271 is prime, and is known to be a factor of 1010100+27. This seemingly amazing fact is actually quite easy to prove, using power-tower modulo reduction. Alpern 94 lists many such factors.
The smallest 9-digit number that, when written in three rows of 3 (as in one block of a Sudoku puzzle) forms a 3×3 magic square. There are 7 others: 294753618, 438951276, 492357816, 618753294,
672159834, 816357492, and 834159672.
An approximation to the speed of light hypothesised to be in Sayana's commentary on the Rigveda; see 2202 for details. See also 309467700.0.
299792458 is the speed of light in meters per second. In 1983 by international agreement, the meter was redefined in terms of the speed of light, and as a result the constant for the speed of light
is now exactly 299792458 meters per second. The second, in turn, is defined as precisely 9192631770 times the frequency of photons in a Caesium maser-based atomic clock. See also 2.54, 8.987552×1016,
1.6160×10-35 and 5.390×10-44.
The speed of light was first calculated from astronomical measurements in 1710 by Ole Romer, but had to be expressed as a ratio to the speed of Earth in its orbit (or equivalently, in terms of
certain unknown Solar System distances and known light travel times) because the size of the astronomical unit had not yet been determined to sufficient accuracy; this would not come until the late
1700's (see 149597870691 for more).
A meter is also just about equal to the length of a pendulum with a period of precisely two seconds (a seconds pendulum, the length is close to 994 millimeters). In fact, this definition was proposed
as the standard unit of length over 100 years before the original Metric system became official, and for most of the 18th century it was one of two competing proposals. The other proposal (based on
the size of the Earth) was chosen because the period of a pendulum depends on where it is measured. (See 20003931.4585 for more about the meridian measurement and its errors).
It is a strange coincidence that the gravitational acceleration at Earth's surface (9.8 meters per second2) times the length of Earth's year (about 31557600 seconds) is about 310000000 meters per
second, just a little bit bigger than the speed of light. There is no significance to this coincidence, it's just kind of cool. See also 3.14187.
The mean acceleration due to gravity on the Earth's surface, times the number of seconds in a mean tropical year. This happens to be only a few percent larger than the speed of light. This serves as
a guideline to some basic limits on long-duration manned space flight. Since astronauts would probably need to experience no more than about 1.1 or 1.2 times normal gravity during their trip, it
would take a few years (even from the astronaut's own relativistic frame of reference) to make the trip even to the nearest stars.
This is the smallest number that can be expressed as a×ba in three distinct ways: 344373768 = 8×98 = 3×4863 = 2×131222. See also 648.
The Mahler-Popken complexity metric (a sort of Kolmogorov complexity) counts how many 1's it takes to create an expression with a certain integer value, using just addition and multiplication (and
parentheses); see OEIS sequence A5245. For example MP(7)=6 because 7=(1+1)×(1+1+1)+1, which uses 6 1's R.K. Guy conjectured that for any prime p, MP(p)=1+MP(p-1). This turns out to be false, the
first counterexample being the prime 353942783: it is the sum of two composites, 2×3+37×9566021, which together add up to a complexity score of 63, the same as 353942782 = 18379×19258. Domotro has
made a great video about the Mahler-Popken metric.
This 9-digit number contains one each of the digits 1 through 9, and has the additional property that the first two digits (38) are a multiple of 2, the first 3 digits (381) are a multiple of 3, and
so on up to the whole thing being a multiple of 9. You can see a bit of symmetry in the digits: the first three digits (381) plus the last 3 (729) add up to 10×111, and the middle 3 (654) plus itself
in reverse (456) also adds up to 10×111. This type of number is called polydivisible, and this one is also pandigital in that it contains each digit (except 0) exactly once. There are lots of such
numbers if you don't care about having one each of the digits 1 through 9. See also 3816547290, 30000600003, and 3608528850368400786036725.
This is the largest number you can express with just two digits and possibly one symbol (99, 9 ^ 9 or 9③9). See also 4.28...×10369693099 and 101.0979×1019.
The number of angels as calculated by Alonso de Espina in his book Fortalitium Fidei, according to the derivation: 9 orders (or "choirs") of angels (according to Pseudo-Dionysius the Areopagite) each
consisting of 6666 legions, each containing 6666 individuals: 9×6666×6666 = 399920004. See also 7405926, 44435622, 133306668, and 1758064176.
This is Hω2+ω*2+2(4) and Hω2*2(3), where Hn() represents the n-indexed function in the Hardy hierarchy, one of the function hierarchies that arise in the study of large numbers and how to name them.
See 402653211 for more.
This number appears in the value of G(4), where G(n) represents the base of the Goodstein's Theorem iteration when the Goodstein sequences iteration ("strong" variant) reaches the value zero: G(4) =
3×2402653211 - 2. See that number for more.
456790123 has the "370-property": it is equal to the average of all possible permutations of its digits. Since there are 9 digits, there are 9! = 362880 permutations. That would take a really long
time to add up to take an average, but we can save a lot of work by noting that each digit occurs in each position an equal number of times. For example, the digit "4" will appear in each position in
exactly 1/9 of the permutations. This effectively means that we can compute the average much more quickly just by using one representative permutation with each digit in each possible position. In
this case, that can be done by computing:
(456790123 + 567901234 + 679012345 + 790123456 + 901234567 + 012345679 + 123456790 + 234567901 + 345679012) / 9
where the 9 terms are the original number rotated into all possible positions (like the multiples of 142857). If you take this sum (on a 10-digit calculator) you'll find that the average is equal to
the original number, 456790123. These numbers of this type (first pointed out to me by reader Claudio Meller) are discussed more fully on their own page.
A "self-describing" number, like 1210 and 42101000; see 6210001000 for more.
535252535 is a palindrome, and the 535252535th prime 11853735811 is also a palindrome. This is similar to 8114118 and was discovered by Giovanni Resta. The prime is a member of A46941 and its index
is in A46942 35
This is a power of 2, and a 9-digit number in which all 9 digits are different. There is no 10-digit power of an integer in which each of the digits 0 through 9 appears once. See also
Length of Earth's orbit in miles, based on this definition of the astronomical unit.
The smallest number expressible as the sum of two 4th powers in two different ways: 635318657 = 594+1584 = 1334+1344. It is sometimes called a Generalized taxicab number because of its shared
property with 1729. See also 50, 65, 1729, and 588522607645608.
The (false) Polya conjecture stated that positive integers with an odd number of prime factors always outnumber those with an even number of prime factors. In this case, the "number of prime factors"
is sequence A001222, in which the same prime can be counted twice (so for example 8=23, 12=22×3 and 30=2×3×5 are all counted as having 3 prime factors). But the conjecture turns out to be false in a
small region starting at 906150257 and extending up to 906488079.
The number of seconds from the 1st January 1970 until the 1st January 2001. This is 11323 days, i.e. (365×31+8)×86400 seconds, because 2001 is 31 years after 1970 and there were 8 leap years during
that period. The number appears as an offset in time/date calculations when converting between the UNIX epoch and the epoch used in the MacOS Cocoa framework ("Core Foundation"), and in application
that use it (such as sqlite3 running on a Mac). Both epochs use 00:00:00 GMT as the moment the counting starts, and ignore leap seconds. Cocoa defines the constant kCFAbsoluteTimeIntervalSince1970
equal to 978307200.0L
This is (1000-1)3 = 10003-3×10002+3×1000-1, and its reciprocal 1/997002999 = 0.000000 001 003 006 010 015 021 028 036 045 055... gives us the triangular numbers. This happens because the generating
function of that sequence is 1/(x-1)3. For more on this, see my separate article Fractions with Special Digit Sequences; see also 89, 99.9998, 199, 998, 9801, and 9899.
A billion in the short scale system used in the United States, and adopted by the UK and other English-speaking countries in the late 20th century. Many other countries use the short scale but have
milliard or a transliteration (such as Arabic milyar) as their name for 109. Other countries and languages (including Afrikaans, Farsi/Persian, most of continental Europe, and countries with earlier
history as European colonies) use the "long scale" in which a "billion" is 1012 and a "trillion" is 1018. Those with no Chuquet-derived names at all include the languages of India, China, and
Southeast Asia (some examples of non-Chuquet names are at 100000, 107, 1011, 1044, and 1059.)
The difference in meaning of "billion" (109 versus 1012) came into being at a time when it didn't matter to most people. But thanks to many factors (population growth, inflation, prosperity,
technology, and education) numbers in the billions are now very common in the news and in everyday speech. The reputation (whether good or bad) associated with the millionaire of the early 1900's now
belongs to the billionaire. We often hear of costs and deficits in the billions; many of our computers have billions of bytes of storage capacity and perform billions of operations per second.
109 is an estimate of the processing power (in floating-point operations per second) embodied in a human retina. The retinas perform image processing to detect such things as edge movement and
boundary direction. The figure is based on a resolution of roughly 106 pixels, a speed of 10 changes per second, and 100 FLOPs per pixel. See also 1018.
Most of the numbers of the form 10n+1 can be factored in simple and pretty ways; this one happens to have two such factorisations.66 Here are most of the simpler patterns:
form examples 103n+1 1001=11×91 1000001=101×9901 1000000001=1001×999001 1000000000001=10001×99990001 105n+1 100001=11×9091 10000000001=101×99009901 1000000000000001=1001×999000999001 107n+1 10000001=
11×909091 100000000000001=101×990099009901 102n+1+1 1001=11×91 100001=11×9091 10000001=11×909091 1000000001=11×90909091 104n+2+1 1000001=101×9901 10000000001=101×99009901 100000000000001=
101×990099009901 106n+3+1 1000000001=1001×999001 1000000000000001=1001×999000999001 108n+4+1 1000000000001=10001×99990001 100000000000000000001=10001×9999000099990001
As you can see, there are two different sets of patterns. As long as n is a multiple of an odd number, 10n+1 fits at least one of the patterns. The numbers excluded by this are of the form 102i+1:
11, 101, 10001, 100000001, 10000000000000001, etc. (Sloane's A80176, the "base 10 Fermat numbers"). There is no easy factorisation pattern for them. ([152] pp. 137-138)
A square in which each digit appears exactly once. (Contributed by Cyril Soler). See also 3816547290, 6210001000, 2504730781961, and 295147905179352825856.
This is the second example in a series of near-misses to Fermat's last theorem discovered by Ramanujan, of which 1729 is the famous first example. 1030301000 is 10103, and is just 1 greater than the
sum of 7913 and 8123. See this article and the 336365328016955757248 entry for details.
This is the decimal value of the hexadecimal integer constant 0x5f3759df that comprises the central mystery to the following bit of code, which is mildly famous among bit-bummers and purports to
compute the function f(x) = 1/√x:
This code actually works. It performs four floating-point multiplys, one floating-point add, an integer shift, an integer subtract, and two register moves (FP to Int and Int back to FP). It generates
the correct answer for the function to within three decimal places for all valid (non-negative) inputs except infinity and denormals.
The hex value 0x5f3759df is best understood as an IEEE floating-point number, in binary it is 0.10111110.01101110101100111011111. The exponent is 101111102, which is 190 in decimal, representing 2
(190-127) which is 263. The mantissa (after adding the hidden or implied leading 1 bit) is 1.011011101011001110111112, which is 1.43243014812469482421875 in decimal. So the magic constant 0x5f3759df
is 1.43243014812469482421875×263, which works out to the integer 13211836172961054720, or about 1.3211...×1019. This is (to a first-order approximation) close to the square root of 2127, which is
about 1.3043...×1019. The reason that is significant is that exponents in 32-bit IEEE representation are "excess-127". This, combined with the fact that the "exponent.mantissa" floating-point
representation crudely approximates a fixed-point representation of the logarithm of the number (with an added offset), means that you can approximate multiplication and division just by adding and
subtracting the integer form of floating-point numbers, and take a square root by dividing by two (which is just a right-shift). This only works when the sign is 0 (i.e. for positive floating-point
Here are some example values of numbers from 1.0 to 4.0 in IEEE single-precision:
0.10000001.00000000000000000000000 = 4.0 0.10000000.10000000000000000000000 = 3.0 0.10000000.00000000000000000000000 = 2.0 0.01111111.10000000000000000000000 = 1.5 0.01111111.00000000000000000000000
= 1.0
Here I have shown the sign, exponent and mantissa separated by dots. Since the logarithm of 1 is zero, the value for 1.0 (0.01111111.00000000000000000000000) can be treated as the "offset". If you
subtract this offset you get these values, which approximate the logarithm of each number:
0.00000010.00000000000000000000000 = 10.02 = 2.0; log2(4)=2 0.00000001.10000000000000000000000 = 1.12 = 1.5; log2(3)≈1.585 0.00000001.00000000000000000000000 = 1.02 = 1.0; log2(2)=1
0.00000000.10000000000000000000000 = 0.12 = 0.5; log2(1.5)≈0.585 0.00000000.00000000000000000000000 = 0.02 = 0.0; log2(1)=0
From this it is easy to see how a right-shift of the value for 4 yields the value for 2, which is exactly the square root of 4, and a right shift of the value for 2 gives the value for 1.5, which is
a bit higher than the square root of 2. Over a full range of input values, the right-shift and addition of the magic constant gives a "piecewise linear" approximation of 1/√x.
The constant "0x5f3759df" is most commonly cited as being found in the Q_rsqrt function of "game/code/q_math.c" in the source code of the videogame Quake III. It is attributed to John Carmack, but
the same hack appears in several earlier sources going as far back as 1974 PDP-11 UNIX.
David Eberly wrote a paper[178] describing how and why the approximation works.
Chris Lomont[182] followed up with investigation into its origins, getting as far as a claimed credit to Gary Tarolli of Nvidia. He thoroughly analyzes the piecewise linear approximation for odd and
even exponents and proposes 0x5f375a86 as being slightly better, and a similar constant 0x5fe6ec85e7de30da for use with 64-bit IEEE double precision.
David Eberly then wrote a longer explanation[213] analyzing the constant 0x5f3759df along with some other candidates (like 0x5f375a86 and 0x5f37642f). It describes efforts to discover why and how
this value originally got chosen; with inconclusive results.
An earlier example of code calculating the square root in this way (approximation via a single shift, possibly with an add or subtract, no conditional testing; but with no Newton iteration) was
described by Jim Blinn in 1997, where we find the following code: (see [165]).
This is actually pretty weird. We are shifting the floating-point parameter — exponent and fraction — right one bit. The low-order bit of the exponent shifts into the high-order bit of the fraction.
But it works. - Jim Blinn ([165] page 83)
The same article discusses several similar functions including ones that include one iteration of Newton's method. Here are his inverse square root functions:
If these are combined together into a single function with the inlines expanded, we get:
A much older example is found in the UNIX library sqrt function for the PDP-11, dating back to June 1974 (see [142]):
which is effectively performing an integer right-shift on the 16 high bits of the input value, then adding a constant similar to the constants in the above examples, and putting the result back into
a floating-point register before proceeding with the Newton's method calculations. Only the upper part of the mantissa is being shifted, but that's good enough. A man page from Feb 1973 (Third
Edition UNIX) suggests that the routine existed as early as then.
The number of demons in hell, by a later version of Johann Weyer's book Pseudomonarchia Daemonum (see 44435622). Here the organisation is 6 legions each with 66 cohorts each with 666 companies each
with 6666 members. See also 666, 7405926, 133306668, and 399920004.
4th in the "paperfolding sequence iteration interpreted as a growing sequence of binary numbers"; see 27876.
The number of seconds from the 1st January 1904 until the 1st January 1970. This is 24107 days, i.e. (365×66+17)×86400 seconds, because 1970 is 66 years after 1904 and there were 17 leap years during
that period (including 1904 itself). It is the offset between the UNIX epoch and the epoch used in the old "Classic" MacOS; see 978307200 and 3061152000 for more.
This number is associated with the UNIX epoch, which (on 32-bit systems) will "roll over" on 2038 Jan 18th. Numberphile has a video on it here: End of time (2147483647)
The number of seconds from the 1st January 1904 until the 1st January 2001. This is 35430 days, i.e. (365×97+25)×86400 seconds, because 2001 is 97 years after 1904 and there were 25 leap years during
that period (including 1904 itself). The number appears as an offset in time/date calculations when converting between the UNIX epoch and the epoch used in the old "Classic" MacOS. Both use 00:00:00
GMT as the moment the counting starts, and ignore leap seconds. The Cocoa / Core Foundation framework defines the constant kCFAbsoluteTimeIntervalSince1904 equal to 3061152000.0L
In late 2014 a Twitter friend and I undertook a challenge to find the smallest (integer, not starting with any 0's) number that does not appear in any Google search results (or, at the very least,
try to estimate how many digits it would have). The agreed rules stipulated that we should back up our claim with an actual number that (by demonstration) actually returns zero results from Google
Search (with the understanding that, once we revealed our result by e.g. Tweeting it publicly, it would soon lose its non-Google-able-number status).
Using Fermi Estimation (see Randall Munroe's what-if 84), I estimated that: there are 1010 people, each has 1 webpage, each with 1000 words; but only 1% of these are devoted to long lists of unique
numbers (like invoice numbers, telephone numbers, etc.), and probably 90% of them are either small and duplicate each other somewhat, or are big and leave gaps. Answer: the smallest integer not
indexed by Google is probably 10 digits long.
He and I spent a while trying numbers, and pretty quickly found that the 10-digit numbers seem to be almost all taken. 11-digit examples were easy to find. After just 10 minutes or so we had gotten
down to the very low 11 digits (my best was 10826746091, his was 11170063270).
He kept looking for 10-digit numbers, and noticed that there seem to be extensive lists of primes, but not of composites. He discovered that 6255626957 = 109×3803×15091 was unknown to Google, and
soon after found that the Marshall Islands have country code +625. (The islands have 7-digit phone numbers but only enough people to use a small fraction of them, thus offering a possible
explanation). Shortly after this, he and another had found 3112066128 = 24×3×64834711. (Internationally, +31 is The Netherlands but 9 digits must be added; within the U.S. 311 is an N11 code; so
there are no 10-digit telephone numbers starting with 311).
Clearly this number would be indexed soon after appearing on this page (and that indeed happened), so I would call it a "likely upper bound" for whatever number is actually the smallest positive
integer not in any Google result. Within a few years (of our contest, i.e. a few years after late 2014), perhaps all 10-digit numbers will have appeared somewhere.
3432948736 is the smallest number N such that N = 2N mod 10K, where K=10. In other words, 2 to the power of 3432948736 ends in the digits 3432948736. This is a member of a sequence (Sloane's A121319)
that is thought to be endless. It has the nice property that each member of the sequence adds a digit to the previous one. For example, 28736 ends in 8736, 248736 ends in 48736, 2948736 ends in
948736, and so on.
The only 10-digit pandigital polydivisible number in base 10: For each n from 1 to 10, the first n digits of this number, taken as an n-digit number, are divisible by n. For example, the first 3
digits are 381, and 381 is divisible by 3. The whole thing is divisible by 10 since it ends in 0, and any permutation of the 10 digits would be divisible by 9 since the sum the 9 digits is 45 which
is a multiple of 9. But the other divisibility requirements impose tight constraints. See 381654729 for more about the pattern in these digits. See also 6210001000, 30000600003,
3608528850368400786036725, and 101.845773452536×1025.
This is 6403202/96 and appears in the Chudnovsky series approximation of pi.
The Human population of the Earth according to the Arecibo message, which was transmitted in 1974. A more modern estimate is 6771000000. This is possibly the most dangerous number anyone has ever
sent in any communication, because as Cassiday notes77, "Aliens who correctly interpret this will know how large an army to send".
Number of base-pairs in the Human genome, as given77 by the Arecibo message. A more modern estimate is 5941000000.
The theoretical number of 32-bit IP addresses; the actual number is a few percent lower because some values are reserved for special purposes. See also 281474976710656.
First composite Fermat number. See here for more on these numbers; see also 17, 257, 641, (2222+1).
The number of years in the Hindu manvantara or "day of Brahma". See 1260 and 622080000000000.
This is e④π, where ④ is the higher-valued form of the hyper4 operator, using my (somewhat arbitrary and speculative) generalisation of tetration to real arguments based on the error function erf(x)).
See also eπ, 4979.003621... and 11058015.34616.
This number is equal to the sum of the 10th powers of each of its digits, and is unique in being the only 10-digit number to meet this requirement. Such numbers are called Armstrong numbers, Plus
Perfect numbers, or narcissistic numbers. See also 153, 1634, 3816547290, 6210001000, and 115132219018763992565095597973971522401.
The number of base-pairs in a haploid human genome counting 46 chromosomes (23 from each parent) and assuming that there is one X and one Y chromosome (i.e. a male individual).
The "self-describing number" described by Numberphile's James Grime in the video Maths Puzzle: The self descriptive number. It is the unique ten-digit number in which the first digit (6) tells how
many zeros the number has; the second digit (2) tells how many 1's, etc., viz.:
The digits in 6210001000 comprise 6 zeros, 2 ones, 1 two, 0 threes, 0 fours, 0 fives, 1 six, 0 sevens, 0 eights, and 0 nines."
One might think that searching for such a number would require checking all 9,000,000,000 ten-digit numbers; but that's not needed because the digits must sum up to 10. As James mentions in the
solution video, the search can be reduced even further by realising any solution must be one of the partitions of 10, of which there are only 42.
6210001000 isn't entirely unique in this regard: there are self-describing numbers with fewer digits: 1210, 2020, 21200, 3211000, 42101000, and 521001000.
Even more exotic is the "amicable pair" of 10-digit numbers: 6300000100 and 7101001000, found by Katie Steckles. Each describes the other.
13 factorial, the number of ways to rearrange 13 distinguishable objects. This number appears in some playing-card probabilities, such as 635013559600 and 2.235197...×1027. See also 1716.
Because 13 is 2×7-1, 13! is the magic constant for this "multiplicative" 7×7 magic square:
which is built on the principle of doing an elementwise multiplication (Hadamard product) on the following two components:
5 6 7 1 2 4 3 o 11 12 13 1 8 9 10
both of which satisfy the row, column, and diagonal requirements, but with repeated numbers. Is is quite efficient, in the sense that it uses 53.8% of the numbers from 1 to 7×13=91, or 67% of those
that remain after casting out all primes greater than 13.
Similarly to the "self-describing number" 6210001000, this number's digits describe the digits in 7101001000, whose digits similarly describe the digits in this number. (Found by Katie Steckles).
This is 29 primorial, 2×3×5×7×11×13×17×19×23×29 and has a really easy-to-remember digit pattern: 646 969 323 0. The pattern results from the properties of 1001=7×11×13 and 2001=3×667=3×23×29, which
multiplied together give 2003001, and 323=17×19.
World population as of 2009 July 16th, as estimated by the U.S. Census Bureau, from the Wikipedia page. Another somewhat higher estimate is given by this site.
Similarly to the "self-describing number" 6210001000, this number's digits describe the digits in 6300000100, whose digits similarly describe the digits in this number. (Found by Katie Steckles).
The first 10-digit prime number that appears as 10 consecutive digits of e:
e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003050353 5475945713 8217852516 6427427466 ...
This is the answer to a puzzle that appeared on billboards in 2004. The billboards stated:
{first 10 digit prime in consecutive digits of e} . com
This little bit of nerd sniping led the solver to another, harder puzzle also involving digits of e. That puzzle, if solved, brought the user to a website soliciting resumes, potentially resulting in
a call from someone at Google.
Alternate answer to the "first prime number in alphabetical order" question (see 8018018851).
This is the first prime number in alphabetical order in the English language: "eight billion eighteen million eighteen thousand eight hundred and fifty-one". It was found by Donald Knuth. All other
numbers that occur earlier in alphabetical order (like 8 and 8018018881) are composite. ([152] p. 15 footnote)
Neil Copeland has suggested32 that 8000000081 is the alphabetically first prime, based on the spelling "eight billion and eighty-one". The use of and is common outside the U.S. (I have confirmed
reports from the UK and New Zealand). Knuth, consistent with his statement in [147], does not use and.
The sixth perfect number. The even perfect numbers (it is not known if there are any odd perfect numbers) can all be expressed in the forms:
where P is a prime and N = P+1. In this example, P is 17. Also, for the number to be perfect, 2P-1 must be prime, and is called a Mersenne prime. See here for a complete list of known perfect
As discovered and described by Marius A. Burtea, this number is one of an infinite class of numbers that are both triangular and have the property of that any digit can be "moved to the denominator"
and the result is an integer (see 742). The triangular construction begins with a number n of the form 10(b+2) + (10b-1)*100/3 + 27. In this example b=3 and n=133327; it is always a 1 followed by b
3's followed by 27. Then we can make the triaigular number n(n+1)/2 which always comes out to (b+1) 8's followed by (b+1) 1's followed by 28. Any number ending in 128 is divisible by 8 (because 128
and 1000 are both divisible by 8), and the same is true for anything ending in 112, and you always get an even number when removing just one digit, so that makes it satisfy the "742" property. Burtea
also found two other infinite classes of numbers like this.
This is "Coulomb's constant", also called the "electric force constant" or "electrostatic constant", and is c2/107 N/A2 where c is the speed of light in metres per second, N and A are the units
newton and ampere. Since c is defined to be precisely 299792458, Coulomb's_constant is precisely 8987551787.3681764 N m2/(A2s2); the units are equivalent to metres per farad.
Frequency (in Hz) of microwave radiation used as the basis of the Caesium-133 atomic clock. This number is part of the official definition of the second (the basic unit of time). The atomic clock
technology was developed in the early 1950's and this number was adopted in 1967, with the wording "The second is the duration of 9,192,631,770 periods of the radiation corresponding to the
transition between the two hyperfine levels of the ground state of the caesium-133 atom." As of 1st May 2019, the wording was changed but the number stayed the same.
The length of the second is originally derived from the rotation of the Earth and time-division decisions by the Babylonians, among other things (see 86400). It was recognized during the 18th century
that the rotation rate of the Earth keeps changing. For example, using the period during 1750-1820 to define the average speed of Earth's rotation, and defining the second based on that, atomic
clocks would be about 60 or 70 seconds out of sync with the Earth after another 100 years49. This is about enough to account for a variation of about 100 in this number 9192631770, depending on when
and how the "standard second" is/was defined. Similarly, the number 299792458 that was for many years used to define the meter in terms of the speed of light would vary by about 2 or 3.
In 2019 the International System of Units (SI) was updated to define its seven base units in a way that defines all seven of them in terms of observable properties of nature, which are given
arbitrary numerical values in terms of the base units. As mentioned above, the second had already been defined this way (i.e. arbitrary unit second is defined in terms of a natural phenomenon of
Caesium-133). For an example of one that changed, see 1.602...×10-19.
Ten billion. This number appears in a Schoolhouse Rock! song; see 101010. See also 525600, 8675309, 1011, 0118 999 881 999 119 725 3, and 101010.
The upper limit of certain slide rule LL scales; see 22026.465794806.
The largest number that can be formed from the digits 1, 2 and 3 using the ordinary functions addition, multiplication and/or exponents. It slightly edges out 231=2147483648 because log(3)/log(2) is
greater than 31/21. The next number in this sequence is 101.0979×1019.
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Could old tide gauges help estimate past atmospheric variability?
Articles | Volume 20, issue 10
© Author(s) 2024. This work is distributed under the Creative Commons Attribution 4.0 License.
Could old tide gauges help estimate past atmospheric variability?
The surge residual is the non-tidal component of coastal sea level. It responds to the atmospheric circulation, including the direct effect of atmospheric pressure on the sea surface. Tide gauges
have been used to measure the sea level in coastal cities for centuries, with many records dating back to the 19th century or even earlier to times when direct pressure observations were scarce.
Therefore, these old tide gauge records may be used as indirect observations of sub-seasonal atmospheric variability that are complementary to other sensors such as barometers. To investigate this
claim, the present work relies on the tide gauge record of Brest, western France, and on the members of NOAA's 20th Century Reanalysis (20CRv3), which only assimilates surface pressure observations
and uses a numerical weather prediction model. Using simple statistical relationships between surge residuals and local atmospheric pressure, we show that the tide gauge record can help to reveal
part of the 19th century atmospheric variability that was uncaught by the pressure-observations-based reanalysis, advocating for the use of early tide gauge records to study past storms. In
particular, weighting the 80 reanalysis members based on tide gauge observations indicates that a large number of members seem unlikely, which induces corrections of several tens of hectopascals in
the Bay of Biscay. Comparisons with independent pressure observations shed light on the strengths and limitations of the methodology, particularly for the case of wind-driven surge residuals. This
calls for the future use of a mixed methodology between data-driven tools and physics-based modeling. Our methodology could be applied to use other types of independent observations (not just tide
gauges) as a means of weighting reanalysis ensemble members.
Received: 12 Dec 2023 – Discussion started: 02 Jan 2024 – Revised: 06 Aug 2024 – Accepted: 19 Aug 2024 – Published: 10 Oct 2024
Understanding the atmospheric system requires an understanding of all scales of variation from daily to centennial. This cannot be done unless long observation records allow the disentanglement of
these scales. The 20th Century Reanalysis Project, hereafter “20CR” (Compo et al., 2011), which is now in its third version, hereafter “20CRv3” (Slivinski et al., 2019), is the only atmospheric
reanalysis that runs through the 19th century. It relies on the International Surface Pressure Databank (Compo et al., 2019), the largest historical global collection of surface pressure
observations, and the NCEP Global Forecast System (GFS) coupled atmosphere–land model.
Because it is the longest atmospheric reanalysis available, 20CR is used to study possible long-term trends in atmospheric dynamics (Rodrigues et al., 2018) or for extreme events (Alvarez-Castro
et al., 2018). However, although the 20th century part of 20CR has been compared with other reanalyses (Wohland et al., 2019) and observations (Krueger et al., 2013), comparisons with independent
observations in the 19th century (Brönnimann et al., 2011) are scarce. The present work is an effort to compare this reanalysis with tide gauge observations. More generally, to the best of our
knowledge, this paper is the first attempt to use old tide gauges as indirect observations of the atmosphere. However, the opposite direction has been taken by Tadesse and Wahl (2021), who extended
storm surge reconstructions into the past using different atmospheric reanalysis products in order to estimate past unobserved extreme storm surges.
Tide gauges are used primarily to measure the tide, which is the largest contributor to sea level variations in many coastal cities. The astronomical tide is the result of gravitational attraction of
the Sun and Moon on the ocean combined with Earth's rotation. It results in the periodic rise and fall of the water level (Melchior, 1983) and has been predicted through harmonic decomposition for
centuries. Other physical phenomena impact the water level: a low atmospheric pressure results in a high sea level, a well-known approximation of which is the “inverse barometer effect” (Roden and
Rossby, 1999; Woodworth et al., 2019), and wind stress transport towards (away from) the coast leads to increased (decreased) sea level. These conditions are usually associated with storms, which is
why the associated sea level variations are called storm surges. For instance, in Brest (France), the amplitude of tidal variations is close to 4m, and storm surges can amount to as much as 1.5m.
Tide gauges are numerous, forming a dense global network in recent years and a sparser one over the past few centuries. As an example and from the GESLA-3 sea level database (Haigh et al., 2023),
10 coastal tide gauge records start before 1907 on the eastern coast of North America, while 20 start before 1900 in Europe. Old tide gauges have varying observation frequencies, from hourly (
Wöppelmann et al., 2006) to daily averages (Marcos et al., 2021). Although the sea level measured by tide gauges is only an indirect tracer of atmospheric pressure variability, the scarcity of direct
sea level pressure measurements motivates the use of tide gauges to study past atmospheric fluctuations. Indeed, even when pressure measurements exist, they are often not yet digitized and even less
available in global repositories (Brönnimann et al., 2019).
It is possible to link sea level variations with atmospheric phenomena using physical laws and models (Lazure and Dumas, 2008) or using statistical tools (Quintana et al., 2021; Pineau-Guillou et al.
, 2023; Harter et al., 2024). This work adopts the second approach, but the underlying physical phenomena will often be used to motivate and interpret the statistical models. Local linear regression
(LLR) will be used to relate the surge residual (see definition in Gregory et al., 2019) to local mean sea level pressure. Hidden Markov models (HMMs) will allow us to perform time smoothing of
probabilities given to members of 20CRv3, taking advantage of the time continuity of each member. The use of a hidden Markov model to smooth the weighting of individual members of a reanalysis based
on independent observations (here, tide gauge observations) was not reported elsewhere in the scientific literature. This general methodology could be used for other problems in order to assess and/
or enhance available reanalysis products.
Note that a recent study by Hawkins et al. (2023) used tide gauge records to check the ability of the 20CR reanalysis to correctly model storms, in particular with the addition of recently digitized
pressure observations. The study used a physics-based coastal model to estimate the storm surges associated with each member of the reanalysis compared to real observations. One conclusion of the
study is that the crude spatiotemporal resolution of the reanalysis is responsible for a systematic underestimation of the observed storm surges when using a direct physical coastal model forced by
20CR members. This justifies the use of statistical methods to quantify uncertainties in the relationship between reanalyzed pressures and real observed sea levels. The present study is thus a first
step towards using statistical models to assess reanalysis from tide gauge data.
The data and preprocessing are detailed in Sect. 2. Section 3 outlines the local linear regression and hidden Markov model used in this study. Section 4 shows the global consequences of applying our
methodology, while Sect. 5 focuses on four specific events and compares with independent pressure observations. Conclusions on the proposed methodology and experiments are drawn in Sect. 6, along
with potential applications of this work.
2.1The 20th Century Reanalysis version 3 (20CRv3)
The 20th Century Reanalysis Project (Compo et al., 2011) aims at producing a global atmospheric reanalysis ending in 2015 and extending back to the 19th century. The present paper uses the latest
version, 20CRv3 (Slivinski et al., 2019), which extends up to 1806. It is an atmospheric reanalysis with 80 members, using an ensemble Kalman filter data assimilation scheme (Evensen, 2003). It has a
temporal resolution of 3h and uses a spectral triangular model in space with truncation of T254 (approximately 75km at the Equator). There are 64 vertical levels up to 0.3mb. It assimilates
surface pressure observations from ships and fixed stations and analyzes cyclone-related IBTrACs data. These surface pressure observations are taken from the International Surface Pressure Databank
(ISPD), which was created for the 20CR project but also exists as an independent product (Compo et al., 2019). In 20CR, the sea surface temperature and sea ice cover are prescribed as boundary
conditions. Sea surface temperature and sea ice cover both benefit from satellite observations from 1981 to 2015 (the end of the reanalysis), allowing more precise boundary conditions.
The surface pressure observation density is considerably lower in the 19th century than in the late 20th century. An online platform (https://psl.noaa.gov/data/20CRv3_ISPD_obscounts_bymonth, last
access: July 2024) allows us to consult the monthly observation count per 2°×2° box. Figure 1 shows yearly averages of the number of surface pressure observations per day, comparing the years 1870
and 2000. The maximum value was set to 24 observations per day, although in 2000 this value is mostly exceeded. In 1870, approximately half of Europe's land surface has no observations at all, and
fewer than 10 points have more than 10 observations per day. Observations coming from ships allow us to raise the number of observations to approximately one per day in densely trafficked areas.
Conversely, in 2000 virtually all of western Europe's land area has more than 24 observations per day. Taking a spatial average over the whole map from Fig. 1 gives approximately 1 observation every
3d in 1870 versus 44 observations per day in 2000. The number of available observations is also highly variable through time, especially in the 19th century. For instance, in the 2°×2° box centered
on 49° latitude, −5° longitude, the number of monthly observations in 1870 ranges from 2 (January 1870) to 85 (May 1870), while in 2000 it ranges from 2152 (June 2000) to 3242 (May 2000).
2.2Preprocessing of mean sea level pressure
In this work, we only use the mean sea level pressure (MSLP) variable from 20CRv3. We perform two different preprocessing steps on this variable.
The first preprocessing step is used for the statistical relationship between the local pressure and the surge residual. As the latter is driven in part by a physical phenomenon called the inverse
barometer effect, which will be introduced in the next section, we consider the difference between the MSLP interpolated at the city of Brest (48.3829°N, 4.49504°W) and the MSLP averaged over all
members of 20CRv3 and over the North Atlantic Ocean (using the reanalysis' land mask and averaging from 0 to 69°N and from 98°W to 12°E), in a similar fashion to Ponte (1994). This spatially
averaged pressure is denoted as ${\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)$ and depends only on time. Note that there is a small variability in ocean-averaged pressure
between 20CRv3 members in the 19th century. However, we have checked that this variability is 1 order of magnitude smaller than the inter-member variability of MSLP at the city of Brest, which
justifies our approximation of using simply the member average of the ocean-averaged pressure as a reference.
A second preprocessing step of MSLP is used to compute the probability of transition from one member of the reanalysis to another in the hidden Markov model (HMM) presented in Sect. 3.2. For this
purpose, we consider seasonal anomalies of MSLP with respect to a climatology computed from the period 1847–1890 because the HMM is run only for those years. The reference MSLP climatology for
calendar day d and hour h is given by the average over days between d−30 and d+30, hours between h−3 and h+3, and all years 1847–1890. This reference MSLP is denoted as ${\stackrel{\mathrm{‾}}{\
mathrm{MSLP}}}^{\mathrm{clim}}$ and depends on latitude and longitude.
2.3Tide gauge of Brest (France)
In this study, the tide gauge of Brest is used as indirect tracer of atmospheric circulation through surge residuals. The Brest sea level record is taken from the GESLA-3 database starting in 1846
with hourly sampling. Apart from a few large gaps, the record is mostly continuous during periods 1847–1945 and 1953–present. This combination of historical and modern records is at the foundation of
the methodology explored in the next section.
2.4Preprocessing of sea level
As mentioned earlier, the part of the sea level that responds to atmospheric processes is the surge residual (see definition in Gregory et al., 2019). To access the surge residual, one has to remove
the tidal part of the signal. Following this, as we are interested in sub-seasonal variations, we also remove the yearly variations in the mean sea level (at interannual and decadal scales), such as
sea level rise (Cazenave and Llovel, 2010). In this work, we also use moving averages and differences in the surge residual. All of these steps are exemplified in Fig. 2.
We first compute the tidal constituents of the raw sea level (blue curve, Fig. 2a) using U-Tide (Codiga, 2011), which performs harmonic (Fourier) decomposition with prescribed frequencies
corresponding to planetary movements. The tidal constituents are computed over two different periods: 1847–1890 and 1981–2015. Removing the tidal part of the signal gives the surge residual (dashed
orange line in Fig. 2a), which has a temporal average value of ∼4m for the Brest tide gauge.
Following this, we remove the yearly median value of the sea level ( dashed orange line in Fig. 2b). We choose to remove the median and not the mean because the mean can in principle be influenced by
the number and magnitude of extremes in a given year, which can be linked to the number and magnitude of storms passing in a given year. This second step allows us to access the zero-median surge
residual which is denoted as h(t) in the following equation:
$\begin{array}{}\text{(1)}& h\left(t\right)=H\left(t\right)-{\mathrm{Tide}}_{H}\left(t\right)-\mathrm{median}\left[H\left({t}^{\prime }\right),{t}^{\prime }\in \mathrm{year}\left(t\right)\right],\end
where H(t) denotes the raw sea level, Tide[H](t) is the tidal part of the signal computed from H, and year(t) is the year in which time t is found.
Note from Fig. 2b that the surge residual fluctuates at an hourly scale, part of which is due to oscillations that are not due to variations in atmospheric pressure. For instance, these oscillations
can be due to tide–surge interactions (Horsburgh and Wilson, 2007) or measurement errors in the 19th century leading to phase shifts. Such 12h oscillations can dominate the surge residual signal in
Brest, where the tidal amplitude is large (see, for instance, 29 and 30 January 2014, Fig. 2). Furthermore, tide–surge interactions lead to stronger surge residuals at low tide and weaker surge
residuals at high tide (Horsburgh and Wilson, 2007). As these phenomena are not linked to atmospheric processes, we chose to filter them out with a simple 12h average (solid green curve in Fig. 2).
Given the spatial resolution of 20CRv3, smaller-scale events are likely to not be represented in the MSLP fields used in this study. In the following, we denote the 12h average of the zero-median
surge residual as ${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, which is calculated using the following equation:
$\begin{array}{}\text{(2)}& {\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)=\frac{\mathrm{1}}{\mathrm{12}}\sum _{{t}^{\prime }=-\mathrm{6}}^{{t}^{\prime
}=+\mathrm{6}}h\left(t+{t}^{\prime }\right).\end{array}$
Finally, note that if atmospheric pressure variations are faster than the typical time of adjustment of sea level, one expects deviations from the inverse barometer approximation (Bertin, 2016).
Therefore, fast time variations in the surge residual are also expected, statistically speaking, to be associated with deviations from the inverse barometer approximation. To allow the model
described in Sect. 3.1 to capture this effect, we compute the difference between the surge at time t and at time t−12h, choosing the 12h interval again to filter out oscillations at a period close
to 12h. Furthermore, since the reanalysis is run at 3h resolution, we perform a 3h moving average of the surge residual before computing the difference. This difference is denoted $\mathrm{\Delta
}{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ and defined by the following equation:
$\begin{array}{}\text{(3)}& \mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right):=\frac{\mathrm{1}}{\mathrm{3}}\sum _{{t}^{\prime }=-\mathrm
{2}}^{{t}^{\prime }=+\mathrm{1}}\left[h\left(t+{t}^{\prime }\right)-h\left(t-\mathrm{12}+{t}^{\prime }\right)\right].\end{array}$
2.5Independent historical pressure observations
In Sect. 5, we use pressure observations for the city of Brest and compare them with 20CRv3 and our estimate of pressure based on tide gauge observations and our statistical model (local linear
regression, LLR). We downloaded these observations from a repository (https://github.com/ed-hawkins/weather-rescue-data/tree/main, last access: July 2024) gathering historical pressure observations.
These pressure observations come from the EMULATE project (Ansell et al., 2006) for 1860–1880 and from Météo France archives for 1858–1860 and from 1880 on. The EMULATE dataset has a daily sampling,
while the Météo France archive dataset has a daily to thrice-daily sampling. These observations were not included in 20CRv3, and we did not use them to tune our model, they thus provide an
independent validation dataset.
We have found a shift in average pressure between the EMULATE and Météo France datasets. To overcome this issue, and since we are only interested in sub-seasonal atmospheric variability, we added a
constant value of ∼0.22hPa for the period 1860–1880 (EMULATE dataset) to each value of the independent pressure observation datasets so that the average pressure is equal between the independent
observed pressure and the 20CRv3 mean pressure linearly interpolated at the city of Brest. We did the same operation for the period covered by the Météo France dataset that we are using (1855–1859
and 1881–1894), adding a value of ∼7.18hPa.
3.1Local linear regression (LLR) between surge residuals and mean sea level pressure
To estimate the statistical relationship between surge residuals in Brest and 20CRv3 mean sea level pressure, we use the period 1981–2015, during which satellite data are used in 20CRv3 to constrain
sea surface temperature and sea ice cover, and the large number of pressure observations gives us high confidence in the 20CRv3 fields of mean sea level pressure (MSLP).
The filtered surge residuals described in Sect. 2.4 respond to sub-seasonal variations in atmospheric pressure. First, the sea level is sensitive to pressure variations. An approximation called the
“inverse barometer effect” (Roden and Rossby, 1999) states that an increase (decrease) of 1hPa in pressure at the mean sea level leads to a decrease (increase) in sea level of approximately 1cm.
This approximation is valid only for slow variations in atmospheric pressure compared to the typical time of dynamic adjustment of the sea level (Bertin, 2016).
Moreover, the piling up of water due to wind blowing perpendicular to the coast is responsible for positive (negative) surge residuals when the wind stress transport is directed towards (away from)
the coast. This effect depends nonlinearly on the wind amplitude and direction (Bryant and Akbar, 2016; Pineau-Guillou et al., 2018). Statistical anticorrelation observed in most regions between
wind-driven and pressure-driven sea level fluctuations causes regressions of surge residual versus atmospheric pressure to deviate from the inverse barometer approximation (Ponte, 1994).
Since wind is not included in our model, the relationship between the filtered surge residuals and the atmospheric pressures from 20CRv3 should not be deterministic. It is also likely that typical
wind conditions depend on the amplitude of the MSLP anomaly, meaning that the average value of MSLP anomaly for a given value of surge residual in Brest may be a nonlinear function. As shown by
Hawkins et al. (2023), using a physical coastal model forced by the values of pressure (and wind) from the 20CR can lead to biases in the estimation of associated surges due to the resolution of the
reanalysis. A statistical model can thus be used as a tool to correct such biases and represent uncertainties. In our case, since we want to estimate pressure based on the surge residuals only, the
effect of unknown wind or other processes must also be taken into account through uncertainty quantification.
Since our predictor variable is the sea level measured by the tide gauge, we will use two proxies to estimate the conditional probability distribution of pressure: ${\stackrel{\mathrm{‾}}{h}}^{\
mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ and $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$. We expect that
corresponding atmospheric pressure variations should be slow and moderate for low absolute values of these two predictors and that winds should be of low intensity, meaning that the inverse barometer
approximation should hold. For larger absolute values of $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, indicating rapidly changing
surge residuals and thus likely also rapidly changing atmospheric conditions, we expect deviations from the inverse barometer due to the dynamical adjustment of the sea level. Similarly, the largest
absolute values of ${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ are likely to be caused by the added contribution of wind to the effect of pressure,
meaning that deviations from the inverse barometer are expected as well.
To model all of these effects, we use a local linear regression (LLR in the following; see, e.g., Fan, 1993; Hansen, 2022), also called kernel regression (Takeda et al., 2007). More precisely, we
borrow our LLR from Lguensat et al. (2017). In such a model, we will search for similar values (neighbors) of the two predictor variables ${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule
{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ and $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ in the whole dataset and compute a linear
regression on this subset of the dataset. The predicted variable is $\mathrm{MSLP}\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)$, where MSLP(t) is the value of
the MSLP linearly interpolated at the city of Brest from the reanalysis.
We will assume that, conditionally to the values of ${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$ and $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\
mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, the predicted variable $\mathrm{MSLP}\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)$ follows a
Gaussian distribution:
$\begin{array}{}\text{(4)}& \mathrm{MSLP}\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)\sim \mathcal{N}\left(m\left(t\right),\mathrm{var}\left(t\right)\right).\
We then assume, following Lguensat et al. (2017), that the average m(t) and variance var(t) of this distribution can be estimated at each time step based on a local linear regression. To perform this
local regression, we search for the k-nearest neighbors of [${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, $\mathrm{\Delta }{\stackrel{\mathrm{‾}}
{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$] in the satellite era (1981–2015), where k is an integer set to 200 (other values have been tested and did not yield
improvement on the results). The nearest-neighbor criterion is the Euclidean distance in the two-dimensional space of values of [${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\
mathrm{h}}\left(t\right)$, $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$]. For each time t at which we want to estimate $\mathrm{MSLP}
\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)$, we thus find the set of times $\mathit{\left\{}{t}_{i},i\in I\left(t\right)\mathit{\right\}}$ where I(t) is an
ensemble of size K, for which the following distance is minimal:
{0.125em}{0ex}}\mathrm{h}}\left({t}_{i}\right)\right)+{\left(\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)-\mathrm{\Delta }{\stackrel{\
We attach a weight ω[i](t) to each index i∈I(t) according to the following formula:
$\begin{array}{}\text{(5)}& {\mathit{\omega }}_{i}\left(t\right)=\frac{\mathrm{exp}\left(-\mathrm{dist}{\left(t,{t}_{i}\right)}^{\mathrm{2}}/\mathit{\lambda }\left(t{\right)}^{\mathrm{2}}\right)}{\
sum _{j\in I\left(t\right)}\mathrm{exp}\left(-\mathrm{dist}{\left(t,{t}_{j}\right)}^{\mathrm{2}}/\mathit{\lambda }\left(t{\right)}^{\mathrm{2}}\right)},\end{array}$
where $\mathit{\lambda }\left(t\right):=\mathrm{median}\mathit{\left\{}\mathrm{dist}\left(t,{t}_{i}\right),i\in I\left(t\right)\mathit{\right\}}$ is defined as the median of the local values of the
distances to the nearest neighbors (Lguensat et al., 2017).
Using this subset of the whole dataset, we compute a weighted linear regression between the subset of regressors ${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left
({t}_{i}\right)$, $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left({t}_{i}\right)$ and of the predicted variable $\mathrm{MSLP}\left({t}_{i}\right)
-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left({t}_{i}\right)$ using the weights ω[i](t). This regression has two linear coefficients denoted as α(t) and β(t) and one intercept
(constant value) denoted as γ(t). Following this, the average is given by applying the local weighted linear model to the actual value of the predictors:
$\begin{array}{}\text{(6)}& m\left(t\right)=\mathit{\alpha }\left(t\right){\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)+\mathit{\beta }\left(t\right)\
mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)+\mathit{\gamma }\left(t\right),\end{array}$
while the variance is given by the weighted variance of the prediction error from the weighted linear model over the set of nearest neighbors:
To test the accuracy of this model on the 1980–2015 period, we apply it for all times t∈[1980–2015], searching for neighboring times t[i] in the same period but with the condition that there is a
minimum of 2 weeks between t and t[i] (i.e., excluding the interval [t−14d, t+14d]). This is called the leave-one-out procedure, ensuring that the data that are used to fit the model do not include
the true values. Following this, we compare the average m(t) with the true value $\mathrm{MSLP}\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)$ in a scatterplot
(Fig. 3a). This figure shows that the LLR is able to predict good average values m(t) for moderate absolute values of pressure difference, although it consistently underestimates the most extreme
values: this behavior is expected as the method is limited by the observations it has seen previously. However, as will be seen in Sect. 5, this simple model is still able to capture storms.
Following this, we test the adequacy of our variability estimate with the parameter var(t) by checking that the following shifted and rescaled variable follows a standard Gaussian distribution with
an average of 0 and variance of 1:
$\begin{array}{}\text{(8)}& \frac{\mathrm{MSLP}\left(t\right)-{\stackrel{\mathrm{‾}}{\mathrm{MSLP}}}^{\mathrm{ocean}}\left(t\right)-m\left(t\right)}{\sqrt{\mathrm{var}\left(t\right)}}.\end{array}$
To do so, we compare the empirical histogram of this variable with the probability density function of a standard Gaussian distribution, as shown in Fig. 3b. Although the shape of the histogram
slightly differs from a Gaussian probability density function (it is more peaked and has heavier tails), the agreement is satisfying enough for the purpose of this article. This shows that the
estimate of variance through var(t) is consistent with the real variability of the estimation process, which is the reason why we advocate for using a statistical method in the first place.
3.2Hidden Markov model (HMM)
In the 19th century, the spread between 20CRv3 members is much larger than in the period 1981–2015. One of the aims of this work is to estimate conditional probabilities of each member of the
reanalysis based on surge residuals in Brest. Note that in the reanalysis the members are assumed to have uniform probabilities, i.e., a probability of $\mathrm{1}/\mathrm{80}$, as we have
80 members.
One can estimate conditional probabilities of each member at time t based on the values of [${\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, $\mathrm{\
Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$]. To do that, we use the satellite-era-derived local linear regression expressed in Sect. 3.1. The
average m(t) and variance var(t) are estimated based on the procedure described in Sect. 3.1 and using the dataset from the period 1981–2015 to search for neighbors of [${\stackrel{\mathrm{‾}}{h}}^{\
mathrm{12}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$, $\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)$] and compute the LLR.
To differentiate these member probabilities from the ones we will derive later on using a hidden Markov model, we use the notation ${p}_{\overline{)\mathrm{HMM}}}\left(i,t\right)$ for the probability
of member i at time t.
We also use the convention that, in the absence of surge residuals observations, all members are given equal probabilities ${p}_{\overline{)\mathrm{HMM}}}\left(i,t\right)=\mathrm{1}/\mathrm{80}$.
Although these probabilities already bear significant information, they have the undesirable property of being time-discontinuous. This is not coherent with the fact that the members of 20CRv3 are
time-continuous: they are propagated in time using a Numerical Weather Prediction (NWP) model. To remedy this issue, we compute smoothed (or reanalyzed) probabilities using a hidden Markov model
(HMM) detailed below, which we write as p[HMM](i,t):
$\begin{array}{}\text{(10)}& {p}_{\mathrm{HMM}}\left(i,t\right):=P\left(\mathrm{member}\left(t\right)=i\mathrm{|}\left[\begin{array}{c}H\left(t=\mathrm{1}\right)\\ \mathrm{⋮}\\ H\left(t=T\right)\end
where one uses an observational record of surge residuals from time index 1 to T, and we use the simple notation $H\left(t\right):=\left[{\stackrel{\mathrm{‾}}{h}}^{\mathrm{12}\phantom{\rule{0.125em}
{0ex}}\mathrm{h}}\left(t\right),\mathrm{\Delta }{\stackrel{\mathrm{‾}}{h}}^{\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{h}}\left(t\right)\right]$ for the vector of surge residual average and
difference. Here, p[HMM](i,t) is a time-smoothed version of ${p}_{\overline{)\mathrm{HMM}}}\left(i,t\right)$ that takes into account past and future values of the surge residual. For this purpose, a
simple hidden Markov model (HMM) is used. The first ingredient of the HMM is the transition matrix T[ij](t) from member i at time t−1 to member j at time t.
$\begin{array}{}\text{(11)}& {\mathbf{T}}_{ij}\left(t\right):=P\left(\mathrm{member}\left(t\right)=j\mathrm{|}\mathrm{member}\left(t-\mathrm{1}\right)=i\right)\end{array}$
To estimate the transition matrix, a strong hypothesis is made:
$\begin{array}{}\text{(12)}& {\mathbf{T}}_{ij}\left(t\right)\propto {K}_{\mathit{\theta }}\left({\mathrm{MSLP}}_{\mathrm{map},j}\left(t\right),{\mathrm{MSLP}}_{\mathrm{map},i}\left(t\right)\right),\
where MSLP[map,i](t) is the ith member's map of mean sea level pressure in a squared box (28°N≤lat≤64°N, 18°W≤long≤18°E) at time t and ${K}_{\mathit{\theta }}\left(\cdot ,\cdot \right)$
is a positive real-valued function that measures the similarity between MSLP[map,i](t) and MSLP[map,j](t) and depends on the parameter θ.
Equation (12) states that transitions from one member to another are more likely if the associated MSLP maps at time t are similar. This prevents abrupt transitions to dissimilar atmospheric states.
The size and location of the map was chosen to cover an area inside which storms and anticyclones that affect the surge residuals in Brest would lie. Ideally, ${K}_{\mathit{\theta }}\left(\cdot ,\
cdot \right)$ should be symmetric and semi-definite. Here, a simple Gaussian kernel of Euclidean distances is used, with normalization factor θ>0, meaning that for two fields X and Y:
$\begin{array}{}\text{(13)}& {K}_{\mathit{\theta }}\left(X,Y\right)=\mathrm{exp}\left\{-\sum _{n\in \mathrm{lons}}\sum _{l\in \mathrm{lats}}\frac{{\left({X}_{nl}-{Y}_{nl}\right)}^{\mathrm{2}}}{{\
mathit{\theta }}^{\mathrm{2}}}\right\},\end{array}$
where the sum over n and l represents a sum over longitudes and latitudes. We then define Θ through the following equation:
$\begin{array}{}\text{(14)}& \frac{\mathit{\theta }}{\mathrm{\Theta }}=\stackrel{\mathrm{‾}}{{\left(\frac{\mathrm{1}}{\mathrm{80}}\sum _{i=\mathrm{1}}^{\mathrm{80}}\frac{\mathrm{1}}{T}\sum _{t=\
where the overbar denotes spatial average. This normalization will allow us to optimize θ through a grid search of Θ for a maximum likelihood of the surge residual observations.
One can compute T[ij](t) by setting a value of θ and using the hypothesis of Eq. (12), along with the fact that for all i and t we have $\sum _{j}{\mathbf{T}}_{ij}\left(t\right)=\mathrm{1}$. This
then allows us to estimate p[HMM](i,t) with the forward–backward algorithm (Rabiner, 1989).
$\begin{array}{}\text{(15)}& & {a}_{i}\left(t\right):=P\left(\left[\begin{array}{c}H\left(\mathrm{1}\right)\\ \mathrm{⋮}\\ H\left(t\right)\end{array}\right],\mathrm{member}\left(t\right)=i\right),\
text{(16)}& & {b}_{i}\left(t\right):=P\left(\left[\begin{array}{c}H\left(t+\mathrm{1}\right)\\ \mathrm{⋮}\\ H\left(T\right)\end{array}\right]\mathrm{|}\mathrm{member}\left(t\right)=i\right).\end
These two quantities can be computed recursively, following the forward procedure:
$\begin{array}{}\text{(17)}& & {a}_{i}\left(\mathrm{1}\right)={p}_{\overline{)\mathrm{HMM}}}\left(i,\mathrm{1}\right),\text{(18)}& & {a}_{i}\left(t+\mathrm{1}\right)={p}_{\overline{)\mathrm{HMM}}}\
left(i,t+\mathrm{1}\right)\sum _{j=\mathrm{1}}^{\mathrm{80}}{a}_{j}\left(t\right){\mathbf{T}}_{ji}\left(t\right),\end{array}$
and the backward procedure:
$\begin{array}{}\text{(19)}& & {b}_{i}\left(T\right)=\mathrm{1},\text{(20)}& & {b}_{i}\left(t\right)=\sum _{j=\mathrm{1}}^{\mathrm{80}}{b}_{j}\left(t+\mathrm{1}\right){\mathcal{T}}_{ij}\left(t\right)
Finally, this allows us to estimate p[HMM](i,t) by noting that
$\begin{array}{}\text{(21)}& {p}_{\mathrm{HMM}}\left(i,t\right)=\frac{P\left(\mathrm{member}\left(t\right)=i,\left[\begin{array}{c}{H}_{\mathrm{1}}\\ \mathrm{⋮}\\ {H}_{T}\end{array}\right]\right)}{P\
left(\left[\begin{array}{c}{H}_{\mathrm{1}}\\ \mathrm{⋮}\\ {H}_{T}\end{array}\right]\right)},\end{array}$
which gives, in terms of a[i](t) and b[i](t):
$\begin{array}{}\text{(22)}& {p}_{\mathrm{HMM}}\left(i,t\right)=\frac{{a}_{i}\left(t\right){b}_{i}\left(t\right)}{\sum _{j=\mathrm{1}}^{\mathrm{80}}{a}_{j}\left(t\right){b}_{j}\left(t\right)},\end
while keeping in mind that Eq. (22) implicitly relies on hypothesis (Eq. 12) and a fixed form of K[θ].
Comparing p[HMM](i,t) with the uniform distribution $p\left(i,t\right)=\frac{\mathrm{1}}{\mathrm{80}}$ allows us to see if the surge residual observations are coherent with the MSLP fields of 20CRv3
(Sect. 4) and to select the most relevant members given surge residual data (Sect. 5).
To choose the parameter θ, we performed a grid search of its normalized form Θ and computed the log likelihood of the surge residual observations as an output of the algorithm. Indeed, the log
likelihood l[θ](0…T) is expressed as follows:
$\begin{array}{}\text{(23)}& {l}_{\mathit{\theta }}\left(\mathrm{0}\phantom{\rule{0.125em}{0ex}}\mathrm{\dots }\phantom{\rule{0.125em}{0ex}}T\right)=\mathrm{log}\left(\sum _{i=\mathrm{1}}^{\mathrm
Figure 4 shows variations in this quantity with Θ for 1 year (1885) of surge residual observations in Brest (i.e., t=0 is 1 January 1885, while T is 1 January 1886). The curve shows a distinct
maximum around Θ≈0.09 and plateaus for higher values. According to Fig. 4, the difference in log likelihood between the model without HMM ($\mathit{\theta }=+\mathrm{\infty }$) and with HMM is close
to 1000. The introduction of one extra parameter in the filtering model compared to the static one is thus clearly justified if the two models are compared using standard criteria such as the Akaike
information criterion (AIC), the Bayesian information criterion (BIC), or likelihood ratio tests (Zucchini et al., 2017).
Note that in the limit $\mathrm{\Theta }=+\mathrm{\infty }$, we have a constant transition probability ${\mathbf{T}}_{ij}\left(t\right)=\frac{\mathrm{1}}{\mathrm{80}}$, and p[HMM](i,t) reduces to $
{p}_{\overline{)\mathrm{HMM}}}\left(i,t\right)$. Figure 4 thus supports the use of the HMM to estimate probabilities of the MSLP map conditioned by surge residual observations.
The choice of restricting the estimation of log likelihood to one arbitrary year (1885) is supported by the fact that estimation of T[ij](t) is computationally expansive. We assume that the optimal
value of θ generalizes well to other years. A better optimization of θ would necessitate further work that is out of the scope of this study. Setting Θ=0.09 will already enable us to find interesting
features of p[HMM](i,t).
4Modification of 20CRv3 ensemble when accounting for surge residuals
This section is devoted to the study of δμ[HMM](t), the difference between weighted and unweighted ensemble average, defined by
$\begin{array}{}\text{(24)}& \mathit{\delta }{\mathit{\mu }}_{\mathrm{HMM}}\left(t\right):=\sum _{i=\mathrm{1}}^{\mathrm{80}}\left({p}_{\mathrm{HMM}}\left(i,t\right)-\frac{\mathrm{1}}{\mathrm{80}}\
where MSLP[map,i](t) is a short notation for the sea level pressure field of 20CRv3's ith member. Here, $\mathit{\delta }{\mathit{\mu }}_{\overline{)\mathrm{HMM}}}\left(t\right)$ is defined
equivalently using ${p}_{\overline{)\mathrm{HMM}}}\left(i,t\right)$. This quantity shows how strong the average deviation is when taking into account surge residual observations. It will also
sometimes be normalized by σ[20CR](t), the estimated standard deviation of the unweighted ensemble:
$\begin{array}{}\text{(25)}& \begin{array}{rl}& {\mathit{\sigma }}_{\mathrm{20}\mathrm{CR}}\left(t\right):=\\ & {\left[\frac{\mathrm{1}}{\mathrm{79}}\sum _{i=\mathrm{1}}^{\mathrm{80}}{\left({\mathrm
{MSLP}}_{\mathrm{map},i}\left(t\right)-\frac{\mathrm{1}}{\mathrm{80}}\sum _{i=\mathrm{1}}^{\mathrm{80}}{\mathrm{MSLP}}_{\mathrm{map},i}\left(t\right)\right)}^{\mathrm{2}}\right]}^{\mathrm{1}/\mathrm
Note that in this definition σ[20CR](t) depends on time, latitude, and longitude. Therefore, at each grid point and for each time step the quantity δμ[HMM](t) will be normalized by a different value,
indicating the strength of the reanalysis ensemble spread at this location in time and space.
To further interpret the result of our HMM algorithm, we introduce the filtered effective ensemble size ν[HMM](t) (Liu, 1996):
$\begin{array}{}\text{(26)}& {\mathit{u }}_{\mathrm{HMM}}\left(t\right):=\frac{\mathrm{1}}{\sum _{i=\mathrm{1}}^{\mathrm{80}}{p}_{\mathrm{HMM}}\left(i,t{\right)}^{\mathrm{2}}},\end{array}$
and we equivalently define ${\mathit{u }}_{\overline{)\mathrm{HMM}}}\left(t\right)$. These quantities are estimates of the number of ensemble members that can be retained according to surge residual
observations, assuming one discards very unlikely members.
In Fig. 5, the variables δμ[HMM], $\mathit{\delta }{\mathit{\mu }}_{\mathrm{HMM}}/{\mathit{\sigma }}_{\mathrm{20}\mathrm{CR}}$, and ν[HMM] are shown as a function of time for the period 1846–1890.
All these quantities show a strong seasonality. This is due to a much stronger MSLP variability in winter and a corresponding stronger response of the surge residuals. The figure shows that the
amount of correction δμ[HMM] and the decrease in ensemble size ν[HMM] are much stronger using smoothed probabilities with HMM rather than probabilities without HMM. Showing the deviation δμ[HMM] in
the Bay of Biscay, where the standard deviation of $\mathit{\delta }{\mathit{\mu }}_{\mathrm{HMM}}/{\mathit{\sigma }}_{\mathrm{HMM}}$ is strongest (see Fig. 6), substantial absolute values of ∼600Pa
are obtained in early 1850s winters, even after averaging over 3 months. These large deviations correspond to more than 1 standard deviations of the ensemble size. Using probabilities without HMM,
deviations are weaker but still non-negligible (∼500Pa, ∼0.7σ). The slow decrease in δμ[HMM] with time is coherent with slowly increasing observations used in 20CRv3, albeit with substantial decadal
variations. However, $\mathit{\delta }{\mathit{\mu }}_{\overline{)\mathrm{HMM}}}/{\mathit{\sigma }}_{\mathrm{HMM}}$ and $\mathit{\delta }{\mathit{\mu }}_{\mathrm{HMM}}/{\mathit{\sigma }}_{\mathrm
{HMM}}$ do not show a clear trend, indicating a persisting gain in information from surge residual observations throughout the 19th century.
In terms of effective size, Fig. 5 shows that the smoothing HMM algorithm imposes a strong member selection, with mostly only one member retained at each time step in winter and before 1880.
Probabilities without HMM mostly retain more than half of the members, although peak low values of ${\mathit{u }}_{\overline{)\mathrm{HMM}}}\left(t\right)$ show that even without the HMM sometimes
more than half of the ensemble members are highly unlikely. Filtered effective ensemble size reaches very low yearly and seasonal average values, indicating that many 20CR members are highly unlikely
with respect to surge residual estimates from tide gauge observations. A strong increase in ν[HMM](t) is witnessed around year 1880. This can be explained by the availability of a large number of
weather station data in eastern Europe and Russia from 1880 on and by an intensification of maritime traffic around 1880.
The spatial structure of δμ is examined in Fig. 6. The analysis of time standard deviation of $\mathit{\delta }{\mathit{\mu }}_{\overline{)\mathrm{HMM}}}$ and $\mathit{\delta }{\mathit{\mu }}_{\
mathrm{HMM}}/{\mathit{\sigma }}_{\mathrm{20}\mathrm{CR}}$ shows that the area of greatest influence of the corrections from surge residual smoothing from Brest tide gauge is in the Bay of Biscay.
This can be explained by the passage of strong storms in the Bay of Biscay, which can cause high surge residuals in Brest, and by the sparsity of direct pressure measurements (ship logs) in this area
in the 19th century. The standard deviation of δμ[HMM] shows the largest values to the northwest of the map, which is where strong storms travel. Indeed, the variability of MSLP shows a great
northwest gradient, as can be seen from maps of time standard deviation of 20CRv3 mean MSLP (not shown). Noticeably, the size of the area of influence of δμ[HMM] is smaller in 1880–1890, which can be
explained by a greater conditioning of 20CRv3 members by observations both offshore and inland. In cases of very sparse observations used in 20CRv3, the area of influence of these corrections widens
due to continuity of MSLP fields. Note also that the area of influence is greater for δμ[HMM] then for $\mathit{\delta }{\mathit{\mu }}_{\overline{)\mathrm{HMM}}}$ because of the time propagation of
corrections thanks to the smoothing HMM algorithm. Finally, Fig. 6 confirms the large difference in amplitude of deviations between pre-1880 and post-1880 corrections already witnessed in Fig. 5.
Similar spatial footprints can be witnessed from maps of high and low quantiles of δμ but with different values (not shown). Similarly, computing the time standard deviations as in Fig. 6 but
restricting the times used for computation to April–September rather than October–March shows the same spatial pattern but with much lower values (not shown).
These corrections also have a strong decadal variation, with non-trivial yearly averages persisting for several years, as shown in Fig. 7. The same behavior can be witnessed for the surge residual,
which is strongly anti-correlated to these deviations (Fig. 7). This can be explained by the fact that 20CRv3 smooths MSLP values in areas of sparse measurements and that surge residual filtering
corrections allow us to retrieve more realistic intense values (either positive or negative). This interannual variability is related to the variability in storminess (Bärring and Fortuniak, 2009).
5Focus on four 19th-century events
One of the aim of this study is to show that old tide gauge data can be used to better understand past severe storms. In this section, two storms and one mild situation are studied for illustrative
To better understand the more general context of the three events studied in this section, we first look at longer time periods (100d) surrounding the events and compare the results of the simple
LLR based on surge residuals with 20CRv3 and independent observations (when available). These are plotted in Fig. 8. One can thus see that the uncertainties associated with the surge-residual-based
LLR do not vary from year to year, while those of 20CRv3 decrease. More precisely, the ratio of the standard deviation of the LLR divided by that of 20CRv3 has an average value of 1.22 in 1847,
1.54 in 1865, 1.95 in 1876, and 2.45 in 1888. That same ratio has a minimum value of 0.47 both in 1847 and in 1865, while its minimum value is of 0.69 in 1876 and 0.94 in 1888. This shows that on
average the reanalysis has lower uncertainty than the surge-residual-based pressure estimate (albeit with fluctuations), and in 1888 the uncertainty of the reanalysis is always smaller. Comparison
with observations also confirms the better precision of the reanalysis.
In 1865 (Fig. 8b), although the surge-residual-based reconstruction is sometimes more consistent with observations than the reanalysis, there are as many occasions where it is the reanalysis that is
more consistent with the independent observations. In 1876 (Fig. 8c), biases of ∼5hPa between the LLR and 20CRv3 are found most of the time. For all four periods shown, the reanalysis and the LLR
pressure estimates show consistent variations in time, albeit with persistent biases (either positive or negative) that last from a few days to ∼15d. We attribute these biases to different
atmospheric conditions that cannot be estimated from the surge residuals with our simple LLR model, in particular wind directions and intensity. These examples show that the results of our algorithm
must be interpreted with care and that a more in-depth analysis is needed to understand the specifics of an individual event.
Our claim that the wind variations are responsible for the persistent biases between the LLR pressure estimation and the reanalysis is supported by Fig. 12f, where we also show the direction and
amplitude of the 10m wind intensity as given by the average over all reanalysis members and interpolated at the city of Brest. In March 1876, two low-pressure systems passed to the north of Brest's
tide gauge, the first around 10 March and second around 12 March, as indicated by the reanalysis members and the independent pressure observations (Fig. 12e). However, the first low-pressure system
did not induce a surge residual as strong as the second one. One key difference between the two events is the wind amplitude, which reached 15ms^−1 during the first event and then decreased to
5–10ms^−1 during the second event, with almost steady wind direction. Although wind intensity and direction estimated from the reanalysis must be taken with care, the value of 15ms^−1 is rarely
exceeded (only 7 in 1000 times in the period 1981–2015, not shown), indicating exceptional wind intensity during the event and justifying the inaccuracy of the LLR, which is based on already observed
events and therefore biased towards typical wind conditions. Our interpretation relies on the fact that the effect of wind on extreme surge residuals acts at small timescales (daily or sub-daily),
which is backed by recent work (Pineau-Guillou et al., 2023).
To aid the interpretation of Figs. 10–13, in Fig. 9 we also show the number of observations assimilated in 20CRv3 in the months of the studied events. In November 1847, observations mostly come from
ground stations, indicated by green–blue squares (more than 1 observation per day). In November 1865, some more stations are available, and at this point the observation density from maritime traffic
also grows. In March 1876 and August 1888, the number of observations surrounding Brest increases with respect to 1865 mostly due to an intensification of maritime traffic, although some new stations
also constrain the reanalysis, but these are not in the direct vicinity of Brest.
One common feature of Fig. 10–12 is that the HMM algorithm tends to be very selective compared to the weighing without HMM. This is the consequence of our optimization of the parameter θ with the
objective of maximizing the likelihood of the surge residual observations on the 20CRv3 ensemble. Having a low theta value allows us to give a high weight to the ensemble member that has the highest
probability according to the surge-residual-based LLR model. However, as is obvious from comparing Fig. 10a with b, Fig. 11a with b, and Fig. 12a with b, this does not always have a strong influence
on the average MSLP field. In the case of Fig. 13, the variability between members of the reanalysis is smaller, and therefore the selection of ensemble members is less acute, with more reasonable
effective ensemble sizes. However, these figures again show that small effective ensemble sizes should not be interpreted as a justification for discarding members with low probability according to
the HMM algorithm but rather as a means to quantify the relative agreement of individual members with the surge residual observations according to the LLR statistical relationship.
The fact that one member is often much more coherent with the series of surge residuals is the result of (1) the high variability of the ensemble and the LLR pressure estimation, (2) the high
dimensionality (or complexity) of the problem, (3) the low size of the ensemble (80 members), and finally (4) the systematic biases between 20CRv3 and the LLR caused by unmodeled atmospheric
conditions (winds). Indeed, in the case of data scarcity, the variability of the reanalysis is large (point 1), and a fixed-size ensemble (point 2) may struggle to correctly span the whole space of
possible atmospheric circulations (point 3), meaning that a few members are actually much closer to the true atmospheric circulation than all other members. Such a problem is called filter degeneracy
(Snyder et al., 2008) and is a common issue in ensemble-based data assimilation schemes. Secondly, since our LLR estimation of pressure experiences time-correlated biases with respect to the 20CRv3
because of unmodeled other variables (winds), this causes the HMM to select the one member that is closest to the biased pressure estimate from the LLR applied to the surge residual signal. All these
issues may remain for other climate science applications if one uses a similar approach of merging independent observations with a HMM algorithm to weight ensemble members.
6Conclusion and perspectives
This study is a proof of concept for the use of century-old tide gauge data as a means of understanding past atmospheric subseasonal variability. Surge residuals of Brest allow us to assess part of
the atmospheric variability that was uncaught in global 20CR reanalyses based on pressure observations. Weighing 20CR members according to surge residual observations reduces the effective ensemble
size and implies significant deviations in member-averaged sea level pressure in the Bay of Biscay. Through the second half of the 19th century these deviations diminish and the effective ensemble
size rises; however, they remain non-negligible. Independent pressure observations in the city of Brest are coherent with pressure estimations from the reanalysis and the surge-residual-based local
linear relationship. Such comparisons also show that the reconstruction of pressure based on surge residuals is ambiguous due to the influence of winds, meaning that biases between the
surge-residual-based and reanalysis-based pressure estimates can last for several days.
This work has several potential applications. First, replicating this work with other tide gauges could help us to validate reanalyses like 20CRv3 against independent data and to potentially identify
anomalous trends or incorrect estimations of specific events. Combining our statistical approach with the physics-based approach of Hawkins et al. (2023) could allow us to have both a precise
estimate from a high-fidelity coastal model and a good quantification of uncertainties. Second, tide gauges could be used to constrain regional-scale atmospheric simulations in order to better
estimate the magnitude and spatial extent of known past severe storms. Third, tide gauge records could be combined with direct observations of atmospheric pressure to give statistical estimates of
atmospheric fluctuations in the 19th century without the use of an NWP model, such as the optimal interpolation of Ansell et al. (2006) based on direct pressure observations only or the analogue
upscaling of Yiou et al. (2014) for the short period 1781–1785 of dense observations in western Europe. Finally, this work could be replicated in a more general context using other types of variables
and observations, learning the relationship between observations and large-scale features using recent observations and precise reanalyses, and applying these statistical relationship in the past to
uncover past large-scale events. In particular, the hidden Markov model algorithm outlined here could be replicated to weigh ensemble members according to independent observations.
The codes used to produce the figures in this article are available upon request from the authors.
Conceptualization: PP and BC. Methodology and software: PP and PA. Investigation: PP, BC, PA, and PT. Writing – original draft preparation: PP. Writing – review and editing: PP, PA, PT, and BC.
Funding acquisition: BC.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation
in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
This work has greatly benefited from the suggestions of two anonymous referees, who we thank here. We thank Simon Barbot for discussions on surge residuals. We thank Jean-Marc Delouis for fruitful
discussions and suggestions regarding our work. Finally, many thanks are owed to Ed Hawkins for contacting us during the open-discussion phase, providing information on newly available pressure data,
and giving us feedback on the idea of using tide gauges as external validation data for atmospheric reanalysis. Support for the 20th Century Reanalysis Project version 3 dataset is provided by the
US Department of Energy, Office of Science Biological and Environmental Research (BER); by the National Oceanic and Atmospheric Administration Climate Program Office; and by the NOAA Physical
Sciences Laboratory.
This research has been supported by the European Research Council (grant no. 856408).
This paper was edited by Alessio Rovere and reviewed by two anonymous referees.
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C Program to Convert Binary to Octal
• C Programming Examples
• C if-else & Loop Programs
• C Conversion programs
• C Pattern Programs
• C Array Programs
• C String Programs
• C File Programs
• C Misc Programs
• C Programming Tutorial
C Program to Convert Binary to Octal
In this article, we will learn how to create a program in C that can convert any given binary number (provided by the user at run-time) into its equivalent octal number. At last, we have also created
a function named BinToOct() that does the same job.
But before going through the program, if you are not aware of
• Binary Number
• Octal Number
• Binary to Octal Conversion
then refer to the step-by-step process for binary to octal conversion. Now let's move on to the program.
Binary to octal in C
The question is, "Write a program in C that converts any given binary number into its equivalent octal value." The answer to this question is:
int main()
int bin, oct=0, i=0, mul=1, count=1, rem, octnum[20];
printf("Enter any Binary Number: ");
scanf("%d", &bin);
rem = bin%10;
oct = oct + (rem*mul);
octnum[i] = oct;
mul = 1;
oct = 0;
count = 1;
mul = mul*2;
bin = bin/10;
octnum[i] = oct;
printf("\nEquivalent Octal Value = ");
for(i=i; i>=0; i--)
printf("%d", octnum[i]);
return 0;
As the above program was written in the Code::Blocks IDE, here is the first snapshot of the sample run:
Now supply any binary number, say 1101110, as input and press the ENTER key to see its equivalent octal value as shown in the second snapshot of the sample run given below:
Here is the final snapshot of another sample run:
Program Explained
• Receive any binary number from the user, say 1101110, at program runtime.
• Create a while loop that runs until the value of the binary number (bin) becomes 0.
• bin (1101110) is not equal 0 on the first run, so program flow continues inside the loop.
• And bin%10 or 1101110%10 or 0 gets initialized to rem, oct + (rem*mul) (we have initialized 0 to oct and 1 to mul at the start of the program) or 0 + (0*1) or 0 gets initialized to oct.
• We have initialized 1 to count at the start of the program, therefore at first run, the value of count is not equal to 3, therefore while dividing the value of count or 1 by 3, we will not get a
remainder of 0, therefore program flow goes inside the else block (as the condition of the if statement evaluates to false at first run of the while loop), and inside the else block, mul*2 or 1*2
or 2 gets initialized to mul, and the value of count gets incremented and becomes 2.
• At last, bin/10, 1101110/10, or 110111 gets initialized to bin, and again the program flow goes back to the while loop's condition.
• Therefore, bin!=0 or 110111!=0 evaluates to true, so the program flow again goes inside the loop.
• And bin%10 or 110111%10 or 1 gets initialized to rem, and oct + (rem*mul) or 0 + (1*2) or 2 gets initialized to oct.
• Again, the condition of the if statement, that is, count%3==0 or 2%3==0, evaluates to false, therefore program flow goes inside the else block, and mul*2 or 2*2 or 4 gets initialized to mul, and
the value of count gets incremented and becomes 3.
• At last, bin/10, 110111/10, or 11011 gets initialized to bin, and again program flow goes back to the loop's condition.
• Therefore, bin!=0 or 11011!=0 evaluates to true, so the program flow again goes inside the loop.
• And bin%10 or 11011%10 or 1 gets initialized to rem, and oct + (rem*mul) or 2 + (1*4) or 6 gets initialized to oct.
• Now the condition of the if statement, that is count%3==0 or 3%3==0, evaluates to true, therefore program flow goes inside the if block, and oct or 6 gets initialized to octnum[i] (we have
initialized 0 to i at the start of the program) or octnum[0], and 1, 0, and 1 get initialized to mul, oct, and count, respectively. And the value of i gets incremented and becomes 1.
• At last, bin/10 or 11011/10 or 1101 gets initialized to bin, and again the program flow goes back to the loop's condition.
• Therefore, bin!=0 or 1101!=0 evaluates to true, and the program flow again goes inside the loop and follows the same procedure as described above.
• After exiting the while loop, check whether the value of count is not equal to 1 or not; if it is, then the value of oct gets initialized to the last index of the octnum[] array.
• Finally, print the value of octal digit one by one from the last index to the first index.
Binary to Octal in C using a User-Defined Function
The question is, "Write a program in C that converts binary to octal using a user-defined function named BinToOct()." Here we have declared the variable i and the array octnum[20] as global variables
(out of both the functions) to make them known inside both the main() and BinToOct() functions.
void BinToOct(int bin);
int i=0;
int octnum[20];
int main()
int binnum;
printf("Enter any Binary Number: ");
scanf("%d", &binnum);
printf("\nEquivalent Octal Value = ");
for(i=i; i>=0; i--)
printf("%d", octnum[i]);
return 0;
void BinToOct(int bin)
int oct=0, mul=1, count=1, rem;
rem = bin%10;
oct = oct + (rem*mul);
octnum[i] = oct;
mul = 1;
oct = 0;
count = 1;
mul = mul*2;
bin = bin/10;
octnum[i] = oct;
Here is the final snapshot of the above program's sample run:
The same program in different languages
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Harvard University Algebra Worksheet - Course Help Online
Do the following word problems no work needed just answer.
Easy algrbra.
no work needed i just need the answer
e: 14.29%, 1 of 7 pts
Question Help
Since 2004, the amount of money spent at restaurants in a certain country has increased at a rate of 6% each year. In 2004, about $300 billion was spent at restaurants. If the trend continues, about
how much will be spent at restaurants in 2014?
About $ billion will be spent at restaurants in 2014 if the trend continues.
(Round to the nearest whole number as needed.)
Enter your answer in the answer box and then click Check Answer.
Check Answer
Clear All
All parts showing
MacBook Pro
Find the balance in the account after the given period.
$3000 principal earning 5% compounded annually, 4 years
The balance after 4 years will be $ .
(Round to the nearest cent as needed.)
ਰਹੀ ਹੈ।
ili www.
Suppose that when your friend was born, your friend’s parents deposited $2000 in an account paying 3.2% interest compounded quarterly. What will the account balance be after 14 years
The balance after 14 years will be $
(Round to the nearest cent as needed.)
nter your answer in the answer box and then click Check Answer.
Clear All
I parts showing
MacBook Pro
و لے
Find the balance in the account after the given period.
$2,500 deposit earning 3.6% compounded monthly, after 4 years.
The balance after 4 years will be $(Round to the nearest cent as needed.)
Clear All
Enter your answer in the answer box and then click Check Answer.
All parts showing
MacBook Pro
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Fact Sheet #82: Fluctuating Workweek Method of Computing Overtime Under the Fair Labor Standards Act (FLSA) / “Bonus Rule” Final Rule
Print Fact Sheet
Fact Sheet #82: Fluctuating Workweek Method of Computing Overtime Under the Fair Labor Standards Act (FLSA) / “Bonus Rule” Final Rule
July 2020
NOTICE: The U.S. Department of Labor final rule, Defining and Delimiting the Exemptions for Executive, Administrative, Professional, Outside Sales, and Computer Employees, takes effect on July 1,
2024. The final rule updates and revises the regulations issued under section 13(a)(1) of the Fair Labor Standards Act implementing the exemption from minimum wage and overtime pay requirements for
executive, administrative, and professional (EAP) employees. Revisions include increases to the standard salary level and the highly compensated employee total annual compensation threshold, and a
mechanism that provides for the timely and efficient updating of these earnings thresholds to reflect current earnings data. The information on this webpage will be updated shortly.
Many employees have work schedules that vary from week to week. As a result, the total number of hours an employee works may increase or decrease from one week to the next. This type of schedule is
called a “fluctuating workweek.”
The FLSA requires that employers pay most employees in the United States at least the federal minimum wage for each hour they work. It also requires that they receive overtime pay at a rate of at
least time and one-half their regular rate for each hour they work over 40 in a workweek. Some employees are paid hourly, while other employees are paid on a different basis, such as salary,
commission, or piece rate. An employee’s regular rate is calculated by dividing the employee’s total pay (except for certain statutory exclusions) in any workweek by the total number of hours
actually worked in that week. Fact Sheet #56A provides additional information about determining the regular rate.
Fluctuating Workweek Method
Many employers simply pay an hourly rate and overtime at time and one-half that hourly rate for each hour over 40. But there are also other allowable ways to compensate nonexempt employees and to
calculate the overtime pay they are owed. Under the fluctuating workweek method, which is explained at 29 CFR 778.114, nonexempt employees receive a set weekly salary no matter how many hours they
work, plus additional overtime pay when they work more than 40 hours in one workweek. In other words, the employee’s weekly salary does not change whether the employee works 30 hours, 40 hours, or
more. In weeks when the employee works more than 40 hours, the employee receives additional overtime pay for each hour of work over 40.
Under the fluctuating workweek method, overtime pay is based on the average hourly rate produced by dividing the employee’s fixed salary and any non-excludable additional pay (e.g., commissions,
bonuses, or hazard pay) by the number of hours actually worked in a specific workweek. The average hourly rate will change from week to week depending on how many hours the employee actually worked.
The employee then receives at least an additional 0.5 times (or additional “half time”) that rate for each hour worked beyond 40 in the workweek.
One condition for using this method is that the employer and employee agree that the set salary is compensation (apart from overtime premiums and any additional non-excludable pay) for all of their
hours worked each workweek, whether they work few or many hours. To use the fluctuating workweek method, employees’ hours actually must change on a week-to-week basis, and employees must receive the
agreed-upon fixed salary even when they work less than their regularly scheduled hours.
Note: The fluctuating workweek method cannot be used if the employee’s salary is understood to be compensation for a specific, fixed number of hours per workweek. For example, the fluctuating
workweek method would not apply to employees of public agencies engaged in law enforcement or fire protection
activities who receive a salary as compensation for working specific, fixed hours within a work period (up to 28 days) under Section 207(k) of the FLSA. (For these employees, the basic principles of
calculating the regular rate that apply to a “workweek” also apply in the same way to a “work period.” See 29 CFR 553.233; WHD Opinion Letter FLSA 1216, 1986 WL 1171126 (Nov. 19, 1986).)
“Bonus Rule”/Final Rule
Effective August 7, 2020, the Fluctuating Workweek/“Bonus Rule” Final Rule clarifies that employers can pay bonuses or other incentive-based pay, such as commissions or hazard pay, above and beyond
workers’ fixed salaries when they are paid using the fluctuating workweek method. It also states that such payments must be included when calculating the regular rate unless they are excludable for
some other reason explained in the law. See Section 207(e)(1) - (8) of the FLSA for information about payments that can be excluded from the regular rate.
When an employee receives non-excludable incentive pay such as commissions or hazard pay, it is added to the weekly salary to determine the total straight-time pay for the week. To calculate the
average hourly rate, divide the total straight-time pay (salary plus non-excludable incentive pay) by the number of hours the employee actually worked that week. The employee then receives an
additional 0.5 times (or additional “half time”) of that rate for each hour worked beyond 40 in the workweek.
In the examples below, an employee works hours that change from week to week and is paid a fixed weekly salary of $600.00. The employee understands this fixed weekly salary will not change if hours
of work increase or decrease.
Examples: Fixed Salary for Fluctuating Hours With and Without a Production Bonus
Workweek 1:
Employee’s Salary $600
No Bonus + $ 0
Total Straight-Time Pay $600
Hours Worked 48
In workweek 1, the employee has earned the fixed weekly salary of $600 with no bonus pay. The employee worked 48 hours, and is due the weekly salary plus additional overtime pay at 0.5 times the
average hourly rate, or “half-time,” for the 8 overtime hours worked. To determine the overtime due and total compensation owed the employee, complete the following steps:
1. Determine the employee’s average hourly rate. Divide the salary, $600, by the number of hours worked, 48 hours. The result is the average hourly or regular rate of $12.50. [$600 ÷ 48 = $12.50]
2. Next, multiply the average hourly rate, $12.50, by .5 to determine the half-time rate. In this case, the half-time rate is $6.25. [$12.50 x .5 = $6.25]
3. Multiply the half-time rate of $6.25 by the number of overtime hours worked, 8, to determine the total amount of overtime due, $50.00. [$6.25 x 8 = $50]
As a result, in workweek 1, the employee is due the $600 salary plus $50 in overtime pay for total compensation in the amount of $650. [$600 + $50 = $650]
Workweek 2:
Employee’s Salary $600
Production Bonus + $100
Total Straight-Time Pay $700
Hours Worked 45
In workweek 2, the employee has earned the fixed weekly salary of $600 plus an additional production bonus. The employee worked 45 hours, and is due the weekly salary and the bonus plus additional
overtime pay at 0.5 times the average hourly rate, or “half-time,” for the 5 overtime hours worked. To determine the overtime due and total compensation owed the employee, follow the steps below:
1. Determine the total straight-time pay for the week. Add the fixed salary, $600, and the production bonus, $100, for a total of $700. [$600 + $100 = $700]
2. Determine the employee’s average hourly rate. Divide the total straight-time pay, $700, by the number of hours worked, 45 hours. The result is the average hourly or regular rate of $15.56.
[$700 ÷ 45 = $15.56]
3. Next, multiply the average hourly rate, $15.56, by 0.5 to determine the half-time rate. In this case, the half-time rate is $7.78. [$15.56 x .5 = $7.78]
4. Multiply the half-time rate of $7.78 by the number of overtime hours worked, 5, to determine the total amount of overtime due, $38.90. [$7.78 x 5 = $38.90]
As a result, in workweek 2, the employee is due the $600 salary and $100 production bonus plus $38.90 in overtime pay for total compensation in the amount of $738.90. [$600 + $100 + $38.90 = $738.90]
Deductions from the Fixed Salary
The fluctuating workweek method can be used only if the salary does not change even when the number of hours worked increases or decreases. However, employers may take occasional deductions from the
employee’s salary for disciplinary reasons such as willful absences or tardiness or for infractions of major work rules, as long as the deductions do not violate the minimum wage or overtime pay
requirements of the FLSA
Where to Obtain Additional Information
For additional information, visit our Wage and Hour Division Website: http://www.dol.gov/agencies/whd and/or call our toll-free information and helpline, available 8 a.m. to 5 p.m. in your time zone,
1-866-4USWAGE (1-866-487-9243).
This publication is for general information and is not to be considered in the same light as official statements of position contained in the regulations.
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The other kind of math rock.
The other kind of math rock.
February 23, 2006 9:52 PM Subscribe
Looking for songs about math! My girlfriend's sister just got into a Ph.D. program for mathematics and we're making her a mix CD to help celebrate. Poppy indie rock especially good.
I've done the song title searches on allmusic.com, and I've gone through these threads on
educational songs
. I've perused
I've got Kate Bush's "Pi", Schoolhouse Rock Rocks ('natch), Modest Mouse's "Never Ending Math Equation", Kraftwerk's "Pocket Calculator", and I know about
this one
. What else?
It's the group name rather than the song title, but
Math and Physics Club
is definitely poppy indie rock; I love "When We Get Famous."
posted by occhiblu at 9:55 PM on February 23, 2006
I'm sure They Might Be Giants will have
posted by rachelpapers at 9:55 PM on February 23, 2006
New Math by Tom Leher is the best I can think of:
(To the equation 342-173)
"You can't take three from two, two is less than three, so you look at the four in the tens place. Now that's really four tens, so you make it three tens, regroup and you change the ten to ten ones
and you add them to the two and get twelve, and you take away three, that's nine."
Mentioned briefly in one of the other threads, but it's absolute necessity.
(He goes over it again in base eight towards the end. Seriously.)
Email me if you're having trouble finding it.
posted by disillusioned at 9:59 PM on February 23, 2006
"One is the loneliest number" - Three Dog Night
"One and one make five" - Pet Shop Boys
posted by occhiblu at 10:01 PM on February 23, 2006
4 out of 5 - Soul Coughing
"Her knees thrust in one direction like a symbol of math, a symbol meaning Greater Than.
I come recommended by four out of five
I'm a factor in the whole plan
Four and five therefore nine
Nine and nine therefore eighteen
Eighteen and eighteen therefore thirty-six
Four and five therefore nine"
posted by karmaville at 10:02 PM on February 23, 2006
(Lord It's Hard to Be Happy When You're Not) Using the Metric System - Atom & His Package
posted by lemuria at 10:10 PM on February 23, 2006
Don't know much Trigonometry
Don't know much about Algebra
Don't know what a slide rule is for
But I know that one and one is two
And if this one could be with you
What a wonderful world this would be
posted by weapons-grade pandemonium at 10:11 PM on February 23, 2006
damn it, lemuria! my stupid 64 megs of ram refused to load mefi any faster. i was just about to post
atom and his packageposted by nadawi at 10:20 PM on February 23, 2006
greatest Math song ever recorded
is The Math Song by The Darkest of the Hillside Thickets (
iTunes link
When I played it for a math PhD friend, she solved for y.
posted by mathowie at 10:21 PM on February 23, 2006
Oh, and that band does some great tongue-in-cheek indie rock from Canada, to boot.
posted by mathowie at 10:21 PM on February 23, 2006
You need "New Math" by Tom Lehrer (who happened to be a mathematician in addition to a musician)
posted by Robot Johnny at 10:26 PM on February 23, 2006
Finite Simple Group (of Order Two) by the Klein Four Group (
) is fantastic. They've got
some others
as well.
posted by The Pusher Robot at 10:27 PM on February 23, 2006
Deathray Davies
hailing from Dallas (if not denton), TX have a side project called "I Love Math", but good luck finding the recordings.
posted by nadawi at 10:32 PM on February 23, 2006
2Gether: "U + Me = Us (Calculus)"
posted by granted at 10:35 PM on February 23, 2006
God Bows To Math by The Minutemen!
posted by joeblough at 10:36 PM on February 23, 2006
"Adding Up Numbers" by Kompressor should work well, and maybe "5546 That's My Number" by Sublime.
Also I second Soul Coughing, and third Tom Lehrer.
posted by anarcation at 10:39 PM on February 23, 2006
Not math, so it doesn't really qualify, but you should check out "The Method" on We Are Scientists' first album. You can download it for free from nebulizemymind.com.
"I watched you watching me from labotory five.
The breathing mask and safety goggles accentuate your eyes."
posted by pomchkn at 10:46 PM on February 23, 2006
I'm sure They Might Be Giants will have something...
You'd think so, but not really. They tend to lean more toward English/history-major nerdery than math. They do have a couple of number-related songs (Number Three, Four of Two, 32 Footsteps leap to
mind), but math isn't really a component of any of those.
I'd suggest Ladytron's 'True Mathematics' and Spoon's 'My Mathematical Mind'.
posted by toddshot at 10:47 PM on February 23, 2006
You need Freezepop's song "Science Genius Girl". You might recognize the chorus:
"three point one four one five nine two six five three five eight nine seven nine three two three eight four six two six four three three eight three two seven nine"
posted by barnacles at 10:58 PM on February 23, 2006
It's 5! by the Architecture in Helsinki
posted by GilloD at 11:02 PM on February 23, 2006
"Inchworm," from the Hans Christian Anderson film, sung by Danny Kaye.
Two plus two is four. Four plus four is eight …
posted by Astro Zombie at 11:02 PM on February 23, 2006
"Algebra" by Soul Hooligans isn't really about math, but Algebra and Calculus are in the chorus.
"Music is Math" by Boards of Canada?
"16 Military Wives" by The Decemberists? Maybe not.
posted by solid-one-love at 11:03 PM on February 23, 2006
oh man, BoC! how could i forget? almost every BoC song is itself... well.. math!
_the smallest weird number_ definitely fits the bill; very self-referential as well since music70 is their label.
A is to B as B is to C is another BoC track that comes to mind.
posted by joeblough at 11:19 PM on February 23, 2006
Murder By Numbers, by the Police
The Number One Song in Heaven by The Sparks
Sea of Numbers by Stylex
anything by either The Numbers or The Atomic Numbers
Microphone Mathematics by Quasimoto
anything by Cotton Mather (if you want to stretch it)
9 to 5 by Dolly Parton
Add It Up by the Violent Femmes
Fifty Two Percent by Glen Baxter
6/8 by Superdrag
The New Face of Zero and One by the New Pornographers
C30 C60 C90 Go! by Bow Wow Wow
My 2600 by Captain Funkaho
(I have a lot more general # songs if you're interested).
posted by klangklangston at 11:32 PM on February 23, 2006
"Carry the Zero" by Built to Spill is a good tune.
posted by lazywhinerkid at 11:33 PM on February 23, 2006
First of all, anything by the
will work here. They're an indie pop/electronic outfit from upstate New York with members such as Pete Pythagorus and Albert Gorithim IV. Granted, they're a novelty act, but the music is real catchy
and danceable!
Second of all, "Never-Ending Math Equation" is a great start!
Bikeride - "That's Math"
(this is essential - cute flute/woodblock/piano pop all about math and love)
Silver Jews - "Inside the Golden Days of Missing You"
(featuring the lyric, "what if life is just some hard equation
on a chalkboard in a science class for ghosts?")
Cass McCombs - "Subtraction"
(upbeat pop with organ)
Mark Mothersbaugh - "Hardest Geometry Problem in the World"
, classical-sounding instrumental)
Clint Mansell - "Pi R Squared"
(creepy chemical brothers theme to
- good intro?)
Built to Spill - "Carry the Zero"
(the best actual song in this list)
The Pixies - "Distance Equals Rate Times Time"
("looking into the sun" sounds like "like an integral... SIGN!")
Jamie Lidell - "Multiply"
The White Stripes - "Black Math"
(classroom rebellion!)
The Shins - "Your Algebra"
(creepy, hymnlike dirge from a usually peppy pop band)
65 Days of Static - "The Fall of Math"
(heavy electronic post-rock)
The Decemberists - "Of Angels and Angles"
(short cute acoustic pop guitar ditty)
Some equation ones:
Radiohead - "2 + 2 = 5" Pavement - "5 - 4 = Unity" Shining - "31=300=20 (It is by Will Alone I Set My Mind in Motion)" Broken Social Scene - "Cause = Time"
Hmmm... I smell a MusicMobs playlist...
posted by themadjuggler at 11:49 PM on February 23, 2006 [1 favorite]
I see Tom Lehrer's "New Math" has been nominated. I hope you'll also include his "Lobachevsky".
posted by orthogonality at 12:01 AM on February 24, 2006
I almost forgot!
Mogwai - "Sine Wave" The Weakerthans - "Uncorrected Proofs" Grandaddy - "Chartsengrafs"
(granted, many of these examples are more mathematical in spirit, the Mathematicians/Bikeride/Pi soundtrack excepted)
posted by themadjuggler at 12:05 AM on February 24, 2006
boards of canada - constants are changing
the campfire headphase built to spill - carry the zero
keep it like a secret
-- a great indie rock song.
dj shadow - the number song
(another great track! and great number samples)
kraftwerk - pocket calculator
a classic track
computerworld pet shop boys - two divided by zero
pleaseposted by ori at 1:11 AM on February 24, 2006
anything by Caribou/Manitoba, aka Dan Snaith (same guy, name changed due to legalities), who happens to hold a Ph.D in math.
posted by heeeraldo at 2:00 AM on February 24, 2006
The Aislers Set - Long Division (legally downloadable for free
), it's math(s) as metaphor and totally dancable.
posted by featherboa at 2:09 AM on February 24, 2006
I think Tom Lehrer's Lobachevsky is a great song for teaching you what a math PhD is really like.
I am never forget the day I am given first original paper to write. It was on analytic and algebraic topology of locally Euclidean parameterization of infinitely differentiable Riemannian
Bozhe moi!
This I know from nothing.
posted by grouse at 3:26 AM on February 24, 2006
Mathematics, by Mos Def
Math Prof Rock Star, by Jim's Big Ego is my own personal theme song
posted by gleuschk at 3:45 AM on February 24, 2006
Can't believe someone would mention "Three is magic number" from Schoolhouse Rock, and not "Little twelvetoes".
Come on, people, it's about counting in base-12!
One, two, three, four, five, six, seven, eight, nine,
dek, ell
, ten, "except my ten would be different from yours!"
posted by gregvr at 4:44 AM on February 24, 2006
Can't believe no one's mentioned the classic
Eleven Twelve
. OK, so it's not so much poppy indie rock, but there might be the retro appeal.
posted by d-no at 5:31 AM on February 24, 2006
Love and Mathematics
by Broken Social Scene, and maybe the second track on Aphex Twin’s
posted by misteraitch at 5:31 AM on February 24, 2006
"One on One" by Hall & Oates
"Rock Around the Clock"
by Bill Hailey & the Comets (an oldie)
"Eight Days a Week" by the Beatles
"Ray's Dad's Cadillac"
by Joni Mitchell
Ray's dad teaches math
I'm a dunce
I'm a decimal in his class
Last night's kisses won't erase
I just can't keep the numbers in their place...
When it comes to mathematics
I got static in the attic
"No sir, nothin's clear!"
I'll be blackboard blind on Monday
Dreamin' of blue runways
On the edge of here
A little atmosphere
Congratulations to your GF on her new adventure. Good luck in the coming years.
posted by bim at 5:43 AM on February 24, 2006
Check out a band from Albany NY called The mathematicians. every song is about numbers!
posted by webtom at 6:10 AM on February 24, 2006
Some will remember a TV show from the 80s called Square One... they had lots of 80s-style MTV music videos that were sendups or original songs. While extremely corny for a modern mix, I think you are
unlikely to find any other song about combinatorics... if you can find anything at all.
, at least, are some realaudio files.
posted by cacophony at 6:26 AM on February 24, 2006
There was a "Mathlete Rap" on the Mean Girls soundtrack...
Yo, yo, yo,
All you sucka MC’s ain’t got nuthin’ on me
Not my grades, not my life
You can’t touch Kevin G
I’m a mathlete, so nerds of the earth
But forget what you heard
I’m like James Bond the Third
Sh-sh-shaken not stirredposted by blackkar at 6:55 AM on February 24, 2006
Sifl and Olly
, sockpuppets who had a show on mtv, had a stupid/funny
Math Song
. I have it if you're interested (email in profile).
posted by Flamingo at 7:33 AM on February 24, 2006
There are lots here.
Some are a bit childish, but that could be part of the fun. Also, a lot of
nerdcore hip-hop
is math oriented.
posted by wackybrit at 7:42 AM on February 24, 2006
There was a mashup of Kanye West and the Beach Boys going around ("West Sounds") that had the producer, Lushlife, singing on a track ("Through the Wire", the 8th track, I think) with some clever
math-terminology-laden lyrics.
posted by milkrate at 8:39 AM on February 24, 2006
Less obvious: "Patterns" by Simon and Garfunkel.
posted by Aknaton at 8:51 AM on February 24, 2006
You can find the surprisingly catch "Mandelbrot Set" at
Jonathan Coulton's song page
Matthew Matics
isn't quite as musically talented as Mr. Coulton, but he is undeniably mathematic. Plus, you can get his songs in Swedish.
posted by yankeefog at 8:54 AM on February 24, 2006
(be sure to post the final tracklisting here!)
posted by ori at 9:06 AM on February 24, 2006
"I've got 1, 2, 3, 4, 5 . . . senses working overtime" by XTC
PS: Song is called "Senses Working Overtime" Seems like there might also be a hidden "X" factor in the song, if you know what I mean and I think you do.
posted by jfwlucy at 9:30 AM on February 24, 2006
Hilary Duff "The Math"
You're always trying to figure out
What I am all about
If you don't know what the answer is
Then just shut up and kiss
It shouldn't take forever
To put it all together
If you can't do the math
Then get out of the equation
I am calling you back
This is * 69
Is it a minus or a plus
Does enough equal enough
If you can't do the math
Then nothing adds up
Tell me why I'm here
Sure I want someone to understand
But I don't need the stress
I'm not about being analyzed
Like it's some kind of test
Don't have to be a genius
To figure what's between us
You can spend your whole life analyzing
Justifying, quantifying, and dividing
'Till there's nothing anymore
Why don't you just close your eyes
And kiss my lips and let it go
Just let it flow
It's what I'm waiting for
Don't have to be a genius
To figure what's between us
[CHORUS 2X]posted by cillit bang at 9:44 AM on February 24, 2006
New Math by Tom Lehrer
18 Wheels on a Big Rig by Trout Fishing in America
Saved by Zero by The Fixx
Oopportunities by the Pet Shop Boys ("majored in mathmatics... studied at the Sorbonne....")
Adding Up Numbers by Kompressor
The entire number section of Schoolhouse Rock
that 1-2-3-
-5... 6-7-8-
-10... 11-12 song from Sesame Street.
One is the Lonliest Number by 3 Dog Night
posted by ilsa at 11:00 AM on February 24, 2006
Oopportunities by the Pet Shop Boys ("majored in mathmatics... studied at the Sorbonne....") Doctored
in math'matics, better yet. (He could've been a don.)
posted by Aknaton at 11:37 AM on February 24, 2006
18 Wheels on a Big Rig was written by Heywood Banks, and his is the definitive version.
posted by klangklangston at 12:21 PM on February 24, 2006
"Numbers" by Kraftwerk -- seems even more appropriate than "Pocket Calculator," mentioned above (it's just a recitation of numbers in various languages).
Also "Let X=X" by Laurie Anderson. Not particularly about math (though the title is in the lyrics), but a groovy song nonetheless.
posted by ROTFL at 2:25 PM on February 24, 2006
The Decemberists - "Of Angels and Angles"
(short cute acoustic pop guitar ditty)
...which has nothing to do with math.
Carry the Zero by Built to Spill is a great song, though.
posted by ludwig_van at 2:41 PM on February 24, 2006
The Butchies, "Forget Your Calculus"
("forget your calculus who will forget us")
Blue Dogs, "River Material"
("Is there nothing unpredictable something to hold on to except for math and science and all they know is true?")
posted by smash at 3:29 PM on February 24, 2006
Science Vs. Romance (Rilo Kiley) "Zeros and Ones"
Your Algebra (The Shins)
Three is a Magic Number, My Hero Zero, Figure Eight (School House Rock)
posted by Packy_1962 at 5:18 PM on February 24, 2006
And for when her head starts spinning from all the numbers, Jimmy Buffett says it all in
Math Sux
posted by MiamiDave at 6:23 PM on February 25, 2006
Response by poster:
Thanks for your posts, everybody! Lot and lots of good stuff here.
For those keeping score at home, here are a few more we dug up:
A-Camp - Algebra
Andrew Bird's Bowl of Fire - What's Your Angle?
Andrew Bird's Bowl of Fire - Beware
Beaumont - Girls and Maths
Bruce Stringbean And The S Street Band (Sesame Street) - Born to Add
Jets to Brazil - Bad Math
Nicolette - Wicked Mathematics
Supernova - Math
Beachwood Sparks - Pi and a Bee
Todd Clark - Mathematics Don't Mean a Thing
George Clinton - Mathematics
Drive Like Jehu - New Math
Ex-Models - Other Mathematics
Tom Glazer & Dotie Evan - How Do We Measure Energy?
For the cover we, er, appropriated Hiroshi Sugimoto's
of plaster casts of mathematical forms.
posted by hydrophonic at 9:03 PM on March 8, 2006
This thread is closed to new comments.
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Compreensão dos riscos geotécnicos – uma estrutura para incertezas
Lyceum 2021 | Together Towards Tomorrow
Knowing the subsurface conditions is critical for successful geotechnical analysis and design.
However, geomechanics properties are inherently variable and difficult to obtain, resulting in uncertainty. How then can we properly understand geotechnical risk? This session presents a means to
characterize risk as a function of the relationship between uncertainty – provides divisions to the continuum of tools available to geoscience and geological engineering practitioners to address
uncertainty in risk assessment – and proposes a framework that matches tools to risk character in order to improve risk assessment outcomes.
Ray Yost
Principal Geotechnical Engineer – Advisian
Chris Kelln
Director of Geotechnical Analysis, GeoStudio – Seequent
Video transcript
(upbeat music)
<v ->Hello and welcome to this presentation</v>
on understanding geotechnical risk.
My name is Chris Kelin,
I’m the director of Geotechnical Analysis
for the GeoStudio business unit here at Seequent.
And I have the pleasure of introducing our speaker,
Dr. Ray Yost.
Ray has nearly 20 years of experience
working in the fields of geology, hydrogeology,
and geotechnical engineering
for the civil and mining sectors.
His career has included tenures
at Oregon Department of Transportation,
Rio Tinto minerals, Teck Resources
and more recently, as a principal geotechnical engineer
at Advisian.
In this role,
Ray serves as a subject matter expert
for a wide range of engineering applications,
including underground mining,
surface mine design, tailing storage facilities,
geo-hazard management, and much more.
Today Ray will talk to us
about a framework for understanding risk
in geotechnical engineering.
Ray, over to you.
<v ->Thank you, Chris.</v>
So my talk is about understanding geotechnical risk
and the corresponding uncertainty we often face.
It’s a structure for understanding uncertainty.
The next slide, please.
It starts us with this idea
that small data sets and the corresponding uncertainty
that comes with them
are a common circumstance in geological engineering.
And by small data sets,
I mean, either actual the small number of values
that we might have,
or small in a sense of a sample to volume ratio.
We’re trying to characterize a very large volume of ground
with a very small number of data points.
And it creates two problems these small data sets
in understanding risk.
The first is pretty immediate.
I mean, we have an analysis to do,
we only have a few data points to choose from,
and we have to pick an appropriate point
that we think represents the ground conditions
or wherever else we’re trying to characterize.
The second problem is less immediate,
but ultimately it’s a lot more important.
And it’s the focus of this talk really.
Because in selecting this value,
what we’re doing is we’re making some assumptions
about that range of data.
And that goes into our analysis,
and that goes into our risk quantification.
And ultimately that goes to our resource allocation
that we have for mitigating that risk.
And we have this now line
between the inherent uncertainty that we’re dealing with
from these small datasets,
all the way to the end,
where we’re actually allocating resources
to mitigating that problem.
So it’s really important to understand
how we think about uncertainty
so that when we get to this resource allocation,
we’re actually applying optimal levels
of mitigation to a problem.
Next slide.
So one of the things
its not that when we say uncertainty,
it’s not just this big black box of unknowns, this void.
One of the advantages we have in geomechanics
is that a lot of our data sets,
or rather, the types of data and information we use
are fairly quantitative.
And so, because of that,
we can develop this relationship
between the little things that we know
that the small data set that we have,
and this larger uncertainty
about what the possible range could be.
We have this idea that variation
is the range of what we know, whatever that range is.
And the uncertainty is what we don’t know.
And given that it’s quantitative often
we can have an open door or closed end to that uncertainty.
A lot of times the minimum value is often zero.
The other end, it can be open in certain circumstances,
Q values, compressive strength, things like that.
But at a certain point, it doesn’t matter anymore.
Once you get past a certain data point
or restraint value or whatever,
it doesn’t matter if it’s 350 MPa or 325 MPa,
it’s strong enough, basically.
So when we start to overlay these two,
we see that this is a useful building block now
for understanding risk,
because we have this chunk of certainty or knowns
in the middle,
and then a chunk of a uncertainty around the sides.
Go to the next slide, please.
It’s a simple diagram,
and it’s going to be the basis for what I’m talking about
with respect to risk.
But I’ve started off right away
with this very idealistic version of what this looks like.
I’ve got this range of variation that we know in the middle
bounded by this equal ranges of uncertainty on either end.
Chances are going to be a lot better actually
that there’s an asymmetry involved.
Either there’s going to be a lot more uncertainty
on one end or the other.
And this is going to, again,
influence how we think about risk
as a function of uncertainty.
Now I’ve talked about this being quantitative information.
So it’s easy to think about this
in terms of a number line and zero at the left side
and whatever the maximum value is at the right side.
And that’s okay to think about it that way.
Since we’re talking about risk though,
and sometimes low values can be lower risk
or high values can be lower risk.
It’s best not to think about it necessarily as numbers
just as relative better or worse
in terms of where this certainty lies.
There’s also a possibility where it could be gapped.
We could have some sort of chunk of what we know
and the uncertainty, another chunk of what we know again,
and then uncertainty on either side of that.
For the purposes of this talk
and just to simplify matters a little bit,
I’m going treat this as basically a bi-modal variation.
And we just have the same sort of circumstance.
There’s a range of certainty that we know about or we know,
and then a range of uncertainty on either side of it.
Next slide please.
So now we want to talk about upside or downside asymmetry.
So I’ve talked about this idea
that we can have significantly
more uncertainty on one side or the other
of our range of what we know.
And to do that,
we want to think about this critical value,
this concept of a critical value.
This is the value at which
if you have an input value,
you’re going to get an output value.
And if you put in a lower input value,
you will get a worse answer.
Or anything to the left of that will be worse.
Anything to the right is better.
So this is the value.
I mean, probably the easiest way to think about it
is, say a stability analysis,
and you need a certain compressive strength
to produce a factor of safety.
So if you have a less compressive strength
or a lower compressive strength,
you’ll get a worse factor of safety,
and a higher compressive strength
is a better factor of safety.
So it’s this critical value.
And now we can start to see where does our variation lie
and versus where does our uncertainty lie
relative to this critical value?
So we can have asymmetric downside risk,
we’re basically what we don’t know makes the problem worse,
or asymmetric upside risk,
what we don’t know makes the problem better
relative to this critical value.
Next slide, please.
Now we want to talk about magnitude of uncertainty.
How big is this range?
We can have of course,
significant downside uncertainty
in the case that I’ve shown.
There’s a lot of uncertainty below this critical value,
or we can have minor downside uncertainty.
There’s just a little bit.
Again, if we think about
a lot of different geomechanical data,
the minimum value is zero.
So if the far lowest known point is slightly more than zero,
yeah, there’s some uncertainty,
maybe there’s a value that would fit into that range,
but it’s a pretty small range
between zero and whatever our minimum value is.
So again, for the purposes of this talk
and to keep things simple,
if we have minor downside uncertainty,
I’m not even going to think about that as uncertainty,
it’s just treated with extending your variation
a little bit more.
Really the purpose of this
and talking about risk and uncertainty
is talking about circumstances
where we have significant
either downside or upside uncertainty,
where we have a lot of unknowns on one side or the other
of that critical value.
Next slide, please.
Now, of course,
there’s a sensitivity too that we have to consider.
This is how sensitive is the output value
to a change in the input value.
We can have circumstances
where our output is insensitive and reasonably linear
as we make these changes and gradual changes
in putting in a higher or lower values
relative to this critical value,
we don’t see much change or an answer,
or we can have very sensitive
and potentially non-linear answers relative to inputs.
We can start to see
that we either get a significant change
in the slope of that output,
or we have just a very significant sensitivity
at the end of the day.
So either one is a cause for concern in this case.
Next slide, please.
And then of course, risk.
We have all of the different things around the probability
and the range of inputs,
and essentially what that value is going to look like.
And the other half of risk is the of course, consequences.
And our consequences can like sensitivity
be low to moderate.
As we’re changing that input value,
we don’t really see a change in the consequence that much.
So say again factor of safety in a stability analysis
is our example.
And we’re reducing that input material strength,
and we’re getting a failure.
But the size of the failure
is not really changing,
the run-out distance isn’t changing.
We’re not really seeing huge differences
in the consequences of that,
even though the factor of safety might be dropping,
it’s not really having an effect
on what the impacts of that would be.
So we can have this lower,
and again, linear consequences
as we go down this potential range of uncertainty,
or we can have very high consequences,
and even non-linear consequences again.
Now one note on the consequences,
we have both downside consequences.
These are often going to take
the form of unmitigated liability.
And why I say liability instead of risk,
is that it’s sort of the next piece.
Things could be worse than we assume,
higher risks, and then these risks
haven’t been attenuated or mitigated
because we aren’t aware of them,
and that’s going to create a liability.
So that’s the downside consequences,
this unmitigated liability.
And the upside consequences are going to be more
in the form of opportunity costs.
Essentially we could have had a leaner, meaner construction
of whatever sort.
We didn’t have to have a to have slope angle that was that shallow.
We didn’t have to have an embankment that was that big.
We dedicated resources to something
that we didn’t need to necessarily
to achieve our desired outcome in terms of safety.
Next slide, please.
So given this construct with sensitivity,
greater or lesser than,
the asymmetry in the outcome upside or downside,
and then the consequences either higher or lower.
We have this box of possibilities
in terms of these risks scenarios now,
and uncertainty scenarios
that we’ve got eight different circumstances
that we can look at
in terms of all of these different ways
we can think about risk as a function of uncertainty.
Next slide, please.
So we’ve talked about now
the first two pieces of that flow diagram
that I showed in the earlier slide
with uncertainty and assumptions.
That’s how we start to think about risk.
Now we talk about the analysis and the risk mitigation,
and this is through the tools that we use.
These are all these different tools
that are available to us as geotechnical engineers
to address this uncertainty.
How do we think about uncertainty?
I won’t say that this is the definitive way
to think about these tools,
but I will say that there is a continuum of sorts
between all these different tools that we have.
And this slide isn’t meant to capture all the tools
that are available to us as geotechnical engineers.
But to talk about them in terms of these broad categories,
where we have tools
that are based in inductive reasoning and inference,
these are things like the first picture
where we have something about A that we don’t know.
This could be again, a material strength.
We know a lot about A, or something about A,
we tend to know a lot more about B
including that material strength
this target thing we want to know.
And A and B share enough characteristics
that we can assume that whatever material strength B has,
A has the same strength or similar strength.
We have tools that fall into this
sort of proportional relationship.
A is somehow relative to B,
think about, we have a few compressive strength samples
where we’ve incurred the higher cost
to do the compressive strength testing
and a whole lot of point load samples.
And there’s a proportionality between those two.
So we can look at the range
of compressive strength variation
as a function of the range of point load strength variation.
There’s a lot more, of course,
again, these are not meant to be exhaustive lists
of all these different tools,
just to get the sense of what an inference
or inductive reasoning type of tool looks like.
We have parametric tools, or these are basically
this one of the kitchen-sink approach.
We’re throwing a lot of different things
in sampling from them into a bin.
This can be a lot of different types of variables.
And we’re trying to come up
with some sort of parametric analysis
based on Monte Carlo, Latin Hypercube sampling, whatever
that produces this range of outcomes.
And we can start to look at that range of outcomes
and make some conclusion from that.
There’s a lot of things around say subjective probability
that might fall into this as well.
A lot of different tools
where we’re basically just looking at
what the distributions are
across a lot of different ranges of our variables.
And then we have these direct
or deductive reasoning types of tools
where we’re just either looking at
what information we have, this is the variation,
or maybe we’re extrapolating from something.
A lot of times frequency or recurrence interval
might fall into these types of deductive reasoning tools.
We have a bunch of data from a time history,
and we’re going to extrapolate that out a little bit
and pretty much this is what we can assume
about the circumstance from the information we have.
We could also assume some cases, just the minimum value.
If we know that the range is bound in a certain way,
it starts at zero, it goes to a hundred,
maybe we pick one or the other
as far as an upper or lower bound to what that would be.
So these things as well,
I’m going to argue, fall on a continuum.
There’s not any necessarily hard lines between them but–
Next slide, please.
We will talk about the relative strengths
and weaknesses of them.
And again, not meant to be an exhaustive list
just to illustrate that each of these has a place and a use
in terms of addressing uncertainty.
With inference and inductive reasoning,
a lot of times we’re using a lot of our knowledge
and understanding as geotechnical engineers
to relate one thing to another,
or use some other bit of data to modify
or increase the precision of our estimate.
We could say, if we don’t know a material strength,
we can assume it’s zero, that’s pretty conservative.
We don’t want to do that necessarily,
so we’re using inference
to increase the precision of that estimate.
Of course, the weakness of this,
it it’s really based on knowledge
from practitioner to practitioner, that can vary.
I might be really good at estimating material strength
from all these other materials strengths that I’m aware of,
the next person has maybe more of a limited expertise
in that area.
And you’re going to get very different answers
from inference and inductive reasoning
from practitioner to practitioner,
probably the basis of a lot of arguments
that we have as geological engineers.
The direct or deductive reasoning,
the strength there is that
since you’re assuming either from what you know,
or from more importantly from some end value of this,
you sort of covered all the bases.
You’re not going to be surprised
by something that wasn’t captured in your assumption.
The weakness of course,
is that these can be fairly conservative estimates.
With the parametric tools,
the strength is that it’s actually
kind of drawing on the strengths
of both inference and inductive reasoning
as well as direct and deductive reasoning.
And so it’s pulling the best of each of those.
The weakness is that
this can require considerable time expertise.
You would have to pull from a lot of different people.
You’re going to have to deal
with some of those issues around,
again, both of the weaknesses of each method.
The other one that it can cause
is that you end up with this range of possible outcomes
that’s going to vary
from some extreme adverse outcome to extreme good outcome,
and you’re going to have to make some decisions
around which one’s going to be the appropriate outcome.
How do you decide?
Do you have a cluster of outcomes around a central value?
And that’s a good thing.
Or do you have these long tails
that you have to make some decisions about?
It can sort of solve some of the problems
of inference or deductive reasoning on the front end,
but cause more problems on the backend.
So no one tool is perfect,
but they all have their advantages and disadvantages.
So next slide.
So now we’ve compartmentalized
all these different circumstances of uncertainty and risk,
and now we have the different tools that we apply.
And we’re going to start talking about
how each of those tools fits
each of these different circumstances.
So we have that box of possibilities
from the previous slide,
and we split it,
and we’re looking at the downside on the left side,
and the upside on the right side.
We can start to look at how these tools
apply to these different circumstances
as a function of sensitivity and consequence,
and then upside or downside risk.
So I’d like to talk about these for a little bit,
and I’ll start with the downside risk
in the lower left-hand corner.
We have a situation where we have pretty low consequences,
pretty low sensitivity, or in sensitivity,
it’s downside risk, but essentially you’re not going to,
because of this insensitivity and lower consequences,
you can assume fairly extreme values
without really any cost in terms of allocation of resources.
So that’s a pretty good place for that tool to sit.
In this middle band,
we have either the higher consequences
with the higher sensitivity,
(coughing) excuse me.
Some of these inductive tools are going to be more important
because now either we have to think about consequences
or we have to think about that sensitivity.
We do want a little bit more precision
in how we approach this.
We want to be aware of implausible or extreme values
and how those might affect our answer,
but we don’t want to let them influence our answer too much
because they could lead to such an extreme outcome
in our risk assessment
that we’re, again, misallocating resources.
So we want to start using some of these inference tools
to increase the precision of our input assumptions.
And then finally, when we get to the upper right side
on the upper right quadrant on the downside risk,
it really speaks to,
we have a lot of sensitivity, high consequences.
We need to look at this parametric approach
because we want to capture potentially some relationships
between different, either within a variable
or due to non-linear responses,
or maybe some combination of variables
that may not be intuitive.
We really want to see what that full range of outcomes
looks like.
So for the upside,
we have a similar set of different tools
that are going to be applied to these different compartments,
but a slight difference.
If we start in the lower right-hand corner of this time,
we have low sensitivity, low consequences,
but because it’s more of a matter of opportunity costs,
we want to use this parametric approach
to understand those a little bit better.
There’s some value in looking at those
so that we’re understanding
that we’re again, allocating resources appropriately.
For these middle two boxes
where we have the higher consequences, but less sensitivity,
these again, these indirect and inductive reasoning methods
are important to increase that precision around our answer.
But once we get into the lower consequences
with greater sensitivity,
the direct and deductive approaches
are more important to use.
And of course, when we get up
into the upper left quadrant there
with greater sensitivity and higher consequences,
we want to use those parametric approaches
again, to understand
if there’s some sort of non-intuitive outcome
that we can experience,
or to look at whether those outcomes are clustered
around some sort of central value
or have these longer tails
that might be important to consider.
Again, it speaks to
how do we start to look at the tools
versus the circumstance
to properly mitigate risk
or allocate resources to mitigate risk.
Next slide, please.
So where do we come to with all this?
We can use this relationship between what we do know
and the range of what we may not know or don’t know
to think about how to characterize uncertainty
relative to risk.
And you can agree or disagree
with any parts of this discussion
or any parts of my presentation.
What it does come down to
again, is this fundamental idea
that we can discuss this and go on and on
and talk about our different approaches and whatnot.
But we do have to think at the end of the day
about that allocation of resources.
And so the purpose of all this
is just to highlight
that there is this structure to uncertainty.
There are impacts that the tools
that we use as geological engineers have
to how we think about that.
And when we start to marry those two
and look at the circumstances of uncertainty
and the tools that we have for addressing it,
we really want to make sure that that’s a good marriage
in terms of producing this optimal allocation of resources
at the end of the day.
And that’s really the message of this entire talk.
Next slide, please.
Thank you for your time and attention.
(upbeat music)
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BoTorch · Bayesian Optimization in PyTorch
Source code for botorch.acquisition.predictive_entropy_search
#!/usr/bin/env python3
# Copyright (c) Meta Platforms, Inc. and affiliates.
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
Acquisition function for predictive entropy search (PES). The code utilizes the
implementation designed for the multi-objective batch setting.
NOTE: The PES acquisition might not be differentiable. As a result, we recommend
optimizing the acquisition function using finite differences.
from __future__ import annotations
from typing import Optional
from botorch.acquisition.multi_objective.predictive_entropy_search import (
from botorch.models.model import Model
from botorch.utils.transforms import concatenate_pending_points, t_batch_mode_transform
from torch import Tensor
class qPredictiveEntropySearch(qMultiObjectivePredictiveEntropySearch):
r"""The acquisition function for Predictive Entropy Search.
This acquisition function approximates the mutual information between the
observation at a candidate point `X` and the optimal set of inputs using
expectation propagation (EP).
(i) The expectation propagation procedure can potentially fail due to the unstable
EP updates. This is however unlikely to happen in the single-objective setting
because we have much fewer EP factors. The jitter added in the training phase
(`ep_jitter`) and testing phase (`test_jitter`) can be increased to prevent
these failures from happening. More details in the description of
(ii) The estimated acquisition value could be negative.
def __init__(
model: Model,
optimal_inputs: Tensor,
maximize: bool = True,
X_pending: Optional[Tensor] = None,
max_ep_iterations: int = 250,
ep_jitter: float = 1e-4,
test_jitter: float = 1e-4,
threshold: float = 1e-2,
) -> None:
r"""Predictive entropy search acquisition function.
model: A fitted single-outcome model.
optimal_inputs: A `num_samples x d`-dim tensor containing the sampled
optimal inputs of dimension `d`. We assume for simplicity that each
sample only contains one optimal set of inputs.
maximize: If true, we consider a maximization problem.
X_pending: A `m x d`-dim Tensor of `m` design points that have been
submitted for function evaluation, but have not yet been evaluated.
max_ep_iterations: The maximum number of expectation propagation
iterations. (The minimum number of iterations is set at 3.)
ep_jitter: The amount of jitter added for the matrix inversion that
occurs during the expectation propagation update during the training
test_jitter: The amount of jitter added for the matrix inversion that
occurs during the expectation propagation update in the testing
threshold: The convergence threshold for expectation propagation. This
assesses the relative change in the mean and covariance. We default
to one percent change i.e. `threshold = 1e-2`.
def forward(self, X: Tensor) -> Tensor:
r"""Evaluate qPredictiveEntropySearch on the candidate set `X`.
X: A `batch_shape x q x d`-dim Tensor of t-batches with `q` `d`-dim
design points each.
A `batch_shape'`-dim Tensor of Predictive Entropy Search values at the
given design points `X`.
return self._compute_information_gain(X)
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s of
Our users:
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Seth Lore, IA
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among
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Methods for calculating multidimensional, transient free surface flows past bodies
Numerical methods for calculating multidimensional, transient, free surface flows interacting with general curved boundaries are discussed. To model a free surface effectively, three problems must be
resolved compatibly: the surface must be numerically defined, a prescription must be provided to advance it in time, and appropriate boundary conditions must be applied at the location of the
surface. Basic motions of Lagrangian and Eulerian finite difference representations are reviewed first. All the free surface schemes discussed are couched in one basic solution algorithm, a direct
extension of the Marker-and-Cell method. A detailed description, including advantages and disadvantages, is given of free surface computational schemes that make use of the surface height function,
surface marker particles, and the volume fraction and variable density schemes. An illustration is given of the added mass and damping coefficients computed for rectangular cylinders undergoing
forced oscillation.
NASA STI/Recon Technical Report N
Pub Date:
□ Multiphase Flow;
□ Surface Properties;
□ Boundary Conditions;
□ Boundary Layer Flow;
□ Cylindrical Bodies;
□ Euler-Lagrange Equation;
□ Fluid Mechanics and Heat Transfer
|
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Distributed models for sparse attack construction and state vector estimation in the smart grid
Two distributed attack models and two distributed state vector estimation methods are introduced to handle the sparsity of smart grid networks in order to employ unobservable false data injection
attacks and estimate state vectors. First, Distributed Sparse Attacks in which attackers process local measurements in order to achieve consensus for an attack vector are introduced. In the second
attack model, called Collective Sparse Attacks, it is assumed that the topological information of the network and the measurements is available to attackers. However, attackers employ attacks to the
groups of state vectors. The first distributed state vector estimation method, called Distributed State Vector Estimation, assumes that observed measurements are distributed in groups or clusters in
the network. The second method, called Collaborative Sparse State Vector Estimation, consists of different operators estimating subsets of state variables. Therefore, state variables are assumed to
be distributed in groups and accessed by the network operators locally. The network operators compute their local estimates and send the estimated values to a centralized network operator in order to
update the estimated values.
Publication series
Name 2012 IEEE 3rd International Conference on Smart Grid Communications, SmartGridComm 2012
Other 2012 IEEE 3rd International Conference on Smart Grid Communications, SmartGridComm 2012
Country/Territory Taiwan, Province of China
City Tainan
Period 11/5/12 → 11/8/12
All Science Journal Classification (ASJC) codes
• Computer Networks and Communications
• Communication
• Smart grid security
• attack detection
• distributed optimization
• false data injection
• sparse models
Dive into the research topics of 'Distributed models for sparse attack construction and state vector estimation in the smart grid'. Together they form a unique fingerprint.
|
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Dyonic black holes
Written 29th of July 2023 Last updated 29th of July 2023
I will here give an overview of the work I did on the inspiral of dyonic black holes in order to model its gravitational waveform. Dyonic black holes carry both electric and magnetic charges. The
work resulted in two papers: this one and this. The work is covered in detail in the thesis.
Magnetic monopoles have been around as a theoretical concept for a long time, and there are several motivations to why they should exist, among other: They would bring symmetry to the Maxwell's
equations; They would explain why electric charge is quantised, as was shown by Dirac in this paper; They are a generic prediction of supersymmetric theories, see these notes, that historically have
provided the leading theoretical framework to extend the standard model of particle physics to contain dark matter; And they have been linked to a solution to the confinement problem of quantum
chromodynamics. However, in spite of this, they have never been observed experimentally, which puts their existence under strain. Still, if they were produced in the early universe, while the
universe was supersymmetric, they might have been so dilluted by now that we do not expect to see them.
Nontheless, in this work, we set out to consider the existence of magnetic monopoles agnostically and phenomenologically and ask the question of what we could observe in the mergers of black holes if
they could carry both magnetic and electric charges.
The Poincaré cone
The first step is to set up the force equation for either of the black holes, that we consider here as simple point particles without any structure. The position vector \(r_2^i\) receives
accelerations from both the magnetic and electric fields, as $$ \begin{eqnarray} m_2 \ddot r_2^i &=& \mu \ddot R^i = C \frac{R^i}{R^3} - D \epsilon_{ijk}\frac{R^j }{R^3} v^k,\label{dyonforce} \\ C &=
& \left(-\mu M + q_1 q_2 + g_1 g_2 \right),\quad D = \left( q_2 g_1 - g_2 q_1\right). \end{eqnarray} $$ We have used \(R^i=r_2^i-r_1^i\) to indicate their separation vector. \(q\) and \(g\) are the
electric and magnetic charge respectively, while we use the lower index to indicate which point we they belong to. \(\mu=m_1 m_2/M\) is the reduced mass while \(M=m_1+m_2\) is the total mass. If
integrate their accelerations numerically in a stationary center of mass system, we find the orbits and we notice that the orbit is no longer confined to a plane, but instead traces a conic shape. We
will understand this below.
An interesting consequence of this setup, is that the usual anguar momentum conservation no longer applies, \(\rm d /\rm d t\enspace \tilde L \neq 0\), but instead there is a generalised conserved
angular momentum
$$ \begin{eqnarray} L^i &\equiv& \tilde L^i - D\hat{R}^i,\label{genangmem}\\ \dot L^i &=& 0 , \end{eqnarray} $$ whose derivative is zero. Since \(\tilde L^i \perp \hat R^i\) and \(D\) is a constant,
the norm of \(\tilde L^i\) must be a constant too. Therefore it is only the direction of the angular momentum that is not conserved. The direction of the angular momentum will rotate so that the
orbit traces out a cone. Using our new constants of motion, we can perform a derivation of the radial separation similar to what is done with Keplerian orbits. We find $$ \begin{eqnarray} R &=& \frac
{a(1-e^2)}{1 + e \cos (\phi \sin \theta)},\label{magmonR}\\ E &=& \frac{C}{2a},\quad \tilde L^2 = \mu|C| a (1-e^2), \end{eqnarray} $$ where \(a,e\) are the semi-major axis and eccentricity
respectively, \(\phi\) is the azimuth and the angle \(\cos\theta=-D/L\) is the opening angle of the cone. We therefore see that there is a precessing motion where the maxmum of the separation occurs
every \(2\pi /\sin \theta\) radians. Using the radial expression, we can find the 3D orbit, knowing that the zenith angle \(\theta\) is constant $$ R^i = R \begin{pmatrix} \sin\theta\cos\phi \\ \sin\
theta\sin\phi \\ \cos\theta \end{pmatrix} . $$ We then have obtained the closed-form expression for the motion of the particles when the energy and generalised angular momentum are conserved. The
acceleration can easily be found by differentiation $$ \begin{eqnarray} \ddot{R}^i &=& \frac{|C|}{\mu R^2} g_E^i,\\ g_E^i &=& e\cos x \begin{pmatrix}(\sin\theta-1/\sin\theta)\cos\phi\\(\sin\theta-1/\
sin\theta) \sin\phi\\\cos\theta \end{pmatrix} - \frac{1}{\sin\theta}\begin{pmatrix} \cos\phi\\\sin\phi\\ 0\end{pmatrix} , \end{eqnarray} $$ and we notice that we obtain Newton's gravitational law for
In order to find the orbital evolution like one does in the slow inspiral modelling of black hole mergers, one needs to include the back-reaction of the radiation fields on the orbit. The binary will
radiate in both gravitational waves and electromagnetic waves, that will drain the system of energy and angular momentum. For example, the expression for the leading order flux of emitted energy in
the electromagnetic channe, by a binary that is only electrically charged, is given by the charge dipole \(Q^i\) like $$ \begin{eqnarray} \dot E &=& -\frac{2 }{3}\langle \ddot Q^i \ddot Q_i \rangle,
\\ Q^{i} &=& \int \rm d^3 x\, j^t x^i \end{eqnarray} $$ where \(j^t=\sum_i q_i\delta (\vec x - \vec x_i)\) is the charge density. In order to extend this expression to calculate the emission for
dyonic binaries that also carry magnetic charge, we make use of the duality transform of electromagnetism. According to it, electromagnetism is invariant under the transformation $$ \begin{eqnarray}
\mathbf E_2 &=& \mathbf E_1 \cos\alpha - \mathbf B_1 \sin\alpha,\\ \mathbf B_2 &=& \mathbf E_1 \sin\alpha + \mathbf B_1 \cos\alpha,\\ q_e &\rightarrow& q_e \cos\alpha + q_g\sin\alpha,\\ q_g &\
rightarrow& q_g\cos\alpha - g_e \sin\alpha . \end{eqnarray} $$ This applies to some transformation parameter \(\alpha\). By considering \(\alpha=\pi/2\), we can consider a purely electrically charged
system, and then find the analog expression for a purely magnetically charged system. We then wish to superimpose the two emissions, and we realise that we are able to do so, since \(\vec E_1\perp\
vec B_1\) and \(\vec E_1 \parallel \vec B_2\implies E_1\perp E_2\) and so on, and the energy \(E\sim \vec E\cdot \vec E + \vec B\cdot \vec B \). In the end, we therefore find the total emissions $$ \
begin{eqnarray} \dot E_{\text{tot}} &=& - \frac{ C^2 \left[\left(\Delta\sigma_e\right)^2 + \left(\Delta\sigma_g\right)^2\right]}{24 a^4 (1-e^2)^{5/2} \sin^2\theta}\left( 16 + 28 e^2 + 3 e^4 + (20+ 3
e^2 ) e^2 \cos (2\theta)\right), \\ \dot J_{\text{tot}} &=& -\frac{2\left[\left(\Delta\sigma_e\right)^2+\left(\Delta\sigma_g\right)^2\right]}{3 }\sqrt{\frac{|C|^3 \mu}{a(1-e^2)}}\left\langle \sqrt{\
left(\epsilon^i_{\,\,jk} g_J^j g_E^k\right)^2}\,/R^2\right\rangle, \end{eqnarray} $$ where we have defined \(\Delta \sigma_e = q_1/m_1-q_2/m_2\) and \(q\rightarrow g\) for \(\Delta\sigma_g\). See the
full derivation for more details.
In the end we obtained the orbital motion and expressions for the energy and angular momentum emissions of binary dyonic black holes. The orbit may now be modelled quasi-statically, meaning that the
orbital energy and angular momentum changes sufficiently slowly that the the emission is well described by that of a binary moving in a constant energy orbit like the one we found. Then we can couple
the emissions to the expressions for \(E\) and \(\tilde L\) and find how the semi-major axis and eccentricity change with time. Finally the gravitational wave emission may be included similarly, and
the waveform can be found now that the orbital evolution is known. This is done in this paper and this one for the eccentric orbit.
This is the first step to get templates for this kind of binary. In the future one ought to do the analysis for the merger stage, and the ringdown stage in order to have complete enough template
waveforms to extract potential signals from the gravitational detector data.
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Ok, a what Transform now??
In the early 1800s, Jean-Baptiste Joseph Fourier, a French mathematician and physicist, introduced the transform in his study of heat transfer. The idea seemed preposterous to many mathematicians at
the time, but it has now become an important cornerstone in mathematics.
So, what exactly is the Fourier Transform? The Fourier Transform is a mathematical transform that decomposes a function into its sine and cosine components. It decomposes a function depending on
space or time into a function depending on spatial or temporal frequency.
Before diving into the mathematical intricacies of the Fourier Transform, it is important to understand the intuition and the key idea behind it. The main idea of the Fourier Transform can be
explained simply using the metaphor of creating a milkshake.
Imagine you have a milkshake. It is hard to look at a milkshake and understand it directly; answering questions such as “What gives this shake its nutty flavour?” or “What is the sugar content of
this shake?” are harder to answer when we are simply given the milkshake. Instead, it is easier to answer these questions by understanding the recipe and the individual ingredients that make up the
shake. So, how exactly does the Fourier Transform fit in here? Given a milkshake, the Fourier Transform allows us to find its recipe to determine how it was created; it is able to present the
individual ingredients and the proportions at which they were combined to make the shake. This brings up the questions of how does the Fourier transform determine the milkshake “recipe” and why would
we even use this transform to get the “recipe”? To answer the former question, we are able to determine the recipe of the milkshake by running it through filters that then extract each individual
ingredient that makes up the shake. The reason we use the Fourier Transform to get the “recipe” is that recipes of milkshakes are much easier to analyze, compare, and modify than working with the
actual milkshake itself. We can create new milkshakes by analyzing and modifying the recipe of an existing milkshake. Finally, after deconstructing the milkshake into its recipe and ingredients and
analyzing them, we can simply blend the ingredients back to get the milkshake.
Extending this metaphor to signals, the Fourier Transform essentially takes a signal and finds the recipe that made it. It provides a specific viewpoint: “What if any signal could be represented as
the sum of simple sine waves?”.
By providing a method to decompose a function into its sine and cosine components, we can analyze the function more easily and create modifications as needed for the task at hand.
A common application of the Fourier Transform is in sound editing. If sound waves can be separated into their “ingredients” (i.e., the base and treble frequencies), we can modify this sound depending
on our requirements. We can boost the frequencies we care about while hiding the frequencies that cause disturbances in the original sound. Similarly, there are many other applications of the Fourier
Transform such as image compression, communication, and image restoration.
This is incredible! An idea that the mathematics community was skeptical of, now has applications to a variety of real-world applications.
Now, for the fun part, using Fourier Transform in a sentence by the end of the day:
Example 1:
Koby: “This 1000 puzzle is insanely difficult. How are we ever going to end up with the final puzzle picture?”
Eng: “Don’t worry! We can think of the puzzle pieces as being created by taking the ‘Fourier transform’ of the puzzle picture. All we have to do now is take the ‘inverse Fourier Transform’ and then
we should be done!”
Koby: “Now when you put it that way…. Let’s do it!”
Example 2:
Grace: “Hey Rohan! What’s the difference between a first-year and fourth-year computer science student?
Rohan: “… what?”
Grace: “A Fouri-y-e-a-r Transform”
Rohan: “…. (╯°□°)╯︵ ┻━┻ ”
I’ll see you in the blogosphere…
Parinita Edke
The MiDATA Word of the Day is…”clyster”
Holy mother of pearl! Do you remember when the first Pokémon games came out on the Game Boy? Never heard of Pokémon? Get up to speed by watching this short video. Or even better! Try out one of the
games in the series, and let me know how that goes!
The name of the Pokémon in this picture is Cloyster. You may remember it from Pokémon Red or Blue. But! Cloyster, in fact, has nothing to do with clysters.
In olden days, clyster meant a bunch of persons, animals or things gathered in a close body. Now, it is better known as a cluster.
You yourself must identify with at least one group of people. What makes you human; your roles, qualities, or actions make you unique. But at the same time, you fall into a group of others with the
same characteristics.
You yourself fall into multiple groups (or clusters). This could be your friend circle or perhaps people you connect with on a particular topic. At the end of the day, you belong to these groups. But
is there a way we can determine that you, in fact, belong?
Take for example Jack and Rose from the Titanic. Did Jack and Rose belong together?
If you take a look at the plot to the right, Jack and Rose clearly do not belong together. They belong to two separate groups (clusters) of people. Thus, they do not belong together. Case closed!
But perhaps it is a matter of perspective? Let’s take a step back…
Woah! Now, you could now say that they’re close enough, they might as well be together! Compared to the largest group, they are more similar than they are different. And so, they should be together!
For the last time, we may have been looking at this completely wrong! From the very beginning, what are we measuring on the x-axis and on the y-axis of our graph?
Say it was muscle mass and height. That alone shouldn’t tell us if Rose and Jack belong together! And yet, that is exactly what we could have done. But if not those, then what..?
Now for the fun part (see the rules here), using clyster in a sentence by the end of the day:
Serious: Did you see the huge star clysters last night? I heard each one contained anywhere from 10,000 to several million stars…
Less serious: *At a seafood restaurant by the beach* Excuse me, waiter! I’d like one of your freshest clysters, please. – “I’m sorry. We’re all out!”
…I’ll see you in the blogosphere.
Stanley Hua
Stanley Hua in ROP299: Joining the Tyrrell Lab during a Pandemic
My name is Stanley Hua, and I’ve just finished my 2^nd year in the bioinformatics program. I have also just wrapped up my ROP299 with Professor Pascal. Though I have yet to see his face outside of my
monitor screen, I cannot begin to express how grateful I am for the time I’ve been spending at the lab. I remember very clearly the first question he asked me during my interview: “Why should I even
listen to you?” Frankly, I had no good answer, and I thought that the meeting didn’t go as well as I’d hoped. Nevertheless, he gave me a chance, and everything began from there.
Initially, I got involved with quality assessment of Multiple Sclerosis and Vasculitis 3D MRI images along with Jason and Amar. Here, I got introduced to the many things Dmitrii can complain about
taking brain MRI images. Things such as scanner bias, artifacts, types of imaging modalities and prevalence of disease play a role in how we can leverage these medical images in training predictive
My actual ROP, however, revolved around a niche topic in Mauro and Amar’s project. Their project sought to understand the effect of dataset heterogeneity in training Convolutional Neural Networks
(CNN) by cluster analysis of CNN-extracted image features. Upon extraction of image features using a trained CNN, we end up with high-dimensional vectors representing each image. As a preprocessing
step, the dimensionality of the features is reduced by transformation via Principal Component Analysis, then selecting a number of principal components (PC) to keep (e.g. 10 PCs). The question must
then be asked: How many principal components should we use in their methodology? Though it’s a very simple question, I took way too many detours to answer this question. I looked at the difference
between standardization vs. no standardization before PCA, nonlinear dimensionality reduction techniques (e.g. autoencoder) and comparisons of neural network image representation (via SVCCA) among
other things. Finally, I proposed an equally simple method for determining the number of PCs to use in this context, which is the minimum number of PCs that gives the most frequent resulting value
(from the original methodology).
Regardless of the difficulty of the question I sought to answer, I learned more about practices in research, and I even learned about how research and industry intermingle. I only have Professor
Pascal to thank for always explaining things in a way that a dummy such as me would understand. Moreover, Professor Pascal always focused on impact; is what you’re doing meaningful and what are its
I believe that the time I spent with the lab has been worthwhile. It was also here that I discovered that my passion to pursue data science trumps my passion to pursue medical school (big thanks to
Jason, Indranil and Amar for breaking my dreams). Currently, I look towards a future, where I can drive impact with data; maybe even in the field of personalized medicine or computational biology.
Whoever is reading this, feel free to reach out! Hopefully, I’ll be the next Elon Musk by then…
Transiently signing out,
Stanley Bryan Z. Hua
Jessica Xu’s Journey in ROP299
Hello everyone! My name is Jessica Xu, and I’ve just completed my second year in Biochemistry and Statistics at the University of Toronto. This past school year, I’ve had the wonderful opportunity to
do a ROP299 project with Dr. Pascal Tyrrell and I’d like to share my experience with you all!
A bit about myself first: in high school, I was always interested in life sciences. My favourite courses were biology and chemistry, and I was certain that I would go to medical school and become a
doctor. But when I took my first stats course in first year, I really enjoyed it and I started to become interested in the role of statistics in life sciences. Thus, at the end of my first year,
while I was looking through the various ROP courses, I felt that Dr. Tyrrell’s lab was the perfect opportunity to explore my budding interest in this area. I was very fortunate to have an interview
with Dr. Tyrrell, and even more fortunate to be offered a position in his lab!
Though it may be obvious, doing a research project when you have no research experience is very challenging! Coming into this lab having taken a statistics course and a few computer science courses
in first year, I felt I had a pretty good amount of background knowledge. But as I joined my first lab meeting, I realized I couldn’t be more wrong! Almost every other word being said was a word I’d
never heard of before! And so, I realized that there was a lot I needed to learn before I could even begin my project.
I then began on the journey of my project, which was looking at how two dimension reduction techniques, LASSO and SES, performed in an ill-posed problem. It was definitely no easy task! While I had
learned a little bit about dimension reduction in my statistics class, I still had a lot to learn about the specific techniques, their applications in medical imaging, and ill-posed problems. I was
also very inexperienced in coding, and had to learn a lot of R on my own, and become familiar with the different packages that I would have to use. It was a very tumultuous journey, and I spent a lot
of time just trying to get my code to work. Luckily, with help from Amar, I was able to figure out some of the errors and issues I was facing in regards to the code.
I learned a lot about statistics and dimension reduction in this ROP, more than I have learned in any other courses! But most importantly, I had learned a lot about the scientific process and the
experience of writing a research paper. If I can provide any advice based on my experience, it’s that sometimes it’s okay to feel lost! It’s not expected of you to have devised a perfect plan of
execution for your research, especially when it’s your first time! There will be times that you’ll stray off course (as I often did), but the most valuable lesson that I learned in this ROP is how to
get back on track. Sometimes you just need to take a step back, go back to the beginning and think about the purpose of your project and what it is you’re trying to tell people. But it’s not always
as easy to realize this. Luckily Dr. Tyrrell has always been there to guide us throughout our projects and to make sure we stay on track by reminding us of the goal of our research. I’m incredibly
grateful for all the support, guidance, and time that Dr. Tyrrell has given this past year. It has been an absolute pleasure of having the experience of working in this lab.
Now that I’ve taken my first step into the world of research, with all the new skills and lessons I’ve learned in my ROP, I look forward to all the opportunities and the journey ahead!
Jessica Xu
Today’s MiWORD of the day is… Lasso!
Wait… Lasso? Isn’t a lasso that lariat or loop-like rope that cowboys use? Or perhaps you may be thinking about that tool in Photoshop that’s used for selecting free-form segments!
Well… technically neither is wrong! However, in statistics and machine learning, Lasso stands for something completely different: least absolute shrinkage and selection operator. This term was coined
by Dr. Robert Tibshirani in 1996 (who was a UofT professor at that time!).
Okay… that’s cool and all, but what the heck does that actually mean? And what does it do?
Lasso is a type of regression analysis method, meaning it tries to estimate the relationship between predictor variables and outcomes. It’s typically used to perform feature selection or
Regularization is a way of reducing overfitting of a model, ie. it removes some of the “noise” and randomness of the data. On the other hand, feature selection is a form of dimension reduction. Out
of all the predictor variables in a dataset, it will select the few that contribute the most to the outcome variable to include in a predictive model.
Lasso works by applying a fixed upper bound to the sum of absolute values of the coefficient of the predictors in a model. To ensure that this sum is within the upper bound, the algorithm will shrink
some of the coefficients, particularly it shrinks the coefficients of predictors that are less important to the outcome. The predictors whose coefficients are shrunk to zero are not included at all
in the final predictive model.
Lasso has applications in a variety of different fields! It’s used in finance, economics, physics, mathematics, and if you haven’t guessed already… medical imaging! As the state-of-the-art feature
selection technique, Lasso is used a lot in turning large radiomic datasets into easily interpretable predictive models that help researchers study, treat, and diagnose diseases.
Now onto the fun part, using Lasso in a sentence by the end of the day! (see rules here)
Serious: This predictive model I got using Lasso has amazing accuracy for detecting the presence of a tumour!
Less serious: I went to my professor’s office hours for some help on how to use Lasso, but out of nowhere he pulled out a rope!
See you in the blogosphere!
Jessica Xu
Jacky Wang’s ROP399 Journey
My name is Jacky Wang, and I am just finishing my third year at the University of Toronto, pursuing a computer science specialist. Looking back on this challenging but incredible year, I was honoured
to have the opportunity to work inside Dr. Tyrrell’s lab as part of the ROP399 course. I would love to share my experience studying and working inside the lab.
Looking back, I realize one of the most challenging tasks is getting onboard. I felt a little lost at first when surrounded by loads of new information and technologies that I had little experience
with before. Though feeling excited by all the collision of ideas during each meeting, having too many choices sometimes could be overwhelming. Luckily after doing more literature review and with the
help of the brilliant researchers in the lab (a big thank you to Mauro, Dimitri, and of course, Dr. Tyrrell), I start to get a better view of the trajectories of each potential project and further
determine what to get out from this experience. I did not choose the machine learning projects, though they were looking shiny and promising as always (as a matter of fact, they turned out to be
successful indeed). Instead, I was more leaning towards studying the sample size determination methodology, especially the concept of ill-posed problems, which often occur when the researchers make
conclusions from models trained on limited samples. It had always been a mystery why I would get different and even contrasting results when replicating someone else’s work on smaller sample sizes.
From there, I settled the research topic and moved onto the implementation details.
This year the ROP students are coming from statistics, computer science and biology etc. I am grateful that Dr. Tyrrell is willing to give anyone who has the determination to study in his lab a
chance though they may have little research experience and come from various backgrounds. As someone who studies computer science with a limited statistics background, the real challenge lies in
understanding all the statistical concepts and designing the experiments. We decided to apply various dimension reduction techniques to study the effect of different sample sizes with many features.
I designed experiments around the principal component analysis (PCA) technique while another ROP student Jessica explored the lasso and SES model in the meantime. It was for sure a long and memorable
experience with many debugging when implementing the code from scratch. But it was never more rewarding than seeing the successful completion of the code and the promising results.
I feel lucky and grateful that Dr. Tyrell helped me complete my first research project. He broke down the long and challenging research task into clear and achievable subgoals within our reach. After
completing each subgoal, I could not even believe it sent us close to the finished line. It felt so different taking an ROP course than attending the regular lessons. For most university courses,
most topics are already determined, and the materials are almost spoon-fed to you. But sometimes, I start to lose the excitement of learning new topics, as I am not driven by the curiosity nor the
application needs but the pressure of being tested. However, taking the ROP course gives me almost complete control of my study. For ROP, I was the one who decides what topics to explore, how to
design the experiment. I could immediately test my understanding and put everything I learned into real applications.
I am so proud of all the skills that I have picked up in the online lab during this unique but special ROP experience. I would like to thank Dr. Tyrrell for giving me this incredible study experience
in his lab. There are so many resources out there to reach and so many excellent researchers to seek help from. I would also like to thank all members of the lab for patiently walking me through each
challenge with their brilliant insights.
Jacky Wang
MiWord of the Day Is… dimensionality reduction!
Guess what?
You are looking at a real person, not a painting! This is one of the great works by a talented artist Alexa Meade, who paints on 3D objects but creates a 2D painting illusion. Similarly in the world
of statistics and machine learning, dimensionality reduction means what it sounds like: reduce the problem to a lower dimension. But only this time, not an illusion.
Imagine a 1x1x1 data point living inside a 2x2x2 feature space. If I ask you to calculate the data density, you will get ½ for 1D, ¼ for 2D and 1/8 for 3D. This simple example illustrates that the
data points become sparser in higher dimensional feature space. To address this problem, we need some dimensional reduction tools to eliminate the boring dimensions (dimensions that do not give much
information on the characteristics of the data).
There are mainly two approaches when it comes to dimension reduction. One is to select a subset of features (feature selection), the other is to construct some new features to describe the data in
fewer dimensions (feature extraction).
Let us consider an example to illustrate the difference. Suppose you are asked to come up features to predict the university acceptance rate of your local high school.
You may discard the “grade in middle school” for its many missing values; discard “date of birth” and “student name” as they are not playing much role in applying university; discard “weight > 50kg”
as everyone has the same value; discard “grade in GPA” as it can be calculated. If you have been through a similar process, congratulations! You just performed a dimension reduction by feature
What you have done is removing the features with many missing values, the least correlated features, the features with low variance and one of the highly correlated. The idea behind feature selection
is that the data might contain some redundant or irrelevant features and can be removed without losing too much loss information.
Now, instead of selecting a subset of features, you might try to construct some new features from the old ones. For example, you might create a new feature named “school grade” based on the full
history of the academic features. If you have been through a thought process like this, you just performed a dimensional reduction by feature extraction
If you would like to do a linear combination, principal component analysis (PCA) is the tool for you. In PCA, variables are linearly combined into a new set of variables, known as the principal
components. One way to do so is to give a weighted linear combination of “grade in score”, “grade in middle school” and “recommend letter” …
Now let us use “dimensionality reduction” in a sentence.
Serious: There are too many features in this dataset, and the testing accuracy seems too low. Let us apply dimensional reduction techniques to reduce overfit of our model…
Less serious:
Mom: “How was your trip to Tokyo?”
Me: “Great! Let me just send you a dimensionality reduction version of Tokyo.”
Mom: “A what Tokyo?”
Me: “Well, I mean … photos of Tokyo.”
I’ll see you in the blogosphere…
Jacky Wang
IMPACT-MED 2024: Innovations in Medical Precision, Accessibility and Collaborative Technology
The IMPACT-MED 2024 Conference is a one-day online event scheduled for Wednesday, October 30th, 2024. It aims to advance global health innovation and entrepreneurship. The conference is supported by
the University of Toronto’s Global Classrooms initiative and the Institute of Medical Science’s (IMS) two specialized courses: MSC1114H: Artificial Intelligence in Medicine and MSC1122H: Startups in
the Medical Sciences. The conference will be hosted on the Zoom platform, offering interactive and engaging experiences for participants. Key activities include expert lectures, practical workshops,
and breakout sessions, covering topics such as AI applications in healthcare, medical startup ecosystems, and global healthcare system comparisons. This event fosters international collaboration and
provides students with valuable insights into both AI and entrepreneurship within medical sciences.
The conference is free to attend.
You can register online here, via the Zoom registration page.
Yan Qing Lee’s ROP299 Journey
Hi! I’m Yan Qing Lee, an incoming 3rd-year Computer Science and Psychology double major undergraduate student. This past summer, I was given the opportunity to embark on my first research project in
the field of artificial intelligence, and I’m excited to share my experience.
My research topic investigated if individuals who receive a false-positive mammogram result by an AI model have a higher risk of receiving a breast cancer diagnosis later on. Past studies have found
that receiving a false-positive mammogram result from radiologists is associated with a higher risk of future breast cancer, but no studies have yet investigated if this holds true for AI breast
cancer detection models. In this project, I used a longitudinal dataset of breast cancer mammograms, and ran a trained AI breast cancer classifier, made of an ensemble of 4 Convnext-small models, to
obtain false-positive and true-negative results. Cox proportional hazards models were then used to investigate the hazard ratio of receiving a false-positive result, from both the AI model, and from
As a student who entered the Computer Science major out-of-stream, I started the ROP feeling really out of place. Although I’ve known I wanted to pursue AI, I had no real experience in neither AI nor
medical imaging, and I wondered if I was too under-qualified for this experience. Still, I was determined to put in as many hours as I needed to succeed.
I first began by familiarizing myself with ML terms, and choosing an area of interest (breast cancer mammography) to formulate a research question upon. As I’m sure other ROP students would agree,
this process was extremely challenging; as weeks passed by, I found that my research questions were always either over-ambitious or not feasible. Over time, however, I realized that my difficulty
with creating a research question stemmed from my lack of knowledge in exactly how ML models work, and the existing literature and gaps within the field of breast cancer mammography. As I dug deeper
into existing literature, the one interesting finding regarding radiologists’ false-positives caught my eye, and this finally led me to my research question.
Once I began working on my project, the many challenges of research revealed themselves to me. This included difficulties of downloading and parsing through a large dataset, of installing packages
and working around incompatible versions of libraries to set up a working environment, and, worst of all, of finding out an AI breast cancer detection model you originally centered your project
around is not as replicable as you assumed it would be. Despite that I made sure to set up my research question to be relatively simple, the process of setting up, debugging preprocessing code,
training and running an AI breast cancer classification model and obtaining undesirable training results was nothing short of complicated. Still, with the weekly lab meetings keeping me on track, and
the support of Dr. Tyrrell, Mauro and the other students in the lab, I slowly but surely overcame every obstacle, and learned immense amounts every week to successfully complete my project. Even
though I had to find a new AI model to use near the end, and redo my experimentation, I found that with my experience with the previous AI model, I was now able to independently set up and run the
new model much more efficiently than before. It was proof of how much I’d learned, and I’m glad to now be able to look back and be proud of how much I’ve accomplished in the span of a few months.
At the end of it all, I have to thank Dr. Tyrrell for fostering my passion towards AI and its applications in fields as impactful and important as breast cancer mammography. This experience only made
me more excited to delve into the applications of AI in other fields in the future, and I can’t thank the MiData lab enough for this experience.
MiWord of the Day is… Region of Interest!
Look! You’ve finally made it to Canada! You gloriously take in the view of Lake Ontario when your friend beside you exclaims, “Look, they have beaver tails!” You excitedly scan the lake, asking,
You see no movement from the lake. It isn’t until your friend pulls you to the front of a storefront says “BeaverTails” with a picture of delicious pastries that you realize they didn’t mean actual
beavers’ tails. It turns out you were looking at the wrong place the whole time!
Often times, it’s easy for us to quickly identify objects because we know the context of where things should be. These are the kinds of things we take for granted, until it’s time to hand the same
tasks over to machines.
In medical imaging, experts label what are called Regions of Interests (ROIs), which are specific areas of a medical image that contain pathology, such as the specific area of a lesion. Having
labelled ROIs are important, as it can help prevent extra time from being wasted on analyzing non-relevant areas of an image, especially since medical images contain complex structures that take time
to interpret. But in the area of machine learning (ML) in medical imaging, having labelled ROIs is also useful because it can help with training ML models that classify whether a medical image
contains a pathology or not; with ROIs identified, cropping can be done during the preprocessing of images so that only relevant areas of images are compared for the model to learn differences
between positive and negative images faster.
In fact, having ROIs is so important, there is an entire field in artificial intelligence dedicated to it: Computer Vision. The field of computer vision focuses on automating the extraction of ROIs
in images or videos, which plays a critical role in the mechanization of tasks like object detection and tracking for things like self-driving cars. In object detection, for example, things like ROI
Pooling can be utilized; this is where multiple ROIs are used to obtain input feature maps, from which maximum values are used to detect the presence of features, giving rise to the ability to
identify many objects at once – this is extremely useful, especially once you’re on the road and there are 10 other cars around you!
Now, the fun part: using Region of Interest in a sentence!
Serious: The coordinates of ROIs are given for the positive mammogram images in the dataset I’m using. Maybe I could use Grad-CAM to see if the ML breast cancer classification model I’m using uses
the same regions of the image to arrive at its classification decision; this way, I can see if its decision making aligns with the decision making of radiologists.
Less serious: I forced my friend to watch my favorite movie with me, but I can’t lie – I think the attractive male lead was her only region of interest!
See you in the blogosphere,
Yan Qing Lee
Yuxi Zhu’s ROP Journey
Hi, I am Yuxi Zhu, a Bioinformatics and Computational Biology specialist and Molecular Genetics Major who just finished my second year. Like most people, this is my first formal research experience.
Professor Tyrrell warned me from the start that I would need to be independent in this lab, but my genuine interest in ML and its applications gave me the confidence to take on the challenge.
Overall, this summer’s ROP journey in the MiDATA lab was filled with both excitement and challenges.
The first challenge was finding a research question. I’m incredibly grateful to Daniel, a volunteer and former ROP student, who introduced me to the concept of “adversarial examples” and helped me
formulate my research question from the start. During the first two months of the literature review, I often found myself diving too deeply into theoretical aspects that were less applicable to
Medical Imaging, or exploring questions that, while feasible, didn’t capture my interest. Luckily, I was able to settle down with understanding the differential effects between random perturbations
(like random noise and loss of resolution) and non-random adversarial perturbations on the model.
As the project progressed, I encountered a series of obstacles and bugs that required constant problem-solving and debugging. For example, my initial findings showed very low performance, all under
50%. Professor Tyrrell pointed out that the accuracy of a binary classifier should never drop below 50%, as that would mean it’s performing worse than a random model. I quickly realized there were
bugs in my code and implementation. Additionally, after obtaining results, I thought interpreting them would be straightforward. However, when Professor Tyrrell asked me why adversarial perturbations
led to accuracies below 50% while the others didn’t, I found myself at a loss for words. In the end, with Professor Tyrrell’s guidance, I was able to interpret the results correctly and articulate
them in my report.
Despite the stress I felt before presenting my findings at our weekly meetings, these sessions became invaluable learning experiences. Professor Tyrrell would scrutinize my work with questions and
critiques, pushing me to think more deeply and critically about every aspect of my research. The other lab members also provided very helpful insights and shared their work. These meetings not only
allowed me to understand what others were working on but also gave me the chance to get involved in or observe lively discussions that often took place.
Looking back on the last few months, this experience has been invaluable. I am deeply thankful to Professor Tyrrell who offered me this wonderful opportunity in ML and guided me through my research
project. I especially appreciate how we weren’t just taught to implement a given research project or conduct a specific experiment; we were taught how to find gaps and how to conduct research. I also
want to express my gratitude to Daniel for his support and insights when I was in doubt, and to Atsuhiro for his helpful suggestions. Completing my first-ever research project was challenging yet
rewarding, and I am grateful for all the guidance and help I received. I’m confident that what I have learned will stay with me in my future research and career.
Today’s MiWORD of the day is… Adversarial Example!
According to the dictionary, the term “adversarial” refers to a situation where two parties or sides oppose each other. But what about the “adversarial example”? Does it imply an example of two
opposing sides? In a way, yes.
In machine learning, an example is one instance of the dataset. Adversarial examples are examples with calculated and imperceptible perturbation that tricks the model into the wrong prediction but
look the same to humans. So “adversarial”, in this case, indicates opposition between something (or human) and the model. The adversarial examples are intentionally crafted to trick the model by
exploiting its vulnerabilities.
How it works? There are many ways to find weak spots and generate adversarial examples, but FGSM is one classic way, and the goal is to make small changes to a picture such that it outputs the wrong
prediction. First, we input the model with the picture. Assume the model outputs the correct prediction, so the loss function, which represents the difference between the prediction and the true
label, will be low. Second, we compute the gradient of the loss function to tell us whether we should add or subtract a certain value epsilon to each pixel to make the loss bigger. Epsilon is
typically very small, resulting in a tiny change to the value. Now, we have a picture that looks the same as the original but will trick the model into the opposite prediction!
One exciting property of adversarial examples is their transferability. It is known that adversarial examples created for one model can also trick other unknown models. This might be due to inherent
flaws in the pattern recognition mechanisms of all models and, sometimes, model similarities, allowing these adversarial examples to exploit common vulnerabilities and lead to incorrect predictions.
Now, use “adversarial example” in a sentence by the end of the day:
Kinda Serious: “Oh I can’t believe my eyes. I am seeing a dog right here and the model says it’s a cupcake…So you’re saying it might be an adversarial image? What even is that? The model is just
Less Serious: Apparently, the movie star has an adversarial relationship with the media, but which stars have a good relationship with the media nowadays?
See you in the blogosphere,
Yuxi Zhu
MiWord of the Day is… Learned Perceptual Image Patch Similarity (LPIPS)!
Imagine you’re trying to compare two images—not just any images, but complex medical images like MRIs or X-rays. You want to know how similar they are, but traditional methods like simply comparing
pixel values don’t always capture the whole picture. This is where Learned Perceptual Image Patch Similarity, or LPIPS, comes into play.
Learned Perceptual Image Patch Similarity (LPIPS) is a cutting-edge metric for evaluating perceptual similarity between images. Unlike traditional methods like Structural Similarity Index (SSIM) or
Peak Signal-to-Noise Ratio (PSNR), which rely on pixel-level analysis, LPIPS utilizes deep learning. It compares images by passing them through a pre-trained convolutional neural network (CNN) and
analyzing the features extracted from various layers. This approach allows LPIPS to capture complex visual differences more closely aligned with human perception. It is especially useful in
applications such as evaluating generative models, image restoration, and other tasks where perceptual accuracy is critical.
Why is this important? In medical imaging, where subtle differences can be crucial for diagnosis, LPIPS provides a more accurate assessment of image quality, especially when images have undergone
various types of degradation, such as noise, blurring, or compression.
Now, let’s use LPIPS in sentences!
Serious: When evaluating the effectiveness of a new medical imaging technique, LPIPS was used to compare the generated images to the original scans, showing that it was more sensitive to perceptual
differences than traditional metrics.
Less Serious: I used LPIPS to compare my childhood photos with recent ones. According to the metric, I’ve definitely “degraded” over time!
See you in the blogosphere!
Jingwen (Lisa) Zhong
Jingwen (Lisa) Zhong’s ROP299 Journey
Hi all! My name is Jingwen (Lisa) Zhong. I’m a Data Science Specialist and Actuarial Science Major at UofT, graduating in 2026. I’m really happy and honored to have joined Prof. Tyrrell’s lab in the
summer of 2024 as an ROP299 student. This was my first research project, and it has truly exercised many of my research and scientific skills, such as literature review, critical thinking, and the
ability to get familiar with a brand-new field.
Coming into the lab, I had no research experience and no prior knowledge of medical imaging. As a student just finishing my second year of study, I felt curious about machine learning and artificial
intelligence because these topics are so widely discussed. However, I still can’t forget how uneasy I felt during the first few weeks as I tried to think of a research question related to medical
images and machine learning. I’m incredibly thankful to Prof. Tyrrell, who ‘relentlessly’ pointed out issues during each lab meeting, and to the lab volunteers, Daniel and Atshuhiro, who were always
willing to help and guide me through the process. I couldn’t have gotten my project ready for implementation without their support. After a month of struggle, I finally settled on my research topic:
investigating whether LPIPS is a better metric for assessing the similarities of medical images compared to PSNR and SSIM under various degradation conditions.
Having a research question is just the beginning; implementing it is another huge mountain to climb. I remember how excited I was when my research question was finally approved. I worked hard that
week to implement almost all the code for my project. If I could go back, I would approach this differently. Instead of diving straight into coding, I would first take the time to design the entire
study process—splitting the dataset, testing the code on a smaller dataset, figuring out how to use the GPU, then applying the code to the full dataset, and finally choosing the appropriate
statistical analysis. I say this because I stumbled at each of these steps. After completing my code, I found that it ran so slowly that it would take several days to get results. So, I began the
process of figuring out how to set up the environment to run on the lab’s GPU. This process took me almost two weeks, but with the help of other ROP students, I finally got the code running on the
Once the GPU problem was solved, my results came in much faster. However, the next obstacle was interpreting these results. As a Data Science student, it’s hard to admit, but I hadn’t yet learned
ANOVA. Initially, I turned to ChatGPT for help, but the results weren’t ideal. Prof. Tyrrell suggested that I use SAS to perform ANOVA, which provided me with ideal and comprehensive results. So, I
learned how to use SAS—a very powerful statistical analysis tool compared to Python.
Through this ROP experience, I learned the importance of communication and teamwork. Although we worked on different projects, the weekly lab meetings were incredibly helpful. It was a place where
everyone’s intelligence came together, and I always left with new insights and a clear plan in mind.
Overall, this journey has been a steep learning curve but an immensely rewarding one. I am grateful for the opportunity to work with such a supportive team, and I know that the skills and lessons
I’ve learned will continue to guide me in my future research endeavors.
AI in Drug & Biological Product Development: FDA & CTTI Workshop
AI holds great potential to transform how drugs are developed, manufactured, and utilized. As with any innovation, AI use in drug development creates new and unique challenges that require both
careful management and a risk-based regulatory framework that is built on sound regulatory science approaches that support innovation. Join us as we explore guiding principles that are being applied
by innovators and regulators to promote the responsible use of AI in the development of safe and effective drugs. Learn from experts about applying principles for good machine learning practices to
ensure responsible use of AI in the development of drugs. Drawing on real case examples, we will discuss the rationale for particular approaches, how success was evaluated, what challenges were
encountered, options for scaling and wider applicability, and considerations for moving forward.
On Tuesday August 6, 2024 from 10 AM to 5:30 PM EDT, FDA and the Clinical Trials Transformation Initiative are hosting a free hybrid public workshop on artificial intelligence (AI) in drug and
biological product development.
Please join us as we explore the guiding principles being applied by innovators and regulators to ensure AI is used responsibly. AI holds great potential to transform how drugs are developed,
manufactured, and used.
Participants may attend virtually or in-person in the FDA Great Room located at 10903 New Hampshire Avenue, Silver Spring, MD 20993.
Registration information can be found here.
The Small Business and Industry Assistance (SBIA) program in the Center for Drug Evaluation and Research provides guidance, education and updates for regulated industry.
MiWord of the Day Is… Volume Rendering!
Volumetric rendering stands at the forefront of visual simulation technology. It intricately models how light interacts with myriad tiny particles to produce stunningly realistic visual effects such
as smoke, fog, fire, and other atmospheric phenomena. This technique diverges significantly from traditional rendering methods that predominantly utilize geometric shapes (such as polygons in 3D
models). Instead, volumetric rendering approaches these phenomena as if they are composed of an immense number of particles. Each particle within this cloud-like structure has the capability to
absorb, scatter, and emit light, contributing to the overall visual realism of the scene.
This is not solely useful for generating lifelike visual effects in movies and video games; it also serves an essential function in various scientific domains. Volumetric rendering enables the
visualization of intricate three-dimensional data crucial for applications such as medical imaging, where it helps in the detailed analysis of body scans, and in fluid dynamics simulations, where it
assists in studying the behavior of gases and liquids in motion. This technology, thus, bridges the gap between digital imagery and realistic visual representation, enhancing both our understanding
and our ability to depict complex phenomena in a more intuitive and visually engaging manner.
How does this work?
Let’s start by talking about direct volume rendering. Instead of trying to create a surface for every object, this technique directly translates data (like a 3D array of samples, representing our
volumetric space) into images. Each point in the volume, or voxel , contains data that dictates how it should appear based on how it interacts with light.
For example, when visualizing a CT scan, certain data points might represent bone, while others might signify soft tissue. By applying a transfer function—a kind of filter—different values are mapped
to specific colors and opacities. This way, bones might be made to appear white and opaque, while softer tissues might be semi-transparent.
The real trick lies in the sampling process. The renderer calculates how light accumulates along lines of sight through the volume, adding up the contributions of each voxel along the way. It’s a
complex ballet of light and matter, with the final image emerging from the cumulative effect of thousands, if not millions, of tiny interactions.
Let us make this a bit more concrete. We first have transfer functions, a transfer function maps raw data values to visual properties like color and opacity. Let us represent the color assigned to
some voxel as C(v) and the opacity as α(v). For each pixel in the final image, a ray is cast through the data volume from the viewer’s perspective. For this we have a ray equation:
Where P(t) is a point along the ray at parameter 𝑡, P[0] is the ray’s origin, and
You probably used Volumetric Rendering
Volumetric rendering transforms CT and MRI scans into detailed 3D models, enabling doctors to examine the anatomy and functions of organs in a non-invasive manner. A specific application includes
most of the modern CT viewers. Volumetric rendering is key in creating realistic simulations and environments. In most AR applications, it is used under the hood to overlay interactive,
three-dimensional images on the user’s view of the real world, such as in educational tools that project anatomical models for medical students.
Now for the fun part (see the rules here), using volume rendering in a sentence by the end of the day:
Serious: The breakthrough in volumetric rendering technology has enabled scientists to create highly detailed 3D models of the human brain.
Less Serious: I tried to use volumetric rendering to visualize my Netflix binge-watching habits, but all I got was a 3D model of a couch with a never-ending stream of pizza and snacks orbiting around
…I’ll see you in the blogosphere.
MiWord of the Day is… KL Divergence!
You might be thinking, “KL Divergence? Sounds exotic. Is it something to do with the Malaysian capital (Kuala Lumpur) or a measurement (kiloliter)?” Nope, and nope again! It stands for
Kullback-Leibler Divergence, a fancy name for a metric to compare two probability distributions.
But why not just compare their means? After all, who needs these hard-to-pronounce names? Kullback… What was it again? That’s a good point! Here’s the catch: two distributions can have the same mean
but look completely
different. Imagine two Gaussian distributions, both centered at zero, but one is wide and flat, while the other is narrow and tall. Clearly, not similar!
So, maybe comparing the mean and variance would work? Excellent thinking! But what if the distributions aren’t both Gaussian? For example, a wide and flat Gaussian and a uniform distribution (totally
flat) might look similar visually, but the uniform distribution is not parametrized by a mean or variance. So, what do we compare?
Enter KL Divergence!
KL Divergence returns a single number that tells us how similar two distributions are, regardless of their types. The smaller the number, the more similar the distributions. But how do we calculate
it? Here’s the formula (don’t worry, you don’t have to memorize it!).
Notice, if the distribution q has probability mass where p doesn’t, the KL Divergence will be large. Good, that’s what we want! But, if q has little mass where p has a lot, the KL Divergence will be
small. Wait, that’s not what we want! No, it’s not, but luckily KL Divergence is asymmetric! KL(q || p) returns a different value than KL(p || q), so
we can compute both! Why are they different? I’ll leave that up to you to figure out!
KL Divergence in Action
Now, the fun part: using KL Divergence in a sentence!
Serious: Professor, can we approximate one distribution with another by minimizing the KL Divergence between them? That’s a great question! You’ve just stumbled on the idea behind Variational
Less Serious: Ladies and gentlemen, the KL Divergence between London and Kuala Lumpur is large, and so our flight time today will be 7 hour and 30 minutes. Please remember to stow your hand luggage
in the overhead bins above you, fold your tray tables, and fasten your seatbelts.
See you in the blogosphere,
Benedek Balla
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Algorithms and
Algorithms and Datastructures - Conditional Course
Undergraduate Course - Winter Term 2019/20
Fabian Kuhn
Course Format
We provide recordings by Prof. Backofen and an exercise sheet every week. The solutions to the exercises and questions about the lecture are discussed in the exercise lessons.
Announcement: There will be no exercise leson on January 8.
Our regular meeting takes place each Wednesday from 12:15 pm to 2:00 pm in building 101 room SR 01-016.
Exam Review: Due to the current situation with COVID-19, exam reviews have to be conducted digitally. If you want to review your graded exam, please contact me (philipp.schneider(at)
Time: The exam will take place on March 9 at 3 pm (9. März, 15 Uhr). The exam will take (at most) 120 minutes.
Place: Building 101 Room 026.
Mode: The exam will be a written exam.
Language: The exam questions will be given in English (it is OK to write answers in German).
Open Book: Everyone is allowed to bring anything that is written or printed on paper (books, cheat sheets, dictionary, etc.). Electronic devices are not allowed!
Student ID: Please remember to bring your student ID with you to the exam!
Slides and Recordings
Slides and recordings are taken from lectures from different years and differ slightly.
No. Topic Slides Recordings
1 Introduction, Sorting (You can ignore the part on organisation) PDF MP4
2 Runtime Analysis MinSort and HeapSort, Induction PDF MP4
3 O-Notation, L'Hopital PDF MP4
4 Runtime Complexity, Associative Arrays PDF MP4
5 Hash Maps, Universal Hashing PDF MP4
6 Treatment of Hash Collisions, Priority Queues PDF MP4
7 Dynamic Arrays, Amortized Analysis PDF MP4
8 Cache Efficiency, Divide and Conquer PDF MP4
9 Divide and Conquer, Master Theorem PDF MP4
10 Linked Lists, Binary Search Trees PDF MP4
11 Balanced Trees PDF MP4
12 Graphs, BFS, DFS PDF MP4
13 Shortest Paths, Dijkstra PDF MP4
14 Edit Distance, Dynamic Programming PDF MP4
Exercises are voluntary, but we highly recommend doing them. If you want feedback for your solution, you can submit them to philipp.Schneider(at)cs.uni-freiburg.de (documents prepared with (La)TeX
preferred, documents generated with MS Word or similar "Wysiwyg" editors are ok, if you send a scan of a handwritte solution make sure it is easily readable, i.e., no photos). Or bring it as hard
copy to the subsequent exercise class (on the Wednesday after the exercise is assigned). No feedack will be given on late submissions (after due date).
Important Note: If you submit the exercises via email, *please* use the subject matter "AD EX [number]" (for [number] insert the current exercise sheet number). This allows me to filter out your
submissions and ensures I do not overlook any. If you want to contact me per mail, use an email seperate from your exercise submission.
Topic(s) Assigned Date Due Date Exercises Solution Files
Sorting 23.10.2019 30.10.2019 Exercise 01 Solution 01 b_sort, c_sort, s_test
O-Notation 30.10.2019 06.11.2019 Exercise 02 Solution 02 -
Avg. Runtime, Hashing 06.11.2019 13.11.2019 Exercise 03 Solution 03 -
Hashing, Amort. Analysis 13.11.2019 20.11.2019 Exercise 04 Solution 04 -
Divide & Conquer 20.11.2019 27.11.2019 Exercise 05 Solution 05 -
Div & Con, Heaps, BSTs 27.11.2019 04.12.2019 Exercise 06 Solution 06 -
BST-, AVL-, (a,b)-Trees 04.12.2019 11.12.2019 Exercise 07 Solution 07 -
Graphs, DFS, BFS 11.12.2019 18.12.2019 Exercise 08 Solution 08 -
Shortest Paths 20.12.2019 15.01.2020 Exercise 09 Solution 09 -
Dyn. Programming 15.01.2020 22.01.2020 Exercise 10 Solution 10 -
Practice Lesson 23.01.2020 - Exercise 11 Solution 11 -
If you have a question to the lecture or the exercises, please use the forum (click on the word forum for the link) instead of writing an email. Then all of us and also your colleagues see the
question and can answer to it. Feel free to also use the forum to discuss anything related to the topics and organization of the lecture.
Old Informatik 2 Exams
Please read this important disclaimer first: These exams originate from another course ("Informatik 2"). The exams are therefore in German. Topics and difficulty level may (and certainly will) differ
from the actual exam of this course. Never the less (some of) the exercises may offer you additional practice for the upcoming exam. I put the material online since I was asked for it. You may use it
at your own discretion.
Klausur Winter 2018/19,
Klausur Sommer 2018,
Klausur Winter 2016/17,
Klausur Sommer 2016,
Klausur Winter 2014/15,
Klausur Sommer 2014
• Introduction to Algorithms (3rd edition); T. Cormen, C. Leiserson, R. Rivest, C. Stein; MIT Press, 2009
• Algorithms and Data Structures; K. Mehlhorn und P. Sanders; Springer, 2008, available online
• Lectures on MIT Courseware:
Introduction to Algorithms 2005 und Introduction to Algorithms 2011
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Algebraic Problem Solving Strategies
Learning outcome
• Use a problem-solving strategy to set up and solve word problems
The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do
I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems.
Previously, you translated word phrases into algebraic equations using some basic mathematical vocabulary and symbols. Since then you’ve increased your math vocabulary as you learned about more
algebraic procedures. You’ve also solved some word problems applying math to everyday situations. This method works as long as the situation is familiar to you and the math is not too complicated.
Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.
Pete bought a shirt on sale for $[latex]18[/latex], which is one-half the original price. What was the original price of the shirt?
Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary
or on the Internet.
• In this problem, do you understand what is being discussed? Do you understand every word?
Step 2. Identify what you are looking for. It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
• In this problem, the words “what was the original price of the shirt” tell you what you are looking for: the original price of the shirt.
Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.
• Let [latex]p=[/latex] the original price of the shirt
Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.
Step 5. Solve the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.
Write the equation. [latex]18=\Large\frac{1}{2}p[/latex]
Multiply both sides by 2. [latex]\color{red}{2}\cdot18=\color{red}{2}\cdot\Large\frac{1}{2}\normalsize p[/latex]
Simplify. [latex]36=p[/latex]
Step 6. Check the answer in the problem and make sure it makes sense.
• We found that [latex]p=36[/latex], which means the original price was [latex]\text{\$36}[/latex]. Does [latex]\text{\$36}[/latex] make sense in the problem? Yes, because [latex]18[/latex] is
one-half of [latex]36[/latex], and the shirt was on sale at half the original price.
Step 7. Answer the question with a complete sentence.
• The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was [latex]\text{\$36}[/latex].”
If this were a homework exercise, our work might look like this:
Try it
We list the steps we took to solve the previous example.
Problem-Solving Strategy
1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary
or on the internet.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem. Make sure it makes sense.
7. Answer the question with a complete sentence.
Let’s use this approach with another example.
Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[/latex] apples to the picnic. How many bananas did he bring?
Show Answer
Try it
In the next example, we will apply our Problem-Solving Strategy to applications of percent.
Nga’s car insurance premium increased by [latex]\text{\$60}[/latex], which was [latex]\text{8%}[/latex] of the original cost. What was the original cost of the premium?
Show Answer
Try it
Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don’t usually
arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference, of, and and.
The difference of a number and six is [latex]13[/latex]. Find the number.
Step 1. Read the problem. Do you understand all the words?
Step 2. Identify what you are looking for. the number
Step 3. Name. Choose a variable to represent the number. Let [latex]n=\text{the number}[/latex]
[latex]n-6\enspace\Rightarrow[/latex] The difference of a number and 6
Step 4. Translate. Restate as one sentence. [latex]=\enspace\Rightarrow[/latex] is
Translate into an equation. [latex]13\enspace\Rightarrow[/latex] thirteen
Step 5. Solve the equation. [latex]n-6=13[/latex]
Add 6 to both sides. [latex]n-6\color{red}{+6}=13\color{red}{+6}[/latex]
Simplify. [latex]n=19[/latex]
Step 6. Check:
The difference of [latex]19[/latex] and [latex]6[/latex] is [latex]13[/latex]. It checks.
Step 7. Answer the question. The number is [latex]19[/latex].
The sum of twice a number and seven is [latex]15[/latex]. Find the number.
try it
Watch the following video to see another example of how to solve a number problem.
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How do I find the partial fraction decomposition of (x^4)/(x^4-1) ? | HIX Tutor
How do I find the partial fraction decomposition of #(x^4)/(x^4-1)# ?
Answer 1
By Partial Fraction Decomposition, we can write
Let us look at some details.
Let us find the partial fractions of
by factoring out the denominator,
by splitting into the partial fraction form,
by taking the common denominator,
by simplifying the numerator,
Since the numerator is originally 1, by matching the coefficients,
(1) #A+B+C=0# (2) #-A+B+D=0# (3) #A+B-C=0# (4) #-A+B-D=1#
(5) #2A+2B=0#
(6) #-2A+2B=1#
(7) #B=1/4#
By plugging (7) into (5),
(8) #A=-1/4#
By plugging (7) and (8) into (1),
(9) #C=0#
By plugging (7) and (8) into (2),
(10) #D=-1/2#
By (5), (6), (9), and (10),
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Answer 2
To find the partial fraction decomposition of (\frac{x^4}{x^4-1}), first factor the denominator as ((x^2-1)(x^2+1)). Then, express (\frac{x^4}{x^4-1}) as (\frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}
{x^2+1}), where (A), (B), (C), and (D) are constants. Then solve for (A), (B), (C), and (D).
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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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Advanced Formatting: Float, Round, and Percent
String Manipulation in Python
String Manipulation in Python
Within a simple {} block, we can round a float number with the necessary precision, or represent a number as a percent.
Let's consider the pattern we will use in this chapter:
{:[thousands separator].[number][type]}
Please note, that like in the previous chapters, we don't need to place square brackets (I placed it for convenience).
• [thousands separator] - the symbol used to separate every thousand (possible values are , and _).
• [number] - is the precision, number of decimal places (used for rounding number).
• [type] - type of number representing (e - scientific notation, % - percentage (will multiply number by 100), g - general format, f - fixed-point notation). You can dive deeper into possible
options in Python documentation.
print("Original number: {0}, formatted number: {0:.2f}".format(255/8))
print("Original number: {0}, formatted number: {0:.2%}".format(15/48))
print("Original number: {0}, formatted number: {0:,.2f}".format(35*6327))
As of 2020, the population of the USA was 331002651. The total land area is 9147420 sq.km. Population density is the population-to-area ratio. Your tasks are:
1. Format the first string so the population and area will be printed in format: 9,147,420, and insert variables in the correct order.
2. Within the second .format function calculate the population density and format the number in format 28.45.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Within a simple {} block, we can round a float number with the necessary precision, or represent a number as a percent.
Let's consider the pattern we will use in this chapter:
{:[thousands separator].[number][type]}
Please note, that like in the previous chapters, we don't need to place square brackets (I placed it for convenience).
• [thousands separator] - the symbol used to separate every thousand (possible values are , and _).
• [number] - is the precision, number of decimal places (used for rounding number).
• [type] - type of number representing (e - scientific notation, % - percentage (will multiply number by 100), g - general format, f - fixed-point notation). You can dive deeper into possible
options in Python documentation.
print("Original number: {0}, formatted number: {0:.2f}".format(255/8))
print("Original number: {0}, formatted number: {0:.2%}".format(15/48))
print("Original number: {0}, formatted number: {0:,.2f}".format(35*6327))
As of 2020, the population of the USA was 331002651. The total land area is 9147420 sq.km. Population density is the population-to-area ratio. Your tasks are:
1. Format the first string so the population and area will be printed in format: 9,147,420, and insert variables in the correct order.
2. Within the second .format function calculate the population density and format the number in format 28.45.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Within a simple {} block, we can round a float number with the necessary precision, or represent a number as a percent.
Let's consider the pattern we will use in this chapter:
{:[thousands separator].[number][type]}
Please note, that like in the previous chapters, we don't need to place square brackets (I placed it for convenience).
• [thousands separator] - the symbol used to separate every thousand (possible values are , and _).
• [number] - is the precision, number of decimal places (used for rounding number).
• [type] - type of number representing (e - scientific notation, % - percentage (will multiply number by 100), g - general format, f - fixed-point notation). You can dive deeper into possible
options in Python documentation.
print("Original number: {0}, formatted number: {0:.2f}".format(255/8))
print("Original number: {0}, formatted number: {0:.2%}".format(15/48))
print("Original number: {0}, formatted number: {0:,.2f}".format(35*6327))
As of 2020, the population of the USA was 331002651. The total land area is 9147420 sq.km. Population density is the population-to-area ratio. Your tasks are:
1. Format the first string so the population and area will be printed in format: 9,147,420, and insert variables in the correct order.
2. Within the second .format function calculate the population density and format the number in format 28.45.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Within a simple {} block, we can round a float number with the necessary precision, or represent a number as a percent.
Let's consider the pattern we will use in this chapter:
{:[thousands separator].[number][type]}
Please note, that like in the previous chapters, we don't need to place square brackets (I placed it for convenience).
• [thousands separator] - the symbol used to separate every thousand (possible values are , and _).
• [number] - is the precision, number of decimal places (used for rounding number).
• [type] - type of number representing (e - scientific notation, % - percentage (will multiply number by 100), g - general format, f - fixed-point notation). You can dive deeper into possible
options in Python documentation.
print("Original number: {0}, formatted number: {0:.2f}".format(255/8))
print("Original number: {0}, formatted number: {0:.2%}".format(15/48))
print("Original number: {0}, formatted number: {0:,.2f}".format(35*6327))
As of 2020, the population of the USA was 331002651. The total land area is 9147420 sq.km. Population density is the population-to-area ratio. Your tasks are:
1. Format the first string so the population and area will be printed in format: 9,147,420, and insert variables in the correct order.
2. Within the second .format function calculate the population density and format the number in format 28.45.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Switch to desktop for real-world practiceContinue from where you are using one of the options below
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Adam2392's Blog
This week, I am finishing my final PR to fix the Coiteration algorithm: https://github.com/hameerabbasi/xsparse/pull/31.
This will enable coiterations over levels that are nested within other levels. As such, we can define multiple Coiterate objects that co-iterate over each dimension respectively.
With this, the ground-work will be laid to very easily implement the MergeLattice Coiterator, which is just an abstraction on top of this idea of calling multiple coiterators.
The final PR now shows a unit-test that Co-iterates over two sets of nested levels, which each together define a CSR matrix. It is done with a conjunctive merge, so the unit-test defines how the API
must be specifically defined. There were a few errors that I ran into that were hard to debug, but it turns out they were consequences of how I was leveraging the Coiterate API, which is slightly
unforgiving right now.
View Blog Post
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TTL tester circuit
Using electronic diagram below can be made a very simple TTL tester. One of the three LEDs will light depending on the applied voltage at test point (TP). This voltage is transmitted first to two
comparators (A1 and A2). At the other input of each comparator voltage reference is brought from the voltage divider R4/R5/R6. Values for this components gives thresholds at 0.8 and 2.4 V and the
range between these two values is the TTL "prohibited area". If the voltage at TP is less than 0.8 V output of A2 swings in state "0" logic and results in ignition red LED (D6). If measured voltage
is greater than 2.4 V, the output of A1 will flip the logic state "0", so the green LED (D5) will light. When the voltage will be between 0.8 and 2.4 V and then no output of A1 or A2's output will be
in state "0" logic. When TP is not connected to any circuit, the same situation is generated due to the action of R1 and R3. Inverting input of A3 is in this situation, in "1" through R9 so that the
yellow LED (D7) lights.
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graphvix v1.1.0
Graphvix.Graph (graphvix v1.1.0)
Models a directed graph that can be written to disk and displayed using Graphviz notation.
Graphs are created by
• adding vertices of various formats to a graph
• connecting them with edges
• grouping them into subgraphs and clusters,
• providing styling to all these elements
They can then be
• written to disk in .dot format
• compiled and displayed in any number of image formats (Graphvix defaults to .png)
Group a set of vertices into a cluster in a graph.
Add an edge between two vertices in a graph.
Group a set of vertices into a subgraph within a graph.
Writes the graph to a .dot file and compiles it to the specified output format (defaults to .png).
Destructures the references to the ETS tables for vertices, edges, and neighbours from the Graph struct.
Creates a new Graph struct.
Sets a property for a vertex or edge that will apply to all vertices or edges in the graph.
Adds a top-level graph property.
Write a graph to file, compile it, and open the resulting image in your system's default image viewer.
Converts a graph to its representation using .dot syntax.
Writes a Graph to a named file in .dot format
@type t() :: %Graphvix.Graph{
digraph: digraph(),
global_properties: keyword(),
graph_properties: keyword(),
subgraphs: list()
Group a set of vertices into a cluster in a graph.
In addition to the graph and the vertex ids, you can pass attributes for node and edge to apply common styling to the vertices included in the cluster, as well as the edges between two vertices both
in the cluster.
The difference between a cluster and a subgraph is that a cluster can also accept attributes to style the cluster, such as a border, background color, and custom label. These attributes can be passed
as top-level attributes in the final keyword list argument to the function.
iex> graph = Graph.new()
iex> {graph, v1id} = Graph.add_vertex(graph, "start")
iex> {graph, v2id} = Graph.add_vertex(graph, "end")
iex> {_graph, cid} = Graph.add_cluster(
...> graph, [v1id, v2id],
...> color: "blue", label: "cluster0",
...> node: [shape: "triangle"],
...> edge: [style: "dotted"]
...> )
iex> cid
In .dot notation a cluster is specified, as opposed to a subgraph, by giving the cluster an ID that begins with "cluster" as seen in the example above. Contrast with Graphvix.Graph.add_subgraph/3.
Add an edge between two vertices in a graph.
It takes 3 required arguments and one optional. The first argument is the graph, the second two arguments are the tail and head of the edge respectively, and the fourth, optional, argument is a list
of layout attributes to apply to the edge.
The arguments for the ends of the edge can each be either the id of a vertex, or a tuple of a vertex id and a port name to attach the edge to. This second option is only valid with Record or
HTMLRecord vertices.
iex> graph = Graph.new()
iex> {graph, v1id} = Graph.add_vertex(graph, "start")
iex> {graph, v2id} = Graph.add_vertex(graph, "end")
iex> {_graph, eid} = Graph.add_edge(graph, v1id, v2id, color: "green")
iex> eid
[:"$e" | 0]
Add a vertex built from a Graphvix.HTMLRecord to the graph.
iex> graph = Graph.new()
iex> record = HTMLRecord.new([
...> HTMLRecord.tr([
...> HTMLRecord.td("start"),
...> HTMLRecord.td("middle"),
...> HTMLRecord.td("end"),
...> ])
...> ])
iex> {_graph, rid} = Graph.add_html_record(graph, record)
iex> rid
[:"$v" | 0]
See `Graphvix.HTMLRecord` for details on `Graphvix.HTMLRecord.new/2`
and the complete module API.
Add a vertex built from a Graphvix.Record to the graph.
iex> graph = Graph.new()
iex> record = Record.new(["a", "b", "c"])
iex> {_graph, rid} = Graph.add_record(graph, record)
iex> rid
[:"$v" | 0]
See `Graphvix.Record` for details on `Graphvix.Record.new/2`
and the complete module API.
Group a set of vertices into a subgraph within a graph.
In addition to the graph and the vertex ids, you can pass attributes for node and edge to apply common styling to the vertices included in the subgraph, as well as the edges between two vertices both
in the subgraph.
iex> graph = Graph.new()
iex> {graph, v1id} = Graph.add_vertex(graph, "start")
iex> {graph, v2id} = Graph.add_vertex(graph, "end")
iex> {_graph, sid} = Graph.add_subgraph(
...> graph, [v1id, v2id],
...> node: [shape: "triangle"],
...> edge: [style: "dotted"]
...> )
iex> sid
Adds a vertex to graph.
The vertex's label text is the argument label, and additional attributes can be passed in as attributes. It returns a tuple of the updated graph and the :digraph-assigned ID for the new vertex.
iex> graph = Graph.new()
iex> {_graph, vid} = Graph.add_vertex(graph, "hello", color: "blue")
iex> vid
[:"$v" | 0]
Writes the graph to a .dot file and compiles it to the specified output format (defaults to .png).
The following code creates the files "graph_one.dot" and "graph_one.png" in your current working directory.
iex> Graph.compile(graph, "graph_one")
This code creates the files "graph_one.dot" and "graph_one.pdf".
iex> Graph.compile(graph, "graph_one", :pdf)
filename works as expected in Elixir. Filenames beginning with / define an absolute path on your file system. Filenames otherwise define a path relative to your current working directory.
Destructures the references to the ETS tables for vertices, edges, and neighbours from the Graph struct.
iex> graph = Graph.new()
iex> Graph.digraph_tables(graph)
Creates a new Graph struct.
A Graph struct consists of an Erlang digraph record, a list of subgraphs, and two keyword lists of properties.
iex> graph = Graph.new()
iex> Graph.to_dot(graph)
~S(digraph G {
iex> graph = Graph.new(graph: [size: "4x4"], node: [shape: "record"])
iex> Graph.to_dot(graph)
~S(digraph G {
node [shape="record"]
Sets a property for a vertex or edge that will apply to all vertices or edges in the graph.
NB :digraph uses vertex to define the discrete points in a graph that are connected via edges, while Graphviz and DOT use the word node. Graphvix attempts to use "vertex" when the context is
constructing the data for the graph, and "node" in the context of formatting and printing the graph.
iex> graph = Graph.new()
iex> {graph, vid} = Graph.add_vertex(graph, "label")
iex> graph = Graph.set_global_property(graph, :node, shape: "triangle")
When the graph is drawn, the vertex whose id is vid, and any other vertices added to the graph, will have a triangle shape.
Global properties are overwritten by properties added by a subgraph or cluster:
{graph, subgraph_id} = Graph.add_subgraph(graph, [vid], shape: "hexagon")
Now when the graph is drawn the vertex vid will have a hexagon shape.
Properties written directly to a vertex or edge have the highest priority of all. The vertex created below will have a circle shape despite the global property set on graph.
{graph, vid2} = Graph.add_vertex(graph, "this is a circle!")
Adds a top-level graph property.
These attributes affect the overall layout of the graph at a high level. Use set_global_properties/3 to modify the global styling for vertices and edges.
iex> graph = Graph.new()
iex> graph.graph_properties
iex> graph = Graph.set_graph_property(graph, :rank_direction, "RL")
iex> graph.graph_properties
rank_direction: "RL"
Write a graph to file, compile it, and open the resulting image in your system's default image viewer.
The following code will write the contents of graph to "graph_one.dot", compile the file to "graph_one.png" and open it.
iex> Graph.show(graph, "graph_one")
filename works as expected in Elixir. Filenames beginning with / define an absolute path on your file system. Filenames otherwise define a path relative to your current working directory.
Converts a graph to its representation using .dot syntax.
iex> graph = Graph.new(node: [shape: "triangle"], edge: [color: "green"], graph: [size: "4x4"])
iex> {graph, vid} = Graph.add_vertex(graph, "a")
iex> {graph, vid2} = Graph.add_vertex(graph, "b")
iex> {graph, vid3} = Graph.add_vertex(graph, "c")
iex> {graph, eid} = Graph.add_edge(graph, vid, vid2)
iex> {graph, eid2} = Graph.add_edge(graph, vid, vid3)
iex> {graph, clusterid} = Graph.add_cluster(graph, [vid, vid2])
iex> Graph.to_dot(graph)
~S(digraph G {
node [shape="triange"]
edge [color="green"]
subgraph cluster0 {
v0 [label="a"]
v1 [label="b"]
v0 -> v1
v2 [label="c"]
v1 -> v2
For more expressive examples, see the .ex and .dot files in the examples/ directory of Graphvix's source code.
Writes a Graph to a named file in .dot format
iex> Graph.write(graph, "my_graph")
will write a file named "my_graph.dot" to your current working directory.
filename works as expected in Elixir. Filenames beginning with / define an absolute path on your file system. Filenames otherwise define a path relative to your current working directory.
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Canonical decompositions in monadically stable and bounded shrubdepth
graph classes
Canonical decompositions in monadically stable and bounded shrubdepth graph classes
We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class
C, and using a first-order formula ϕ with parameters we are able to define, in every graph G in C, a relation R that satisfies some hereditary first-order assertion ψ. Then we are able to find a
first-order formula ϕ' that has the same property, but additionally is finitary: there is finite bound k such that in every graph G in C, different choices of parameters give only at most k different
relations R that can be defined using ϕ'. We use the Finitary Substitute Lemma to derive two corollaries about the existence of certain canonical decompositions in classes of well-structured graphs.
- We prove that in the Splitter game, which characterizes nowhere dense graph classes, and in the Flipper game, which characterizes monadically stable graph classes, there is a winning strategy for
Splitter, respectively Flipper, that can be defined in first-order logic from the game history. Thus, the strategy is canonical. - We show that for any fixed graph class C of bounded shrubdepth,
there is an O(n^2)-time algorithm that given an n-vertex graph G in C, computes in an isomorphism-invariant way a structure H of bounded treedepth in which G can be interpreted. A corollary of this
result is an O(n^2)-time isomorphism test and canonization algorithm for any fixed class of bounded shrubdepth.
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First Place
Analyzing Impact of Twitter Sentiment on Stock Market Dynamics Using Spectral Clustering and Deep Learning
Rahul Gupta ’26, Vagmin Viswanathan ’26
Tristan Wells ’25, Sam Orientale ’25, Alec Kong ’23, Eric Strawn ’25
First Place, Solo Poster
Translating Neurophysiological Recordings into Dynamic Estimates of Conceptual Knowledge and Learning
Daniel Carstensen ’24
Second Place
Meghan O'Keefe ’24, Stephanie Lee ’24, Annabel Gerber ’24, Julia Binder ’24
Mary Bocock ’27, Emma Elsbecker ’24, O’Connell ’23, Jose Salina ’25
Ben Shaman ’26, Agastya Nashier ’26, Josh Lee ’26, Sumant Sharma ’26
First Place, Applied Mathematics
Conner Boehm ’24, Isaiah Bradner ’26, Catherine Chu ’26, Annie Tang ’25, Wade Williams ’24, Geoffrey Yang ’25
Second Place, Applied Mathematics
Sourjyamoy Barman ’26, Joshua Piesner ’26, Vedant Tapiavala ’26
Third Place, Applied Mathematics
Zimehr Abbasi ’23, Min Hur ’24, Austin Hyun ’23, Tate McDowell ’23, Jessica Wang ’23, Ryan Xu ’23
Fourth Place, Applied Mathematics
Richard W. Dionne ’19
First Place, Pure Mathematics
Mario Tomba Morillo ’25
Second Place, Pure Mathematics
Sofia Goncalves ’25
Due to COVID-19 restrictions we were unable to host an in-person poster session in recent years. However, in 2020 we were pleased to share the team project presentations from Math 70, Elements of
Multivariate Statistics and Statistical Learning, taught by Professor Eugene Demidenko this spring. Data scientists are among the most in-demand technical jobs, and Dartmouth offers a Mathematical
Data Science major with Math 70 as the culminating course. The course combines theoretical mathematics empowered by proficient programming in R for solving real-life problems, such as the analysis of
COVID-19 dynamics and its prediction, and prepares students for a career in data analysis and statistical problem solutions.
Team projects from Math 70
Osman Khan, Tudor Muntianu, Joe Gyorda, Sri Yenamandra, Oliver Levy
Tanli Su, Sophie Wang, Saunak Badam, Aaron Lee, Alexander (Sasha) Kokoshinskiy
Clayton Bass, Thomas Brown, Kayshav Prakash, Bruce Zou, Allison Tong
First Place, Applied Mathematics
For Whom the Bell Tolls: Modeling Wind Chimes with the Classical Wave Equation
Louisa Gao ’22, Matthew Sawicki ’20
Second Place, Applied Mathematics
Modeling Spillovers of Emerging Infectious Diseases with Intermediate Hosts
Katherine Royce ’19
Third Place, Applied Mathematics
Varying Measles Vaccination Rates in a Theoretical Prison Population
Addison Green, Erika Hernandez, Kenna Vansteyn
First Place, Pure Mathematics
Continued Fractions and abc-Triples
Ethan Goldman
Second Place, Pure Mathematics
$\alpha\beta\gamma$ Conjecture for Gaussian Integers
Jared Hodes, Liam Morris, Tanish Raghavan, May Fahrenthold, Dylan Burke
First Place, Applied Mathematics
Exploring Health Policies to Prevent Another SARS Outbreak in Hong Kong
Ray Guo, Matthew Yung, Hailey Jiang, Eva Wang
Second Place, Applied Mathematics
Spread of the Renaissance through Publication Networks
Brian Chekal, Jason Cheal
Third Place, Applied Mathematics
Sentiment-Based Prediction of Alternative Cryptocurrency Price Fluctuations Using a Gradient Boosting Tree Mode
Anup Chamrajnagar, Xander Fong, Tianyu “Ray” Li, Nick Rizik
First Place, Pure Mathematics
Embedding the Complete Bipartite Graph
Jacob Marchman
First Place, Applied Mathematics
A Model for Self-Sustaining Litopenaeus Vannamei Farm Alternatives
Anup Chamrajnagar, Jason Cheal, John Glance, Xander Fong
Second Place, Applied Mathematics
Mathematical Modeling of Cancer Growth
Yixuan He
Third Place, Applied Mathematics
Modeling Solid Fuel Rocket Launch and Orbit
Jon Chu, Annika Roise, Ethan Isaacson
First Place, Pure Mathematics
Lying on the Fermat Primality Test
Jared Duker Lichtman
First Place, Applied Mathematics
Analyzing in vitro Monolayer and 3-D Spheroid Culture Response of Non-Small Cell Lung Cancer Cell Line H358 to Bortezomib Drug Treatment
Yixuan He
Shared Second Place, Applied Mathematics
Rumor Spreading in Social Networks
Ke Li, Min Hyung Kang, Qi Wei
Shared Second Place, Applied Mathematics
Antibiotics: Pharmaceutical Incentives in Monopoly
Ryan Schiller, Daniel Salas
First Place, Pure Mathematics
Hemipolyhedra and their connection to $\boldsymbol{RP} ^{2}$ and the torus
Julio Resendiz
Second Place, Pure Mathematics
The Plastic Number and Padovan Cuboid Spiral
Helena Caswell
First Place, Applied Mathematics
Transmission of Bubonic Plague along the Via Francigena in the 14th Century
Christine Lu, Mary Rogers, Anita Kodali
Second Place, Applied Mathematics
Universal Voluntary Testing and Antiretroviral therapy: A Comparison of 2 HIV Models
Jennifer Jin, Eileen Jin
First Place, Pure Mathematics
Transplantation of Eigenfunctions on Isospectal Domians
Feynman Liang
Second Place, Pure Mathematics
The Complement of the Figure 8 Knot in the 3-Sphere
Molly McBride
First Place, Applied Mathematics
Michelle Chen, Paula Chen, Milan Huynh, Evan Rheingold, Ann Dnham
Shared Second Place, Applied Mathematics
Ivan Antoniv
Shared Second Place, Applied Mathematics
Andi Leibowitz, Emily Hoffman
First Place, Pure Mathematics
Annie Laurie Mauhs-Pugh
Shared Second Place, Pure Mathematics
Hanh Nguyen
Shared Second Place, Pure Mathematics
Adenrele Adewusi
Shared Second Place, Pure Mathematics
Ethan Thomas
First Place
John Conley
Second Place
Pawan Dhakal
Third Place
Gabriel Dorfsman-Hopkins
First Place
Boundary Methods for Variable Coefficient Helmholtz Boundary Value
Brad Nelson
Second Place
Reading with new eyes: Single word network analysis of the representation of biofuels in contemporary media
Anna Morenz
Third Place
The Spectral Structure of the Credit Default Swap Market
Philip Winsor
Special poster session:
2010 Mathematical Biology Student Poster Session
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8572 Square Inches to Square Meters
Do you want to know how much is 8572 square inches converted to square meters? With our free square inches to square meters conversion tool, you can determine the value in square meters of 8572
square inches.
8572 square inches = 5.53031152 square meters
Convert 8572 square meters to square inches
How to convert 8572 square inches to square meters?
Note: in^2 is the abbreviation of square inches and m^2 is the abbreviation of square meters.
1 square inches is equal to 0.00064516 square meters:
1 in^2 = 0.00064516 m^2
In order to convert 8572 in^2 to m^2 you have to multiply 8572 by 0.00064516:
8572 in^2 x (0.00064516 m^2 / 1 in^2) = 8572 x 0.00064516 m^2 = 5.53031152 m^2
So use this simple rule to calculate how many square meters is 8572 square inches.
8572 square inches in other area units
Want to convert 8572 square inches to other area units? Use these links below:
Recent square inches to square meters conversions:
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Bekenstein Bound
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Bekenstein Bound
From: Jim choate <[email protected]>
> The problem I see with this is that there is no connection between a black holes
> mass and surface area (it doesn't have one). In reference to the 'A' in the
> above, is it the event horizon? A funny thing about black holes is that as the
> mass increases the event horizon gets larger not smaller (ie gravitational
> contraction).
Actually black holes do have a defined surface area, which is basically, as
you suggest, the area of the event horizon. And of course this is larger
for more massive black holes, as you say.
I believe the Bekenstein bound is based on reasoning that suggests that
if the state density of a region exceeds that bound, it will essentially
collapse into a black hole and be inaccessible to the rest of the universe.
The surface area in that context can be the conventionally defined area.
To bring this back to crypto a bit, the point of this discussion was that
there can be only a finite amount of processing done in finite time by
a finite-sized machine, even when QM is taken into consideration. Note,
though, that this result appears to require bringing in quantum gravitation,
a very poorly understood theory at present.
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PHYS102: Introduction to Electromagnetism
A potential point of confusion exists when you try to apply Kirchhoff's first rule for the current: if you do not know all the currents in a circuit, how do you choose the direction of the currents,
as Kirchhoff's first rule dictates?
The short answer is: do not worry about it. It will work out fine no matter which direction you draw the arrows labeled by the currents $I_{1}$, $I_{2}$, etc. in a circuit diagram like figure 21.25.
There is a long answer for those who want to understand why it works. It is okay to skip ahead to the next section if you are not interested in the details!
The confusion about current directions in Kirchhoff's first law is removed if you think about the blue arrows in figure 21.25 as coordinate axes, not as currents. Remember that current through any
imaginary surface in the circuit has a direction. This is still true here. And just as for velocities, we can specify the direction of a current by using plus or minus signs relative to a coordinate
axis that we can choose arbitrarily. We make this choice of coordinate axis once and for all in each segment of the circuit that goes from one junction to another. That is what the blue arrows in
figure 21.25 really mean. If a current of positive charge in a given segment is moving in the direction of the coordinate arrow for that segment, then that current counts as positive, and otherwise
it counts as negative.
To be specific, let's look at the junction labeled "a" in figure 21.25. Kirchhoff's rule now tells us to sum up all the currents in the wires whose arrows point into the junction, and set that total
equal to the sum of all the currents in the wires whose arrows point out of the junction. At junction "a", there is only one arrow pointing toward it, and that is labeled by $I_{1}$. The other two
arrows are pointing away from junction "a", and we decide to label them by the variables I2 and I3. Then their corresponding currents satisfy the equation $I_{1}=I_{2}+I_{3}$.
The variable $I_{1}$ is the current in the horizontal wire, counted as positive if the flow of positive charge is in the same direction as the arrow we drew there – pointing toward "a". Now we could
have equally well chosen to draw all arrows so that they point away from junction "a" – but then the same current that was previously described by the value $I_{1}$ (positive charges flowing toward
"a") would have to have its sign reversed and be called $-I_{1}$. This sign change is caused only by the change in the reference direction represented by the coordinate arrow, but the actual current
it describes is the same as before.
In Kirchhoff's first law, we now have only arrows pointing out of the junction, so we must total up all the currents on one side of the equation, and set the result equal to zero – because none of
the currents belong to arrows pointing into the junction. The total corresponding to the outgoing arrows again contains $I_{2}+I_{3}$ as before, but it also includes the value $-I_{1}$ labeling the
arrow we just reversed. We include the minus sign so that we are still describing the same physical current as before the arrow was reversed. This makes the equation for Kirchhoff's Law $-I_{1}+I_{2}
+I_{3}=0$. But this equation is mathematically equivalent to the equation we got with the original choice of arrow direction, as you can see by bringing $I_{1}$ to the other side.
This is why the set of simultaneous equations that you obtain by applying Kirchhoff's first law is not substantially affected by the choice of coordinate arrow directions in the individual branches
of the circuit. You can make that choice completely at random, provided you remember that some of the currents that are found as solutions to the equations can turn out negative, which means they
describe flow of positive charge opposite to the chosen arrow direction.
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π ΠΠ°Π»Π΅Π½Π΄Π°ΡΡ ΡΠΎΠ±ΡΡΠΈΠΉ ΠΈΠ· ΠΌΠΈΡΠ° ΠΊΡΠΈΠΏΡΠΎΠ²Π°Π»ΡΡ - coincalendar.io
$2.56 (34.39 %)
ΰΈΏ0.00003664 (38.62 %)
Π Ρ Π½ΠΎΡ Π½Π°Ρ ΠΊΠ°ΠΏΠΈΡ Π°Π»ΠΈΠ·Π°Ρ ΠΈΡ
Π Π±Ρ Π΅ΠΌ (24h)
$3 719.58
9 999 996(Max)
0 (Available)
Manage the events related to Bitcoin
FEATHER is a utility token within the Night Crows game on the WEMIX3.0 blockchain. Backed by the in-game "Piece of the Sky", it's essential for crafting the Aircraft Toolbox, a crucial item in the
game. Players can mint FEATHER within the game's "TOKEN" menu after reaching level 45. With a fixed supply of 10 million tokens and a 1:1 value with its in-game counterpart, FEATHER can also be
traded on the PNIX DEX. Created with Unreal Engine 5, Night Crows is a blockchain game on the WEMIX3.0 mainnet, launched on March 12, 2024. Accessible on both PC and mobile devices(iOS and Android)
and over 1M downloads, Night Crows has quickly gained global popularity. Within 3 days of its global launch, it surpassed $10 million in cumulative revenue and reached a concurrent player count of
230,000, a number that continues to climb. To accommodate the growing player base, the number of servers has increased from 24 at launch to 54 today.
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What is a Measure in Tableau? Understanding The Basics
So, you’ve started using Tableau and seen “Measure,” but unsure about what it is and how to use it?
Well, now it’s time to learn the basic fundamentals of the software and get a good understanding, so you can create some killer data visualizations.
In Tableau, a measure is a numerical value that represents a quantifiable metric in your data, such as sales, profit, or quantity. Measures are used to perform calculations and aggregate data. They
are typically displayed as continuous, quantitative values in Tableau visualizations.
In this article, we’ll provide an in-depth explanation of measures in Tableau, including their key characteristics and their use in calculations and aggregations, and show you how to work with them
in your data visualizations.
Let’s get started!
What is The Difference Between Dimension and Measure
One term that frequently comes up with measures is dimension.
In Tableau, a dimension is a field or category used to organize, segment, and categorize your data. You can think of dimensions as the descriptive parts of your data, such as names, dates, or
geographical data. These are qualitative values in your dataset.
On the other hand, measure values are fields that contain quantifiable data, which can be measured and subjected to mathematical operations.
Think of measures as the numbers or metrics in your data, such as sales amounts, profit margins, or quantities sold. Measures can be both continuous and discrete fields in your data set.
4 Ways How to Use Measures in Tableau
Measures in Tableau are central to performing data analysis and creating visualizations.
Following are some of the ways you can use measures in Tableau:
1. Basic Aggregations
2. Creating Calculated Fields
3. Trend Lines and Forecasting
4. Creating Bins and Histograms
1. How to Use Measures For Basic Aggregation
You can perform basic mathematical operations like sum, average, median, count, and more.
Tableau allows you to drag and drop a measure value into a view, and then you can choose the type of aggregation you want.
For example, let’s say you are working with a sample dataset from a superstore that includes a measure named Sales. If you want to understand the average sales made by the superstore, you can drag
the Sales measure into your visualization pane.
Then, you can choose the Average aggregation to calculate the average sales.
2. How to Use Measure For Creating Calculated Fields
You can create new data from existing measures by defining your calculations.
This is useful when you need to create ratios, differences, or custom aggregations that are not directly available in your dataset.
Let’s say you want to find the profit ratio from the measures in your dataset. You can do this by creating a calculated field.
Right-click in the data pane on the left side and select “Create Calculated Field.”
In the calculation editor, you can write your formula. For the profit ratio, the formula would be:
[Profit] / [Sales]
This formula divides the Profit measure by the Sales measure for each row in your dataset.
Drag your newly created calculated field, “Profit Ratio,” into your view just like you would with any other field.
3. How to Use Measure For Trend Lines and Forecasting
You can use measures to create trend lines or perform forecasting on your visualizations.
This can help in understanding the direction of your data points or predicting future values.
Drag the time dimension values (e.g., ‘Order Date’) to the columns shelf and the measure (e.g., ‘Sales’) to the rows shelf. This will create a line chart showing sales over time.
You can add a trend line by right-clicking on the visualization and selecting ‘Trend Lines’ > ‘Show Trend Lines’. Tableau will automatically add the best-fit line for your data.
If you want to forecast future sales, you can right-click on the visualization and select ‘Forecast’ > ‘Show Forecast’.
4. How to Use Measures For Creating Bins and Histograms
You can use measures to create bins, which can then be used to build histograms. This is useful for understanding the distribution of your data.
Let’s say you have a dataset from a superstore, and you’re interested in analyzing the distribution of sales amounts, which is an integer data type.
Right-click on the ‘Sales’ measure in the data pane and select “Create Bins.”
In the “Create Bins” dialog, specify the size of each bin. For example, if you set the bin size to 100, Tableau will create bins like $0-$100, $100-$200, and so on.
Click “OK” to create the bins.
Drag the newly created ‘Sales (bin)’ field to the Columns shelf.
Drag the measure you want to count, like ‘Sales’, to the Rows shelf. Tableau will automatically aggregate this measure as a count.
The above are just four use cases of measures in Tableau. They form the basis of all data analysis and visualization in Tableau.
Learn the future of Data Tech by watching the following video:
Final Thoughts
Understanding measures in Tableau is essential for anyone looking to make the most of this powerful data visualization tool. As we’ve explored in this article, measures represent the quantitative
values in your dataset, such as sales, profit, or quantity.
They can be aggregated, used in calculations, and displayed as continuous fields in your visualizations. By grasping the concept of measures and their role in Tableau, you’ll be better equipped to
create insightful and meaningful visualizations that drive informed decision-making.
Whether you’re a beginner or an experienced user, mastering measures will undoubtedly enhance your Tableau journey and enable you to unlock the full potential of your data.
Frequently Asked Questions
In this section, you will find some frequently asked questions you may have when working with measures in Tableau.
What are the differences between a dimension and a measure in Tableau?
Dimensions are qualitative data fields that provide context and can be used to segment and categorize data. They are often discrete and are typically represented by blue pills in Tableau.
Measures, on the other hand, are quantitative data fields that provide numerical values for analysis.
They can be aggregated and are often continuous, represented by green pills in Tableau.
How are measures used in Tableau visualizations?
Measures are used to create quantitative visualizations in Tableau. When measures are placed on the Rows, Columns, Pages, or Marks card, they are displayed as continuous axes, allowing for the
creation of various chart types, such as line charts, bar charts, and scatter plots.
What are some examples of measures in Tableau?
Common examples of measures in Tableau include:
• Sales: The total revenue generated from product sales.
• Profit: The net income derived from sales after accounting for costs.
• Quantity: The number of units sold or purchased.
• Discount: The percentage or amount of money deducted from the original price.
• Shipping Cost: The expenses associated with shipping and delivering products.
• Customer Count: The number of unique customers or transactions.
How to create a measure in Tableau?
To create a measure in Tableau, follow these steps:
1. Connect to your data source.
2. In the Data pane, locate the field that contains your quantitative data, such as sales or profit.
3. Drag the field from the Dimensions area to the Measures area in the Data pane.
How can I calculate the percentage change in Tableau?
To calculate the percentage change in Tableau, you can use the following formula in a calculated field:
(LOOKUP(SUM([Measure]),-1) / SUM([Measure])) – 1
Replace [Measure] with the name of the measure you want to calculate the percentage change for. This formula uses the LOOKUP function to compare the current value with the previous one and calculates
the percentage change.
In Tableau, data sources are the foundation for all your visualizations and analyses. They allow you to...
How to Replace Data Source in Tableau in 3 Simple Steps
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5,314 research outputs found
In this note, we show that any continued fraction convergent of the number $e = 2.71828...$ can be derived by approximating some integral $I_{n, m} := \int_{0}^{1} x^n (1 - x)^m e^x d x$ $(n, m \in \
mathbb{N})$ by 0. In addition, we present a new way for finding again the well-known regular continued fraction expansion of $e$.Comment: 7 pages, To appea
In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $\leq n$, we show that the smallest
positive integer $c_n$ satisfying the property: $\forall P \in E_n, c_n P' \in E_n$ is $c_n = \mathrm{lcm}(1 , 2 , \dots , n)$. As an application, we deduce an easy proof of the well-known inequality
$\mathrm{lcm}(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two
divisibility properties for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest
positive integer $\lambda_n$ satisfying the property: $\forall P \in E_n$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in E_n$. In particular, we show that: $\lambda_n = \prod_{p \text{ prime}} p
^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$).Comment: 17 page
In this paper, we present new structures and results on the set \M_\D of mean functions on a given symmetric domain \D of $\mathbb{R}^2$. First, we construct on \M_\D a structure of abelian group in
which the neutral element is simply the {\it Arithmetic} mean; then we study some symmetries in that group. Next, we construct on \M_\D a structure of metric space under which \M_\D is nothing else
the closed ball with center the {\it Arithmetic} mean and radius 1/2. We show in particular that the {\it Geometric} and {\it Harmonic} means lie in the border of \M_\D. Finally, we give two
important theorems generalizing the construction of the \AGM mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regular conditions, there
exists a unique mean $M$ satisfying the functional equation: $M(M_1, M_2) = M$.Comment: 23 pages. To appea
In this paper, we introduce an analog of the Al-Karaji arithmetic triangle by substituting in the formula of the binomial coefficients the products by the least common multiples. Then, we give some
properties and some open questions related to the obtained triangle.Comment: 10 page
In this paper, we find the closed sums of certain type of Fibonacci related convergent series. In particular, we generalize some results already obtained by Brousseau, Popov, Rabinowitz and
others.Comment: 14 page
It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was
extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty
set of digits. Actually, such series are known to be all convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of the reciprocal of the positive integers whose the decimal
representation contains the digit 9 exactly $r$ times, the impressive obtained result is that $S^{(r)}$ tends to $10 \log{10}$ as $r$ tends to infinity!Comment: 5 pages, to appear in (The) American
Mathematical Monthl
We show among others that the formula: $\lfloor n + \log_{\Phi}\{\sqrt{5}(\log_{\Phi}(\sqrt{5}n) + n) -5 + \frac{3}{n}\} - 2 \rfloor (n \geq 2),$ (where $\Phi$ denotes the golden ratio and $\lfloor \
rfloor$ denotes the integer part) generates the non-Fibonacci numbers.Comment: 5 page
In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several
important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.Comment: To appea
In this paper, we present a way to measure the intelligence (or the interest) of an approximation of a given real number in a given model of approximation. Basing on the idea of the complexity of a
number, defined as the number of its digits, we introduce a function noted $\mu$ (called a measure of intelligence) associating to any approximation $\mathbf{app}$ of a given real number in a given
model a positive number $\mu(\mathbf{app})$, which characterises the intelligence of that approximation. Precisely, the approximation $\mathbf{app}$ is intelligent if and only if $\mu(\mathbf{app}) \
geq 1$. We illustrate our theory by several numerical examples and also by applying it to the rational model. In such case, we show that it is coherent with the classical rational diophantine
approximation. We end the paper by proposing an open problem which asks if any real number can be intelligently approximated in a given model for which it is a limit point.Comment: 22 page
A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed that the series $\sum_{a \in \mathscr{A}} \
frac{1}{a \log a}$, where $\mathscr{A}$ is a primitive sequence different from $\{1\}$, are all convergent and their sums are bounded above by an absolute constant. Besides, he conjectured that the
upper bound of the preceding sums is reached when $\mathscr{A}$ is the sequence of the prime numbers. The purpose of this paper is to study the Erd\H{o}s conjecture. In the first part of the paper,
we give two significant conjectures which are equivalent to that of Erd\H{o}s and in the second one, we study the series of the form $\sum_{a \in \mathscr{A}} \frac{1}{a (\log a + x)}$, where $x$ is
a fixed non-negative real number and $\mathscr{A}$ is a primitive sequence different from $\{1\}$. In particular, we prove that the analogue of Erd\H{o}s's conjecture for those series does not hold,
at least for $x \geq 363$. At the end of the paper, we propose a more general conjecture than that of Erd\H{o}s, which concerns the preceding series, and we conclude by raising some open
questions.Comment: 11 page
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