content stringlengths 86 994k | meta stringlengths 288 619 |
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@article{1849, keywords = {Animals, Signal Transduction, Humans, Entropy, Models, Neurological, Neurons, Synaptic Transmission, Nerve Net, Visual Perception, Probability, Automatic Data Processing,
Decision Making, Learning, Nervous System Physiological Phenomena, Normal Distribution, Space Perception}, author = {Tatyana Sharpee and William Bialek}, title = {Neural decision boundaries for
maximal information transmission.}, abstract = {
We consider here how to separate multidimensional signals into two categories, such that the binary decision transmits the maximum possible information about those signals. Our motivation comes from
the nervous system, where neurons process multidimensional signals into a binary sequence of responses (spikes). In a small noise limit, we derive a general equation for the decision boundary that
locally relates its curvature to the probability distribution of inputs. We show that for Gaussian inputs the optimal boundaries are planar, but for non-Gaussian inputs the curvature is nonzero. As
an example, we consider exponentially distributed inputs, which are known to approximate a variety of signals from natural environment.
}, year = {2007}, journal = {PLoS One}, volume = {2}, pages = {e646}, language = {eng}, } | {"url":"https://lsi.princeton.edu/bibcite/export/bibtex/bibcite_reference/1849","timestamp":"2024-11-03T15:20:46Z","content_type":"application/x-bibtex-text-file","content_length":"4158","record_id":"<urn:uuid:1e5e91fd-13e2-44ce-a5f1-aa757d0fe36f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00827.warc.gz"} |
Question ID - 156198 | SaraNextGen Top Answer
A pipe of $0.7 \mathrm{~m}$ diameter has a length of $6 \mathrm{~km}$ and connects two reservoirs $\mathrm{A}$ and $\mathrm{B}$. The water level in reservoir $\mathrm{A}$ is at an elevation $30 \
mathrm{~m}$ above the water level in reservoir $\mathrm{B}$. Halfway along the pipe line, there is a branch through which water can be supplied to a third reservoir $\mathrm{C}$. The friction factor
of the pipe is $0.024 .$ The quantity of water discharged into reservoir $\mathrm{C}$ is $0.15 \mathrm{~m}^{3} / \mathrm{s}$. Considering the acceleration due to gravity as $9.81 \mathrm{~m} / \
mathrm{s}^{2}$ and neglecting minor losses, the discharge (in $\mathrm{m}^{3} / \mathrm{s}$ ) into the reservoir $\mathrm{B}$ is | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=156198","timestamp":"2024-11-06T21:39:27Z","content_type":"text/html","content_length":"15367","record_id":"<urn:uuid:2bdf718a-3e2b-4396-bf2a-05fc8b5eee61>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00561.warc.gz"} |
Levinthal paradox
Levinthal's Paradox is named after scientist Cyrus Levinthal, who surmised that it should take an eternity for proteins to fold because of the huge number of accessible conformations it has to
explore before finding the native state. The fact that it does not take an eternity to occur argues that folding is directed and not random. | {"url":"https://m.everything2.com/title/Levinthal+paradox","timestamp":"2024-11-13T06:26:34Z","content_type":"text/html","content_length":"32988","record_id":"<urn:uuid:c2f75a4a-30ac-4ce8-b4b6-988e72b09e83>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00494.warc.gz"} |
Common structured patterns in linear graphs: Approximation and combinatorics
A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (square subset) or crossing (between). Given a
family of linear graphs, and a non-empty subset R ⊆ {<, square subset, between}, we are interested in the MCSP problem: Find a maximum size edge-disjoint graph, with edge-pairs all comparable by one
of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. In this paper, we generalize the framework of Davydov and Batzoglou by considering patterns comparable by
all possible subsets T ⊆ {<, square subset, between}. This is motivated by the fact that many biological applications require considering crossing structures, and by the fact that different
combinations of the relations above give rise to different generalizations of natural combinatorial problems. Our results can be summarized as follows: We give tight hardness results for the MCSP
problem for {<, between}-structured patterns and {square subset, between}-structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2 ℋ(k) for {<, between}
-structured patterns, (ii) k^1/2 for {square subset, between}-structured patterns, and (iii) Ο(y√k lg k) for {<, square subset, between}-structured patterns, where k is the size of the optimal
solution and ℋ(k) = Σ[i=1]^k 1/i is the k-th harmonic number.
Original language English
Title of host publication Combinatorial Pattern Matching - 18th Annual Symposium, CPM 2007, Proceedings
Publisher Springer Verlag
Pages 241-252
Number of pages 12
ISBN (Print) 9783540734369
State Published - 1 Jan 2007
Externally published Yes
Event 18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007 - London, ON, Canada
Duration: 9 Jul 2007 → 11 Jul 2007
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 4580 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007
Country/Territory Canada
City London, ON
Period 9/07/07 → 11/07/07
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Common structured patterns in linear graphs: Approximation and combinatorics'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/common-structured-patterns-in-linear-graphs-approximation-and-com-2","timestamp":"2024-11-10T17:49:26Z","content_type":"text/html","content_length":"61560","record_id":"<urn:uuid:afc6a532-4a5b-4fa4-96b9-26a1b5f86e56>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00141.warc.gz"} |
Reimplementing the PyTorch training loop in simple Python | Aman
Reimplementing the PyTorch training loop in simple Python
Understand the internal abstractions of a training loop in PyTorch
To master some framework, you need to understand how it is built. Let’s try to understand and reimplement the internal abstractions of a training loop in PyTorch. I’ll start with a bare training loop
which doesn’t use PyTorch’s dataloaders or optimizers. Then I’ll reimplement Dataset and Dataloader in Python. To update models, PyTorch relies on torch.nn.Parameters and torch.optim. I’ll show how
to reproduce them in simple Python.
You can follow along by running this notebook on Google Colab.
Bare Training Loop
The training loop shown below doesn’t use any abstractions over training data or model updates. I manually index through the list of X and Y values to iterate over batches of training examples. I
update the model by going through the layers one by one and updating the weights and biases with their loss gradient.
for epoch in range(epochs):
num_batches = (n-1)//bs
for i in range(num_batches):
# TODO: training data abstraction
start_idx, end_idx = bs*i, bs*(i+1)
xb = x_train[start_idx:end_idx]
yb = y_train[start_idx:end_idx]
yb_pred = model(xb)
loss = loss_fn(yb_pred, yb)
# TODO: model update abstraction
with torch.no_grad():
for layer in model.layers:
if hasattr(layer, "weight"):
layer.weight -= lr * layer.weight.grad
if hasattr(layer, "bias"):
layer.bias -= lr * layer.bias.grad
PyTorch Data Abstractions
Now I’ll rebuild the PyTorch abstractions over data. First, the Dataset is a joint list over X and Y values.
class Dataset():
def __init__(self, x, y):
self.x, self.y = x, y
def __len__(self):
return self.x.shape[0]
def __getitem__(self, i):
return self.x[i], self.y[i]
The Dataloader builds over a Dataset. Its core functionality is to iterate over batches of X and Y items fetched from the Dataset.
class DataLoader():
def __init__(self, dataset, batch_size, sampler, collate_fn):
self.ds, self.bs = dataset, batch_size
self.sampler, self.collate_fn = sampler, collate_fn
def __iter__(self):
for batch_idxs in self.sampler:
yield self.collate_fn([self.ds[idx] for idx in batch_idxs])
You can also customize how you want to sample the X and Y items (sampler) to form batches (collate_fn). I’ll reimplement the random sampler that comes built-in with PyTorch.
class Sampler():
def __init__(self, data_size, batch_size, shuffle):
self.n, self.bs, self.shuffle = data_size, batch_size, shuffle
def __iter__(self):
sample_idx = torch.randperm(self.n) if self.shuffle else list(range(self.n))
for i in range(0, self.n, self.bs):
yield sample_idx[i : i+self.bs]
Now I can rewrite the training loop to use my Dataloader.
for epoch in range(epochs):
# dataloader abstraction
for xb,yb in train_dl:
yb_pred = model(xb)
loss = loss_fn(yb_pred, yb)
# TODO: optimizer abstraction
with torch.no_grad():
for layer in model.layers:
if hasattr(layer, "weight"):
layer.weight -= lr * layer.weight.grad
if hasattr(layer, "bias"):
layer.bias -= lr * layer.bias.grad
PyTorch Training Abstractions
Let’s reproduce the way you use PyTorch to update models at every training step.
Instead of manually looping over the layers of a model to get its parameters, PyTorch maintains a key-value store for all the parameters as they are defined. This can be achieved by using the
__setattr__ method courtesy of the Python Data Model. In the code snippet below, I use a _modules dictionary to store all the layers as they are defined.
class OurModule():
def __init__(self, x_dim, y_dim, h_dim):
self._modules = {}
self.l1 = nn.Linear(x_dim, h_dim)
self.l2 = nn.Linear(h_dim, y_dim)
def __setattr__(self, k, v):
if not k.startswith('_'):
self._modules[k] = v
super().__setattr__(k, v)
def __repr__(self):
return f"{self._modules}"
def parameters(self):
for m in self._modules.values():
for p in m.parameters():
yield p
def __call__(self, x):
x = F.relu(self.l1(x))
return self.l2(x)
Now let’s update our training loop with the model.parameters() functionality.
for epoch in range(epochs):
# dataloader abstraction
for xb,yb in train_dl:
yb_pred = model(xb)
loss = loss_fn(yb_pred, yb)
# TODO: optimizer abstraction
with torch.no_grad():
for p in model.parameters():
p -= lr * p.grad
PyTorch lets you quickly construct a model by defining a list of layers. Here’s how you can reproduce the nn.Sequential functionality in plain Python.
class Sequential(nn.Module):
def __init__(self, *layers):
self.layers = self.module_list(layers)
def module_list(self, layers):
for i, l in enumerate(layers):
self.add_module(name = f'{i}',
module = l)
def forward(self):
for l in self.layers:
x = l(x)
return x
In PyTorch, you can use different optimization functions to update the model. Here’s how you can reimplement a basic SGD optimizer from scratch.
class Optimizer():
def __init__(self, params, lr):
self.params, self.lr = list(params), lr
def step(self):
with torch.no_grad():
for p in self.params:
p -= lr * p.grad
def zero_grad(self):
for p in self.params:
Lets finally rewrite the entire training loop using the custom dataloader and optimizer.
for epoch in range(epochs):
# dataloader abstraction
for xb,yb in train_dl:
yb_pred = model(xb)
loss = loss_fn(yb_pred, yb)
# optimizer abstraction
There you go. We have rebuilt the PyTorch training loop in plain Python. | {"url":"https://amanhussain.com/post/rebuild-pytorch/","timestamp":"2024-11-07T22:53:50Z","content_type":"text/html","content_length":"31926","record_id":"<urn:uuid:321d50bd-1c8a-4551-8d99-31132bc48cf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00493.warc.gz"} |
How Many Ion Pairs Will a Beta Particle Produce Inside a Helium-Filled Geiger Tube?
How can we calculate the number of ion pairs produced by a beta particle passing through a helium-filled Geiger tube? The beta particle is estimated to produce approximately 4689 ion pairs inside the
helium-filled Geiger tube. To calculate this, we can use the concept of ionization energy and the energy of the beta particle.
Ionization Energy in Helium Gas
The energy required to produce an ion pair (W) is typically around 33.97 electron volts (eV) in helium gas. This means that for every ion pair produced, 33.97 eV of energy is needed. To convert this
value to megaelectron volts (MeV), we divide by 1 million: W = 33.97 eV / 1,000,000 = 0.00003397 MeV.
Calculating Number of Ion Pairs
Given that the kinetic energy of the beta particle is 0.159 MeV, we can calculate the number of ion pairs produced as follows: Number of ion pairs = (Kinetic energy of the beta particle) /
(Ionization energy per ion pair). Therefore, Number of ion pairs = 0.159 MeV / 0.00003397 MeV/ion pair ≈ 4689 ion pairs.
Therefore, the beta particle is estimated to produce approximately 4689 ion pairs inside the helium-filled Geiger tube. | {"url":"https://www.brundtlandnet.com/chemistry/how-many-ion-pairs-will-a-beta-particle-produce-inside-a-helium-filled-geiger-tube.html","timestamp":"2024-11-07T15:40:49Z","content_type":"text/html","content_length":"22662","record_id":"<urn:uuid:094ed15a-a689-46b6-b428-04bea5f59abe>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00809.warc.gz"} |
Polynomial Ring - (Algebraic K-Theory) - Vocab, Definition, Explanations | Fiveable
Polynomial Ring
from class:
Algebraic K-Theory
A polynomial ring is a mathematical structure formed by the set of all polynomials in one or more variables with coefficients from a specified ring. This structure allows for the addition and
multiplication of polynomials, making it a fundamental object in algebra and important in various areas, including algebraic geometry and commutative algebra. Understanding polynomial rings is
crucial when exploring concepts like ideals, roots of polynomials, and their applications in other mathematical theories.
congrats on reading the definition of Polynomial Ring. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The polynomial ring in one variable over a ring R is denoted as R[x], where x represents the variable and R is the coefficient ring.
2. Polynomial rings are commutative, meaning the order of multiplication does not affect the result, which is essential for factorization.
3. In polynomial rings, the degree of a polynomial is the highest power of the variable present, and it plays a key role in determining properties like divisibility.
4. The Quillen-Suslin theorem states that every vector bundle over a certain type of space can be trivialized, which can be connected to polynomial rings through their role in defining vector
5. Every polynomial ring over a field is a unique factorization domain (UFD), meaning every polynomial can be factored into irreducible elements uniquely up to ordering.
Review Questions
• How do polynomial rings relate to ideals, and what implications does this relationship have for understanding vector bundles?
□ Polynomial rings have a strong connection to ideals since every ideal in a polynomial ring corresponds to certain algebraic sets defined by the polynomials. This relationship is important for
understanding vector bundles because the properties of these ideals can help determine how vector bundles behave over spaces represented by polynomial rings. In particular, by examining the
structure of these ideals, one can glean insights into whether vector bundles can be trivialized or not.
• Discuss how the structure of polynomial rings supports the concept of unique factorization domains (UFD) and its significance in algebra.
□ Polynomial rings are unique factorization domains (UFDs), meaning that every non-zero polynomial can be expressed uniquely as a product of irreducible polynomials. This property is
significant because it simplifies many algebraic processes, such as finding roots or analyzing solutions to polynomial equations. The UFD structure also connects to various areas in algebra,
as it ensures that methods used in integer factorization can be similarly applied to polynomials.
• Evaluate how the properties of polynomial rings influence the applications in algebraic geometry and relate to the Quillen-Suslin theorem.
□ The properties of polynomial rings are pivotal in algebraic geometry as they provide a framework for defining varieties through zero sets of polynomials. This directly ties into the
Quillen-Suslin theorem, which asserts that certain vector bundles can be trivialized over affine spaces represented by polynomial rings. By establishing this link between the structures of
polynomial rings and vector bundles, one can further understand how geometrical properties emerge from algebraic foundations.
"Polynomial Ring" also found in:
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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/algebraic-k-theory/polynomial-ring","timestamp":"2024-11-08T22:09:36Z","content_type":"text/html","content_length":"145293","record_id":"<urn:uuid:fee0fa5c-99f6-492e-b140-6c01d7b795c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00350.warc.gz"} |
Standard deviation
Normally you would also have to replace all the commas in my example formulas with semi-colon. Do you need a complete list of all the function names in
The standard deviation s (V ) calculated using the formula 3.3 is the standard deviation of an individual pipetting result (value). When the mean value is
Let S = {4, 6, 8, 2, 5} Step 1: Find the mean of S = {4, 6, 8, 2, 5} mean =. 4 + 6 + 8 + 2 + 5. We can find the standard deviation of a set of data by using the following formula: Where: Ri – the
return observed in one period (one observation in the data set) Standard Deviation Formula. The standard deviation formula is similar to the variance formula. It is given by: σ = standard deviation.
What is Standard Deviation? Standard deviation is a number that tells you how far numbers are from their mean. 1. For example, the numbers below have a mean (average) of 10. Explanation: the numbers
are all the same which means there's no variation. As a result, the numbers have a standard deviation of zero. The STDEV function is an old function.
where x takes on each value in the set, x is the average (statistical mean) of the set of values, and n is the number of values in the set.. If your data set is a sample of a population, (rather than
an entire population), you should use the slightly modified form of the Standard Deviation, known as the Sample Standard Deviation. The equation for this is:
Standard Deviation Formula. The standard deviation formula is similar to the variance formula. It is given by: σ = standard deviation.
Standard deviation is a formula used to calculate the averages of multiple sets of data. Standard deviation is used to see how closely an individual set of data is to the average of multiple sets of
data. There are two types of standard deviation that you can calculate:
Population standard deviation looks at the square root of the variance of the set of numbers. It's used to determine a confidence interval for drawing conclusions (such as accepting or rejecting a
hypothesis). A slightly more complex calculation is called sample standard deviation.
This is a crucial step in any type of statistical calculation, even if it is a simple figure like the mean or 20 Aug 2012 Follow these two formulas for calculating standard deviation. The first
formula is for calculating population data and the latter is if you're To understand how to do the calculation, look at the table for the number of days per week a men's soccer team plays soccer. To
find the standard deviation, add Well, the standard deviation is a calculated number that represents the dispersion of data from the mean of the data. It is calculated as the square root of
Heimstaden danmark
2019-11-15 If you’re wondering, “What is the formula for standard deviation?” it looks like this: In order to determine standard deviation: Determine the mean (the average of all the numbers) by
adding up all the data pieces ( xi ) and dividing by the number of pieces of data ( n ).
But knowing the true mean and standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured. The steps to calculate
mean & standard deviation are: 1) Process the data. For ungrouped data, sort and tabulate the data in a table. For grouped data, obtain the mid-value of each intervals.
New snowtam format
The burning rate BR20-80 is therefore calculated by the following equation: where The mean , variance and standard deviation are presented as procedures for summarizing a set of scores . Formulas
for calculating these statistics are shown A general formula ( Eq . 9 : 1 ) expressing how many samples ( n ) are required If the relative standard deviation is 25.4 % , which it is for chlorophyll
- a in our Definition av standardavvikelse. Med standardavvikelsen menar vi ett mått på den genomsnittliga avvikelsen från medelvärdet i en serie observationsvärden.
Performance guarantee
Se hela listan på wallstreetmojo.com
What is Standard Deviation? Standard deviation is a number that tells you how far numbers are from their mean. 1. For example, the numbers below have a mean (average) of 10. | {"url":"https://investeringariojmqcn.netlify.app/37859/54459","timestamp":"2024-11-07T23:10:52Z","content_type":"text/html","content_length":"9022","record_id":"<urn:uuid:3b9eba9b-542f-4208-81a2-e117a4536295>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00619.warc.gz"} |
Areas of Composite Figures
Lesson Video: Areas of Composite Figures Mathematics
In this video, we will learn how to find the area of a composite figure consisting of two or more shapes including polygons, circles, half circles, and quarter circles.
Video Transcript
In this video, we will learn how to find the area of composite figures consisting of two or more shapes, including polygons, circles, half circles, and quarter circles.
A composite figure is simply a figure or a two-dimensional shape that’s constructed from two or more geometric figures. For example, if we had a decorative window and we wanted to find the area of
this shape, we could notice that it’s divided into a semicircle and a rectangle. And therefore, the area of this composite figure would be the sum of these areas.
We might often find that there’s more than one way to calculate the area of a composite figure. Let’s say we wanted to calculate the area of this composite shape. One way we could calculate this is
by calculating the area of the two rectangles internally. Or we could also calculate the area by finding the area of the larger rectangle and then subtract the area of the smaller orange rectangle
that would give us the area of the blue shape. So when we’re finding the area of a composite figure, sometimes we might need to add the areas, sometimes subtract, and sometimes even a mixture of
As we’ll be looking at the areas of polygons and other shapes, let’s remind ourselves of some of the key formulas. The area of a rectangle is the length multiplied by the width. The area of a
triangle is half times the base times the perpendicular height. The area of a circle is equal to 𝜋 times the radius squared. And we can remember that it’s just the radius that’s squared, and it
doesn’t include the 𝜋.
Knowing this formula will also allow us to work out the area of parts of a circle. For example, the area of a semicircle is equal to 𝜋𝑟 squared over two. And if we wanted to find, for example, the
area of a quarter circle, we would have 𝜋𝑟 squared divided by four. And finally, the area of a trapezoid is equal to 𝑎 plus 𝑏 over two multiplied by ℎ, where 𝑎 and 𝑏 are the lengths of the two
parallel sides and ℎ is the perpendicular height.
We’ll now look at some questions where we find the area of a composite figure.
Using 3.14 as an estimate for 𝜋, find the area of this shape.
We notice that this composite figure is made up of a triangle and a quarter circle. So to find the area of the shape, we need to calculate the area of the triangle and the area of the quarter circle
and add them together. Beginning with the triangle, we can recall that the area of a triangle is equal to half times the base times the height. In the triangle, we can see that we have a length of
28.5, which we can take as the base. For the height then, as we can see that this is part of the quarter circle with a radius of 14 centimeters, then this means that the height of our triangle will
also be 14 centimeters. So, we’ll be calculating a half times 28.5 times 14. We can simplify this calculation to 28.5 multiplied by seven, which evaluates as 199.5. And as we’re working with an area,
we’ll have the square units of square centimeters.
Next, we can find the area of the quarter circle. And we can recall that the area of a circle is 𝜋𝑟 squared. So therefore, the area of a quarter circle would be a quarter of this, which is 𝜋𝑟 squared
over four. Substituting in the value of 14 for the radius, we’ll then have 𝜋 times 14 squared over four. As 14 squared is 14 times 14, we’ll have 196𝜋 over four. This simplifies to 49𝜋. And as we
were told to use 3.14 as an estimate for 𝜋, then we can evaluate 49 times 3.14 as 153.86. And our units here will still be in square centimeters. We can now use these two areas to find the total
area. So we add the area of our triangle, 199.5, to the area of our quarter circle, which was 153.86. And so our final answer is 353.36 square centimeters.
In the next two examples, we’ll see how you must subtract the areas within a composite figure to find the remaining area.
Determine, to the nearest tenth, the area of the shaded region.
We begin by noticing that this circle is inscribed within a square. So to find the area of the shaded region, we would begin by finding the area of the square and then subtract the area of the circle
within it. To find the area of a square, we multiply the length by the length. But what would this length be? Well, as the circle sits exactly within the square, this means that the lengths of all
the sides of the square would be the same as the diameter of the circle, which is 19.7 yards. And so our area is 19.7 times 19.7, 388.09. And as it’s an area, our units will be squared, so we’ll have
square yards. Then, to find the area of the circle, we recall that this is equal to 𝜋 times the radius squared. We’ll need to be careful when we’re plugging in the values here as the radius is not
19.7 because that’s the diameter.
To find the radius, we half the diameter, so we’ll be calculating 𝜋 times 9.85 squared. We can then put that directly into our calculator, being careful just to square the 9.85 and not the 𝜋 as well,
giving us a value of 304.805173 and so on square yards. We won’t round this decimal value yet until we reach the final stage of the question. Putting these together then to find the area of the
shaded region, we’ve got the area of our square and the area of our circle. So we have 388.09 subtract 304.805 and so on, which gives us 83.284826 and so on square yards. And since we’re asked to
round it to the nearest tenth, that means we check our second decimal digit and see if it’s five or more. And as it is, then our answer rounds up to 83.3 square yards for the area of the shaded
Using 3.14 as an approximation for 𝜋, find the area of the shaded shape.
If we look at the diagram, we can see from the shading that this shape in orange is excluded from the area. This orange shape is a semicircle. As this shaded figure or shape is composed of two
geometric figures, then we can call this a composite figure. We can calculate the area of this shaded region by finding the area of the large rectangle and then subtracting the area of the
semicircle. Let’s begin by calculating the area of this rectangle. And as we find the area of a rectangle by multiplying the length by the width, we’d have 52 times 23. We can evaluate this without a
calculator to give us the value 1196. And the units here as it’s an area will be squared centimeters.
Now let’s focus on the area of the semicircle. We can use the fact that the area of a circle is equal to 𝜋𝑟 squared to establish that the area of a semicircle would be half of that. In other words,
it’s 𝜋𝑟 squared divided by two. Before we can use this formula, however, we need to work out the value of the radius of this semicircle. The length of our rectangle is 52 centimeters. And therefore,
we can see that the diameter of this circle can be found by subtracting the other two lengths of 15 from 52, which gives us the diameter of 22 centimeters. And so the length of our radius will be
half of that. Half of 22 will give us 11 centimeters. Plugging this value into our formula for the area of a semicircle gives us 𝜋 times 11 squared over two, which is 𝜋 times 121 over two.
In order to evaluate this, we were told to use 3.14 as an approximation for 𝜋. We could put this directly into our calculator if we wished. But if we weren’t using a calculator, then we could
simplify the calculation to 1.57 multiplied by 121 and then use any method of multiplication to work this out. Here, I’ve used the grid or area method to find the value of 189.97 square centimeters
for the area of the semicircle. Now that we’ve found the two important pieces of information, the area of the rectangle and the area of the semicircle, we can work out the area of the shaded shape.
Remembering that as this shaded shape does not include the area of the semicircle, this means we must subtract it from the area of the rectangle. Therefore, we’ll have the calculation 1196 subtract
189.97, which gives us 1006.03 square centimeters. And that is our final answer for the area of the shaded shape.
In these questions, so far, we have seen how two shapes can make up a composite figure. In the final question, we’ll see an example of a more complex composite figure, which is made up of four
Determine, to the nearest tenth, the area of the given figure.
We can see from the diagram that this figure is a composite figure. That simply means that it’s made up of two or more geometric figures. Let’s see if we can work out what these different shapes are.
Well, the three shapes at the top are semicircles. Notice that these three semicircles are all the same size or congruent. We can say this because the marking on the radii shows that these are all
the same lengths in the three semicircles. Looking at the shape on the lower part of this figure, we could say that it’s composed of a rectangle and two triangles. However, there is a much easier
shape. And that is that we can identify that this is a trapezoid.
And so to find the area of the entire figure, we’ll need to find the areas of the three semicircles and the area of the trapezoid and add them together. Let’s begin by finding the area of the
semicircles. We can recall that the area of a circle is equal to 𝜋 times the radius squared. And so therefore, the area of a semicircle would be half of that, 𝜋𝑟 squared over two. Before we use this
formula, we need to find out the value of the radius as this value of 10 inches refers to the diameter. As the radius is half of the diameter, this means that our radius is five. And so we have the
calculation 𝜋 times five squared over two. We can write this as 25𝜋 over two. As this is an area, our units will be squared, so we’ll have square inches.
We can keep our answer in terms of 𝜋 as we’ll use it in the final calculation. But if we did change it into a decimal, then we wouldn’t round that value yet. In a moment, we’ll be able to use this
value for the area of a semicircle to find the area of three semicircles. But let’s move on to finding the area of the trapezoid. The area of a trapezoid is calculated by 𝑎 plus 𝑏 over two times ℎ,
where 𝑎 and 𝑏 are the two parallel sides and ℎ is the perpendicular height. Filling in the values then for our trapezoid, we have a length of 20. And the other parallel length is formed of three lots
of 10 inches.
We can say this because we know that we had three congruent semicircles that all had a diameter of 10 inches. And we also have a perpendicular height of 20 inches. We then have a calculation of 50
over two multiplied by 20, which simplifies to give us the answer of 500 square inches. We’ve now found enough information to help us calculate the area of the entire figure. As we calculated that
the area of one of these semicircles is 25𝜋 over two, then as we established that our semicircles are congruent. To find the area of three semicircles, we would multiply the area of one semicircle by
three. And to this we add the area of our trapezoid, which was 500 square inches. We can use our calculator to evaluate this as 617.8097245 and so on square inches.
But to complete our answer, we must round to the nearest tenth. And as our second decimal digit is not five or more, then our answer stays as 617.8 square inches. And so we found the area of this
figure by adding together the individual areas of each of the shapes within it.
We can now look at some of the important things that we learned in this video. We learned firstly that a composite figure is made up of two or more geometric figures. To find the area of a composite
figure, we separate it into simpler shapes whose area can be found. And finally, when we’re finding the area of a composite figure, sometimes we need to add the individual areas, and sometimes we
need to subtract them. | {"url":"https://www.nagwa.com/en/videos/737175716921/","timestamp":"2024-11-05T03:02:11Z","content_type":"text/html","content_length":"265895","record_id":"<urn:uuid:f45a4c96-6ff9-419b-89a9-18a5022b1cab>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00089.warc.gz"} |
Design Shear Forces for Polygonal Cross-Section Walls
• In polygonal walls meeting the H [w] / l [w] > 2.0 condition, the design shear force, V [e] , which will be taken as basis in calculating the transverse reinforcement in any section considered ,
is automatically calculated with Equation (7.16) .
• Design Shear Force , V [e] for each point of each branch of polygonal walls , is automatically calculated according to the shear force diagram given in Figure 7.12 (c) .
• The shear force dynamic magnification coefficient in Equation (7.16) is determined automatically. β [v] = 1.5 is taken as β [v] = 1.0 in buildings where all of the earthquake load is carried by
reinforced concrete walls .
• In all sections of polygonal walls with H [w ]/ l [w] ≤ 2.0, the condition that the design shear forces are taken equal to the shear forces calculated according to Section 4 is automatically
D = Strength Excess Coefficient
f [ck ]= Characteristic cylinder compressive strength of concrete
f [yk ]= Characteristic yield strength of longitudinal reinforcement
H [w] = Total curtain height measured from the top of the foundation or the ground floor slab
H [cr] = Curtain critical height
l [w] = Curtain or tie-beam curtain part length in plan
( M [p] ) [t] = f[ ck at the] base section of the curtain, f[ yk]and the moment capacity calculated by considering the strength increase of the steel
( M [d] ) [t] = The moment calculated under the combined effect of the vertical loads and earthquake loads multiplied by the load coefficients at the base section of the wall
R = The load-bearing system behavior coefficient
V [d ]= The vertical loads and earthquake loads multiplied by the load coefficients Shear force calculated under the joint effect
V [e ]= Shear force based on transverse reinforcement calculation at column, beam, junction area and curtain
β [v ]= Shear force dynamic magnification coefficient in curtain
Design Shear Forces of Polygon Walls under H [w] / l [w] > 2 and H [w] / l [w] <= 2
Polygon (group) screens are modeled with shell finite elements. The shear section is defined in the center of gravity so as to provide the three-dimensional rigid body movement condition of the
polygon curtain finite elements, and the values obtained from the curtain finite element results of the relevant loading combination are collected at this center of gravity and the internal forces
of the polygon curtain are obtained. The shear force on each arm of polygon curtains is distributed appropriately in proportion to the shear areas by using the shear force calculated at the center of
gravity, and the shear force coming to each arm is found for polygon curtains. TBDY Article 7.6.6.3 and TBDY Figure 7.12 (c)According to ', there are various increments to calculate the shear forces
of polygon walls. While applying these increments, the shear forces obtained from the analysis results are found by distributing the shear force found as a whole of the curtain as described above to
the arms of the polygon curtain. While calculating the shear strength of polygonal walls, the shear strength of each arm is taken into account.
The design shear force, V [e] , [to] be taken as basis in calculating the transverse reinforcement of each branch of polygon (group) walls , is explained in TBDY Article 7.6.6.3 . In addition to this
item, the shear force plot of the shear design is shown in TBDY Figure 7.12 (c) . The design shear force calculated for group walls is found from the shear force diagram in TBDY Figure 7.12 (c) .
The shear forces in the graph shown in TBDY Figure 7.12 (c) are explained below.
"The shear force diagram found from the solution" is the shear force diagram calculated under the combined effect of vertical loads and earthquake loads.
The diagram shown with the "increased shear diagram" is the shear force diagram resulting from the increment mentioned in the TBDY Article 7.6.6.3 . In this shear diagram, Equation. The shear force
and vertical loads calculated by (7.16) are compared with the shear force value obtained by increasing the shear force calculated from the earthquake according to Section 4 by 1.2D (gapless walls) or
1.4D (bond beam walls) and the smaller shear force is taken into account. taken. This check is done for all loading combinations and the most unfavorable condition is taken into account.
The diagram shown with "Design shear force V [Ed] " is the same diagram as the "Increased shear diagram" in the region of H [w] / 3. H [w] / 3 In the region above the H [w] / 3 points in enhanced
shear-shear force is taken as half the wall base at the highest point of the screen is the shear force generated in combination with a linear line. This check is done for all loading combinations and
the most unfavorable condition is taken into account.
These shear forces are calculated at the center of gravity of the polygon curtain. Then this shear force is distributed to the arms of the polygon curtain in accordance with the shear force
direction, taking into account the shear shear areas.
In TBDY Figure 7.12 (c) , H [w] is the total curtain height measured from the top of the foundation or from the ground floor slab. l [w] is the plan length of one arm of the polygon curtain. Since
the length (l [w] ) of each arm of polygon curtains may be different from each other, H [w] / l [w] values are calculated on the basis of arm.
Considering the TBDY Article 7.6.6.3 and the TBDY Figure 7.12 (c) , three situations arise when the polygonal bulkheads have the design shear force.
• Polygon Shearwalls in the Lower Zone of H [w] / l [w] > 2.0 and H [w] / 3
Design shear force, V [e] , [to] be taken as basis in the transverse reinforcement calculation for walls meeting the condition H [w] / l [w] > 2.0 , of polygon wall arms according to TBDY Article
7.6.6.3 shall be calculated by Equation 7.16 .
In this equation (M [p] ) [t] means the moment capacity calculated by considering the strength increase of f [ck] , f [yk] and steel in the base section of the polygon wall . This value is calculated
by moment curvature analysis for the entire polygon wall. This value is also the moment of power consumption of the group curtain section. (M [d] ) [t ], vertical loads in load multiplied by the base
section and the polygon screen means torque calculated under the combined effect of earthquake loads. Here, the results of the finite elements of the polygon wall are summed at the center of gravity
of the wall section to satisfy the three-dimensional rigid body motion condition, and (M [d] ) [t]value is obtained. V [d] value is the shear force value on each branch of the polygon wall calculated
under the combined effect of vertical loads and earthquake loads multiplied by the load coefficients. The shear force in this equation is taken as the dynamic magnification coefficient β [v] = 1.5. Β
[v] = 1.0 is taken for buildings where all earthquake load is carried by reinforced concrete curtains .
However, with the vertical loads, the shear force obtained by increasing the shear force found in the earthquake calculation by 1.2D (gapless walls) and 1.4D (bond beam walls) creates an upper limit.
Therefore, if the shear force considered with 1.2D or 1.4D is smaller than the shear force value found by TBDY Equation 7.16 , this shear force is used as the Design Shear Force. This check is done
for all loading combinations and the most unfavorable condition is taken into account.
As can be understood from the above explanation, while calculating the moment capacity in polygon walls, the entire polygon curtain is taken into consideration while calculating the design shear
force, the shear strength from the whole polygon wall to a single arm.
• Shearwalls in the upper region of H [w] / l [w] > 2.0 and H [w] / 3
In the polygon walls that meet the condition H [w] / l [w] > 2.0 and located in the upper region of the height H [w] / 3 in the shear force diagram , the linearized design shear force value shown in
TBDY Figure 7.11c is used in addition to the control performed using TBDY Equation 7.16 . In this graph, the shear force value to be used in the upper section of the wall is taken as half of the
design shear force at the base of the curtain. H [w] / 3 The design shear force at the height of the curtain section and the shear force value at the top point of the curtain are combined with a
linear line and the design shear forces in other sections are found. This check is done for all loading combinations and the most unfavorable condition is taken into account.
The design shear force V [e] value of the polygon walls in the upper part of H [w] / 3 is found with the linearized graph described above. After the shear force coming to the center of gravity of the
polygon curtains is found by this method, the Design Shear Force, V [e] , of the polygon curtain arms is distributed to the polygon curtain arms in accordance with the shear force direction, taking
into account the shear areas. This check is done for all loading combinations and the most unfavorable condition is taken into account.
• Shearwalls with H [w] / l [w] ≤ 2.0
Design shear forces of polygonal walls with H [w] / l [w] ≤ 2.0 according to TBDY Article 7.6.6.3 are taken equal to shear forces calculated according to TBDY Section 4 . The shear force value
calculated under the combined effect of the vertical loads and the earthquake loads increased with the Resistance Excess Coefficient D is used as the design shear force, V [e] . These shear forces
are calculated in the center of gravity of the polygon walls and distributed to the polygon curtain arms in accordance with the shear force direction, taking into account the shear areas. This check
is done for all loading combinations and the most unfavorable condition is taken into account. | {"url":"https://help.idecad.com/ideCAD/design-shear-forces-for-polygonal-cross-section-wa","timestamp":"2024-11-10T19:29:54Z","content_type":"text/html","content_length":"47886","record_id":"<urn:uuid:f2b68970-1e8d-43b3-96d9-c8f54753958d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00621.warc.gz"} |
Top Mathematics Courses - Learn Mathematics Online
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The language used throughout the course, in both instruction and assessments.
Explore the Mathematics Course Catalog
• University of Pennsylvania
Skills you'll gain: Mathematics, Calculus, Mathematical Theory & Analysis, Differential Equations, Problem Solving
• Skills you'll gain: Applied Mathematics, Mathematical Theory & Analysis, Mathematics, Estimation, Algorithms, Calculus, Probability & Statistics, Geometry, Natural Language Processing,
Probability Distribution, Strategy and Operations
• University of Pennsylvania
Skills you'll gain: Algebra, Calculus, Mathematics, Differential Equations, Mathematical Theory & Analysis, Problem Solving, Linear Algebra
• Universidad Nacional Autónoma de México
Skills you'll gain: Algebra, Calculus, Linear Algebra, Mathematical Theory & Analysis, Mathematics, Problem Solving, Differential Equations, Operational Analysis, Process Analysis, Critical
• Universitat Autònoma de Barcelona
Skills you'll gain: Mathematics, Calculus, Problem Solving, Algebra, Mathematical Theory & Analysis, Linear Algebra, Angular, Geometry, Operational Analysis
• Skills you'll gain: Computational Logic, Computational Thinking, Critical Thinking, Decision Making, Mathematical Theory & Analysis, Problem Solving
• Skills you'll gain: Mathematics, Computational Logic
• Skills you'll gain: Machine Learning, Calculus, Differential Equations, Mathematics, Machine Learning Algorithms, Regression, Algebra, Algorithms, Artificial Neural Networks
• Skills you'll gain: Data Analysis, Mathematics, Business, General Statistics, Microsoft Excel, Statistical Analysis
• Skills you'll gain: Algebra, Linear Algebra, Mathematics, Problem Solving, Applied Mathematics, Critical Thinking, Plot (Graphics)
• University of Pennsylvania
Skills you'll gain: Calculus, Mathematics, Algebra, Geometry, Probability & Statistics
Mathematics learners also search
In summary, here are 10 of our most popular mathematics courses
Mathematics courses cover a broad range of topics essential for understanding and applying mathematical principles. These include foundational topics such as algebra, geometry, and calculus. Learners
will explore areas like statistics, probability, and discrete mathematics. Advanced courses might cover linear algebra, differential equations, mathematical modeling, and abstract algebra. Practical
exercises and problem-solving sessions help learners apply these concepts to real-world scenarios, enhancing their analytical and critical thinking skills.
Choosing the right mathematics course depends on your current proficiency level and learning objectives. Beginners should look for courses that cover the basics of algebra, geometry, and introductory
calculus. Those with some experience might benefit from intermediate courses focusing on more advanced topics like linear algebra, statistics, and mathematical reasoning. Advanced learners or
professionals seeking specialized knowledge might consider courses on mathematical modeling, abstract algebra. Reviewing course content, instructor expertise, and learner feedback can help ensure the
course aligns with your goals.
A certificate in mathematics can open up various career opportunities in education, research, and applied fields. Common roles include data analyst, statistician, math teacher, and research
scientist. These positions involve analyzing data, conducting research, teaching mathematical concepts, and applying mathematical techniques to solve complex problems. With the increasing importance
of quantitative skills in various industries, earning a certificate in mathematics can significantly enhance your career prospects and opportunities for advancement in fields such as finance,
technology, engineering, and education. | {"url":"https://www.coursera.org/courses?query=mathematics&page=3","timestamp":"2024-11-12T08:42:29Z","content_type":"text/html","content_length":"790834","record_id":"<urn:uuid:edb5e31c-d90a-4450-b909-a9bbb936a7b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00213.warc.gz"} |
Formulas for Geometry
Slide 1
Formulas for Geometry
Mr. Ryan
Slide 2
Don’t Get Scared!!!
Evil mathematicians have created formulas to save you time. But, they always change the letters of the formulas to scare you!
Slide 3
Any shape’s “perimeter” is the outside of the shape…like a fence around a yard.
Slide 4
To calculate the perimeter of any shape, just add up “each” line segment of the “fence”.
Slide 5
Free powerpoint template: www.brainybetty.com
Triangles have 3 sides…add up each sides length.
The Perimeter is 24
Slide 6
A square has 4 sides of a fence
Slide 7
Sometimes, problems may only give you two measurements for a square or rectangle.
No problem…use the formula for squares/rectangles (only!)
Slide 8
Squares ALL sides are equal…so if they give you one side, you know ALL the sides
Length=the Largest side
If they “leave” numbers out, they are equal to their opposite side. If they give you the bottom of a square/rectangle type shape then the top is the same
Slide 9
The Same!!
If the bottom is 15…the top is…
Slide 10
Square/Rectangle Formula
P= 2(25+14) P=2(20+20)
P= 50+28 P=40+40
P= 78 P=80
Slide 11
Other shapes
Just add up EACH segment
8 sides, each side 10 so 10+10+10+10+10+10+10+10=80
Slide 12
Odd shapes
Count ALL sides
Remember if one side blank, it’s equal to its opposite
25+25=50 (for Length)
15+5+15+5=40 (for Width)
Slide 13 | {"url":"https://www.sliderbase.com/spitem-249-1.html","timestamp":"2024-11-12T06:28:51Z","content_type":"text/html","content_length":"15928","record_id":"<urn:uuid:e6ba6104-d4f8-450f-9770-09c02c153740>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00788.warc.gz"} |
maximum-score-of-a-good-subarray | Leetcode
Similar Problems
Similar Problems not available
Maximum Score Of A Good Subarray - Leetcode Solution
LeetCode: Maximum Score Of A Good Subarray Leetcode Solution
Difficulty: Hard
Topics: stack binary-search array two-pointers
Problem Statement:
Given an array of integers nums, a good subarray is a contiguous subarray of nums containing no more than one unique value.
Return the maximum number of good subarrays of nums.
We can solve this problem using the sliding window technique.
We will initialize three pointers: left, right, and uniqueCount.
uniqueCount will keep track of the number of unique integers in the current subarray.
Initially, left and right will both be at the 0th index of the array and uniqueCount will be 1.
We will then iterate through the array using the right pointer and incrementing uniqueCount if we encounter a new integer.
If uniqueCount becomes more than 2, we will increment the left pointer and decrement uniqueCount if the integer that left pointer is pointing to is not present in the rest of the subarray.
At every step of the iteration, we will calculate the length of the current subarray and add it to a sum variable.
We will then return the sum variable.
Below is the implementation of this algorithm:
class Solution:
def maxGoodSubarrays(self, nums: List[int]) -> int:
left = right = good_subarrays = unique_count = 0
while right < len(nums):
if nums[right] not in nums[left:right]:
unique_count += 1
while unique_count > 1:
if nums[left] not in nums[left+1:right]:
unique_count -= 1
left += 1
good_subarrays += right - left + 1
right += 1
return good_subarrays
Time Complexity:
The above algorithm has a time complexity of O(n) as it just iterates over the array once.
Space Complexity:
The space complexity of the algorithm is O(1) as we only use constant extra space.
Input: nums = [1,2,1,3,1] Output: 4 Explanation: There are 4 good subarrays: [1], [2], [1,2,1], [1,3,1].
In this problem, we used the sliding window technique to solve the problem in linear time complexity. We initialized three pointers and iterated over the array while updating the pointers and keeping
track of the count of good subarrays. This is a good example of how the sliding window technique can be used to solve a problem efficiently.
Maximum Score Of A Good Subarray Solution Code | {"url":"https://prepfortech.io/leetcode-solutions/maximum-score-of-a-good-subarray","timestamp":"2024-11-08T07:35:48Z","content_type":"text/html","content_length":"56608","record_id":"<urn:uuid:196013d4-6698-4d57-9553-267329e35bcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00683.warc.gz"} |
applied math and science
An introduction to continuum mechanics and some numerical methods used for simulations of continuous phenomena, with a brief overview of their use in the numerical modeling of impact cratering
phenomena. References are provided for more details.
An examination of the problem of assessing the validity of numerical simulations of continuous-time dynamical systems by treating numerical methods as discrete-time dynamical systems. Simulations of
simple explicit and implicit Runge-Kutta methods, along with their Matlab code, are included. | {"url":"http://phenomenologica.com/index.php/resources/expositions/34-applied-math/31-applied-math","timestamp":"2024-11-06T04:38:40Z","content_type":"text/html","content_length":"16406","record_id":"<urn:uuid:25bec153-2376-40da-b2b9-d237e7625f00>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00432.warc.gz"} |
K5 Math | BJU Press
K5 Math
Laying the Groundwork
Equip your early learners with foundational concepts and skills in math. K5 Math develops students’ number sense grounded in hands-on learning that lays the groundwork for a lifetime of math skills.
Students will progress in their understanding of number sense and place value from manipulatives to abstract math. They will compare measurements, solve addition and subtraction problems, identify
dates and times, and apply biblical worldview themes as they use math to solve problems. This new 5th edition adds STEM lessons throughout and reintroduces a chapter on money.
Results for K5 Math
• K5 Math Worktext, 5th ed.
The BJU Press K5 Math Student Worktext includes practice problems, a colorful layout, STEM activities, manipulatives, differentiated instruction, and review questions.
• K5 Math Trove eWorktext, 5th ed.
The BJU Press K5 Math Student Worktext includes practice problems, a colorful layout, STEM activities, manipulatives, differentiated instruction, and review questions.
• K5 Math Teacher Edition, 5th ed.
The BJU Press K5 Math Teacher Edition includes math problems, demonstrations, manipulatives, videos, and learning targets in each lesson to help students best learn math.
• K5 Math Visuals, 5th ed.
The BJU Press K5 Math visuals packet includes 35 colorful illustrations and manipulatives such as stick puppets, number cards, sign cards, and dot pattern cards.
• K5 Math Manipulatives Packet, 5th ed.
The BJU Press K5 Math Manipulatives include number lines, geometric shapes, money, fractions, and rulers. Manipulatives help students have a deeper understanding of math.
• Math K5 Student Worktext, 4th ed.
The Math K5 Student Worktext, 4th ed. will help your kindergarten student develop an understanding of numbers and how they are used to represent addition, subtraction, measurement, and more. The
program has a farm theme that makes learning fun and incorporates age-appropriate stories, colorful illustrations, and connections to real-world problems.
• Math K5 Teacher's Edition, 4th ed.
The Math K5 Teacher’s Edition, 4th ed. provides instruction for the use of manipulatives to introduce each new math concept for a hands-on approach to learning. Through practice and interactive
questioning by the teacher, the student gradually progresses to an abstract level in which he uses only numbers.
• Math K5 Teacher's Visual Packet, 4th ed.
The first part of the packet contains thirty-five 12 ¼” x 17 ¾” teaching charts with colorful illustrations. These charts are frequently used in the lessons and may be displayed in the classroom.
The second part contains manipulatives for teacher demonstration.
• Math K5 Student Manipulative Packet, 3rd ed.
The BJU Press math program seeks to teach for understanding. One of the best ways to help students understand foundational math concepts is by using manipulatives. The K5 Math Manipulatives
Packet contains workmats, animal counters, geometric shapes, and money that can be used in the lessons. The packet also contains Number Cards, Dot Pattern Cards, a clock, rulers,
• Math K5 Student Worktext, 3rd ed.
The Worktext provides two teacher-directed pages to review the skills taught in the lesson and to give individual help where needed. While emphasizing the farm themes introduced in the lessons,
the book provides a variety of activities. A chapter review, a cumulative review, and a note to the parent are included.
• Math K5 Teacher's Edition, 3rd ed.
The Teacher’s Edition contains 160 lessons divided into 22 chapters. The colorful Student Worktext pages with answer overprint are included for each lesson. Each chapter is preceded by a Chapter
Overview, which explains the chapter theme and contains notes about materials that need to be gathered and prepared for the lessons.
• Math K5 Teacher's Visual Packet (3rd ed.)
The first part of the packet contains thirty-five teaching charts with colorful illustrations. These charts are frequently used in the lessons and may be displayed in the classroom. The second
part contains manipulatives, money, stick puppets, Number Cards, and Dot Pattern Cards for teacher demonstration.
• Math K5-Grade 1 Student Manipulative Packet, 4th ed.
The BJU Press educational materials for math seeks to teach for understanding. One of the best ways to help students understand foundational math concepts is by using manipulatives. The Math K5-1
Student Manipulative Packet contains geometric shapes, fraction pieces, and money to be used in the lessons by the students.
How We Teach K5 Math
Start with Stories
Every chapter starts with a story to awaken learners’ imaginations and frame their math learning in the context of solving real-world problems.
Cyclical Instruction
More than just repetition and memorization, K5 Math implements a cyclical spiral-review approach with multiple chapters per topic, each building on the skills of the last.
Learning Targets
New for the 5th edition, learning targets appear in each lesson and on every worktext page, orienting students and teachers to daily goals and acting as a potential formative assessment opportunity.
Modeling & Application
BJU Press has developed course materials that offer numerous opportunities for teacher modeling and application of mathematical concepts. Complementary learning centers encourage students to make
their own discoveries as well.
K5 Math Educational Materials
Student Worktext
The student worktext uses eye-catching visuals to engage young students. The structure of each lesson has been improved thanks to new learning targets, essential questions, and the integrated
four-step teaching cycle. New chapter openers begin with a story and provide opportunities for introducing biblical worldview shaping concepts. STEM lessons have been added throughout the worktext,
and the total lesson count has increased to 180.
Teacher Edition
The teacher edition provides strategies for direct instruction, discussion, modeling, and facilitating student-guided discovery. This new edition implements our four-step teaching cycle (engage,
instruct, apply, assess), adding strategy and structure to the teacher’s approach. New lesson openers provide opportunities for teachers to engage in biblical worldview shaping.
Student Manipulatives & Visuals
Student manipulatives and visuals enable tactile learning across a range of purposeful activities, including constructing shapes, solving word problems, and representing addition and subtraction | {"url":"https://www.bjupress.com/category/elementary-math/k5-math/5637160391.c","timestamp":"2024-11-12T01:52:24Z","content_type":"text/html","content_length":"618211","record_id":"<urn:uuid:b31d421c-964d-4484-b27b-7abad5b5d935>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00678.warc.gz"} |
Profit and Loss Basic Concepts
Here we discussed the main concepts of quantitative aptitude, which are Profit and Loss. You must know these basic concepts, formulas, and tricks. It helps you to solve problems quickly in your
competitive exams.
Cost Price
The price at which a person purchases an item is called Cost Price(CP).
CP = (100 × SP)÷(100 + Profit%)
CP = (100 × SP)÷(100 – Loss%)
Selling Price
The price at which a person sells the item is called the Selling Price(SP).
SP = (100 + Profit%) × CP/100
SP = (100 – Loss%) × CP/100
If a person sells an item at a price that is more than the actual price, he makes a profit.
Profit = Selling Price(SP) – Cost Price(CP)
Profit% = (Profit/CP) × 100
If a person sells an item that is less than the actual price, he makes a loss.
Loss = Cost Price(CP) – Selling Price(SP)
Loss% = (Loss/CP) × 100
Trick 1
If the selling price of X goods = Cost price of Y goods then,
Profit% or Loss% = (Goods left/ Goods sold)×100 = (Y–X)×100/X
If Y-X is positive, it returns a profit.
If Y-X is negative, it returns a loss.
Trick 2
If CP1=CP2 and half of the goods are sold at an X% profit and the remaining half of the goods at Y% loss then, the overall or net gain/loss percent is (X-Y)/2
Marked Price
Marked price is also named tagged, displayed, or labeled price. They allow a discount on the Marked Price. | {"url":"https://notesjam.com/profit-and-loss-formula-basic-concepts/","timestamp":"2024-11-01T22:23:27Z","content_type":"text/html","content_length":"117649","record_id":"<urn:uuid:291ca198-0c60-40db-a3a6-b83b773dc316>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00712.warc.gz"} |
(PDF) Mathematics Curriculum Development and the Role of Problem Solving
... This may be due to mathematics lessons that neglect linking students' informal knowledge gained through their experiences with their formal knowledge (Gee et al., 2018). In this sense, RME may be
considered a powerful approach to support the implementation of a problem-oriented curriculum (Anderson, 2009), as it involves working with non-routine problems that allow a deeper understanding of
mathematical concepts (Stacey, 2005). Stacey (2005) also argues that to be able to go beyond routine problems, students should be equipped with heuristic strategies, good communication skills, and
the ability to work in cooperative groups. ...
... Namely, a context should be presented in a manner that engages students and should include more authentic than artificial aspects. On the other hand, it can be difficult for teachers to spend the
time and effort to seek real objects or to design real-life problem situations (Anderson, 2009). To address this issue, the current study suggests using virtual learning environments to provide
non-routine problems. ... | {"url":"https://www.researchgate.net/publication/255630930_Mathematics_Curriculum_Development_and_the_Role_of_Problem_Solving","timestamp":"2024-11-03T13:27:08Z","content_type":"text/html","content_length":"550256","record_id":"<urn:uuid:414cfad6-e9b9-4eca-9a99-b2d5ec57b3a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00176.warc.gz"} |
Intermediate Algebra
Learning Outcomes
• Define decimal and scientific notation
• Convert between scientific and decimal notation
Convert Between Scientific and Decimal Notation
Before we can convert between scientific and decimal notation, we need to know the difference between the two. Scientific notation is used by scientists, mathematicians, and engineers when they are
working with very large or very small numbers. Using exponential notation, large and small numbers can be written in a way that is easier to read.
When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a
small number that is between [latex]0[/latex] and [latex]1[/latex]. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.
Word How many thousands Number Scientific Notation
million [latex]1000 x 1000[/latex] = a thousand thousands [latex]1,000,000[/latex] [latex]10^6[/latex]
billion [latex](1000 x 1000) x 1000[/latex] = a thousand millions [latex]1,000,000,000[/latex] [latex]10^9[/latex]
trillion [latex](1000 x 1000 x 1000)[/latex] x 1000 = a thousand billions [latex]1,000,000,000,000[/latex] [latex]10^{12}[/latex]
1 billion can be written as [latex]1,000,000,000[/latex] or represented as [latex]10^9[/latex]. How would [latex]2[/latex] billion be represented? Since [latex]2[/latex] billion is [latex]2[/latex]
times [latex]1[/latex] billion, then [latex]2[/latex] billion can be written as [latex]2\times10^9[/latex].
A light year is the number of miles light travels in one year, about [latex]5,880,000,000,000[/latex]. That is a lot of zeros, and it is easy to lose count when trying to figure out the place value
of the number. Using scientific notation, the distance is [latex]5.88\times10^{12}[/latex] miles. The exponent of [latex]12[/latex] tells us how many places to count to the left of the decimal.
Another example of how scientific notation can make numbers easier to read is the diameter of a hydrogen atom, which is about 0.00000005 mm, and in scientific notation is [latex]5\times10^{-8}[/
latex] mm. In this case, the [latex]-8[/latex] tells us how many places to count to the right of the decimal.
Outlined in the box below are some important conventions of scientific notation format.
Scientific Notation
A positive number is written in scientific notation if it is written as [latex]a\times10^{n}[/latex] where the coefficient a is [latex]1\leq{a}<10[/latex], and n is an integer.
Look at the numbers below. Which of the numbers is written in scientific notation?
Number Scientific Notation? Explanation
[latex]1.85\times10^{-2}[/latex] yes
[latex]-2[/latex] is an integer
[latex] \displaystyle 1.083\times {{10}^{\frac{1}{2}}}[/latex] no [latex] \displaystyle \frac{1}{2}[/latex] is not an integer
[latex]0.82\times10^{14}[/latex] no [latex]0.82[/latex] is not [latex]\geq1[/latex]
[latex]10\times10^{3}[/latex] no [latex]10[/latex] is not <[latex]10[/latex]
Now compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay
close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.
Large Numbers Small Numbers
Decimal Notation Scientific Notation Decimal Notation Scientific Notation
[latex]500.0[/latex] [latex]5\times10^{2}[/latex] [latex]0.05[/latex] [latex]5\times10^{-2}[/latex]
[latex]80,000.0[/latex] [latex]8\times10^{4}[/latex] [latex]0.0008[/latex] [latex]8\times10^{-4}[/latex]
[latex]43,000,000.0[/latex] [latex]4.3\times10^{7}[/latex] [latex]0.00000043[/latex] [latex]4.3\times10^{-7}[/latex]
[latex]62,500,000,000.0[/latex] [latex]6.25\times10^{10}[/latex] [latex]0.000000000625[/latex] [latex]6.25\times10^{-10}[/latex]
Convert from Decimal Notation to Scientific Notation
To write a large number in scientific notation, move the decimal point to the left to obtain a number between [latex]1[/latex] and [latex]10[/latex]. Since moving the decimal point changes the value,
you have to multiply the decimal by a power of [latex]10[/latex] so that the expression has the same value.
Let us look at an example:
Notice that the decimal point was moved [latex]5[/latex] places to the left, and the exponent is [latex]5[/latex].
Write the following numbers in scientific notation.
1. [latex]920,000,000[/latex]
2. [latex]10,200,000[/latex]
3. [latex]100,000,000,000[/latex]
Show Solution
To write a small number (between [latex]0[/latex] and [latex]1[/latex]) in scientific notation, you move the decimal to the right and the exponent will have to be negative, as in the following
You may notice that the decimal point was moved five places to the right until you got to the number [latex]4[/latex] which is between [latex]1[/latex] and [latex]10[/latex]. The exponent is [latex]
Write the following numbers in scientific notation.
1. [latex]0.0000000000035[/latex]
2. [latex]0.0000000102[/latex]
3. [latex]0.00000000000000793[/latex]
Show Solution
In the following video, you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.
Convert from Scientific Notation to Decimal Notation
You can also write scientific notation as decimal notation. Recall the number of miles that light travels in a year is [latex]5.88\times10^{12}[/latex], and a hydrogen atom has a diameter of [latex]5
\times10^{-8}[/latex] mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive, move the decimal point
to the right. If the exponent is negative, move the decimal point to the left.
For each power of [latex]10[/latex], you move the decimal point one place. Be careful here and do not get carried away with the zeros—the number of zeros after the decimal point will always be
[latex]1[/latex] less than the exponent because it takes one power of [latex]10[/latex] to shift that first number to the left of the decimal.
Write the following in decimal notation.
1. [latex]4.8\times10{-4}[/latex]
2. [latex]3.08\times10^{6}[/latex]
Show Solution
Think About It
To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox
to write your ideas before you reveal the solution.
1. You are writing a number whose absolute value is greater than 1 in scientific notation. Will your exponent be positive or negative?
2.You are writing a number whose absolute value is between 0 and 1 in scientific notation. Will your exponent be positive or negative?
3. What power do you need to put on [latex]10[/latex] to get a result of [latex]1[/latex]?
Show Solution
In the next video, you will see how to convert a number written in scientific notation into decimal notation.
Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and
division on large and small numbers is made easier by scientific notation and the rules of exponents. | {"url":"https://courses.lumenlearning.com/intermediatealgebra/chapter/read-writing-scientific-notation-2/","timestamp":"2024-11-06T04:03:45Z","content_type":"text/html","content_length":"63509","record_id":"<urn:uuid:d30daf53-f0d3-45a7-a7b2-909f76953f1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00467.warc.gz"} |
Communications of the ACM
Artificial Intelligence and Machine Learning Review articles
Proving Program Termination
In contrast to popular belief, proving termination is not always impossible.
The program termination problem, also known as the uniform halting problem, can be defined as follows:
Using only a finite amount of time, determine whether a given program will always finish running or could execute forever.
This problem rose to prominence before the invention of the modern computer, in the era of Hilbert’s Entscheidungsproblem:^a the challenge to formalize all of mathematics and use algorithmic means to
determine the validity of all statements. In hopes of either solving Hilbert’s challenge, or showing it impossible, logicians began to search for possible instances of undecidable problems. Turing’s
proof^38 of termination’s undecidability is the most famous of those findings.^b
The termination problem is structured as an infinite set of queries: to solve the problem we would need to invent a method capable of accurately answering either “terminates” or “doesn’t terminate”
when given any program drawn from this set. Turing’s result tells us that any tool that attempts to solve this problem will fail to return a correct answer on at least one of the inputs. No number of
extra processors nor terabytes of storage nor new sophisticated algorithms will lead to the development of a true oracle for program termination.
Unfortunately, many have drawn too strong of a conclusion about the prospects of automatic program termination proving and falsely believe we are always unable to prove termination, rather than more
benign consequence that we are unable to always prove termination. Phrases like “but that’s like the termination problem” are often used to end discussions that might otherwise have led to viable
partial solutions for real but undecidable problems. While we cannot ignore termination’s undecidability, if we develop a slightly modified problem statement we can build useful tools. In our new
problem statement we will still require that a termination proving tool always return answers that are correct, but we will not necessarily require an answer. If the termination prover cannot prove
or disprove termination, it should return “unknown.”
Using only a finite amount of time, determine whether a given program will always finish running or could execute forever, or return the answer “unknown.”
This problem can clearly be solved, as we could simply always return “unknown.” The challenge is to solve this problem while keeping the occurrences of the answer “unknown” to within a tolerable
threshold, in the same way that we hope Web browsers will usually succeed to download Web pages, although we know they will sometimes fail. Note that the principled use of unknown in tools attempting
to solve undecidable or intractable problems is increasingly common in computer science; for example, in program analysis, type systems, and networking.
In recent years, powerful new termination tools have emerged that return “unknown” infrequently enough that they are useful in practice.^35 These termination tools can automatically prove or disprove
termination of many famous complex examples such as Ackermann’s function or McCarthy’s 91 function as well as moderately sized industrial examples such as Windows device drivers. Furthermore, entire
families of industrially useful termination-like properties—called liveness properties—such as “Every call to lock is eventually followed by a call to unlock” are now automatically provable using
termination proving techniques.^12,29 With every month, we now see more powerful applications of automatic termination proving. As an example, recent work has demonstrated the utility of automatic
termination proving to the problem of showing concurrent algorithms to be non-blocking.^20 With further research and development, we will see more powerful and more scalable tools.
We could also witness a shift in the power of software, as techniques from termination proving could lead to tools for other problems of equal difficulty. Whereas in the past a software developer
hoping to build practical tools for solving something related to termination might have been frightened off by a colleague’s retort “but that’s like the termination problem,” perhaps in the future
the developer will instead adapt techniques from within modern termination provers in order to develop a partial solution to the problem of interest.
The purpose of this article is to familiarize the reader with the recent advances in program termination proving, and to catalog the underlying techniques for those interested in adapting the
techniques to other domains. We also discuss current work and possible avenues for future investigation. Concepts and strategies will be introduced informally, with citations to original papers for
those interested in more detail. Several sidebars are included for readers with backgrounds in mathematical logic.
Disjunctive Termination Arguments
Thirteen years after publishing his original undecidability result, Turing proposed the now classic method of proving program termination.^39 His solution divides the problem into two parts:
Termination argument search: Find a potential termination argument in the form of a function that maps every program state to a value in a mathematical structure called a well-order. We will not
define well-orders here, the reader can assume for now that we are using the natural numbers (a.k.a. the positive integers).
Termination argument checking: Proves the termination argument to be valid for the program under consideration by proving that result of the function decreases for every possible program transition.
That is, if f is the termination argument and the program can transition from some state s to state s‘, then f(s) > f(s’).
(Readers with a background in logic may be interested in the formal explanation contained in the sidebar here.)
A well-order can be thought of as a terminating program—in the example of the natural numbers, the program is one that counts from some initial value in the natural numbers down to 0. Thus, no matter
which initial value is chosen the program will still terminate. Given this connection between well-orders and terminating programs, in essence Turing is proposing that we search for a map from the
program we are interested in proving terminating into a program known to terminate such that all steps in the first program have analogous steps in the second program. This map to a well-order is
usually called a progress measure or a ranking function in the literature. Until recently, all known methods of proving termination were in essence minor variations on the original technique.
The problem with Turing’s method is that finding a single, or monolithic, ranking function for the whole program is typically difficult, even for simple programs. In fact, we are often forced to use
ranking functions into well-orders that are much more complex than the natural numbers. Luckily, once a suitable ranking function has been found, checking validity is in practice fairly easy.
The key trend that has led toward current progress in termination proving has been the move away from the search for a single ranking function and toward a search for a set of ranking functions. We
think of the set as a choice of ranking functions and talk about a disjunctive termination argument. This terminology refers to the proof rule of disjunctively well-founded transition invariants.^31
The recent approaches for proving termination for general programs^3,4,9,12,14,26,32 are based on this proof rule. The proof rule itself is based on Ramsey’s theorem,^34 and it has been developed in
the effort to give a logical foundation to the termination analysis based on size-change graphs.^24
The principle it expresses appears implicitly in previously developed termination algorithms for rewrite systems, logic, and functional programs, see refs^10,15,17,24.
The advantage to the new style of termination argument is that it is usually easier to find, because it can be expressed in small, mutually independent pieces. Each piece can be found separately or
incrementally using various known methods for the discovery of monolithic termination arguments. As a trade-off, when using a disjunctive termination argument, a more difficult validity condition
must be checked. This difficulty can be mitigated thanks to recent advances in assertion checking tools (as discussed in a later section).
Example using a monolithic termination argument. Consider the example code fragment in Figure 1. In this code the collection of user-provided input is performed via the function input(). We will
assume the user always enters a new value when prompted. Furthermore, we will assume for now that variables range over possibly negative integers with arbitrary precision (that is, mathematical
integers as opposed to 32-bit words, 64-bit words, and so on). Before reading further, please answer the question: “Does this program terminate, no matter what values the user gives via the input()
function?” The answer is given below.^c
Using Turing’s traditional method we can define a ranking function from program variables to the natural numbers. One ranking function that will work is 2x + y, though there are many others. Here we
are using the formula 2x + y as shorthand for a function that takes a program configuration as its input and returns the natural number computed by looking up the value of x in the memory,
multiplying that by 2 and then adding in y‘s value—thus 2x + y represents a mapping from program configurations to natural numbers. This ranking function meets the constraints required to prove
termination: the valuation of 2x + y when executing at line 9 in the program will be strictly one less than its valuation during the same loop iteration at line 4. Furthermore, we know the function
always produces natural numbers (thus it is a map into a well-order), as 2x + y is greater than 0 at lines 4 through 9.
Automatically proving the validity of a monolithic termination argument like 2x + y is usually easy using tools that check verification conditions (for example, Slam^2). However, as mentioned
previously, the actual search for a valid argument is famously tricky. As an example, consider the case in Figure 2, where we have replaced the command “y := y + 1;” in Figure 1 with “y := input ();
”. In this case no function into the natural numbers exists that suffices to prove termination; instead we must resort to a lexicographic ranking function (a ranking function into ordinals, a more
advanced well-order than the naturals).
Example using a disjunctive termination argument. Following the trend toward the use of disjunctive termination arguments, we could also prove the termination of Figure 1 by defining an argument as
the unordered finite collection of measures x and y. The termination argument in this case should be read as:
x goes down by at least 1 and is larger than 0.
y goes down by at least 1 and is larger than 0
We have constructed this termination argument with two ranking functions: x and y. The use of “or” is key: the termination argument is modular because it is easy to enlarge using additional measures
via additional uses of “or.” As an example, we could enlarge the termination argument by adding “or 2w y goes down by at least 1 and is greater than 1,000.” Furthermore, as we will discuss later,
independently finding these pieces of the termination argument is easier in practice than finding a single monolithic ranking function.
The expert reader will notice the relationship between our disjunctive termination argument and complex lexicographic ranking functions. The advantage here is that we do not need to find an order on
the pieces of the argument, thus making the pieces of the argument independent from one another.
The difficulty with disjunctive termination arguments in comparison to monolithic ones is that they are more difficult to prove valid: for the benefit of modularity we pay the price in the fact that
the termination arguments must consider the transitions in all possible loop unrollings and not just single passes through a loop. That is to say: the disjunctive termination argument must hold not
only between the states before and after any single iteration of the loop, but before and after any number of iterations of the loop (one iteration, two iterations, three iterations, and so on). This
is a much more difficult condition to automatically prove. In the case of Figure 1 we can prove the more complex condition using techniques described later.
Note that this same termination argument now works for the tricky program in Figure 2, where we replaced “y := y + 1;” with “y := input();.” On every possible unrolling of the loop we will still see
that either x or y has gone down and is larger than 0.
To see why we cannot use the same validity check for disjunctive termination arguments as we do for monolithic ones, consider the slightly modified example in Figure 3. For every single iteration of
the loop it is true that either x goes down by at least one and x is greater than 0 or y goes down by at least one and y is greater than 0. Yet, the program does not guarantee termination. As an
example input sequence that triggers non-termination, consider 5, 5, followed by 1, 0, 1, 0, 1, 0, …. If we consider all possible unrollings of the loop, however, we will see that after two
iterations it is possible (in the case that the user supplied the inputs 1 and 0 during the two loop iterations) that neither x nor y went down, and thus the disjunctive termination argument is not
valid for the program in Figure 3.
Argument Validity Checking
While validity checking for disjunctive termination arguments is more difficult than checking for monolithic arguments, we can adapt the problem statement such that recently developed tools for
proving the validity of assertions in programs (such as Slam^2).
An assertion statement can be put in a program to check if a condition is true. For example, assert (y ≥ 1); checks that y ≥ 1 after executing the command. We can use an assertion checking tool to
formally investigate at compile time whether the conditions passed to assertion statements always evaluate to true. For example, most assertion checking tools will be able to prove the assert
statement at line 3 in Figure 4 never fails. Note that compile-time assertion checking is itself an undecidable problem, although it is technically in an easier class of difficulty than termination.^
The reason that assertion checking is so important to termination is the validity of disjunctive termination arguments can be encoded as an assertion statement, where the statement fails only in the
case that the termination argument is not valid. Once we are given an argument of the form T[1] or T[2] or … or T[n], to check validity we simply want to prove the following statement:
Each time an execution passes through one state and then through another one, T[1] or T[2] or … or T[n] holds between these two states. That is, there does not exist a pair of states, one being
reachable from the other, possibly via the unrolling of a loop, such that neither T[1] nor T[2] nor … nor T[n] holds between this pair of states.
This statement can be verified a program transformation where we introduce new variables into the program to record the state before the unrolling of the loop and then use an assertion statement to
check the termination argument always holds between the current state and the recorded state. If the assertion checker can prove the assert cannot fail, it has proved the validity of the termination
argument. We can use encoding tricks to force the assertion checker to consider all possible unrollings.
Figure 5 offers such an example, where we have used the termination argument “x goes down by at least one and x is greater than 0″ using the encoding given in Cook et al.^14 The new code (introduced
as a part of the encoding) is given in red, whereas the original program from Figure 1 is in black. We make use of an extra call to input() to decide when the unrolling begins. The new variables oldx
and oldy are used for recording a state. Note that the assertion checker must consider all values possibly returned by input() during its proof, thus the proof of termination is valid for any
starting position. This has the effect of considering any possible unrolling of the loop. After some state has been recorded, from this point out the termination argument is checked using the
recorded state and the current state. In this case the assertion can fail, meaning that the termination argument is not valid.
If we were to attempt to check this condition in a naïve way (for example, by simply executing the program) we would never find a proof for all but the most trivial of cases. Thus, assertion checkers
must be cleverly designed to find proofs about all possible executions without actually executing all of the paths. A plethora of recently developed techniques now make this possible. Many recent
assertion checkers are designed to produce a path to a bug in the case that the assertion statement cannot be proved. For example, a path leading to the assertion failure is 1 → 2 → 3 → 4 → 5 → 7 → 8
→ 9 → 10 → 11 → 12 → 16 → 17 → 4 → 5 → 6. This path can be broken into parts, each representing different phases of the execution: the prefix-path 1 → 2 → 3 → 4 is the path from the program initial
state to the recorded state in the failing pair of states. The second part of the path 4 → 5 →…5 → 6 represents how we reached the current state from the recorded one. That is: this is the unrolling
found that demonstrates that the assertion statement can fail. What we know is that the termination argument does not currently cover the case where this path is repeated forever.
See Figure 6 for a version using the same encoding, but with the valid termination argument:
x goes down by at least 1 and is larger than 0
y goes down by at least 1 and is larger than 0.
This assertion cannot fail. The fact that it cannot fail can be proved by a number of assertion verification tools.
Finding Termination Arguments
We have examined how we can check a termination argument’s validity via a translation to a program with an assertion statement. We now discuss known methods for finding monolithic termination
Rank function synthesis. In some cases simple ranking functions can be automatically found. We call a ranking function simple if it can be defined by a linear arithmetic expression (for example, 3x
= 2y + 100). The most popular approach for finding this class of ranking function uses a result from Farkas^16 together with tools for solving linear constraint systems. (See Colón and Sipma^11 or
Polelski and Rybalchecko^30 for examples of tools using Farkas’ lemma.) Many other approaches for finding ranking functions for different classes of programs have been proposed (see refs^1,6 8,19,37
). Tools for the synthesis of ranking functions are sometimes applied directly to programs, but more frequently they are used (on small and simplified program fragments) internally within termination
proving tools for suggesting the single ranking functions that appear in a disjunctive termination argument.
Termination analysis. Numerous approaches have been developed for finding disjunctive termination arguments in which—in effect—the validity condition for disjunctive termination arguments is almost
guaranteed to hold by construction. In some cases—for example, Berdine et al.^3—to prove termination we need only check that the argument indeed represents a set of measures. In other cases, such as
Lee et al.^24 or Manolios and Vroon,^26 the tool makes a one-time guess as to the termination argument and then checks it using techniques drawn from abstract interpretation.
Consider the modified program in Figure 7. The termination strategy described in Berdine et al.^3 and Podelski and Rybalchenko^32 essentially builds a program like this and then applies a custom
program analysis to find the following candidate termination argument:
for the program at line 4—meaning we could pass this complex expression to the assertion at line 4 in Figure 7 and know that the assertion cannot fail. We know this statement is true of any unrolling
of the loop in the original Figure 1. What remains is to prove that each piece of the candidate argument represents a measure that decreases—here we can use rank function synthesis tools to prove
that oldx > x + 1 and oldx > 0 . . . represents the measure based on x. If each piece between the ors in fact represents a measure (with the exception of copied ≠ 1 which comes from the encoding)
then we have proved termination.
One difficulty with this style of termination proving is that, in the case that the program doesn’t terminate, the tools can only report “unknown,” as the techniques used inside the abstract
interpretation tools have lost so much detail that it is impossible to find a non-terminating execution from the failed proof and then prove it non-terminating. The advantage when compared to other
known techniques is it is much faster.
Finding arguments by refinement. Another method for discovering a termination argument is to follow the approach of Cook et al.^14 or Chawdhary et al.^9 and search for counterexamples to (possibly
invalid) termination arguments and then refine them based on new ranking functions found via the counterexamples.
In recent years, powerful new termination tools have emerged that return “unknown” infrequently enough that they are useful in practice.
Recall Figure 5, which encoded the invalid termination argument for the program in Figure 1, and the path leading to the failure of the assertion: is 1 → 2 → 3 → 4 → 5 → 7 → 8 → 9 → 10 → 11 → 12 → 16
→ 17 → 4 → 5 → 6. Recall this path represents two phases of the program’s execution: the path to the loop, and some unrolling of the loop such that the termination condition doesn’t hold. In this
case the path 4 → 5 → . . . 6 represents how we reached the second failing state from the first. This is a counterexample to the validity of the termination argument, meaning that the current
termination argument does not take this path and others like it into account.
If the path can be repeated forever during the program’s execution then we have found a real counterexample. Known approaches (for example, Gupta et al.^21) can be used to try and prove this path can
be repeated forever. In this case, however, we know that the path cannot be repeated forever, as y is decremented on each iteration through the path and also constrained via a conditional statement
to be positive. Thus this path is a spurious counterexample to termination and can be ruled out via a refinement to the termination argument. Again, using rank function synthesis tools we can
automatically find a ranking function that demonstrates the spuriousness of this path. In this case a rank function synthesis tool will find y, meaning that the reason this path cannot be repeated
forever is that “y always goes down by at least one and is larger than 0.” We can then refine the current termination argument used in Figure 5:
x goes down by at least 1 and is larger than 0 with the larger termination argument:
x goes down by at least 1 and is larger than 0
y goes down by at least 1 and is larger than 0
We can then check the validity of this termination argument using a tool such as IMPACT on the program in Figure 6. IMPACT can prove this assertion never fails, thus proving the termination of the
program in Figure 1.
Further Directions
With fresh advances in methods for proving the termination of sequential programs that operate over mathematical numbers, we are now in the position to begin proving termination of more complex
programs, such as those with dynamically allocated data structures, or multithreading. Furthermore, these new advances open up new potential for proving properties beyond termination, and finding
conditions that would guarantee termination. We now discuss these avenues of future research and development in some detail.
Dynamically allocated heap. Consider the C loop in Figure 8, which walks down a list and removes links with data elements equaling 5. Does this loop guarantee termination? What termination argument
should we use?
The problem here is that there are no arithmetic variables in the program from which we can begin to construct an argument—instead we would want to express the termination argument over the lengths
of paths to NULL via the next field. Furthermore, the programmer has obviously intended for this loop to be used on acyclic singly linked lists, but how do we know that the lists pointed to by head
will always be acyclic? The common solution to these problems is to use shape analysis tools (which are designed to automatically discover the shapes of data-structures) and then to create new
auxiliary variables in the program that track the sizes of those data structures, thus allowing for arithmetic ranking functions to be more easily expressed (examples include refs^4,5,25). The
difficultly with this approach is that we are now dependent on the accuracy and scalability of current shape analysis tools—to date the best known shape analysis tool^40 supports only lists and trees
(cyclic and acyclic, singly and doubly linked) and scales only to relatively simple programs of size less than 30,000 LOC. Furthermore, the auxiliary variables introduced by methods such as Magill et
al.^25 sometimes do not track enough information in order to prove termination (for example, imagine a case with lists of lists in which the sizes of the nested lists are important). In order to
improve the state of the art for termination proving of programs using data structures, we must develop better methods of finding arguments over data structure shapes, and we must also improve the
accuracy and scalability of existing shape analysis tools.
Bit vectors. In the examples used until now we have considered only variables that range over mathematical numbers. The reality is that most programs use variables that range over fixed-width
numbers, such as 32-bit integers or 64-bit floatingpoint numbers, with the possibility of overflow or underflow. If a program uses only fixed-width numbers and does not use dynamically allocated
memory, then termination proving is decidable (though still not easy). In this case we simply need to look for a repeated state, as the program will diverge if and only if there exists some state
that is repeated during execution. Furthermore, we cannot ignore the fixed-width semantics, as overflow and underflow can cause non-termination in programs that would otherwise terminate, an example
is included in Figure 9. Another complication when considering this style of program is that of bit-level operations, such as left- or right-shift.
Binary executables. Until now we have discussed proving termination of programs at their source level, perhaps in C or Java. The difficulty with this strategy is the compilers that then take these
source programs and convert them into executable artifacts can introduce termination bugs that do not exist in the original source program. Several potential strategies could help mitigate this
problem: We might try to prove termination of the executable binaries instead of the source level programs, or we might try to equip the compiler with the ability to prove that the resulting binary
program preserves termination, perhaps by first proving the termination of the source program and then finding a map from the binary to the source-level program and proving that the composition with
the source-level termination argument forms a valid termination argument for the binary-level program.
Non-linear systems. Current termination provers largely ignore nonlinear arithmetic. When non-linear updates to variables do occur (for example x := y * z;), current termination provers typically
treat them as if they were the instruction x := input();. This modification is sound—meaning when the termination prover returns the answer “terminating,” we know the proof is valid. Unfortunately,
this method is not precise: the treatment of these commands can lead to the result “unknown” for programs that actually terminate. Termination provers are also typically unable to find or check
non-linear termination arguments (x^2, for example) when they are required. Some preliminary efforts in this direction have been made,^1,6 but these techniques are weak. To improve the current power
of termination provers, further developments in non-linear reasoning are required.
Concurrency. Concurrency adds an extra layer of difficulty when attempting to prove program termination. The problem here is that we must consider all possible interactions between concurrently
executing threads. This is especially true for modern fine-grained concurrent algorithms, in which threads interact in subtle ways through dynamically allocated data structures. Rather than
attempting to explicitly consider all possible interleavings of the threads (which does not scale to large programs) the usual method for proving concurrent programs correct is based on
rely-guarantee or assume-guarantee style of reasoning, which considers every thread in isolation under assumptions on its environment and thus avoids reasoning about thread interactions directly.
Much of the power of a rely-guarantee proof system (such as Jones^22 and Misra and Chandy^28) comes from the cyclic proof rules, where we can assume a property of the second thread while proving
property of the first thread, and then assume the recently proved property of the first thread when proving the assumed property of the second thread. This strategy can be extended to liveness
properties using induction over time, for example, Gotsman et al.^20 and McMillan.^27
As an example, consider the two code fragments in Figure 10. Imagine that we are executing these two fragments concurrently. To prove the termination of the left thread we must prove that it does not
get stuck waiting for the call to lock. To prove this we can assume the other thread will always eventually release the lock—but to prove this of the code on the right we must assume the analogous
property of the thread on the left, and so on. In this case we can certainly just consider all possible interleavings of the threads, thus turning the concurrent program into a sequential model
representing its executions, but this approach does not scale well to larger programs. The challenge is to develop automatic methods of finding non-circular rely-guarantee termination arguments.
Recent steps^20 have developed heuristics that work for non-blocking algorithms, but more general techniques are still required.
With fresh advances in methods for proving the termination of sequential programs that operate over mathematical numbers, we are now in the position to begin proving termination of more complex
Advanced programming features. The industrial adoption of high-level programming features such as virtual functions, inheritance, higher-order functions, or closures make the task of proving
industrial programs more of a challenge. With few exceptions (such as Giesl et al.^18), this area has not been well studied.
Untyped or dynamically typed programs also contribute difficulty when proving termination, as current approaches are based on statically discovering data-structure invariants and finding arithmetic
measures in order to prove termination. Data in untyped programs is often encoded in strings, using pattern matching to marshal data in and out of strings. Termination proving tools for JavaScript
would be especially welcome, given the havoc that nonterminating JavaScript causes daily for Web browsers.
Finding preconditions that guarantee termination. In the case that a program does not guarantee termination from all initial configurations, we may want to automatically discover the conditions under
which the program does guarantee termination. That is, when calling some function provided by a library: what are the conditions under which the code is guaranteed to return with a result? The
challenge in this area is to find the right precondition: the empty precondition is correct but useless, whereas the weakest precondition for even very simple programs can often be expressed only in
complex domains not supported by today’s tools. Furthermore, they should be computed quickly (the weakest precondition expressible in the target logic may be too expensive to compute). Recent work
has shown some preliminary progress in this direction.^13,33
Liveness. We have alluded to the connection between liveness properties and the program termination problem. Formally, liveness properties expressed in temporal logics can be converted into questions
of fair termination—termination proving were certain non-terminating executions are deemed unfair via given fairness constraints, and thus ignored. Current tools, in fact, either perform this
reduction, or simply require the user to express liveness constraints directly as the set of fairness constraints.^12,29 Neither approach is optimal: the reduction from liveness to fairness is
inefficient in the size of the conversion, and fairness constraints are difficult for humans to understand when used directly. An avenue for future work would be to directly prove liveness
properties, perhaps as an adaption of existing termination proving techniques.
Dynamic analysis and crash dumps for liveness bugs. In this article we have focused only on static, or compile-time, proof techniques rather than techniques for diagnosing divergence during
execution. Some effort has been placed into the area of automatically detecting deadlock during execution time. With new developments in the area of program termination proving we might find that
automatic methods of discovering livelock could also now be possible. Temporary modifications to scheduling, or other techniques, might also be employed to help programs not diverge even in cases
where they do not guarantee termination or other liveness properties. Some preliminary work has begun to emerge in this area (see Jula et al.^23) but more work is needed.
Scalability, performance, and precision. Scalability to large and complex programs is currently a problem for modern termination provers—current techniques are known, at best, to scale to simple
systems code of 30,000 lines of code. Another problem we face is one of precision. Some small programs currently cannot be proved terminating with existing tools. Turing’s undecidability result, of
course, states that this will always be true, but this does preclude us from improving precision for various classes of programs and concrete examples. The most famous example is that of the Collatz’
problem, which amounts to proving the termination or non-termination of the program in Figure 11. Currently no proof of this program’s termination behavior is known.
This article has surveyed recent advances in program termination proving techniques for sequential programs, and pointed toward ongoing work and potential areas for future development. The hope of
many tool builders in this area is that the current and future termination proving techniques will become generally available for developers wishing to directly prove termination or liveness. We also
hope that termination-related applications—such as detecting livelock at runtime or Wang’s tiling problem—will also benefit from these advances.
The authors would like to thank Lucas Bourdeaux, Abigail See, Tim Harris, Ralf Herbrich, Peter O’Hearn, and Hongseok Yang for their reading of early drafts of this article and suggestions for
Figure 3. Another example program.
Figure 4. Example program with an assertion statement in line 3.
Figure 5. Encoding of termination argument validity.
Figure 6. Encoding of termination argument validity using the program.
Figure 7. Program prepared for abstract interpretation.
Figure 8. Example C loop over a linked-list data-structure with fields next and data.
Figure 9. Example program illustrating nontermination.
Figure 10. Example of multi-threaded terminating producer/consumer program.
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Learning Representation and Control in Markov Decision Processes: New Frontiers
Table of contents:
1: Introduction
2: Sequential Decision Problems
3: Laplacian Operators and MDPs
4: Approximating Markov Decision Processes
5: Dimensionality Reduction Principles in MDPs
6: Basis Construction: Diagonalization Methods
7: Basis Construction: Dilation Methods
8: Model-Based Representation Policy Iteration
9: Basis Construction in Continuous MDPs
10: Model-Free Representation Policy Iteration
11: Related Work and Future Challenges | {"url":"https://nowpublishers.com/article/Details/MAL-003","timestamp":"2024-11-07T18:57:56Z","content_type":"text/html","content_length":"22438","record_id":"<urn:uuid:4d2273ec-078d-4dbf-b15b-624c13a0bc88>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00309.warc.gz"} |
2014-2015 Baylor Undergraduate Lecture Series in Mathematics
Arthur Benjamin
Arthur Benjamin
Professor Arthur Benjamin , Professor of Mathematics at Harvey Mudd College (Claremont, CA), was the seventh speaker in our annual Baylor Undergraduate Lecture Series in Mathematics. It was a short -
but sweet - visit for the much -traveled Dr. Benjamin; indeed, he gave both of his lectures on October 21.
Dr. Benjamin earned his B.S. degree in Applied Mathematics from Carnegie Mellon University in 1983. He then went on to receive his Ph.D. from Johns Hopkins University in 1989. He then immediately
joined the faculty at Harvey Mudd College . He served as Chair of the Department of Mathematics at Harvey Mudd from 2002-2004.
He is the author of more than 90 research papers in mathematics, several books and DVD courses, and has received national awards for his writing and teaching. He has given three TED talks which have
been viewed over 10 million times. He is a past winner of the American Backgammon Tour, and in 2012, he was selected by Princeton Review as one of The Best 300 Professors. Dr. Benjamin has appeared
on many television and radio programs, including The Today Show, CNN, National Public Radio, and The Colbert Report. He has been profiled in The New York Times, Los Angeles Times, USA Today,
Scientific American, Discover, Omni, Esquire, People Magazine, and Reader's Digest.
The titles, and abstracts, for his two lectures are:
Tuesday, October 21, 2014 at 4:00 pm - BSB 109
Combinatorial Trigonometry (and a method to DIE for)
Abstract: Many trigonometric identities, including the Pythagorean theorem, have combinatorial proofs. Furthermore, some combinatorial problems have trigonometric solutions. All of these problems can
be reduced to alternating sums, and are attacked by a technique we call D.I.E. (Description, Involution, Exception). This technique offers new insights to identities involving binomial coefficients,
Fibonacci numbers, derangements,and Chebyshev polynomials.
Tuesday, October 21, 2014 at 7:00 pm - BSB 110
Public Lecture:
Mathemagics! (Alternate titles: Secrets of Mental Math or The Art of Mental Calculation)
Abstract: Dr. Arthur Benjamin is a mathematician and a magician. In his entertaining and fast-paced performance, he will demonstrate and explain how to mentally add and multiply numbers faster than a
calculator, how to figure out the day of the week of any date in history, and other amazing feats of mind. He has presented his mixture of math and magic to audiences all over the world. | {"url":"https://math.artsandsciences.baylor.edu/talks/baylor-undergraduate-lecture-series-mathematics/2014-2015-baylor-undergraduate-lecture-series","timestamp":"2024-11-05T13:27:13Z","content_type":"text/html","content_length":"119719","record_id":"<urn:uuid:4734f736-d8cf-4ff9-a37e-81f4e7e185c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00815.warc.gz"} |
Paper Group AWR 442
Explainability Techniques for Graph Convolutional Networks. Learning from the Past: Continual Meta-Learning via Bayesian Graph Modeling. LUTNet: Learning FPGA Configurations for Highly Efficient
Neural Network Inference. Fast Compressive Sensing Recovery Using Generative Models with Structured Latent Variables. Semi-Supervised Monocular Depth Estim …
Explainability Techniques for Graph Convolutional Networks
Title Explainability Techniques for Graph Convolutional Networks
Authors Federico Baldassarre, Hossein Azizpour
Graph Networks are used to make decisions in potentially complex scenarios but it is usually not obvious how or why they made them. In this work, we study the explainability of Graph
Abstract Network decisions using two main classes of techniques, gradient-based and decomposition-based, on a toy dataset and a chemistry task. Our study sets the ground for future development as
well as application to real-world problems.
Published 2019-05-31
URL https://arxiv.org/abs/1905.13686v1
PDF https://arxiv.org/pdf/1905.13686v1.pdf
PWC https://paperswithcode.com/paper/explainability-techniques-for-graph
Repo https://github.com/baldassarreFe/graph-network-explainability
Framework pytorch
Learning from the Past: Continual Meta-Learning via Bayesian Graph Modeling
Title Learning from the Past: Continual Meta-Learning via Bayesian Graph Modeling
Authors Yadan Luo, Zi Huang, Zheng Zhang, Ziwei Wang, Mahsa Baktashmotlagh, Yang Yang
Meta-learning for few-shot learning allows a machine to leverage previously acquired knowledge as a prior, thus improving the performance on novel tasks with only small amounts of data.
However, most mainstream models suffer from catastrophic forgetting and insufficient robustness issues, thereby failing to fully retain or exploit long-term knowledge while being prone to
cause severe error accumulation. In this paper, we propose a novel Continual Meta-Learning approach with Bayesian Graph Neural Networks (CML-BGNN) that mathematically formulates
Abstract meta-learning as continual learning of a sequence of tasks. With each task forming as a graph, the intra- and inter-task correlations can be well preserved via message-passing and history
transition. To remedy topological uncertainty from graph initialization, we utilize Bayes by Backprop strategy that approximates the posterior distribution of task-specific parameters with
amortized inference networks, which are seamlessly integrated into the end-to-end edge learning. Extensive experiments conducted on the miniImageNet and tieredImageNet datasets demonstrate
the effectiveness and efficiency of the proposed method, improving the performance by 42.8% compared with state-of-the-art on the miniImageNet 5-way 1-shot classification task.
Tasks Continual Learning, Few-Shot Learning, Meta-Learning
Published 2019-11-12
URL https://arxiv.org/abs/1911.04695v1
PDF https://arxiv.org/pdf/1911.04695v1.pdf
PWC https://paperswithcode.com/paper/learning-from-the-past-continual-meta
Repo https://github.com/Luoyadan/BGNN-AAAI
Framework pytorch
LUTNet: Learning FPGA Configurations for Highly Efficient Neural Network Inference
Title LUTNet: Learning FPGA Configurations for Highly Efficient Neural Network Inference
Authors Erwei Wang, James J. Davis, Peter Y. K. Cheung, George A. Constantinides
Research has shown that deep neural networks contain significant redundancy, and thus that high classification accuracy can be achieved even when weights and activations are quantized down
to binary values. Network binarization on FPGAs greatly increases area efficiency by replacing resource-hungry multipliers with lightweight XNOR gates. However, an FPGA’s fundamental
building block, the K-LUT, is capable of implementing far more than an XNOR: it can perform any K-input Boolean operation. Inspired by this observation, we propose LUTNet, an end-to-end
Abstract hardware-software framework for the construction of area-efficient FPGA-based neural network accelerators using the native LUTs as inference operators. We describe the realization of both
unrolled and tiled LUTNet architectures, with the latter facilitating smaller, less power-hungry deployment over the former while sacrificing area and energy efficiency along with
throughput. For both varieties, we demonstrate that the exploitation of LUT flexibility allows for far heavier pruning than possible in prior works, resulting in significant area savings
while achieving comparable accuracy. Against the state-of-the-art binarized neural network implementation, we achieve up to twice the area efficiency for several standard network models
when inferencing popular datasets. We also demonstrate that even greater energy efficiency improvements are obtainable.
Published 2019-10-24
URL https://arxiv.org/abs/1910.12625v2
PDF https://arxiv.org/pdf/1910.12625v2.pdf
PWC https://paperswithcode.com/paper/lutnet-learning-fpga-configurations-for
Repo https://github.com/awai54st/LUTNet
Framework tf
Fast Compressive Sensing Recovery Using Generative Models with Structured Latent Variables
Title Fast Compressive Sensing Recovery Using Generative Models with Structured Latent Variables
Authors Shaojie Xu, Sihan Zeng, Justin Romberg
Deep learning models have significantly improved the visual quality and accuracy on compressive sensing recovery. In this paper, we propose an algorithm for signal reconstruction from
compressed measurements with image priors captured by a generative model. We search and constrain on latent variable space to make the method stable when the number of compressed
Abstract measurements is extremely limited. We show that, by exploiting certain structures of the latent variables, the proposed method produces improved reconstruction accuracy and preserves
realistic and non-smooth features in the image. Our algorithm achieves high computation speed by projecting between the original signal space and the latent variable space in an alternating
Tasks Compressive Sensing
Published 2019-02-19
URL https://arxiv.org/abs/1902.06913v4
PDF https://arxiv.org/pdf/1902.06913v4.pdf
PWC https://paperswithcode.com/paper/fast-compressive-sensing-recovery-using
Repo https://github.com/sihan-zeng/f-csrg
Framework tf
Semi-Supervised Monocular Depth Estimation with Left-Right Consistency Using Deep Neural Network
Title Semi-Supervised Monocular Depth Estimation with Left-Right Consistency Using Deep Neural Network
Authors Ali Jahani Amiri, Shing Yan Loo, Hong Zhang
There has been tremendous research progress in estimating the depth of a scene from a monocular camera image. Existing methods for single-image depth prediction are exclusively based on
deep neural networks, and their training can be unsupervised using stereo image pairs, supervised using LiDAR point clouds, or semi-supervised using both stereo and LiDAR. In general,
semi-supervised training is preferred as it does not suffer from the weaknesses of either supervised training, resulting from the difference in the cameras and the LiDARs field of view, or
Abstract unsupervised training, resulting from the poor depth accuracy that can be recovered from a stereo pair. In this paper, we present our research in single image depth prediction using
semi-supervised training that outperforms the state-of-the-art. We achieve this through a loss function that explicitly exploits left-right consistency in a stereo reconstruction, which has
not been adopted in previous semi-supervised training. In addition, we describe the correct use of ground truth depth derived from LiDAR that can significantly reduce prediction error. The
performance of our depth prediction model is evaluated on popular datasets, and the importance of each aspect of our semi-supervised training approach is demonstrated through experimental
results. Our deep neural network model has been made publicly available.
Tasks Depth Estimation, Monocular Depth Estimation
Published 2019-05-18
URL https://arxiv.org/abs/1905.07542v1
PDF https://arxiv.org/pdf/1905.07542v1.pdf
PWC https://paperswithcode.com/paper/semi-supervised-monocular-depth-estimation
Repo https://github.com/a-jahani/semiDepth
Framework tf
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
Title DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
Authors Lu Lu, Pengzhan Jin, George Em Karniadakis
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single
hidden layer can approximate accurately any nonlinear continuous operator \cite{chen1995universal}. This universal approximation theorem is suggestive of the potential application of neural
networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important
optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively
Abstract small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors $x_i, i=1,\dots,m$ (branch net), and another for encoding the
locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and
demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in
terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe
high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset
Published 2019-10-08
URL https://arxiv.org/abs/1910.03193v1
PDF https://arxiv.org/pdf/1910.03193v1.pdf
PWC https://paperswithcode.com/paper/deeponet-learning-nonlinear-operators-for
Repo https://github.com/lululxvi/deepxde
Framework tf
Tracing Forum Posts to MOOC Content using Topic Analysis
Title Tracing Forum Posts to MOOC Content using Topic Analysis
Authors Alexander William Wong, Ken Wong, Abram Hindle
Massive Open Online Courses are educational programs that are open and accessible to a large number of people through the internet. To facilitate learning, MOOC discussion forums exist
where students and instructors communicate questions, answers, and thoughts related to the course. The primary objective of this paper is to investigate tracing discussion forum posts back
Abstract to course lecture videos and readings using topic analysis. We utilize both unsupervised and supervised variants of Latent Dirichlet Allocation (LDA) to extract topics from course material
and classify forum posts. We validate our approach on posts bootstrapped from five Coursera courses and determine that topic models can be used to map student discussion posts back to the
underlying course lecture or reading. Labeled LDA outperforms unsupervised Hierarchical Dirichlet Process LDA and base LDA for our traceability task. This research is useful as it provides
an automated approach for clustering student discussions by course material, enabling instructors to quickly evaluate student misunderstanding of content and clarify materials accordingly.
Tasks Topic Models
Published 2019-04-15
URL http://arxiv.org/abs/1904.07307v1
PDF http://arxiv.org/pdf/1904.07307v1.pdf
PWC https://paperswithcode.com/paper/tracing-forum-posts-to-mooc-content-using
Repo https://github.com/awwong1/topic-traceability
Framework none
Bayesian Allocation Model: Inference by Sequential Monte Carlo for Nonnegative Tensor Factorizations and Topic Models using Polya Urns
Title Bayesian Allocation Model: Inference by Sequential Monte Carlo for Nonnegative Tensor Factorizations and Topic Models using Polya Urns
Authors Ali Taylan Cemgil, Mehmet Burak Kurutmaz, Sinan Yildirim, Melih Barsbey, Umut Simsekli
We introduce a dynamic generative model, Bayesian allocation model (BAM), which establishes explicit connections between nonnegative tensor factorization (NTF), graphical models of discrete
probability distributions and their Bayesian extensions, and the topic models such as the latent Dirichlet allocation. BAM is based on a Poisson process, whose events are marked by using a
Bayesian network, where the conditional probability tables of this network are then integrated out analytically. We show that the resulting marginal process turns out to be a Polya urn, an
integer valued self-reinforcing process. This urn processes, which we name a Polya-Bayes process, obey certain conditional independence properties that provide further insight about the
Abstract nature of NTF. These insights also let us develop space efficient simulation algorithms that respect the potential sparsity of data: we propose a class of sequential importance sampling
algorithms for computing NTF and approximating their marginal likelihood, which would be useful for model selection. The resulting methods can also be viewed as a model scoring method for
topic models and discrete Bayesian networks with hidden variables. The new algorithms have favourable properties in the sparse data regime when contrasted with variational algorithms that
become more accurate when the total sum of the elements of the observed tensor goes to infinity. We illustrate the performance on several examples and numerically study the behaviour of the
algorithms for various data regimes.
Tasks Model Selection, Topic Models
Published 2019-03-11
URL http://arxiv.org/abs/1903.04478v1
PDF http://arxiv.org/pdf/1903.04478v1.pdf
PWC https://paperswithcode.com/paper/bayesian-allocation-model-inference-by
Repo https://github.com/atcemgil/bam
Framework none
A Unifying Bayesian View of Continual Learning
Title A Unifying Bayesian View of Continual Learning
Authors Sebastian Farquhar, Yarin Gal
Some machine learning applications require continual learning - where data comes in a sequence of datasets, each is used for training and then permanently discarded. From a Bayesian
perspective, continual learning seems straightforward: Given the model posterior one would simply use this as the prior for the next task. However, exact posterior evaluation is intractable
with many models, especially with Bayesian neural networks (BNNs). Instead, posterior approximations are often sought. Unfortunately, when posterior approximations are used, prior-focused
Abstract approaches do not succeed in evaluations designed to capture properties of realistic continual learning use cases. As an alternative to prior-focused methods, we introduce a new approximate
Bayesian derivation of the continual learning loss. Our loss does not rely on the posterior from earlier tasks, and instead adapts the model itself by changing the likelihood term. We call
these approaches likelihood-focused. We then combine prior- and likelihood-focused methods into one objective, tying the two views together under a single unifying framework of approximate
Bayesian continual learning.
Tasks Continual Learning
Published 2019-02-18
URL http://arxiv.org/abs/1902.06494v1
PDF http://arxiv.org/pdf/1902.06494v1.pdf
PWC https://paperswithcode.com/paper/a-unifying-bayesian-view-of-continual
Repo https://github.com/Saraharas/A-Unifying-Bayesian-View-of-Continual-Learning
Framework pytorch
Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport
Title Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport
Authors Tengfei Ma, Jie Chen
Hierarchical abstractions are a methodology for solving large-scale graph problems in various disciplines. Coarsening is one such approach: it generates a pyramid of graphs whereby the one
in the next level is a structural summary of the prior one. With a long history in scientific computing, many coarsening strategies were developed based on mathematically driven heuristics.
Recently, resurgent interests exist in deep learning to design hierarchical methods learnable through differentiable parameterization. These approaches are paired with downstream tasks for
Abstract supervised learning. In practice, however, supervised signals (e.g., labels) are scarce and are often laborious to obtain. In this work, we propose an unsupervised approach, coined
OTCoarsening, with the use of optimal transport. Both the coarsening matrix and the transport cost matrix are parameterized, so that an optimal coarsening strategy can be learned and
tailored for a given set of graphs. We demonstrate that the proposed approach produces meaningful coarse graphs and yields competitive performance compared with supervised methods for graph
classification and regression.
Tasks Graph Classification
Published 2019-12-24
URL https://arxiv.org/abs/1912.11176v1
PDF https://arxiv.org/pdf/1912.11176v1.pdf
PWC https://paperswithcode.com/paper/unsupervised-learning-of-graph-hierarchical-1
Repo https://github.com/matenure/OTCoarsening
Framework pytorch
Relational Graph Learning for Crowd Navigation
Title Relational Graph Learning for Crowd Navigation
Authors Changan Chen, Sha Hu, Payam Nikdel, Greg Mori, Manolis Savva
We present a relational graph learning approach for robotic crowd navigation using model-based deep reinforcement learning that plans actions by looking into the future. Our approach
reasons about the relations between all agents based on their latent features and uses a Graph Convolutional Network to encode higher-order interactions in each agent’s state
Abstract representation, which is subsequently leveraged for state prediction and value estimation. The ability to predict human motion allows us to perform multi-step lookahead planning, taking
into account the temporal evolution of human crowds. We evaluate our approach against a state-of-the-art baseline for crowd navigation and ablations of our model to demonstrate that
navigation with our approach is more efficient, results in fewer collisions, and avoids failure cases involving oscillatory and freezing behaviors.
Published 2019-09-28
URL https://arxiv.org/abs/1909.13165v2
PDF https://arxiv.org/pdf/1909.13165v2.pdf
PWC https://paperswithcode.com/paper/relational-graph-learning-for-crowd
Repo https://github.com/vita-epfl/CrowdNav
Framework none
Limitations of Lazy Training of Two-layers Neural Networks
Title Limitations of Lazy Training of Two-layers Neural Networks
Authors Behrooz Ghorbani, Song Mei, Theodor Misiakiewicz, Andrea Montanari
We study the supervised learning problem under either of the following two models: (1) Feature vectors ${\boldsymbol x}i$ are $d$-dimensional Gaussians and responses are $y_i = f*({\
boldsymbol x}i)$ for $f*$ an unknown quadratic function; (2) Feature vectors ${\boldsymbol x}_i$ are distributed as a mixture of two $d$-dimensional centered Gaussians, and $y_i$'s are the
corresponding class labels. We use two-layers neural networks with quadratic activations, and compare three different learning regimes: the random features (RF) regime in which we only
Abstract train the second-layer weights; the neural tangent (NT) regime in which we train a linearization of the neural network around its initialization; the fully trained neural network (NN)
regime in which we train all the weights in the network. We prove that, even for the simple quadratic model of point (1), there is a potentially unbounded gap between the prediction risk
achieved in these three training regimes, when the number of neurons is smaller than the ambient dimension. When the number of neurons is larger than the number of dimensions, the problem
is significantly easier and both NT and NN learning achieve zero risk.
Published 2019-06-21
URL https://arxiv.org/abs/1906.08899v1
PDF https://arxiv.org/pdf/1906.08899v1.pdf
PWC https://paperswithcode.com/paper/limitations-of-lazy-training-of-two-layers
Repo https://github.com/bGhorbani/Lazy-Training-Neural-Nets
Framework tf
Declarative Question Answering over Knowledge Bases containing Natural Language Text with Answer Set Programming
Title Declarative Question Answering over Knowledge Bases containing Natural Language Text with Answer Set Programming
Authors Arindam Mitra, Peter Clark, Oyvind Tafjord, Chitta Baral
While in recent years machine learning (ML) based approaches have been the popular approach in developing end-to-end question answering systems, such systems often struggle when additional
knowledge is needed to correctly answer the questions. Proposed alternatives involve translating the question and the natural language text to a logical representation and then use logical
Abstract reasoning. However, this alternative falters when the size of the text gets bigger. To address this we propose an approach that does logical reasoning over premises written in natural
language text. The proposed method uses recent features of Answer Set Programming (ASP) to call external NLP modules (which may be based on ML) which perform simple textual entailment. To
test our approach we develop a corpus based on the life cycle questions and showed that Our system achieves up to $18%$ performance gain when compared to standard MCQ solvers.
Tasks Natural Language Inference, Question Answering
Published 2019-05-01
URL http://arxiv.org/abs/1905.00198v1
PDF http://arxiv.org/pdf/1905.00198v1.pdf
PWC https://paperswithcode.com/paper/declarative-question-answering-over-knowledge
Repo https://github.com/OpenSourceAI/sota_server
Framework none
Self-Supervised Monocular Depth Hints
Title Self-Supervised Monocular Depth Hints
Authors Jamie Watson, Michael Firman, Gabriel J. Brostow, Daniyar Turmukhambetov
Monocular depth estimators can be trained with various forms of self-supervision from binocular-stereo data to circumvent the need for high-quality laser scans or other ground-truth data.
The disadvantage, however, is that the photometric reprojection losses used with self-supervised learning typically have multiple local minima. These plausible-looking alternatives to
ground truth can restrict what a regression network learns, causing it to predict depth maps of limited quality. As one prominent example, depth discontinuities around thin structures are
Abstract often incorrectly estimated by current state-of-the-art methods. Here, we study the problem of ambiguous reprojections in depth prediction from stereo-based self-supervision, and introduce
Depth Hints to alleviate their effects. Depth Hints are complementary depth suggestions obtained from simple off-the-shelf stereo algorithms. These hints enhance an existing photometric
loss function, and are used to guide a network to learn better weights. They require no additional data, and are assumed to be right only sometimes. We show that using our Depth Hints gives
a substantial boost when training several leading self-supervised-from-stereo models, not just our own. Further, combined with other good practices, we produce state-of-the-art depth
predictions on the KITTI benchmark.
Tasks Depth Estimation, Monocular Depth Estimation
Published 2019-09-19
URL https://arxiv.org/abs/1909.09051v1
PDF https://arxiv.org/pdf/1909.09051v1.pdf
PWC https://paperswithcode.com/paper/self-supervised-monocular-depth-hints
Repo https://github.com/nianticlabs/depth-hints
Framework none
Learn Stereo, Infer Mono: Siamese Networks for Self-Supervised, Monocular, Depth Estimation
Title Learn Stereo, Infer Mono: Siamese Networks for Self-Supervised, Monocular, Depth Estimation
Authors Matan Goldman, Tal Hassner, Shai Avidan
The field of self-supervised monocular depth estimation has seen huge advancements in recent years. Most methods assume stereo data is available during training but usually under-utilize it
and only treat it as a reference signal. We propose a novel self-supervised approach which uses both left and right images equally during training, but can still be used with a single input
Abstract image at test time, for monocular depth estimation. Our Siamese network architecture consists of two, twin networks, each learns to predict a disparity map from a single image. At test
time, however, only one of these networks is used in order to infer depth. We show state-of-the-art results on the standard KITTI Eigen split benchmark as well as being the highest scoring
self-supervised method on the new KITTI single view benchmark. To demonstrate the ability of our method to generalize to new data sets, we further provide results on the Make3D benchmark,
which was not used during training.
Tasks Depth Estimation, Monocular Depth Estimation
Published 2019-05-01
URL http://arxiv.org/abs/1905.00401v1
PDF http://arxiv.org/pdf/1905.00401v1.pdf
PWC https://paperswithcode.com/paper/learn-stereo-infer-mono-siamese-networks-for
Repo https://github.com/mtngld/lsim
Framework tf | {"url":"https://paper.telematika.org/2019/pwc/paper_group_awr_442/","timestamp":"2024-11-01T19:34:19Z","content_type":"text/html","content_length":"57636","record_id":"<urn:uuid:8638be5c-bf7b-4ee3-b05c-e0214f60c9fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00681.warc.gz"} |
Primitive objects
\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\
kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \
newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left
(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\
textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw
{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw
{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \
newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#
1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\
textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}
{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \
mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin
{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \
newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]
Primitive objects¶
Primitive Integers¶
The language of terms features 63-bit machine integers as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the PrimInt63 module):
Primitive int := #int63_type.
This type can be understood as representing either unsigned or signed integers, depending on which module is imported or, more generally, which scope is open. Uint63 and uint63_scope refer to the
unsigned version, while Sint63 and sint63_scope refer to the signed one.
The PrimInt63 module declares the available operators for this type. For instance, equality of two unsigned primitive integers can be determined using the Uint63.eqb function, declared and specified
as follows:
Primitive eqb := #int63_eq. Notation "m '==' n" := (eqb m n) (at level 70, no associativity) : uint63_scope. Axiom eqb_correct : forall i j, (i == j)%uint63 = true -> i = j.
The complete set of such operators can be found in the PrimInt63 module. The specifications and notations are in the Uint63 and Sint63 modules.
These primitive declarations are regular axioms. As such, they must be trusted and are listed by the Print Assumptions command, as in the following example.
From Coq Require Import Uint63.
[Loading ML file ring_plugin.cmxs (using legacy method) ... done] [Loading ML file zify_plugin.cmxs (using legacy method) ... done] [Loading ML file micromega_plugin.cmxs (using legacy method)
... done] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done]
Lemma one_minus_one_is_zero : (1 - 1 = 0)%uint63.
1 goal ============================ (1 - 1)%uint63 = 0%uint63
Proof. apply eqb_correct; vm_compute; reflexivity. Qed.
No more goals.
Print Assumptions one_minus_one_is_zero.
Axioms: sub : int -> int -> int eqb_correct : forall i j : int, (i =? j)%uint63 = true -> i = j eqb : int -> int -> bool
The reduction machines implement dedicated, efficient rules to reduce the applications of these primitive operations.
The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlInt63 module can be used when extracting to
OCaml: it maps the Coq primitives to types and functions of a Uint63 module (including signed functions for Sint63 despite the name). That OCaml module is not produced by extraction. Instead, it has
to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq.
Literal values (at type Uint63.int) are extracted to literal OCaml values wrapped into the Uint63.of_int (resp. Uint63.of_int64) constructor on 64-bit (resp. 32-bit) platforms. Currently, this cannot
be customized (see the function Uint63.compile from the kernel).
Primitive Floats¶
The language of terms features Binary64 floating-point numbers as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the PrimFloat module):
Primitive float := #float64_type.
This type is equipped with a few operators, that must be similarly declared. For instance, the product of two primitive floats can be computed using the PrimFloat.mul function, declared and specified
as follows:
Primitive mul := #float64_mul. Notation "x * y" := (mul x y) : float_scope. Axiom mul_spec : forall x y, Prim2SF (x * y)%float = SF64mul (Prim2SF x) (Prim2SF y).
where Prim2SF is defined in the FloatOps module.
The set of such operators is described in section Floats library.
These primitive declarations are regular axioms. As such, they must be trusted, and are listed by the Print Assumptions command.
The reduction machines (vm_compute, native_compute) implement dedicated, efficient rules to reduce the applications of these primitive operations, using the floating-point processor operators that
are assumed to comply with the IEEE 754 standard for floating-point arithmetic.
The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlFloats module can be used when extracting to
OCaml: it maps the Coq primitives to types and functions of a Float64 module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or
execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq.
Literal values (of type Float64.t) are extracted to literal OCaml values (of type float) written in hexadecimal notation and wrapped into the Float64.of_float constructor, e.g.: Float64.of_float (0
Primitive Arrays¶
The language of terms features persistent arrays as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the PArray module):
Primitive array := #array_type.
This type is equipped with a few operators, that must be similarly declared. For instance, elements in an array can be accessed and updated using the PArray.get and PArray.set functions, declared and
specified as follows:
Primitive get := #array_get. Primitive set := #array_set. Notation "t .[ i ]" := (get t i). Notation "t .[ i <- a ]" := (set t i a). Axiom get_set_same : forall A t i (a:A), (i < length t) = true ->
t.[i<-a].[i] = a. Axiom get_set_other : forall A t i j (a:A), i <> j -> t.[i<-a].[j] = t.[j].
The rest of these operators can be found in the PArray module.
These primitive declarations are regular axioms. As such, they must be trusted and are listed by the Print Assumptions command.
The reduction machines (vm_compute, native_compute) implement dedicated, efficient rules to reduce the applications of these primitive operations.
The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlPArray module can be used when extracting to
OCaml: it maps the Coq primitives to types and functions of a Parray module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or
execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq (see kernel/parray.ml).
Coq's primitive arrays are persistent data structures. Semantically, a set operation t.[i <- a] represents a new array that has the same values as t, except at position i where its value is a. The
array t still exists, can still be used and its values were not modified. Operationally, the implementation of Coq's primitive arrays is optimized so that the new array t.[i <- a] does not copy all
of t. The details are in section 2.3 of [CF07]. In short, the implementation keeps one version of t as an OCaml native array and other versions as lists of modifications to t. Accesses to the native
array version are constant time operations. However, accesses to versions where all the cells of the array are modified have O(n) access time, the same as a list. The version that is kept as the
native array changes dynamically upon each get and set call: the current list of modifications is applied to the native array and the lists of modifications of the other versions are updated so that
they still represent the same values. | {"url":"https://coq.inria.fr/doc/V8.18.0/refman/language/core/primitive.html","timestamp":"2024-11-12T13:47:47Z","content_type":"text/html","content_length":"147257","record_id":"<urn:uuid:c7845d3f-d10e-4305-9c58-3c3ad9b5c17d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00481.warc.gz"} |
Principles of Distributed Computing (FS 2008)
This page is no longer maintained. Up-to-date versions of lecture and exercise material can be found here.
In the last two decades, we have experienced an unprecedented growth in the area of distributed systems and networks; distributed computing now encompasses many of the activities occurring in today's
computer and communications world. This course introduces the principles of distributed computing, highlighting common themes and techniques. We study the fundamental issues underlying the design of
distributed systems: communication, coordination, fault-tolerance, locality, parallelism, symmetry breaking, synchronization, uncertainty. We explore essential algorithmic ideas and lower bound
techniques, basically the "pearls" of distributed computing. We will cover a fresh topic every week.
Note that this lecture has been revised. Several topics are covered for the first time this year, while other topics treated in previous years are not covered anymore.
Course pre-requisites: Interest in algorithmic problems. (No particular course needed.)
Course language: English or German (depending on the audience).
If you want to review the corrections of your exam, go to our secretary, Monica Fricker (ETZ G88), either on Tuesdays or Thursdays between 9 and 11 AM.
Written exam: 9:00-11:00, August 18, 2008.
Previous exams: SS 03 (Problems 3 & 4 not covered), SS 04 (Problem 3 not covered).
Lecture by Roger Wattenhofer, Fabian Kuhn
Wednesday 8.15-10.00 @ CAB G51.
Exercises by Thomas Locher, Yvonne Anne Oswald, Christoph Lenzen
Wednesday 10.15-11.00 @ CAB G52.
First (short) exercise meeting: February 20, 2008.
Title Lecture Notes (PDF) References
Chapter 0
Introduction Download [peleg] Preface & Chapter 1
Chapter 1
Vertex Coloring Download [peleg] Chapter 7
Chapter 2 [aw] Chapter 3
Leader Election Download [hkpru] Chapter 8
Chapter 3 [peleg] Chapter 3-5
Tree Algorithms Download [hkpru] Chapter 7
Chapter 4
Maximal Independent Set Download [peleg] Chapter 8
Chapter 5
Dominating Set Download ---
Chapter 6 [peleg] Chapter 6 & 25
Synchronizers Download [aw] Chapter 11
Chapter 7
All-to-All Communication Download Slides (from previous year)
Chapter 8 [aw] Chapter 5 & 14.3
Consensus Download Slides (from previous year)
Chapter 9 [leighton] Chapter 1.6 & 3.5
Distributed Sorting Download [clr] Chapter 28
Chapter 10
Peer-to-Peer Computing Download Slides (from talk at P2P)
Chapter 11
Locality Lower Bounds Download [peleg] Chapter 7.5
Chapter 12
Shared Objects Download ---
Chapter 13
Shared Memory Download [aw] Chapter 4
Title Exercise Sample Solution
Exercise 1
Assigned: 2008/02/20 Download Download
Due: 2008/02/27
Exercise 2
Assigned: 2008/02/27 Download Download
Due: 2008/03/05
Exercise 3
Assigned: 2008/03/05 Download Download
Due: 2008/03/12
Exercise 4
Assigned: 2008/03/12 Download Download
Due: 2008/03/19
Exercise 5
Assigned: 2008/03/19 Download Download
Due: 2008/04/02
Exercise 6
Assigned: 2008/04/02 Download Download
Due: 2008/04/09
Exercise 7
Assigned: 2008/04/09 Download Download
Due: 2008/04/16
Exercise 8
Assigned: 2008/04/16 Download Download
Due: 2008/04/23
Exercise 9
Assigned: 2008/04/23 Download Download
Due: 2008/04/30
Exercise 10
Assigned: 2008/04/30 Download Download
Due: 2008/05/14
Exercise 11
Assigned: 2008/05/14 Download Download
Due: 2008/05/21
Exercise 12
Assigned: 2008/05/21 Download Download
Due: 2008/05/28
[aw] Distributed Computing: Fundamentals, Simulations and Advanced Topics
Hagit Attiya, Jennifer Welch.
McGraw-Hill Publishing, 1998, ISBN 0-07-709352 6
[clr] Introduction to Algorithms
Thomas Cormen, Charles Leiserson, Ronald Rivest.
The MIT Press, 1998, ISBN 0-262-53091-0 oder 0-262-03141-8
[hkpru] Dissemination of Information in Communication Networks
Juraj Hromkovic, Ralf Klasing, Andrzej Pelc, Peter Ruzicka, Walter Unger.
Springer-Verlag, Berlin Heidelberg, 2005, ISBN 3-540-00846-2
[leighton] Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes
Frank Thomson Leighton.
Morgan Kaufmann Publishers Inc., San Francisco, CA, 1991, ISBN 1-55860-117-1
[peleg] Distributed Computing: A Locality-Sensitive Approach
David Peleg.
Society for Industrial and Applied Mathematics (SIAM), 2000, ISBN 0-89871-464-8
Articles chapter by chapter:
Chapter 0: Introduction
○ G. Tel. Introduction to Distributed Algorithms. Cambridge University Press, England, 1994.
○ N. Lynch. Distributed Algorithms. Morgan Kaufmann Publishers, Inc., San Mateo, CA, 1995.
○ V.C. Barbosa. An Introduction to Distributed Algorithms. MIT Press, Cambridge, MA, 1996.
Chapter 1: Vertex Coloring
○ R. Cole and U. Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Inform. Comput., volume 70, pages 32-56, 1986.
○ A.V. Goldberg and S. Plotkin. Parallel δ+1 coloring of constant-degree graphs. Inform. Process. Lett., volume 25, pages 241-245, 1987.
○ N. Lineal. Locality in Distributed Graph Algorithms. In SIAM Journal on Computing 21(1), pages 193-201, 1992.
○ K. Kothapalli, M. Onus, C. Scheideler and C. Schindelhauer. Distributed coloring in O(sqrt{log n}) bit rounds. In IEEE International Parallel and Distributed Processing Symposium (IPDPS), 2006.
Chapter 2: Leader Election
○ D. Angluin. Local and global properties in networks of processors. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 82-93, 1980.
○ J.E. Burns. A formal model for message passing systems. Technical Report 91, Indiana University, September 1980.
○ D.S. Hirschberg, and J.B. Sinclair. Decentralized extrema-finding in circular configurations of processors. In Communications of the ACM 23(11), pages 627-628, November 1980.
○ G. LeLann. Distributed systems, towards a formal approach. In IFIP Congress Proceedings, pages 155-160, 1977.
Chapter 3: Tree Algorithms
○ D. Bertsekas and R. Gallager. Data Networks. 2nd Edition. Prentice-Hall International, London, 1992.
○ Y.K. Dalal and R.M. Metcalfe. Reverse path forwarding of broadcast packets. Communications of the ACM, volume 12, pages 1040-1048, 1978.
○ S. Even. Graph Algorithms. Computer Science Press, Rockville, MD, 1979.
○ P. Fraigniaud and E. Lazard. Methods and problems of communication in ususal networks. Discrete Appl. Mathematics, volume 53, pages 79-133, 1994.
○ R. G. Gallager, P. A. Humblet, and P. M. Spira. Distributed Algorithm for Minimum-Weight Spanning Trees. In ACM Transactions on Programming Languages and Systems (TOPLAS), 5(1):66-77, January
Chapter 4: Maximal Independent Set
○ M. Luby. A Simple Parallel Algorithm for the Maximal Independent Set Problem. In SIAM Journal on Computing, November 1986.
○ A. Israeli, A. Itai. A Fast and Simple Randomized Parallel Algorithm for Maximal Matching. In Information Processing Letters volume 22(2), pages 77-80, 1986.
○ N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583, 1986.
Chapter 5: Dominating Set
○ L. Jia, R. Rajaraman, T. Suel. An Efficient Distributed Algorithm for Computing Small Dominating Sets. Distributed Computing 15(4), December 2002.
○ F. Kuhn and R. Wattenhofer. Constant-Time Distributed Dominating Set Approximation. In Principles of Distributed Computing (PODC), Boston, Mssachusetts, USA, July 2003. For a copy, see
Chapter 6: Synchronizers
○ B. Awerbuch. Complexity of Network Synchronization. Journal of the ACM (JACM), 32(4), October 1985.
○ B. Awerbuch and D. Peleg. Network Synchronization with Polylogarithmic Overhead. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science (FOCS), October 1990.
Chapter 7: All-to-All Communication
○ Z. Lotker, B. Patt-Shamir, Elan Pavlov, and David Peleg. MST Construction in O(log log n) Communication Rounds. In Proceedings of the 15th Annual ACM Symposium on Parallel Algorithms and
Architectures (SPAA), 2003.
Chapter 8: Consensus
○ L. Lamport, R. Shostak, and M. Pease. The Byzantine generals problem. ACM Trans. Program. Lang. Syst., 4(3):382-401, July 1982.
○ M. J. Fischer, N. A. Lynch, and M. S. Paterson. Impossibility of distributed consensus with one faulty processor. Journal of the ACM (JACM), 32(2):374-382, April 1985.
○ M. Ben-Or. Another advantage of free choice: Completely asynchronous agreement protocols. In Proceedings of the 2nd Annual ACM Symposium on Principles of Distributed Computing (PODC), pages
27-30, 1983.
○ P. Feldman and S. Micali. Optimal algorithms for Byzantine agreement. In Proceedings of the 20th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 162-172, 1988.
Chapter 9: Distributed Sorting
○ M. Ajtai, J. Komlos, and E. Szemeredi. An 0 (n log n) sorting network. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 1-9, April 1983.
○ J. Aspnes, M.P. Herlihy, and N. Shavit. Counting Networks. In Journal of the ACM, 41(5): pages 1020-1048, September 1994.
○ K. Batcher. Sorting networks and their applications. In Proceedings of the AFIPS Spring Joint Computing Conference, volume 32, pages 307-314, 1968.
○ C. Busch and M. Herlihy. A Survey on Counting Networks. In Proceedings of Workshop on Distributed Data and Structures, Orlando, Florida, March 30, 1998.
○ N. Haberman. Parallel neighbor-sort (or the glory of the induction principle). Technical Report AD-759 248, National Technical Information Service, US Department of Commerce, 5285 Port Royal
Road, Springfield VA 22151, 1972.
○ C. Kaklamanis, D. Krizanc, and T. Tsantilas. Tight bounds for oblivious routing in the hypercube. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, pages
31-36, July 1990.
○ M. Klugerman, C. Greg Plaxton: Small-Depth Counting Networks. In STOC 1992. pages 417-428.
○ K. Sado and Y. Igarashi. Some parallel sorts on a mesh-connected processor array and their time efficiency. In Journal of Parallel and Distributed Computing, volume 3, pages 398-410, September
Chapter 10: Peer-to-Peer Computing
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Rhode Island, August 2001. For a copy, see Publications.
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June 2004. For a copy, see Publications.
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(FOCS), 1997.
Chapter 13: Shared Memory
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○ H. Attiya, A. Fouren, E. Gafni. An Adaptive Collect Algorithm with Applications. Distributed Computing 15(2), 2002.
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○ H. Attiya, F. Kuhn, M. Tolksdorf, and Roger Wattenhofer. Efficient Adaptive Collect using Randomization. In Proceedings of the 18th Annual Conference on Distributed Computing (DISC), pages
159-173, 2004.
○ Y. Afek and Y. De Levie. Space and Step Complexity Efficient Adaptive Collect. In Proceedings of the 19th Annual Conference on Distributed Computing (DISC), pages 384-398, 2005. | {"url":"https://disco.ethz.ch/courses/fs08/distcomp/","timestamp":"2024-11-14T15:30:08Z","content_type":"text/html","content_length":"34538","record_id":"<urn:uuid:865259d5-cabe-4d30-8648-3b2a5cdcc030>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00692.warc.gz"} |
Free online mental arithmetic games for juniors - Solumaths
Thanks to the mental arithmetic games offered on this site, children will be able to develop their ability to control a result and their critical thinking skills. Each sequence of games allows kids
to evaluate themselves and to identify operations that require further training.
This site offers many mental arithmetic games that allow children to practice arithmetic calculations. The online mental arithmetic games for children available on this site will allow them to :
• develop their understanding of the notion of number
• work on the sense of arithmetic operations
• use the properties of arithmetic operations
• develop numeracy skills
• develop techniques to handle more complex calculations | {"url":"https://www.solumaths.com/en/math-games-online/list/mental-calculation","timestamp":"2024-11-02T03:12:25Z","content_type":"text/html","content_length":"37088","record_id":"<urn:uuid:8a8e508d-aa93-4295-8516-d823df79e4aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00766.warc.gz"} |
Codeforces Round #235 (Div. 2) problem D doubt - Codeforces
Here is the link to the problem: http://mirror.codeforces.com/contest/401/problem/D
So I was trying the same DP approach as most people did, but with a slight change in the state.
What most people used as their DP state: < remainder, mask_of_18_bits >
where 'mask_of_18_bits' is used to denote whether i-th digit in number 'n' is used or not.
What I used as DP state: < remainder, zeroes_left, mask_of_18_digits >
Please note that here the mask is entirely different.
where 'zeroes_left' = number of 0s left in the number 'n' to be used.
and 'mask_of_18_digits' is defined in this way 'aabbccddeeffgghhii'
aa = number of 9s left in 'n' to be used,
bb = number of 8s left in 'n' to be used,
ii = number of 1s left in 'n' to be used
Here is the link to my submission: http://mirror.codeforces.com/contest/401/submission/5990314
but with my approach, I used std::map and that gives TLE at test case 80.
Can someone please suggest a workaround using the same idea? I want to be able to define a state in terms of counts of the digits 0-9.
11 years ago, # |
» +3
pzajec Take a look at my solution. I've used the same idea but encoded counts in binary bitmask.
→ Reply
• 11 years ago, # ^ |
← Rev. 2 → 0
» It's amazing how you encoded all that into just one long long variable. Can you please brief me a bit about your encoding and decoding process?
Zzyzx EDIT: I have understood your method. Cool trick! Thanks for sharing!
→ Reply | {"url":"https://mirror.codeforces.com/blog/entry/10848","timestamp":"2024-11-05T04:02:25Z","content_type":"text/html","content_length":"91611","record_id":"<urn:uuid:32d43675-8ae4-48c7-b0bb-97a6886e8ad7>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00451.warc.gz"} |
A New Robust Mathematical Model for the Multi-product Capacitated Single Allocation Hub Location Problem with Maximum Covering Radius
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Paper 2, Section II, G
Suppose the functions $f_{n}(n=1,2, \ldots)$ are defined on the open interval $(0,1)$ and that $f_{n}$ tends uniformly on $(0,1)$ to a function $f$. If the $f_{n}$ are continuous, show that $f$ is
continuous. If the $f_{n}$ are differentiable, show by example that $f$ need not be differentiable.
Assume now that each $f_{n}$ is differentiable and the derivatives $f_{n}^{\prime}$ converge uniformly on $(0,1)$. For any given $c \in(0,1)$, we define functions $g_{c, n}$ by
$g_{c, n}(x)= \begin{cases}\frac{f_{n}(x)-f_{n}(c)}{x-c} & \text { for } x eq c, \\ f_{n}^{\prime}(c) & \text { for } x=c .\end{cases}$
Show that each $g_{c, n}$ is continuous. Using the general principle of uniform convergence (the Cauchy criterion) and the Mean Value Theorem, or otherwise, prove that the functions $g_{c, n}$
converge uniformly to a continuous function $g_{c}$ on $(0,1)$, where
$g_{c}(x)=\frac{f(x)-f(c)}{x-c} \quad \text { for } x eq c$
Deduce that $f$ is differentiable on $(0,1)$. | {"url":"https://questions.tripos.org/part-ib/2010-4/","timestamp":"2024-11-09T22:45:12Z","content_type":"text/html","content_length":"36663","record_id":"<urn:uuid:04e44cc9-83c1-48e4-bfc9-a7bf7e1e3f5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00516.warc.gz"} |
HP Forums
Hello all,
I am wondering if it is possible to return general solutions for fMax. For example, CASIO Classpad gives {MaxValue=1,x=2π*constn(1)} for fMax(cos(x)). Prime, on the other hand, currently only gives a
principal: 0.
Also, in cases when the Prime does provide general solutions, it uses a variable n_integer. You would expect it to be n_1 all the time (or maybe n_2 if there is another solution), but on my
calculator at least, n_ is followed by massive integers, sometimes 12, sometimes 120; Once it even got up to 500. The solutions are still correct, but the way n behaves is so weird and annoying. | {"url":"https://www.hpmuseum.org/forum/archive/index.php?thread-7180.html","timestamp":"2024-11-07T19:24:41Z","content_type":"application/xhtml+xml","content_length":"3693","record_id":"<urn:uuid:40b1d59f-4e27-4ddd-bc43-64cf2f378892>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00254.warc.gz"} |
How do you do QAM modulation in Matlab?
y = qammod( x , M ) modulates input signal x by using QAM with the specified modulation order M . Output y is the modulated signal. y = qammod( x , M , symOrder ) specifies the symbol order.
How do you draw the constellation diagram of QAM in Matlab?
Create 32-QAM Constellation Diagram Use the qammod function to generate the 32-QAM symbols with binary symbol ordering. M = 32; data = 0:M-1; sym = qammod(data,M,’bin’); Plot the constellation. Label
the order of the constellation symbols.
How many bits is 64 QAM?
six bits
64 QAM has six bits per symbol per polarization, so to encode our 12 bits, we break up the 12-bit string into two groups of six, thereby only using only one symbol for each polarization.
What is the symbol rate of 64 QAM?
Tabular difference between 16-QAM, 64-QAM and 256-QAM
Specifications 16-QAM modulation 64-QAM modulation
Number of bits per symbol 4 6
Symbol rate (1/4) of bit rate (1/6) of bit rate
KMOD 1/SQRT(10) 1/SQRT(42)
What QAM 16?
16 Quadrature Amplitude Modulation. This is a modulation technique in which the carrier can exist in one of sixteen different states. As such, each state can represent four bits – 0000 through to
1111, per symbol change.
What is QAM signal?
Key Takeaway: QAM (quadrature amplitude modulation) is a modulation scheme used by network operators when transmitting data. QAM relates to a way of changing the amplitude, or power level, of two
signals. QAM enables an analog signal to efficiently transmit digital information and increases the useable bandwidth.
What is Awgn Matlab?
Description. y = awgn( x , snr ) adds white Gaussian noise to the vector signal x . This syntax assumes that the power of x is 0 dBW. For more information about additive white Gaussian noise, see
What is AWGN?
What is 16-QAM used for?
In the UK, 16 QAM and 64 QAM are currently used for digital terrestrial television using DVB – Digital Video Broadcasting. In the US, 64 QAM and 256 QAM are the mandated modulation schemes for
digital cable as standardised by the SCTE in the standard ANSI/SCTE 07 2000.
Why is it called 256 QAM?
The significance of 256-QAM is the availability of two new data rates (MCS 8 and MCS 9) which achieve higher throughput than the previously highest data rate of MCS 7, in the same amount of airtime.
Needless to say all big number claims for 802.11ac presume 256-QAM is being used.
What QAM 256?
Just what is QAM-256? It refers to Quadrature Amplitude Modulation, which is the means by which a carrier signal, such as an LTE waveform, transmits data and information. Ideally, a waveform/symbol
carries as much data as possible in order to achieve higher data rates and increase spectral efficiency.
How does 64 QAM work?
Sixty-four QAM is a higher order modulation technique, which allows one single radio wave to represent six bits of data by manipulating the amplitude and phase of the radio wave into one of 64
different discrete and measurable states.
What is AWGN channel model?
AWGN is often used as a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of
bandwidth) and a Gaussian distribution of amplitude.
How do you convert QAM to matrix in MATLAB?
Use the de2bi function to convert the data symbols from the QAM demodulator, dataSymbolsOut, into a binary matrix, dataOutMatrix, with dimensions –by–. In the matrix, is the total number of QAM
symbols, and is the number of bits per symbol. For 16-QAM, = 4.
Where can I find full Matlab code for digital simulations?
Full Matlab code available in the book Digital Modulations using Matlab – build simulation models from scratch
How do you convert QAM symbols to binary in Python?
Use the qamdemod function to demodulate the received data and output integer-valued data symbols. Use the de2bi function to convert the data symbols from the QAM demodulator, dataSymbolsOut, into a
binary matrix, dataOutMatrix, with dimensions –by–.
How do you plot a QAM constellation in MATLAB?
Construct the modulator object. Plot the constellation. This example shows how to plot a QAM constellation having 32 points. Use the qammod function to generate the 32-QAM symbols with binary symbol
ordering. Plot the constellation. Label the order of the constellation symbols. | {"url":"https://vivu.tv/how-do-you-do-qam-modulation-in-matlab/","timestamp":"2024-11-10T04:20:32Z","content_type":"text/html","content_length":"125806","record_id":"<urn:uuid:685555dc-6188-48c9-bc7e-4595c7dbda19>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00427.warc.gz"} |
An (n-2)-dimensional Quadratic Surface Determining All Cliques and a Least Square Formulation for the Maximum Clique Problem
Arranging an n-vertex graph as the standard simplex in R^n, we identify graph cliques with simplex faces formed by clique vertices. An unstrict quadratic inequality holds for all points of the
simplex; it turns to equality if and only if the point is on a face corresponding to a clique. This way this equality determines a quadratic surface in R^n characterizing all graph cliques. Since the
standard simplex is a polyhedron located within the hyperplane e'x=1, we may decrease the dimensionality by 1 considering the intersection of this surface with the hyperplane. Therefore, we obtain a
quadratic surface of dimensionality n-2 determining all graph cliques. We call it the clique wrapper. The higher the dimensionality of a standard simplex face, the less the distance from it to the
geometric center of the simplex. Therefore, the maximum clique problem is equivalent to finding a point of the clique wrapper closest to the geometric center of the simplex and lying not outside the
simplex. When the latter is relaxed, all stationary points of such a least square program can be found via roots of one univariate rational equation. If the adjacency matrix of the graph does not
have multiple eigenvalues, the number of the stationary points is not above 2(n-1). If a clique is such that any vertex outside it has the same number of adjacent vertices in the clique, the
geometric center of the clique face is such a stationary point.
Stas Busygin's NP-Completeness Page (http://www.busygin.dp.ua/npc.html), 2002.
View An (n-2)-dimensional Quadratic Surface Determining All Cliques and a Least Square Formulation for the Maximum Clique Problem | {"url":"https://optimization-online.org/2002/08/524/","timestamp":"2024-11-05T06:34:40Z","content_type":"text/html","content_length":"85493","record_id":"<urn:uuid:29e61004-5017-47c4-8ba0-b54372151c41>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00022.warc.gz"} |
Organising the Organisation
I am the chief of the Personnel Division of a moderate-sized com- pany that wishes to remain anonymous, and I am currently facing a small problem for which I need a skilled programmer’s help.
Currently, our company is divided into several more or less independent divisions. In order to make our business more effi- cient, these need to be organised in a hierarchy, indicating which
divisions are in charge of other divisions. For instance, if there are four divisions A, B, C and D we could organise them as in Figure 1, with division A controlling divisions B and D, and division
D controlling division C. One of the divisions is Central Management (division A in the figure above), and should of course be at the top of the hierar- chy, but the relative importance of the
remaining divisions is not determined, so in Figure 1 above, division C and D could equally well have switched places so that C was in charge over division D. One complication, however, is that it
may be impossible to get some divisions to cooperate with each other, and in such a case, neither of these divisions can be directly in charge of the other. For instance, if in the example above A
and D are unable to cooperate, Figure 1 is not a valid way to organise the company. In general, there can of course be many different ways to organise the organisation, and thus it is desirable to
find the best one (for instance, it is not a good idea to let the programming people be in charge of the marketing people). This job, however, is way too complicated for you, and your job is simply
to help us find out how much to pay the consultant that we hire to find the best organisation for us. In order to determine the consultant’s pay, we need to find out exactly how difficult the task
is, which is why you have to count exactly how many different ways there are to organise the organisation. Oh, and I need the answer in five hours. Input The input consists of a series of test cases,
at most 50, terminated by end-of-file. Each test cases beginswiththreeintegersn,m,k(1≤n≤50,1≤m≤n,0≤k≤1500). ndenotesthenumberof divisions in the company (for convenience, the divisions are numbered
from 1 to n), and k indicates which division is the Central Management division. This is followed by m lines, each containing two integers 1 ≤ i,j ≤ n, indicating that division i and division j
cannot cooperate (thus, i cannot be directly in charge of j and j cannot be directly in charge of i). You may assume that i and j are always different. Output For each test case, print the number of
possible ways to organise the company on a line by itself. This number will be at least 1 and at most 1015. Note: The three possible hierarchies in the first sample case Figure 1: Example
2/2 Sample Input 552 31 34 45 14 53 411 14 302 Sample Output 3 8 3 | {"url":"https://ohbug.com/uva/10766/","timestamp":"2024-11-03T15:44:55Z","content_type":"text/html","content_length":"4117","record_id":"<urn:uuid:41ebab9f-f365-4a43-9baf-fccffc16bf65>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00079.warc.gz"} |
Books Archives | Fundamentals of Mathematics and Physics
I wrote about the power of abstraction earlier, and I just came across a beautiful passage on the same subject by one of my favourite authors, the prolific and master expositor, John Stillwell (see
also here). It’s taken from the preface to Elements of Algebra: Geometry, Numbers, Equations, Springer1994: Algebra is abstract mathematics — let … Read more
Black Earth Into Yellow Crocus
Perhaps my favourite joke of all time is actually an anecdote that I read in the wonderful book Thirty Years that Shook Physics, by George Gamow. The book is available in an inexpensive Dover
edition, and would make a fine complement to a course textbook in modern physics, which amounts to introductory quantum mechanics. Gamow … Read more
The Disney World of Good vs. Evil
One of the traditional purposes of culture is to educate. Before books were common, the spoken word was the essential tool for teaching. Stories are memorable, and so telling stories was an effective
way to pass on life lessons, particularly moral lessons. But in recent times, “information media” have been used overwhelmingly often only for … Read more
On Failure
A great line from Gretchen Rubin’s delightful blog (The Happiness Project): If you’re not failing, you’re not trying hard enough. It’s similar to a basketball truism about playing defense (“If you
commit no fouls at all, you’re not trying hard enough.”), and just as true. She has a book by the same name, which I … Read more
Words, Episode 2: compassion
Jian Ghomeshi interviewed Karen Armstrong (her recent book is 12 Steps to a Compassionate Life; for reviews see here, here, and here, for example) yesterday on Q, and she made the point that the
major religions have largely failed at training their members to be compassionate, instead emphasizing doctrine, and rigid adherence to rules of … Read more
Random Humour and the Sufis
I read this while browsing in a store in Ellicottville the other day: An actor included the following provision in his will: When he died he wished to be cremated, and then 10% of his ashes should be
thrown in his agent’s face. Funny, eh? It says so much. The Sufis used humorous stories as … Read more | {"url":"https://fomap.org/category/books/","timestamp":"2024-11-09T09:03:05Z","content_type":"text/html","content_length":"68443","record_id":"<urn:uuid:9a11fd7f-1e0a-4c9c-92b2-3749dcd731c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00024.warc.gz"} |
Dynamic Representations - Trigonometry - Unit Circle
Trigonometry - Unit Circle
Interactive and dynamic unit circle
Move the black point to change the angle and the blue point to change the hypotenuse of the enlarged triangle.
Turn parts of the representation on and off using the control panel to the left. >>
Click on points to reveal the hidden values.
Move the question mark to create your own questions.
Pythagoras Theorem
Ratio Tables
Similar Triangles
The triangle has a constant hypotenuse of 1 unit.
The red and green shorter sides vary with the size of the angle.
Move the point on the circumference of the circle above to change the angle.
You can get the lengths of the shorter sides by reading off the values from the axis or from the coordinates of the point on the circumference.
Protractor on graph template.docx
Download protractor on axis template.
Definitions of sine and cosine
Sine: sinθ is the length of the side opposite angle θ in a standard right angled triangle within the first quadrant of the unit circle
Cosine: cosθ is the length of the side adjacent to angle θ in a standard right angled triangle within the first quadrant of the unit circle. Cosine is also known as sine of the complement.
(cosθ,sinθ) is the coordinate of the point where the hypotenuse meets the circumference of the circle.
This is the only new knowledge that students will need.
This reduces cognitive load and frees up working memory to solve problems with trigonometry.
Use the triangle and graph to find an approximate value for.docx
Students need to be able to see the unit triangle independently of the circle and in different orientations.
The following triangles all have angle.docx
Click and drag the blue point to enlarge the unit triangle. Click the points on the sides to reveal the missing lengths. Click and drag the question mark to set your own problems.
Use similar triangles to find the missing side. See the two methods below
Example 1 - Finding a missing shorter length given the hypotenuse is 3 and the angle is 30.
Draw the two similar triangles (above) and complete a ratio table (below)
Example 2 - Finding the hypotenuse when given the shorter side is 3 and the angle is 30.
Draw the unit triangle to match the problem (left) and complete a ratio table (below)
Example 3 - Finding the shorter side when a given shorter side is 3 and the angle is 30.
Draw the unit triangle to match the problem (left) and complete a ratio table (below)
Finding missing angles given the hypotenuse
Example 1 - Finding a missing angle given the hypotenuse is 3 and the opposite side is 2
Draw the unit triangle to match the problem (left) and complete a ratio table (below)
Tangent: tanθ is the height of the opposite side that is the tangent to the unit circle at the point x=1
tan θ = sinθ / cosθ
Finding missing angles without being given the hypotenuse
Example 1 - Finding a missing angle when given 2 shorter sides lengths using tanθ = sinθ / cosθ
Draw the unit triangle to match the problem (left) and complete a ratio table (below)
Example 2 - Finding a missing angle when given 2 shorter sides lengths using Pythagoras
Draw the unit triangle to match the problem (left).
Use Pythagoras to find the hypotenuse then complete a ratio table (below)
Exact trigonometric values
Finding exact values for 30 and 60 degrees
1) Draw an equilateral triangle with side lengths 1
2) Slice the triangle in half
3) Find the missing sides using Pythagoras
4) Compare with the unit triangle
Finding exact values for 45 degrees
1) Draw an isosceles right angled triangle shorter with side lengths 1
2) Find the missing side using Pythagoras
3) Compare with the unit triangle
4) Use the ratio table to find sin, cos and tan.
Compare the cognitive load needed to solve these problems using the unit circle compared to SOH CAH TOA.
Click the image to access the full interactive tool on Desmos.
Click the points in the ratio table to hide and reveal ratios.
Turn elements on and off using the control panel on the left. >>
Click the circle containing the folder icon to hide and reveal elements of the graph eg protractor and ratio table. | {"url":"https://www.enigmadynamicrepresentations.com/multiplicative-reasoning/trigonometry-unit-circle","timestamp":"2024-11-08T06:11:39Z","content_type":"text/html","content_length":"296671","record_id":"<urn:uuid:e84f1226-b9ca-4c37-a8ce-28c0a441c043>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00752.warc.gz"} |
PCB Impedance Calculator - PCB & PCBA Manufacturer
PCB Impedance Calculator is a software tool for calculating the impedance of transmission lines on PCBs. By using it, engineers can quickly and accurately predict and optimize the impedance
performance of a PCB to ensure reliable signal transmission.
PCB Impedance is the ratio of voltage to current in a circuit and is used to measure how well a circuit impedes a signal. In a circuit board, PCB impedance manifests as the combined effect of
resistance, inductance, and capacitance between components on the PCB board. The impedance magnitude directly affects signal transmission quality and energy loss.
Factors affecting the PCB impedance
Resistance: The resistance of a conductor is directly proportional to the length of the conductor and inversely proportional to the cross-sectional area of the conductor. In printed circuit boards,
the length and width of the conductor have a significant effect on resistance.
Inductance: When the current changes, a magnetic field is generated around the conductor, which will resist the current. In printed circuit boards, the length and number of turns of the wire affect
the inductance.
Capacitance: An electric field exists between two conductors, and the ability of a capacitor to hold a charge is related to its capacitance when the potential difference is certain. In the printed
circuit board, the distance between the wires and the relative position of the capacitance has an impact.
Dielectric: The dielectric of a printed circuit board also has an important effect on impedance. The relative dielectric constant and loss angle tangent of the dielectric affects the magnitude of
capacitance and inductance.
PCB impedance test
Calculation formula for pcb impedance
The impedance calculation of printed circuit boards needs to consider the combined effect of resistance, inductance, and capacitance. Commonly used calculation formula: Z = R + j (ωL-1/ωC)
Among them:
Z for the impedance, the unit is ohms (Ω).
R is the resistance in ohms.
ω is the angular frequency in radians/second (rad / s).
L is the inductance in Henrys (H).
C is the capacitance in Farads (F).
j is the imaginary unit, indicating a 90° phase angle.
Parameter explanation and calculation method
Resistance (R): Resistance is the obstruction of current by a conductor and is determined by the material, length and cross-sectional area of the conductor. In PCB, the resistance of a wire can be
calculated by the following formula:
ρ is the resistivity of the conductor in Ω-m.
L is the length of the wire, in m.
A is the cross-sectional area of the wire, the unit is m².
Inductance (L): Inductance is the coil of the current changes in the obstruction. In PCBs, the inductance of a wire is related to the length, the number of turns and the distance between the wires.
Usually, PCB manufacturers provide inductance values for standard line widths and spacing. If customized line width and spacing is required, it can be calculated by simulation with relevant software
Capacitance (C): Capacitance is the ability of an electric field to store an electric charge. In a PCB board, the capacitance between wires is related to the distance and relative position between
them. The formula for capacitance is as follows:
ε0 is the vacuum capacitance, with a value of 8.854 × 10-¹² F/m.
εr is the relative capacitance rate, related to the PCB sheet.
S is the distance between the two wires, in m.
d is the wire diameter or spacing in m.
Angular Frequency (ω): Angular frequency is a physical quantity that describes the frequency of a simple harmonic vibration or fluctuation, and is related to the period T and frequency f as:
ω = 2πf = 2π/T
f is the frequency in Hz.
T is the period in s.
PCB Impedance Matching and Signal Integrity
PCB Impedance matching is critical in the transmission of high-speed digital signals. When the PCB impedance of the source and the load are perfectly matched, the signal can be transmitted without
loss. To achieve PCB impedance matching, the wiring of the printed circuit board needs to be carefully designed, such as selecting the appropriate line width, line spacing, size, and other
parameters. Signal integrity (SI) issues such as signal reflection, crosstalk, and propagation delays also need to be considered. Evaluating and optimizing the SI performance utilizing simulation and
testing is one of the hotspots in current research.
PCB impedance calculation
Advantages of PCB impedance calculator
Fast and Accurate: The PCB impedance calculator is based on a precise mathematical model and can quickly and accurately calculate PCB impedance value, avoiding the errors and tediousness of manual
Adjustable Parameters: The PCB impedance calculator provides a series of adjustable parameters, such as the width, thickness, material, and spacing of the wires, allowing users to customize the
calculations according to their actual needs.
Predictive capability: With the PCB Impedance Calculator, engineers can predict the impact of different process parameters on PCB impedance, thereby identifying and solving potential problems in
advance and improving the reliability and performance of the design.
Material Selection: The PCB impedance calculator can compare the performance of different materials and media to help engineers select the most suitable PCB materials, reducing costs and risks.
But it also has some disadvantages
Dependency: requires users to input some parameters and data, if these data are inaccurate or incomplete, it will lead to inaccurate calculation results.
Technical requirements: requires some technical knowledge and experience to use correctly, for beginners may have some learning difficulties.
Limitations: provides a certain range of reference values and cannot completely replace the actual test and measurement. Therefore, in the actual application, it is still necessary to carry out
actual testing and verification.
PCB impedance calculator plays an PCB important role in PCB design. Through the use of PCB impedance calculator, engineers can quickly and accurately predict and optimize the PCB impedance
performance, thereby improving the reliability and performance of electronic equipment. | {"url":"https://ipcb.co/pcb-impedance-calculator/","timestamp":"2024-11-10T13:03:40Z","content_type":"text/html","content_length":"190337","record_id":"<urn:uuid:cd952dea-bdb7-4231-bce9-419989e7434b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00761.warc.gz"} |
Exploring Subspace Topology: A Deep Dive into Point-Set Concepts
Written on
Chapter 1: Understanding Topological Spaces
In our previous discussion regarding embeddings and immersions, we referenced various aspects of point-set topology. To ensure clarity and accuracy, it's essential to elucidate some of the
terminology and definitions associated with subspaces and embeddings.
To begin, we must define what constitutes a topology from the perspective of point-set topology. Following this, we will delve into the notions of continuity, compactness, and the Hausdorff
Topological Spaces and Continuity
A topological space, denoted as (X, ?), consists of a set X along with a collection of subsets ? (referred to as open sets), provided that they adhere to the open set axioms.
When considering a mapping f that connects elements from one topological space to another, we can articulate the concept of continuity through the lens of open maps. Specifically, a mapping f: X → Y
is deemed continuous if, for every open subset U in Y, the preimage f^(-1)(U) is an open set in X. The preimage is defined as the collection of all elements in the original space that correspond to
points in the target space.
As we explore embeddings and immersions further, it becomes necessary to understand the subspace topology. Given a topological space X and a subset Y of X, the subspace topology comprises open sets
that are defined as follows:
In this context, the inclusion map assigns X to Y, where the domain is equipped with the subspace topology and the range is characterized by the standard topology. Additionally, the inclusion map is
continuous because ??1(U_Y) = U_Y ∩ X, within ??_X.
Chapter 2: Subspace Topology and Its Applications
The concepts we've discussed are foundational for grasping more complex ideas in topology. For a deeper understanding of subspace topology, let's explore the following video.
This video titled "Subspace Topology 1 (Topology)" provides a detailed examination of subspace topology and its significance within the broader context of topological spaces.
Continuing our exploration, we will now look at an important problem related to the cross-ratio and subspaces.
The second video, "Cross ratio & 4 subspace problem," delves into the complexities of the cross-ratio in relation to subspaces, further enhancing our understanding of these concepts. | {"url":"https://zhaopinxinle.com/exploring-subspace-topology-deep-dive.html","timestamp":"2024-11-04T20:57:52Z","content_type":"text/html","content_length":"11528","record_id":"<urn:uuid:b44d5edb-9ad5-40c6-ba60-30ea69cd4330>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00091.warc.gz"} |
What Voltage My Solar Panel Produces (Calculations + Examples) - Solar Panel Installation, Mounting, Settings, and Repair.
What Voltage My Solar Panel Produces (Calculations + Examples)
The voltage a solar panel produces can vary for a few reasons. Some of the reasons are positive, some are not. The voltage produced by a panel is really only part of a more important question: How
many watts should the panel produce?
There are three factors that impact this question
• Volts
• Amps
• Weather Conditions
Every panel on the market is designed to produce a certain voltage and current under various conditions. These specifications are generally printed on the back of the panel.
Knowing how to assess the specifications of a panel will help you determine if it will provide the power you need.
Solar Panel Voltage
The voltage of a solar panel is the result of individual solar cell voltage, the number of those cells, and how the cells are connected within the panel. Every cell and panel has two voltage ratings.
• Open Circuit Voltage (Voc)
• Voltage at Maximum Power (Vmp)
Open Circuit Voltage
The Voc is the amount of voltage the device can produce with no load at 25º C. This value is a little like the maximum horsepower a car’s engine can put out. It is a lab-produced value that has
little value in the field.
The value will vary due to atmospheric conditions and temperature.
• Maximum potential voltage.
• No Load.
• Zero current.
• Not a working voltage.
See also: Calculate Solar Panel kWp & KWh (KWh Vs. KWp + Meanings)
Voltage at Maximum Power
The Vmp is the voltage the device will produce a maximum power output. This is essentially the working voltage of the device. It is the voltage the panel will supply to a battery or charge
• Maximum working voltage.
• Full load.
• Full current.
• The voltage applied to your electrical system.
How Various Panel Voltages Are Produced
Solar panels can be designed to produce just about any voltage. A panel is a collection of individual solar cells. Individual cells produce between 0.45 and 0.6 volts (Vmp) at 25º C.
The voltage output of the individual cells can vary due to the type and quality of the cell used. Groups of cells are wired together in a panel to produce various voltages.
Number of Cells for Typical Voltage Panels
• 32 cells x 0.46 Voc = 14.72 Vmp (12 volt system.)
• 72 cells x 0.46 volts = 27.60 Vmp (24 volt system.)
• 96 cells x 0.50 volts = 48.0 Vmp (Large commercial arrays.)
This is where we find part of the answer to, “How many volts should my panel put out?” Most 32 cell panels are wired in series to produce voltage for a 12-volt system.
Most 72 cell panels are wired in series to produce 24 volts, but could also have pairs of strings wired in parallel to produce more current at 12 volts.
Vmp to Voc Ratio
When looking at a panel of a given nominal voltage, a good rule of thumb for estimating the Vmp is to add about 20% to the nominal voltage. To estimate the Voc value, add about 80% to the nominal
These will almost never be exactly right but are a good estimate. The certificate on the back of the panel or other manufacturer documentation is the only place to find the exact voltage ratings of a
Estimating Voc and Vmp Value For a Panel
• 24 volt panel
• 24 volts x 0.8 = 18 volts
• 24 volts + 18 volts = 42 Voc
• 24 volt panel
• 24 volts x 0.2 = 4.8 volts
• 24 volts + 4.8 volts = 28.8 Vmp
If you measure the voltage of a panel that is not connected to any load and is in full sun you should measure the Voc value.
As soon as you connect the leads to a load, the voltage will drop to something near the Vmp value. It will vary based on the load applied.
Measuring Panel Voltage
Measuring volts is a fairly simple procedure. A simple Voltmeter or Multi-meter from your local hardware store is all you need. Set the meter to DC Volt in the appropriate range.
Touch the probes of the meter to bare wire at the end of the cables and you can measure the voltage of the panel. Be careful not to let wires touch each other.
Panel Current: Watt – Volts – Amps – Ipm
To calculate the power (watts) provided by a solar panel we need to know the size of the electrical wave (volts) and the force of the current (amps) behind the wave. Most solar panels list two
current values: Maximum Current (Ipm) and Short Circuit Current (Isc).
• Amps = Force.
• Ipm = Amps at Maximum Power.
• Isc = Amps at Short Circuit.
How Various Amp Ratings Are Achieved.
A typical solar cell produces around 30 milliamps per square centimeter or about 187 milliamps per square inch. At that rate, a 4-inch square cell will produce approximately 3 amps. Different cell
materials and cell sizes will produce various current outputs.
Various sized cell output at 187 Milliamps per square inch.
• 3 inch square cell = 1.7 amps.
• 4 inch round cell = 2.2 amps.
• 4 inch square cell = 3.0 amps.
Higher amp ratings are achieved by wiring groups of cells in parallel. This will lower the volt rating of the panel but may increase the overall power (watt) output.
Measuring Amps of a Panel
Measuring current is not as simple as measuring volts. The Current at Maximum Power (Imp) can only be measured while there is power running through the wire attached to the panel. DC Amp Meters are a
little pricey but are available if you have the urge to measure your current.
The Short Circuit Current (Isc) requires shorting the Positive and Negative leads through an appropriate Amp Meter. Short-Circuiting the leads of the panel could damage it, so this practice should be
avoided unless you know what you are doing.
I have lived off-grid for 20 years and have never found it necessary to measure the Isc of a panel. Take the manufacturer’s word for the rated current. It is safer and easier.
Watts is a measure of work. It is the amount of energy the panel can provide to your system at maximum solar exposure at 25º C. It is calculated by multiplying Volts at Maximum Power (Vmp) and the
Current at Maximum Power (Ipm).
This calculation expresses the maximum potential power the panel could provide. Load, atmospheric conditions, and temperature, can all impact this value.
The likelihood that you will experience this output very often is pretty slim.
• V x I = P (Volts x Current = Power in watts)
• Most panels are rated by Watts at some Voltage.
• Only achievable in specific conditions.
As is often the case, a simple question does not have a simple answer. “How many volts should my solar panel put out?” is not as straightforward as one might expect. There are a lot of variables at
Elliot has 20+ years of experience in renewable technology, from conservation to efficient living. His passion is to help others achieve independent off-grid living.
SolVoltaics is an affiliate and an Amazon Associate, we earn from qualifying purchases - at no extra cost to you. | {"url":"https://solvoltaics.com/voltage-solar-panel-produce/","timestamp":"2024-11-02T16:51:58Z","content_type":"text/html","content_length":"69082","record_id":"<urn:uuid:9c08ef8b-e1be-4856-b331-250950b5b822>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00350.warc.gz"} |
Letters by a modern St. Ferdinand III
Scientism - James Webb Telescope and disproving the Banging Religion
The worship of abstract maths is disproved by reality. Yes, reality does bite.
This is a really good 24-minute video on issues with Space-Time and STR, by Roger Penrose. The Scientism of Einstein, as will be elaborated in 2 coming posts and which is analysed here, here and here
, is coming to an end. It might take 50 more years, but the reign is soon over. There is precious little proof that Banging is relevant.
This video is a good overview of what is wrong when one analyses ‘space-time’ a key insight not from Einstein but from his maths teacher Minkowski, which Einstein incorporated into his STR. This
video however misses the obvious fact that Time might well exist outside of space, separate and not integrated into a 4^th dimension as proposed by Einstein. This is far more likely, than a
‘dimensional’ cube of interwoven space and time, an idea which Einstein never bothered to physically prove.
Physics is the physical reality and proofs using objects in the real world. Mathematical theories are not proof.
There are no valid reasons or observations to entangle space-time, unless you are trying to prove the mathematics of the STR.
· Contrary to STR and Banging theology, stellar galaxies should be far less than 1 billion years old but this is not what they have found
· Exploding Supernovae are young
· Galaxies – larger than ours – are newly created
· Super galaxies have formed in short periods of time
· According to STR and Banging, the Universe should collapse on itself and either reform or cease to exist
· Magical dark matter and dark energy are invoked to prevent a collapse, neither has been found or can even be described
· Time as an idea is problematic given that it is a human construct
· Time is quite likely, not a linear product or chronometry
· There may be multiple cycles and multiple universes, no one knows, but ineluctably according to the BB, there was a beginning and an end to the universe
· Einstein’s ‘constant’ which is the same as dark matter, dark energy, preserves the universe’s ‘steady state’ that exists for eternity, and prevents a cosmic implosion due to gravity (which by
itself is a weak force)
· If the universe is flat, which is what the cosmological proofs state, than STR and its space-time theory based on maths and only maths, is irrelevant
· It is becoming more obvious that space time is not Einstein’s smooth fabric
· Contrary to Newton’s idea of gravity as a pool, Einstein viewed gravity as part of space-time curving the universe
· The STR and Banging maintain that space and time are interconnected as an unprovable 4^th dimension
· This means that everything coexists in space and time including the future and past
· There is no proof of this, and it contravenes physical reality
· Others believe that space-time is an artifact of the quantum world
· Quantum theory is where particles exist in multiple places simultaneously (Schrodinger’s cat where the cat is both alive and dead) but this cannot be reconciled with space-time
· Long standing problem in physics is of locality and entanglement, if we have 2 particles far apart, changing one will affect the other, violating the STR
· This means that different observers will have different ideas of locality – for example you can feel closer to someone you love who is far away, than your neighbour that you don’t care about
· STR, GTR maintain that a gravity field cannot be in 2 places or states simultaneously
· Where does the gravitational field reside? No one knows.
· Any theories attempting to merge STR with quantum theory have failed
· String theory is trying to merge quantum and STR/GTR
· Vibrating stings make up molecules and particles
· For this to work the strings must vibrate across 7 dimensions, only 4 are now proposed with the 4^th of space-time unproven and theoretical
· This is an abstract maths-based idea with no physical proofs
· Loop quantum gravity is now proposed to replace string theory, in which space-time is a woven loop or network of complexity, contrary to STR
· These defects in STR can only be viewed in Planck time or a millionth of a millionth of a metre
· It is impossible to test LQG with particle accelerators, we would need an accelerator 1000-trillion times more powerful than those at CERN and the size of the milky way
· For now, it is an abstract maths based theory with no proofs
· Many believe that a quantum world is influenced by gravity, which is an entirely new approach to physics and cosmology.
· This pursuit should have the potential to impact real life, since any changes to space time theories would affect all theories in physics and cosmology including our own ideas of the age of
space, and the Earth.
· In my opinion the ages of the cosmos and Earth are not the same, with space appearing to be much older than the Earth given the differential in clocks between a terrestrial clock and a space
clock (this is probably a valid part of the STR).
Given that all of our devices function according to quantum theory, using this as a basis for a new approach is sensible. Hawking, the rather puerile salesman for Einstein, acknowledged before he
died that quantum mechanics properly destroyed his idea about Black Holes. As reality displaces complex mathematics the end of much of STR is guaranteed.
Next posts: Dingle’s clock paradox, and the many issues of STR as given by hundreds of scientific experiments and observations.
Scientism and the Special Theory of Relativity. Part One (b), A layman’s overview of STR
STR, mathematical models, and their implications.
The first post discussed the theory of STR and what the theory is trying to achieve. There are 2 postulates or Einstein’s ‘laws’, which the science says are infallible and proven.
The first postulate is that the laws of physics apply to all objects universally, as long as an ether or medium is not present. ‘Laws of physics’ refers to the Catholic Galileo’s work in the 17th
century and his law of inertia, summarised as: "Objects move with constant speed in a straight line when no external agent is acting on them”.
The second rule or postulate is that the speed of light or its velocity is constant for all objects regardless of their motion. Again, this is similar to Galileo’s own experimental proofs.
Einstein’s ex-cathedra postulate pronouncements are now deemed infallible and eternal. These postulates surround and protect his STR theory. The postulate and rules by themselves may well be
sensible. I don’t think they prove much of anything if we extend the postulates to the STR itself or the physical world. Postulate one is basically unprovable. Postulate 2 would not hold outside of a
vacuum. STR itself, in toto, is largely premised of course, on mathematics and in particular endless pages of dense equations.
As Einstein supposedly remarked, not more than a dozen people on the planet would understand the theorem. Maybe that was the whole point. The layman with his curiosity and his weathered dirty hand on
a shovel handle, may inquire as to what connection exists between the endless equations with their odd symbols, and the physical world. He is answered with more maths and sometimes ridicule.
This post carries on from the introductory post, looking specifically at the mathematical foundations of the theory.
By 1632 the Catholic Galileo had developed complicated equations which would be the basis of mechanical physics. These equations described the transformation between the coordinates of two inertial
frames (objects on a grid). The equations are the basis of STR theory. In essence Galileo’s transformation model can be summarised as:
x^1 = vt -> where x is a coordinate of an event in one inertial frame (called S usually, with S being an object eg a moving train); x^1 is the coordinates of the same event in another inertial
frame (S') moving at a constant velocity
y^1 = y -> the same as x in the context of y
z^1 = z -> same as x, y, in the context of z
t^1 = t where t = inertial time
In this theorem the transformations in the y and z directions are identity transformations, which means that there is no change along those axes. The above is very similar to what is deployed in the
Using the above maths Galileo describe the transformations between frames of reference at speeds much less than the speed of light. It is not relativistic. The theory cannot describe the relationship
between space and time at relativistic speeds. The Lorentz transformation theory by contrast, can articulate space and time at all relativistic speeds, including at the speed of light.
The Galilean transformation was superseded by the Dutchman Hendrik Lorentz’s equations or the ‘Lorentz transformations’, published in 1895. It is undeniable that much of STR is derived from Lorentz
though Einstein did not credit the Dutchman. It is well-documented that Albert Einstein corresponded with Lorentz, and he had studied Lorentz's work. The Lorentz transformations describe the
mathematical relationship between the coordinates of events in different inertial frames and are the core of Einstein's theory of special relativity.
In their correspondence, Lorentz and Einstein discussed various aspects of electromagnetism, and Einstein was familiar with Lorentz's efforts to reconcile the phenomena of electromagnetism with the
principles of classical Galilean mechanics. Einstein took Lorentz’s work and extended it, adding the 2 postulates analysed in the first post, and E=mc2 or the conversion of energy and mass. This
would be proven in nuclear reactions. E is the energy of the object, m its mass, and c is the speed of light in a vacuum.
In reading both theorems I am not convinced that Einsteins’ theory is that much different. Einstein uses Lorentz whole hog and adds E=mc2, a formulae developed by others though he stated he arrived
at it independently. I don’t find this convincing. For the record, E=mc2 is initially an output of electro-magnetic theory, with Maxwell and others (for example, the British scientist J.J. Thomson,
in 1881, and the Italian Olinto De Pretto, in 1903), being its true inventors, not Einstein. Einstein dispensed with the ‘ether’ (luminiferous particles which interact with light and
electro-magnetism, which is indeed different than the idea of ‘dark matter’ which does not interact with light). He reformed the velocity of light to be a constant in a vacuum as well as proposing
the exchange of energy with mass. Lorentz’s ideas are the real foundation of STR.
Lorentz’s calculations run to many pages. In summary and to simplify some core aspects of the theory found in textbooks and on sites would be the following.
We can see the similarity with, and the use of Galileo’s transformation equations.
Lorentz’s theorem can be summarised in the following way:
1. -t = time and x = space coordinates for one reference object, called S, moving at a certain velocity or v, relative to another reference object (or frame) called S^1
2. v = the relative velocity between two inertial frames of reference or 2 objects within the same or different coordinates
3. -For both objects S and S^1, coordinates or location maps covering both objects, are calculated using the variables, y, z an y^1, z^1 for both objects along the y and z axes
4. -The z axis is a 3^rd dimension axis to imitate space is added to the above 2-dimensional diagram (the z axis was added in 1908, by Einstein’s math professor Hermann Minkowski)
5. -c is the speed of light (for Einstein, this meant only a vacuum)
6. The symbol γ is gamma or the Lorentz factor given below
The full equations and explanations can be found in any physics textbook online. Many sites possess calculators where you can stroll through the theorem against a thought or paper experiment.
The maths is difficult to wade through.
A key difference between the 2 theorems is that Lorentz, like Maxwell and others, believed in an ether, or a medium in space which could interact with and propagate light, whereas Einstein did not.
However, both theories are so similar that they are conflated with each other, with STR categorised as the ‘Lorentz-Einstein model’. Lorentz’s theory in and of itself is not relativistic, but
Einstein’s is. However, proofs offered for STR include the relativistic outputs of Lorentz’s equations. This is both tautological and usually tangential. You cannot use part of a theorem to prove the
The maths is abstract but can be applied to moving objects. In his 1905 paper, Einstein used the analogy of a train and an observer on a train platform. Assume we have an observer on a train
platform, watching a train moving past the station at 60 mph. Let’s put two objects emitting light, one at the beginning of the train, and one at the end of the train. Calculations can be made to
show the separation in time between when the lights flashed as seen by either the platform observer, or a second observer seated in the train in the middle of the car.
Logically the person in the train would never see the flashing lights but that reality is dismissed in favour of trying to prove the relativistic nature of objects in motion and their light signals
as seen by the 2 observers, one static on a platform, the other in a train moving at 60 mph away from the platform. What does it mean?
1. The Lorentz transformation purportedly shows that time and space are not absolute and can be different for observers in relative motion.
2. The Lorentz factor accounts for the relativistic effects of time dilation and length contraction with the length (or mass) of an object becoming smaller as it moves away from an observer.
Lorentz offers the ‘transformation’ equations to show relativity in space and time. He also uses his ‘factor’ to ‘prove’ relativity between objects in time dilation (the twin’s paradox) and length
Based on the above ideas, perhaps the key point in special relativity is that the relationships between space and time coordinates are intertwined. Observers moving relative to each other at a
constant velocity will experience different perceptions of time and space.
As already stated a major addition to Lorentz’s theorem by Einstein, is the formulation of Energy = mass x the speed of light in a vacuum square (E=mc2). This theory states that mass and energy are
interchangeable. Nuclear reactions offer proof of this. Fair enough. For example, a small amount of mass may be converted into a large amount of energy.
Within STR the mainstream narrative offers that this equation is a key element in describing how energy and mass are observed in different inertial reference frames moving relative to each other. I
am not sure this is true. It can lead to theories about time dilation for example, or as already stated, length contraction where an object that is moving away relative to an observer will appear
contracted in the direction of motion (which may be incorrect, more later).
Much of this is only theory and still open to dispute. Though E=mc², is a fundamental expression in the theory of relativity, it is not and cannot be used as a direct proof of time dilation or the
concept of different clocks running at different speeds. This would be tautological. At best it helps describe such phenomena if they exist.
The main areas of importance of STR, cited within the mainstream science literature include:
1. Time-dilation (not as clear cut as presented, scanty proofs, but can be observed),
2. Red-shift calculations (absolutely nothing to do with STR and eviscerated here),
3. GPS systems (if STR did not exist these would still work),
4. Bending of light (this seems correct).
Two out of four. The above are not the vital points about STR. There are two implications of STR in my opinion. More here
Scientism and the Special Theory of Relativity. Part One (a), A layman’s overview of STR
An introduction to a complex but important topic that has many implications if any of its supposed 'proofs' are disproven.
Why is Einstein’s Special Theory of Relativity or STR and its purported defects of importance with all that has gone on and is going on in our world? It is a very complex domain. Those of us who have
spent time studying and investigating this topic know of its importance both in and out of science, especially given the often hysterical and exaggerated claims by its supporters as to the limitless
wonders and knowledge it bestows on humanity. STR has indeed conferred proofs and benefits but much of it is entirely open to dispute though the single narrative of ‘the science’ will never discuss
such issues.
There are many reasons for laypeople to dig into STR besides its practical usage in some technical and scientific applications.
First, anyone who studies STR objectively must notice that STR is one of the best examples in modern history of the cult of ‘science’ or Scientism, in which arcana and esoterica, often unrelated to
physical evidence, become ‘laws’. From these ‘postulates’ one does notice the creation of entire industries and compliant media, replete with funding and power. This is now how ‘modern science’
operates. It often has little to do with physical science.
Second, the elevation of ‘science’ to cult like status, including ‘heroes’ like Einstein who are to be feted and worshipped, provides a cultural backdrop for totalitarianism as evidenced by the
Corona plan-scam-demic in which ‘science’ (fiction only), with its priesthood of ‘scientists’ and ‘experts’ clinging to the only holy gospel of secular truth, were invoked as rationale for the
complete destruction of freedom and rights. Corona can be accurately described as a pilot project of geo-Fascism. Scientism or the religion of science, is essential for a technocratic fascism, with
its exalted prophets, priests, cardinals and bishops, against whom no heresy or heretic can stand.
Third, if there is a disconnect between STR and practical physical reality this must by default, retard and obstruct actual scientific advancement, especially in physics and cosmology. Entire tracts
of both would need to be reassessed when enough brave souls dare to put STR to proper physical testing and assess its tautological inconsistencies. This includes the obvious issues and disproofs of
the Big Bang theology.
The following is Part One of 4 articles on the topic, split into equal measure due to complexity and length.
Part One is split between (a) and (b) to introduce the subject. Part One (a) looks at the theory, its context and its primary focus. Part One (b) delves into the underlying mathematical models.
Part Two discusses the historical context of STR, and key antecedental and contemporary figures of Einstein including Maxwell, and Lorentz along with mathematical theorists like Minkowski who
greatly influenced Einstein.
Part Three will focus on the famed scientist Herbert Dingle’s detailed and never answered critique of STR (the only replies were ad hominins and completing ignoring his maths and proofs). Dingle
was of course slandered, attacked, and degraded but his many objections were never answered by either experimental proofs or logical mathematics.
Part Four will look at other general objections to STR and its many problems outside of Dingle’s complaint. Many of these are detailed, logical and experimentally based. They are of course
dismissed as ‘outside the mainstream’ as if that is a serious defect. Part Four will also summarise the implications of what has been investigated.
In short, any layperson with a curious mind, possessing average logical and mathematical skills can understand STR. The priests and prophets of the ‘science’, many of whom do not understand the
theory, keep it cloaked in mystery and incense to elicit compliance and to evade incisive and debilitating questions. Can you imagine a schoolteacher reacting to a curious student’s statement, ‘yes
teacher E=mc2 is interesting but has little to do with the theory of relativity, or relative motion and using that as proof is circular reasoning since it is embedded in the theorem’.
The abstract, closeted nature of theoretical arcana which are declared to be laws, is precisely why STR should be looked at. It is about time that ‘science’ rejoin reality and be forced to explain
its ‘logic’ in plain terms, to common people. We should encourage questions and debate.
In 1905 Einstein issued his paper on STR which was divided into two parts:
I. Kinematical Section;
II. Electrodynamical Section.
Kinematics is the branch of mechanics concerned with objects in motion, but not with the forces involved and electrodynamics is the study of moving electric charges and their interaction with
magnetic and electric fields. The whole essence of STR is contained in section I on kinematics, which is significant, though I found little commentary on it in the mainstream publications (more
STR is usually portrayed as a revolt against Newtonian mechanics. I am not sure this is correct. What Einstein was trying to do was correct some issues with Newton’s theorems on gravity – in
particular the orbit of Mercury. He was trying to support it, not displace it. He offered the view that each object through gravity, can ‘curve’ the fabric of space and time around them, forming a
sort of a depression, akin to a heavy object resting on your sofa cushion. Objects including your cat or light would fall into the depression.
This is the space-time curve, which Newton did not know about, and which is now attributed to Einstein but it was not his ‘discovery’ but originally that of Minkowski a mathematician (more later).
More here
$cientism: Germ and Virus theory nonsense. Béchamp’s experiments which disprove germ theory
The colossal failure of modern health and medicine and the creation of the criminal Pharma industry - all based on a myth.
It is obvious to anyone who has a functioning cortex, that the Corona plandemic used the mythical scariants of ‘viruses’ as the casus belli for what is rightly termed a poisonous and often lethal
injection marketed as health care, designed to accrue profits and power. As the founder of Merck said, create the disease, sell the cure. Since the days of the quack and fraud Jenner, ‘germs’ and
‘viruses’ have invaded the imaginations of the ‘science’ and its ‘experts’ and our daily lexicon, much to the detriment of health and freedom. As Antoine Béchamp noted in his book ‘Les Microzymas’ p.
819 (1878), on the incoherence of ‘germ’ theory:
‘In all the experiments of recent years, it has been the microzyma proper to an animal and not a germ of the air that has been found to be the seat of the virulence. No one has even been able to
produce with germs obtained from the atmosphere any of the so-called parasitic diseases. Whenever, by inoculation, a typical known malady has been reproduced, it has been necessary to go and take
the supposed parasite from a sick animal; thus to inoculate tuberculosis, the tubercle has to be taken from the subject affected.’
No germ or virus carrying disease or infection has ever been isolated in the atmosphere or even in a human cell. Germ theory wrongly conflates external ‘germs’ with internal bacteria and microzymas
which do produce disease as explained below. The implications of this fraud and incoherence is staggering.
Bacteria exist in their billions inside each human body. What Béchamp discovered was that these ‘microbes’ or microzymas as he named them, can develop into bacteria and are not only vital to clean
out the human body and cell detritus, but they can change shape, form and function if the ‘terrain’ or environment of the body is assaulted with poisons and becomes toxic. This completely upends
Pasteurian myth and our entire modern medical and healthy systems.
Inside each human cell, there is a nucleus which contains DNA genetic material (deoxyribonucleic acid). The nucleus has a membrane which is similar to the cell membrane. There is also a pathway to
the nucleus of each human cell called the endoplasmic reticulum. This has ribosome's connected to it which also contain genetic material called RNA (ribonucleic acid), or the ‘logic’ that transcribes
the ‘plans’ from DNA to produce functionality. This endoplasmic reticulum is also made of cellular membrane material.
Each human cell is therefore a very complicated ecosystem with ‘organelles’ floating in a watery mixture called a cytoplasm which also harbours RNA.
In a case of toxaemia poisoning, a cell, containing the complexity mentioned above, will burst, providing an abundance of cellular membrane material containing the sticky proteins of DNA and RNA
which are now floating in the fluid between the other living cells. This is called the interstitial fluid. ‘Germs’ are simply bacteria which normally do waste management but have now changed their
shape and function to feed on the cellular destruction, spreading the toxicity and detritus. As with any living organism these pleomorphic (many forms) bacterium emit waste, fluids and if the cell is
damaged and toxic, poisons. In simple terms this process is what causes an illness.
The important point is that if the host’s toxicity is pervasive, the bacterial effluence and invasion can damage cells. This is not from an external barbarian-germ or virus invasion, but due entirely
to the detritus and cellular debris now building up and furthering the toxicity of the host.
Immunity is a defence system the body uses to eliminate toxic matter that has built up over a period of time. The immune system is wonderfully complex and starts when the ‘front lines’ of the body’s
defence have not been able to eliminate toxic matter effectively, and the toxicity has breached the circulatory system and organs. Our immune systems will then issue white blood cells which are
designed to remove dead organic matter, or garbage, from the body.
Within the immune system antibodies are also produced. These are specially synthesized body proteins which aid the white blood cells when toxicity reaches more extreme levels and when the body is
trying to eject a surfeit of poison. For the record injecting antibodies in the form of jabs and stabs is not only useless but dangerous, given that the immune system will not recognise the foreign
antibodies as anything other than a foreign toxic agent and the compound proteins they contain cannot be broken down and will directly damage organs and tissues.
F. Harrison, a Professor of Bacteriology at McGill University, wrote a book, ‘Historical Review of Microbiology, published in Microbiology’, in which he says: “Geronimo Fracastorio (an Italian poet
and physician, 1483 – 1553) of Verona, published a work (De Contagionibus et Contagiosis Morbis, et eorum Curatione) in Venice in 1546 which contained the first statement of the true nature of
contagion, infection, or disease organisms, and of the modes of transmission of infectious disease.” This predates Pasteur by 300 years.
Fracastorio divided diseases into different groups, one of which infected the host by either immediate contact, through intermediate agents, or at a distance through the air. Organisms which cause
disease were in his view, composed of viscous or glutinous matter. These particles, too small to be seen, were also capable of reproduction in appropriate media, and became pathogenic through the
action of animal heat. These concepts anticipate Pasteur’s ideas. Fracastorio did not possess a microscope and could not know that these substances might be individual living organisms.
According to Harrison the first compound microscope was made in 1590 in Holland, but it was not until about 1683 that anything was built of sufficient power to show up bacteria. Harrison relates: “In
the year 1683, Antonius van Leenwenhoek, a Dutch naturalist and a maker of lenses, communicated to the English Royal Society the results of observations which he had made with a simple microscope of
his own construction, magnifying from 100 to 150 times. He found in water, saliva, dental tartar, etc., entities he named animalcula.” More here
$cientism and Louis Pasteur as a case study. Part II
Theories and fraud in lieu of real science and proof – Germs, Viruses, Stabbinations
In Part I about Louis Pasteur, it was offered that much of his work was of great scientific value and merit. A long list of real scientific discoveries is certainly attributable to Pasteur and his
associates, many of them beneficial, verifiable, and reproducible. However, when the trained chemist strays into diseases, immunology and poisoned concoctions named after the cow, he is supremely out
of his depth, reflected in the fraud, misuse of data, and even outright mendacity in forwarding claims around the germ theory of disease, flying viruses which spread infections and stabbinations
which provide succour and safety for the afflicted.
In short, Pasteur made the unfortunate but predictable journey from scientist to Scientism. Like the unimpressive venal corrupt quack Edward Jenner, Pasteur became a Public Relations salesman, a
sophist and hand-waver, eager to accrue credit, assets and attract the adoring gaze of a redeemed, sacralised, and saved public. The rush to fame if not fortune prevented the Catholic Pasteur from
entering the narrow gate.
We now turn our attention to a very dense and excitable book on this aspect of Pasteur by Ethel D Hume ‘Béchamp or Pasteur’ first published in 1923. The investigator is very hostile to Pasteur and
his ‘science’, and this should be borne in mind. Transparently she does declare this bias, which is unlike ‘modern science’ which hides their worldviews and sources of finance and the power cliques
which direct their ‘research’. No need for transparency in ‘modern science’.
This book ravages all of Pasteur's work and discoveries which seems rather excessive. In essence, fermentation, silkworm infection, viniculture spoilation, biogenic life creation and other claimed
discoveries by Pasteur are alleged frauds, with Pasteur stealing the ideas, conclusions, and even experimental proofs from Béchamp and others. It is more likely that many researchers were involved in
the same domains, doing different varieties of experimentation, and given that most were writing letters and sharing information it is rather intolerable that we ascribe fraud to all of Pasteur’s
work. It is better to discern the good from the bad. So, we will skip much of the book and focus on the 3 main areas of controversy, germs, viruses, stabbinations and the related fraud and deceit.
Hume’s detail is meticulous and much of it has been confirmed by observations and research since 1923. | {"url":"https://stferdinandiii.com/archive/11/2023","timestamp":"2024-11-07T19:07:38Z","content_type":"application/xhtml+xml","content_length":"1048884","record_id":"<urn:uuid:538cd1cb-9562-4a22-8abe-d2e632d69363>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00730.warc.gz"} |
100 Pips to Dollars: How Much is Your Trading Worth?
For forex traders, understanding the value of pips is essential for profitable trading. Pips, short for "percentage in point," represent the smallest unit of measure for currency movements. In the
foreign exchange market, pips are used to measure the change in the exchange rate between two currencies. Traders use pips to determine their risk-to-reward ratio and calculate their profits and
One common question among forex traders is how much is 100 pips worth in dollars. The answer depends on the currency pair being traded and the size of the position. Let's take a closer look at how to
calculate the value of 100 pips in dollars.
First, it's important to understand the concept of pipettes. A pipette represents one-tenth of a pip. So, if the quote for the EUR/USD currency pair moves from 1.2500 to 1.2501, that's a one-pip
move. However, if the quote moves from 1.2500 to 1.2505, that's a five-pipette move or 0.5 pips.
To calculate the value of a pipette, traders need to know the size of their position and the pip value for the currency pair being traded. The pip value represents the dollar value of a pip and
varies depending on the currency pair and the lot size.
For example, let's say a trader has a standard lot position on the EUR/USD currency pair, which has a pip value of $10. If the price moves from 1.2500 to 1.2600, that's a 100-pip move. To calculate
the dollar value of 100 pips, we multiply the pip value by the number of pips. In this case, 100 x $10 = $1,000. So, a 100-pip move in a standard lot position on the EUR/USD currency pair is worth
However, if the trader has a mini lot position on the same currency pair, which has a pip value of $1, the value of 100 pips would be $100. If the trader has a micro lot position, which has a pip
value of $0.10, the value of 100 pips would be $10.
It's important to note that the value of pips can fluctuate depending on the exchange rate and the size of the position. Traders should always use proper risk management and calculate their pip value
before entering a trade.
In conclusion, the value of 100 pips in dollars depends on the currency pair being traded and the size of the position. Traders can calculate the pip value using the pip value for the currency pair
and the size of their position. By understanding the value of pips, forex traders can make informed trading decisions and manage their risk effectively.
Go up | {"url":"https://towla24.com/how-much-is-100-pips-in-dollars/","timestamp":"2024-11-04T16:57:39Z","content_type":"text/html","content_length":"153171","record_id":"<urn:uuid:7dbcc3fb-3897-4c40-b4d1-982c060c32ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00232.warc.gz"} |
Binary interactive error resilience beyond
Interactive error correcting codesInteractive error correcting codes are codes that encode a two party communication protocol to an error-resilient protocol that succeeds even if a constant fraction
of the communicated symbols are adversarially corrupted, at the cost of increasing the communication by a constant factor. What is the largest fraction of corruptions that such codes can protect
against? If the error-resilient protocol is allowed to communicate large (constant sized) symbols, Braverman and Rao (STOC, 2011) show that the maximum rate of corruptions that can be tolerated is.
They also give a binary interactive error correcting protocol that only communicates bits and is resilient to fraction of errors, but leave the optimality of this scheme as an open problem. We answer
this question in the negative, breaking the barrier. Specifically, we give a binary interactive error correcting scheme that is resilient to fraction of adversarial errors. Our scheme builds upon a
novel construction of binary list-decodable interactive codes with small list size.
Publication series
Name Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume 2020-November
ISSN (Print) 0272-5428
Conference 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
Country/Territory United States
City Virtual, Durham
Period 11/16/20 → 11/19/20
All Science Journal Classification (ASJC) codes
• Communication Complexity
• Error Resilience
• Interactive Coding
Dive into the research topics of 'Binary interactive error resilience beyond'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/binary-interactive-error-resilience-beyond","timestamp":"2024-11-04T07:43:35Z","content_type":"text/html","content_length":"53132","record_id":"<urn:uuid:c0569149-3fcc-4fcf-b737-522ebb2f4c70>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00090.warc.gz"} |
Creating an 'overflow' semicircle donut chart
I'm trying to create a semicircle donut chart in the dashboard to show how much the team overspent towards a budget.
For example, the budget is $100 per month but the team spent $150. Is there a way of creating an 'overflow' semicircle donut chart that shows the extra? Instead of showing a full semicircle donut
chart for those more than 100%.
Attached is a sample for reference.
Best Answer
• The general idea is that we are using a full circle. Since your base amount is half of that circle, the other half will add up to the same amount. This means overage plus white. Since we know
that the base amount is (for example) 100, we know that overage plus white must equal 100, so you would do 100 minus overage to get the white.
Note: You will also need to use metrics widgets to display each of the percentages.
• You would have a full circle chart and make one section white. To get the white section automated, you would use something along the lines of subtracting the overflow from the cap.
It isn't quite exactly what you have in your screenshot, but this is about the closest we can get in Smartsheet:
• Hi Paul,
Thanks for the image. Could you elaborate more on how to get the white section automated?
• The general idea is that we are using a full circle. Since your base amount is half of that circle, the other half will add up to the same amount. This means overage plus white. Since we know
that the base amount is (for example) 100, we know that overage plus white must equal 100, so you would do 100 minus overage to get the white.
Note: You will also need to use metrics widgets to display each of the percentages.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/109452/creating-an-overflow-semicircle-donut-chart","timestamp":"2024-11-13T15:34:37Z","content_type":"text/html","content_length":"412516","record_id":"<urn:uuid:12fb8d57-3e81-4bee-8684-6c0bfa4917be>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00685.warc.gz"} |
Asymptotically Optimum Properties of Certain Sequential Tests
Ann. Math. Statist. 39(4): 1244-1263 (August, 1968). DOI: 10.1214/aoms/1177698250
Let $X_1, X_2, \cdots$ be independent and identically distributed random variables whose common distribution is of the one-parameter Koopman-Darmois type, i.e., the density function of $X_1$ relative
to some $\sigma$-finite nondegenerate measure of $F$ on the real line can be written as $f(x, \theta) = \exp (\theta x - b(\theta))$, where $b(\theta)$ is some real function of the parameter $\
theta$. Consider the hypotheses $H_0 = \{\theta \leqq \theta_0\}$ and $H_1 = \{\theta \geqq \theta_1\}$ where $\theta_0 < \theta_1$ and $\theta_0, \theta_1$ are in $\Omega$, the natural parameter
space. We want to decide sequentially between the two hypotheses. Suppose $l(\theta)$ is the loss for making a wrong decision when $\theta$ is the true parameter and assume $0 \leqq l(\theta) \leqq
1$ for all $\theta$ and $l(\theta) = 0$ if $\theta$ is in $(\theta_0, \theta_1)$, i.e., $(\theta_0, \theta_1)$ is an indifference zone. Let $c$ be the cost of each observation. It is sufficient to
let the decision depend on the sequence $(n, S_n), n \geqq 1$, where $S_n = X_1 + \cdots + X_n$. We shall consider the observed values of $(n, S_n)$ as points in a $(u, v)$ plane. Then, for any test,
the region in the $(u, v)$ plane where sampling does not stop is called the continuation region of the test. A test and its continuation region will be denoted by the same symbol. Schwarz [4]
introduced an a priori distribution $W$ and studied the asymptotic shape of the Bayes continuation region, say $B_W(c)$, as $c \rightarrow 0$. He showed that $B_W(c)/\ln c^{-1}$ approaches, in a
certain sense, a region $B_W$ that depends on $W$ only through its support. Whereas Schwarz's work is concerned with Bayes tests, in this paper the main interest is in characteristics of sequential
tests as a function of $\theta$. In particular, it is desired to minimize the expected sample size (uniformly in $\theta$ if possible) subject to certain bounds on the error probabilities. Our
approach, like Schwarz's, is asymptotic, as $c \rightarrow 0$. It turns out that an asymptotically optimum test--in the sense indicated above, is $B_W \ln c^{-1}$ if $W$ is a measure that dominates
Lebesgue measure. Such a measure will be denoted by $L$ (for Lebesgue dominating) from now on. Thus, Bayes tests, as a tool, will play a significant role in this paper. In order to prove the optimum
characteristic of $B_L \ln c^{-1}$, some other results, of interest in their own right, are established. For any $W$ satisfying certain conditions that will be given later, we show that the stopping
variable $N(c)$ of $B_W(c)$ approaches $\infty$ a.e. $P_\theta$ for every $\theta$ in $\Omega$. This result together with Schwarz's result that $B_W(c) \ln c^{-1}$ approaches a finite region, leads
to the following results: (i) for $B_W(c), E_\theta N(c)/\ln c^{-1}$ tends to a constant for each $\theta$ in $\Omega$ and (ii) the same is true for the stopping variable of $B_W \ln c^{-1}$.
Furthermore, it is shown that for $B_L \ln c^{-1}$ the error probabilities tend to zero faster than $c \ln c^{-1}$. Consequently, the contributions of the expected sample sizes of both $B_L \ln c^
{-1}$ and $B_L(c)$ to their integrated risks, over any $L$-measure, approach 100%. Moreover $B_L \ln c^{-1}$ is asymptotically Bayes. The last result can be shown without (i) since it is sufficient
to show (ii) and to apply the same argument used by Kiefer and Sacks [3] in the proof of their Theorem 1. But we show (i) because of its intrinsic interest and present a different proof using (i).
Kiefer and Sacks assumed a more general distribution for $X_1$, constructed a procedure $\delta^{'I}_c$ and showed that it is asymptotically Bayes. Our $B_L \ln c^{-1}$ is somewhat more explicit than
their $\delta_c'I$. We would also like to point out that an example of $B_L \ln c^{-1}$, when the distribution is normal, is very briefly discussed in their work. We shall restrict ourselves to a
priori distribution $W$ for which $\sup (\mod W)H_0 = \theta_0, \inf (\mod W)H_1 = \theta_1$ and $0 < W(H_0 \cup H_1) < 1$. The phrase "for any $W"$ or "for every $W$" is to be understood in that
sense. Any Lebesgue dominating measure satisfies these conditions and also the following type of $W$ that will be used: the support of $W$ consists of $\theta_0, \theta_1$ and a third point $\theta^\
ast, \theta_0 < \theta^\ast < \theta_1$. Such a $W$ will be called a $\theta^\ast$-measure, and the corresponding $B_W$ denoted by $B_{\theta^\ast}$. From Schwarz's equations for $B_W$ it follows
readily that $B_L \subset B_W$ for every $W$. In particular, $B_L \subset B_{\theta^\ast}$. As a consequence, the statement about the error probabilities as well as others concerning $B_L \ln c^{-1}$
in the last paragraph, remain true when $L$ is replaced by $\theta^\ast$ or any $W$. Those geometric characteristics will be dealt with in Section 2. We shall also show there that $\partial B_{\theta
^\ast}$, the boundary of $B_{\theta^\ast}$ (which consists of line segments), is tangent to $\partial B_L$ at some point, and that if $\theta^\ast$ is such that $b'(\theta^\ast) = (b(\theta_1) - b(\
theta_0))/(\theta_1 - \theta_0)$ then $\max_{(u, \nu)\text{in}B_L} u = \max_{u,v)\text{in}B_\theta^\ast} u$. Let the ray through the origin and with slope equal to $E_\theta X_1$ intersect $\partial
B_L$ at $(m(\theta), m(\theta)E_\theta X_1)$. In Section 3, after proving $\lim_{c\rightarrow 0} N(c) = \infty$ a.e. $P_\theta$, we show $\lim_{c\rightarrow 0} N(c)/\ln c^{-1} = m(\theta)$ a.e. $P_\
theta$ and $\lim_{c\rightarrow 0} E_\theta N(c)/\ln c^{-1} = m(\theta)$. It is shown in Section 4 that $\sup_{\theta \text{in} H_0 \mathbf{\cup} H_1} P_\theta$ (error $\mid B_L \ln c^{-1}) = o(c \ln
c^{-1})$. The main results are given in Section 5. We first show that after dividing by $c \ln c^{-1}$, the difference of the integrated risks of $B_L \ln c^{-1}$ and $B_W(c)$, for any $W$, tends to
zero. It follows from this result that $B_L \ln c^{-1}$ asymptotically minimizes the maximum (over $\theta$ in $\Omega$) expected sample size in $\mathscr{F}(c)$, a family of tests whose error
probabilities are bounded by $\max_{i=0,1} P_{\theta_i}$ (error $\mid B_L \ln c^{-1}$). The precise statement is given in Theorem 5.1. A sharper result under a stronger hypothesis is given in Theorem
5.2 which states that $B_L \ln c^{-1}$ asymptotically minimizes the expected sample size $E_\theta N$ for each $\theta, \theta_0 < \theta < \theta_1$, among all procedures of $\mathscr{F}(c)$ for
which $E_{\theta_0}N/\ln c^{-1}$ and $E_{\theta_1}N/\ln c^{-1}$ are bounded in $c$.
Download Citation
Seok Pin Wong. "Asymptotically Optimum Properties of Certain Sequential Tests." Ann. Math. Statist. 39 (4) 1244 - 1263, August, 1968. https://doi.org/10.1214/aoms/1177698250
Published: August, 1968
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177698250
Rights: Copyright © 1968 Institute of Mathematical Statistics
Vol.39 • No. 4 • August, 1968 | {"url":"https://www.projecteuclid.org/journals/annals-of-mathematical-statistics/volume-39/issue-4/Asymptotically-Optimum-Properties-of-Certain-Sequential-Tests/10.1214/aoms/1177698250.full","timestamp":"2024-11-04T12:40:46Z","content_type":"text/html","content_length":"160105","record_id":"<urn:uuid:e5865722-41c8-4c8b-bd7e-7b3fa02dda6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00541.warc.gz"} |
ent Notes
1. Nonconstant Dividends. Apocalyptica Corporation is expected to pay the following dividends over the next four years: $6, $12, $17, and $3.25. Afterward, the company pledges to maintain a constant
5 percent growth rate in dividends, forever. If the required return on the stock is 11 percent, what is the current share price?
With nonconstant dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the present value of the future stock price, plus the present value of
all dividends during the nonconstant growth period. The stock begins constant growth after the fourth dividend is paid, so we can find the price of the stock at Year 4, when the constant dividend
growth begins, as:
P[4] = D[4] (1 + g) / (R – g)
P[4] = $3.25(1.05) / (.11 – .05)
P[4] = $56.88
The price of the stock today is the present value of the first four dividends, plus the present value of the Year 4 stock price. So, the price of the stock today will be:
P[0] = $6 / 1.11 + $12 / 1.11^2 + $17 / 1.11^3 + $3.25 / 1.11^4 + $56.88 / 1.11^4
P[0] = $67.18
2. Supernormal Growth. Burton Corp. is growing quickly. Dividends are expected to grow at a rate of 25 percent for the next three years, with the growth rate falling off to a constant 6 percent
thereafter. If the required return is 11.5 percent and the company just paid a dividend of $2.50, what is the current share price?
With nonconstant dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the present value of the future stock price, plus the present value of
all dividends during the nonconstant growth period. The stock begins constant growth after the third dividend is paid, so we can find the price of the stock in Year 3, when the constant dividend
growth begins as:
P[3] = D[3] (1 + g[2]) / (R – g[2])
P[3] = D[0] (1 + g[1])^3 (1 + g[2]) / (R – g[2])
P[3] = $2.50(1.25)^3(1.06) / (.115 – .06)
P[3] = $94.11
The price of the stock today is the present value of the first three dividends, plus the present value of the Year 3 stock price. The price of the stock today will be:
P[0] = $2.50(1.25) / 1.115 + $2.50(1.25)^2 / 1.115^2 + $2.50(1.25)^3 / 1.115^3 + $94.11 / 1.115^3
P[0] = $77.35
3. Nonconstant Growth. Metallica Bearings, Inc., is a young start-up company. No dividends will be paid on the stock over the next nine years, because the firm needs to plow back its earnings to fuel
growth. The company will then pay a dividend of $19 per share 10 years from today and will increase the dividend by 5 percent per year thereafter. If the required return on this stock is 13 percent,
what is the current share price?
Here, we have a stock that pays no dividends for nine years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that
point. It is important to remember that the general constant dividend growth formula is:
P[t] = [D[t] × (1 + g)] / (R – g)
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year 9. The dividend growth model is similar to the present value of an annuity and the present
value of a perpetuity: The equation gives you the present value one period before the first payment. So, the price of the stock in Year 9 will be:
P[9] = D[10] / (R – g)
P[9] = $19.00 / (.13 – .05)
P[9] = $237.50
The price of the stock today is simply the PV of the stock price in the future. We simply discount the future stock price at the required return. The price of the stock today will be:
P[0] = $237.50 / 1.13^9
P[0] = $79.06
4. Wesen Corp. will pay a dividend of $3.14 next year. The company has stated that it will maintain a constant growth rate of 4.5 percent a year forever. If you want a return of 12 percent, how much
will you pay for the stock? What if you want a return of 8 percent? What does this tell you about the relationship between the required return and the stock price?
Here, we need to value a stock with two different required returns. Using the constant growth model and a required return of 12 percent, the stock price today is:
P[0] = D[1] / (R – g)
P[0] = $3.14 / (.12 – .045)
P[0] = $41.87
And the stock price today with a required return of 8 percent will be:
P[0] = D[1] / (R – g)
P[0] = $3.14 / (.08 – .045)
P[0] = $89.71
5. Take Time Corporation will pay a dividend of $3.65 per share next year. The company pledges to increase its dividend by 5.1 percent per year, indefinitely. If you require a return of 11 percent on
your investment, how much will you pay for the company’s stock today?
The Toy Chest will pay an annual dividend of $2.64 per share next year and currently sells for $48.30 a share based on a market rate of return of 11.67 percent. What is the capital gains yield?
g = .1167- ($2.64/$48.30) = .0620, or 6.20 percent
6. For the past six years, the price of Slippery Rock stock has been increasing at a rate of 8.21 percent a year. Currently, the stock is priced at $43.40 a share and has a required return of 11.65
percent. What is the dividend yield?
Dividend yield = .1165-.0821 = .0344, or 3.44 percent
7. Village East expects to pay an annual dividend of $1.40 per share next year, and $1.68 per share for the following two years. After that, the company plans to increase the dividend by 3.4 percent
annually. What is this stock’s current value at a discount rate of 13.7 percent?
P[3]= ($1.68 × 1.034) / (.137 – .034)= $16.86
P[0] = $1.40 / 1.137 + $1.68 / 1.137^2 + ($1.68+ $16.86)/ 1.137^3
P[0]= $15.15
8. Toy Mart recently announced that it will pay annual dividends at the end of the next two years of $1.60 and $1.10 per share, respectively. Then, in Year 5 it plans to pay a final dividend of
$13.50 a share before closing its doors permanently. At a required return of 13.5 percent, what should this stock sell for today?
P[0] = $1.60/1.135 + $1.10/1.135^2 + $13.50/1.135^5= $9.43
9. Business Solutions is expected to pay its first annual dividend of $.84 per share in Year 3. Starting in Year 6, the company plans to increase the dividend by 2 percent per year. What is the value
of this stock today, Year 0, at a required return of 14.4 percent?
P[5] = ($.84 ×1.02)/(.144-.02) = $6.91
P[0] =($.84/1.144^3) + ($.84/1.144^4) + [($.84 + 6.91)/1.144^5] = $5.01
10. This morning, you purchased a stock that will pay an annual dividend of $1.90 per share next year. You require a 12 percent rate of return and the dividend increases at 3.5 percent annually. What
will your capital gain be in dollars on this stock if you sell it three years from now?
P[0] = $1.90/(.12 -.035) = $22.35
P[3] = [$1.90 x (1.035)^3]/(.12 -.035) = $24.78
Capital gain = $24.78 -22.35 = $2.43
11. Dry Dock Marina is expected to pay an annual dividend of $1.58 next year. The stock is selling for $18.53 a share and has a total return of 9.48 percent. What is the dividend growth rate?
g =.0948- ($1.58/$18.53) = .0095, or .95 percent | {"url":"https://www.student-notes.net/ewy/","timestamp":"2024-11-11T07:01:21Z","content_type":"application/xhtml+xml","content_length":"50237","record_id":"<urn:uuid:2c95f18d-972c-4288-a8fd-6fc4a8d1116e>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00000.warc.gz"} |
The National Curriculum for Mathematics in Year 6.
Number & Place Value
Our children will be taught to:
• read, write, order and compare numbers up to 10 000 000 and determine the value of each digit
• round any whole number to a required degree of accuracy
• use negative numbers in context, and calculate intervals across 0
• solve number and practical problems that involve all of the above.
Addition, Subtraction, Multiplication & Division
Our children will be taught to:
• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as
appropriate for the context
• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
perform mental calculations, including with mixed operations and large numbers.
• identify common factors, common multiples and prime numbers
use their knowledge of the order of operations to carry out calculations involving the 4 operations
• solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why
• solve problems involving addition, subtraction, multiplication and division
• use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.
Fractions (decimals & percentages)
Our children will be taught to:
• use common factors to simplify fractions; use common multiples to express fractions in the same denomination
• compare and order fractions, including fractions >1
• add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
• multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1/4 × 1/2 = 1/8 ]
• divide proper fractions by whole numbers [for example 1/3 ÷ 2 = 1/6]
• associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8]
• identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers are up to three decimal places
• multiply one-digit numbers with up to 2 decimal places by whole numbers
• use written division methods in cases where the answer has up to 2 decimal places
• solve problems which require answers to be rounded to specified degrees of accuracy
• recall and use equivalences between simple fractions, decimals and percentages,
• including in different contexts.
Ratio & Proportion
Our children will be taught to:
• solve problems involving the relative sizes of two quantities where missing values
• can be found by using integer multiplication and division facts
• solve problems involving the calculation of percentages [for example of measures and such as 15% of 360] and the use of percentages for comparison
• solve problems involving similar shapes where the scale factor is known or can be found
• solve problems involving unequal sharing and grouping using knowledge of fractions and multiples.
Our children will be taught to:
• use simple formulae
• generate and describe linear number sequences
• express missing number problems algebraically
• find pairs of numbers that satisfy an equation with two unknowns
• enumerate possibilities of combinations of 2 variables.
Our children will be taught to:
• solve problems involving the calculation and conversion of units of measure, using decimal notation up to 2 decimal places where appropriate
• use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal
notation to up to 3 decimal places
• convert between miles and kilometres
• recognise that shapes with the same areas can have different perimeters and vice versa
• recognise when it is possible to use formulae for area and volume of shapes
• calculate the area of parallelograms and triangles
• calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³].
Properties of Shape
Our children will be taught to:
• draw 2-D shapes using given dimensions and angles
• recognise, describe and build simple 3-D shapes, including making nets
• compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
• illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
• recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles.
Position & Direction
Our children will be taught to:
• describe positions on the full coordinate grid (all 4 quadrants)
• draw and translate simple shapes on the coordinate plane, and reflect them in the axes.
Our children will be taught to:
• interpret and construct pie charts and line graphs and use these to solve problems
• calculate and interpret the mean as an average. | {"url":"https://salehurst.thebridgefederation.org/mathematics-6/","timestamp":"2024-11-02T01:17:13Z","content_type":"text/html","content_length":"55976","record_id":"<urn:uuid:c731b0c9-871e-4238-b4dd-e67cc4627e1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00867.warc.gz"} |
The AVEDEV Function | SumProduct are experts in Excel Training: Financial Modelling, Strategic Data Modelling, Model Auditing, Planning & Strategy, Training Courses, Tips & Online Knowledgebase
A to Z of Excel Functions: The AVEDEV Function
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the AVEDEV function.
The AVEDEV function
I thought AVEDEV was an Indian cricket captain from the 1980’s. Apparently not. However, the definition of this statistical function may knock you for six: it returns the average of the absolute
deviations of data points from their mean, i.e. AVEDEV measures the variability in a data set.
It’s not quite as complicated as it sounds. If you have a set of numbers and calculated their average – the mean – and then for each value, took the average of how far each value was from the average
(ignoring whether the distance was positive or negative), you would calculate this value. It’s not as statistically robust as calculating the standard deviation, but it is probably a better
understood measure.
The AVEDEV function employs the following syntax to operate:
AVEDEV(number1, [number2], ...)
The AVEDEV function has the following arguments:
• number1, number2, ...: number1 is always required, but subsequent numbers are optional. The function takes between one and 255 arguments for which you wish the average of the absolute deviations.
You can also use a single array or a reference to an array instead of arguments separated by commas assuming this is your function delimiter.
It should be further noted that:
• AVEDEV is influenced by the unit of measurement in the input data
• Arguments must either be numbers or be names, arrays, or references that contain numbers
• Logical values and text representations of numbers that you type directly into the list of arguments are counted
• If an array or reference argument contains text, logical values, or empty cells, those values are ignored; however, cells with the value zero are included
• The equation for average deviation is:
Please see my example below:
We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every other business day. | {"url":"https://www.sumproduct.com/blog/article/a-to-z-of-excel-functions/the-avedev-function","timestamp":"2024-11-10T13:59:42Z","content_type":"text/html","content_length":"21024","record_id":"<urn:uuid:000fb513-278c-4d45-934d-df315bbc9546>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00420.warc.gz"} |
Gouy-Chapman Debye layer
14.5 PASS: Gouy-Chapman Debye layer
J.M. López-Herrera
sh debye.sh
Required files
debye.gfs (view) (download)
debye.sh points analytical
Running time
12 seconds
The Debye layer is the ionic concentration and potential distribution equilibrium structure that appears on the surface of a charged electrode in contact with solvents in which are dissolved ionic
species. Gouy and Chapman proposed a model of a diffuse Debye layer taking into account that the concentrations of these ionic species are governed by the combined effect of its thermal diffusion and
its electrostatic attraction or repulsion.
In the case of a plane electrode within a fully dissolved binary system (ion and counterions of valence z, |z|=1) the model reduces to the following dimensionless equations:
d^2 φ/d x^2 = −(n[+]−n[−])=2 sinh(φ)
with the boundary conditions: φ(0)=φ[o] and φ(x → ∞)=0. In the above equations the concentrations of anions and cations, n[+] and n[−] respectively, have been made dimensionless with the bulk
concentration, n[o], the potential φ with (K[b] T/e) and lengths with the Debye’s length λ[D] = [n[o] e^2/(K[b] T ε)]^1/2 where K[b] is Boltzmann’s constant, T the temperature, e the charge of the
electron and ε the permittivity of the fluid. | {"url":"http://gerris.dalembert.upmc.fr/gerris/tests/tests/debye.html","timestamp":"2024-11-14T12:21:37Z","content_type":"text/html","content_length":"4385","record_id":"<urn:uuid:a168c236-3a51-47cd-a17f-634440d607a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00742.warc.gz"} |
alculation of freezing index
Annex A
Definition and calculation of freezing index
A.1 General
This annex gives the method of calculation of the design freezing index F[d] from meteorological records of daily mean external air temperatures for the locality concerned.
A.2 defines the calculation of the freezing index, F, for one particular winter. The design data given in clauses 8 to 10 are based on F[n], the freezing index which statistically is exceeded once in
n years, e.g. F[10], F[50], F[100]. These values may be obtained from a set of individual values of F calculated for several winters using the statistical treatment described in A.3.
A.2 Calculation of freezing index for one winter
The freezing index is the 24 times sum of the difference between freezing point and the daily mean external air temperature:
is the freezing index for one winter, in K·h;
is equal to 0°C;
is the daily mean external air temperature for day j, in °C;
and the sum includes all days in the freezing season (as defined below).
The daily mean external air temperature may be obtained as the average of several readings, or as the average of the maximum and minimum values, for the day in question.
Both positive and negative differences, within the freezing season, are included in the accumulation of equation (A.1). A negative difference (daily mean temperature above 0 °C) implies some thawing
of the ground, which serves to reduce the frost penetration in the ground.
For the purposes of the summation in equation (A.1) the freezing season starts at the point from which the accumulation remains always positive throughout the winter. With reference to Figure A.1,
there is initially some freezing as a result of the area marked A, followed by complete thawing as a result of the area marked B since this is greater than area A. The accumulation therefore starts
after this. In Figure A.2, area A is greater than area B, so the thawing is not complete and the accumulation starts earlier as indicated on that Figure.
The freezing season ends at the point which results in the largest total accumulation for the winter. If a short thawing period is followed by a larger freezing period both are included, while if a
thawing period is followed by a lesser freezing period neither is included, as illustrated in Figures A.1 and A.2.
• 1 Start
• 2 End
• 3 Autumn
• 4 Winter
• 5 Spring
NOTE Area B > area A, and area C > area D
Figure A.1 — Illustration of the limits of the freezing season (first example)
• 1 Start
• 2 End
• 3 Autumn
• 4 Winter
• 5 Spring
NOTE Area B < area A, and area C < area D
Figure A.2 — Illustration of the limits of the freezing season (second example)
NOTE 1 In the past, freezing indexes have sometimes been calculated including only positive differences in equation (A.1), i.e. ignoring the effect of thawing periods. Tables or maps of freezing
indexes calculated on that basis, which give higher values of F than as defined above and so a greater margin of safety, may be used for the purposes of this standard. On the other hand an
accumulation on the basis of average monthly temperatures can significantly underestimate the true freezing index and such data should not be used.
NOTE 2 An alternative, and equivalent, method of obtaining the freezing index is to plot the cumulative difference between daily mean temperature and freezing point against time for a complete
12-month period (from midsummer to midsummer). The freezing index is then the largest difference between maximum and minimum turning points on this curve.
NOTE 3 Freezing in the ground depends on the ground surface temperature. However, because air temperatures are more readily available than ground surface temperatures, this standard uses the air
freezing index, i.e. the freezing index calculated from external air temperatures, as the design parameter. In most cases the use of air temperatures provides a safety margin because factors such as
the presence of vegetation and snow cover, and solar radiation, result in ground surface temperatures being higher than air temperatures. However the opposite may apply for snow-free surfaces in
permanent sun shadow, for which ground surface temperatures can be lower as a result of radiation to clear skies.
A.3 Statistical determination of design freezing index
The design freezing index, F[n], is the freezing index that statistically is exceeded once in n years. This implies that the probability that the freezing index in any one winter exceeds F[n] is 1/n.
NOTE 1 The appropriate value of n should be decided upon with regard to the level of safety that is required for the building in question. Parameters to consider are the expected lifetime of the
structure, the sensitivity of the type of structure to frost heave, etc. For permanent buildings n is normally chosen as 50 years or 100 years.
NOTE 2 n is referred to as the return period, i.e. the average number of years between successive occurrences of freezing indexes greater than F[n].
The design freezing index for a given location is obtained from a set of freezing indexes F[i], calculated as described in A.2, of m winters at the location. Whenever possible, the value of m should
not be less than 20. The use of data from m consecutive, or nearly consecutive, winters is recommended.
Use a statistical distribution that realistically reflects extreme events. The Gumbel distribution (see A.4) has been found to be suitable for many climates, and is recommended in the absence of
specific information for the locality concerned.
A.4 Use of the Gumbel distribution
Calculate the average freezing index, s[F], using (A.3):
i = 1,2,....., m
The design freezing index is then given by (A.4):
where y denotes the reduced variable in the Gumbel distribution.
Obtain the appropriate values of s[y] from Table A.1 corresponding to the number m of individual values of F[i] used in the calculation.
Obtain the value of y[n] from Table A.2 corresponding to the value of n chosen for the design.
Table A.1 — Values of s[y]
m s[y] m s[y]
10 0,50 0,95 50 0,55 1,16
15 0,51 1,02 60 0,55 1,17
20 0,52 1,06 70 0,55 1,19
25 0,53 1,09 80 0,56 1,19
30 0,54 1,11 90 0,56 1,20
40 0,54 1,14 100 0,56 1,21
Table A.2 — Values of y[n]
n 5 10 20 50 100
y[n] 1,50 2,25 2,97 3,90 4,60
NOTE For further information about the Gumbel distribution, see [1] and [2] in Bibliography. | {"url":"https://geotechnicaldesign.info/iso13793-2001/a-1.html","timestamp":"2024-11-09T08:10:40Z","content_type":"text/html","content_length":"39552","record_id":"<urn:uuid:97590743-1a16-449c-aa8e-4248268366ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00222.warc.gz"} |
Pupil Centroid Y-Coordinate Attribute
C.8.30.3.1.4 Corneal Vertex Location
The Corneal Vertex Location (0046,0202) establishes the reference point for the corneal vertex, the origin of the Ophthalmic Coordinate System. The Ophthalmic Coordinate System is used as the Frame
of Reference that establishes the spatial relationship for the corneal vertex (i.e., used within corneal topography maps) for a set of Images within a Series. It also allows Images across multiple
Series to share the same Frame of Reference. The corneal vertex is the point located at the intersection of the patient's line of sight (visual axis) and the corneal surface. It is represented by the
corneal light reflex when the cornea is illuminated coaxially with fixation.
Since the criteria used to group images into a Series is application specific, it is possible for imaging applications to define multiple Series within a Study that share the same imaging space.
Therefore the images with the same Frame of Reference UID (0020,0052) Attribute value share the same corneal vertex location within the patient's eye.
Figure C.8.30.3.1-3 illustrates the representation of corneal topography. The corneal vertex lies at the center of the rulers. Typical circular grids are 3, 5, 7, and 9 mm diameters centered on the
vertex. The annotations in Figure C.8.30.3.1-3 are R, right; L, left; H = Head; F = Foot.
Figure C.8.30.3.1-3. Representation of Corneal Topography
Numerical position data shall use the Cartesian (i.e., two dimensional rectangular) coordinate system. The direction of the axes are determined by Patient Orientation (0020,0020), see
Section C.7.6.1.1.1 for further explanation.
Devices that internally capture data in polar coordinates will need to convert to Cartesian coordinates, see Figure C.8.30.3.1-4.
Figure C.8.30.3.1-4. Sample Coordinate Data Points
When using the 3 dimensional coordinates (X, Y, Z), the Z axis shall represent corneal elevation. Z shall be measured from the length of a vector normal to the plane that is normal to and intersects
the corneal vertex at the intersection of the x, y, z, axes. It is shown in the diagram as "+" (0.0, 0.0, 0.0). The Z axis shall be positive towards the anterior direction of the eye; (i.e., it is a
right-hand rule coordinate system. Thus the Z values (see Figure C.8.30.3.1-5 and Figure C.8.30.3.1-6) will be predominantly negative, as they are posterior to the plane of the corneal vertex.
Figure C.8.30.3.1-5. Schematic of the 3-Dimensional Representation of Corneal Elevation
Figure C.8.30.3.1-6. Schematic of the Ophthalmic Coordinate System of the 3-Dimensional Representation used in Wide Field Measurements | {"url":"https://dicom.innolitics.com/ciods/corneal-topography-map/corneal-topography-map-analysis/00460204","timestamp":"2024-11-04T08:24:48Z","content_type":"text/html","content_length":"103320","record_id":"<urn:uuid:0b3636c6-bfde-4f8e-9eda-5eb99113a321>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00572.warc.gz"} |
stem and leaf plot calculator
You will just need to interpret the results appropriately. Stem and Leaf. The following diagram shows how to construct a stem-and-leaf plot or stemplot. What is the age of the youngest actor to win
the award? A stem-and-leaf plot is a way of organizing data into a form that makes it easy to see the frequency of different types of values. Data can be shown in a variety of ways including graphs,
charts, and tables. Stem-values represent either the first or first-two significant digits of each value. Cite this content, page or calculator as: Furey, Edward "Stem and Leaf Plot Generator";
CalculatorSoup, Thus, the first row of the plot represents sample values of approximately 80, 82, and 83. A stem-and-leaf plot is a chart tool that has a similar purpose as the histogram, and it is
used to quickly assess distributional properties of a sample (the shape of its probability distribution). Stem-and-Leaf Plots Introduction This procedure generates a stem -and-leaf plot of a batch of
data. High Performance Bass Fishing Boat Speed Calculator, Perches to square meters and square feet Calculator. Generate an online stem and leaf plot, or stemplot, and calculate basic descriptive
statistics for a sample data set with 4 or more values and up to 1000 values, all non-negative. Read more. stem leaf 13 6 14 1 1 4 6 15 3 8 16 5 8 17 2 3 6 18 0 6 7 Sorting data on TI83-84, creating
a frequency distribution, histogram, and step and leaf plot A stem-and-leaf diagram is one way of taking raw data and presenting it inan easy and quick way to help spot patterns in the spread of
data.They are best used when we have a relatively small set of data andwant to find the median or quartiles. Save my name, email, and website in this browser for the next time I comment. The answer
lies with the stem and leaf plot key. {x1 + x2 + x3 + ... + xn}. If n is odd the median is the center value. Stem-and-Leaf Plots for TI-Nspire™ CX and TI-Nspire™ CAS Although TI-Nspire technology
provides a comprehensive set of statistical functions and statistical graphs, it does not have the capability to create stem-and-leaf plots, a commonly-used diagram for displaying quantitative data.
If you really need to work with negative values please. Calculating IQR from stem leaf plot with Even number of scores with step by step illustration Slideshare uses cookies to improve functionality
and performance, and to provide you with relevant advertising. The final digits ("leaves") of each column are then placed in a row next to the appropriate column and sorted in numerical order. Scroll
down the page for more examples and solutions on how to construct and use stem-and-leaf plots. Example: Construct a stem-and-leaf plot for the following set of data. The key on this plot shows that
the stem is the tens place and the leaf is the ones place. Check out the example showing ages at a birthday party. Ordering a data set {x1 ≤ x2 ≤ x3 ≤ ... ≤ xn} from lowest to highest value, the
median is the numeric value separating the upper half of the ordered sample data from the lower half. Decimal In the Stem If n is even the median is the average of the 2 center values. https://
www.calculatorsoup.com - Online Calculators. Stem and Leaf Plots are a great way to organize lists of numbers in an appealing way. Each number in the data is broken down into a stem and a leaf, thus
the name. Info. Suppose that your class had the following test scores: 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, and 90 and you wanted to see at a glance what features were present in the data. They
take long lists of data points and organizing them. A lesson teaching the basics of stem and leaf diagrams with a nice colourful example. To set up a stem-and-leaf plot we follow some simple steps.
The first row has a stem value of 8 and contains the leaf values 0, 2, and 3. Recent Posts. Report a problem. For example "32" is split into "3" (stem) and "2" (leaf). It is a special table, where
each value is split into stem and leaf. All the results produced by this stem and leaf plot calculator are higly reliable. Your browser will open a new window and display the Stem-and-Leaf plot. This
calculator allows you to create a special table where each data value is split into a stem (the first digit or digits) and a leaf (usually the last digit). Like in this example: ©2020 calculator.com.
Does not currently handle values less than 0. Here is how to easily pull out the stem and the leaf: For 12, 2 is the leaf and 1 is the stem. Separate the different values by spaces, commas, or
newlines as you enter them. Then press Calculate. You can also copy and paste lines of data points from documents such as Excel spreadsheets or text documents with or without commas in the formats
shown in the table below. A stem and leaf plot organizes data by showing the items in order using stems and leafs.The leaf is the last digit on the right or the ones digit.The stem is the remaining
digit or digits. To use StemLeafPlot, you first need to load the Statistical Plots Package using Needs ["StatisticalPlots`"]. Stem and Leaf Plot Generator Create stem and leaf plots, or stemplots,
for sets of data values and calculate basic statistics including the minimum, maximum, sum, count, mean, median, mode, standard deviation and variance. Categories & Ages. Stem-and-Leaf Plots (A)
Answer the questions about the stem-and-leaf plot. A stem and leaf plot, or stem plot, is a technique used to classify either discrete or continuous variables. If you need to work with decimals you
can multiply all of your values by a factor of 10 and calculate based on those. 1 3 5 5 5 8 9 2 0 1 1 3 3 2 4 7 4 2 6 5 1 7 Now find the Lower and Upper Quartile by dividing the Stem-Leaf Plot in two
41. This could for instance be the results from a math test taken by a group of students at the Mathplanet School. For the example 12.3, the Key would show that 12 | 3 equals 12.3 units. This free
online software (calculator) computes the Stem-and-leaf Plot for a univariate dataset. If n is odd the median is the value at position p where, If n is even the median is the average of the values at
positions p and p + 1 where. Create a double or two-sided stem and leaf plot in seconds. A stem-and-leaf diagram, also called a stem-and-leaf plot, is a diagram that quickly summarizes data while
maintaining the individual data points. But, as with any plot or graph, some context is needed. When reading a stem and leaf plot, you will want to start with the key. Texas Instruments Ti Nspire
Color Graphing Calculator Nscx2tbl1l1 Office Depot Box And Whisker Plot Mather Com Identifying Outliers From A Stem And Leaf Plot You What Are Stem And Leaf Plots Data Science Pr Stem and leaf plot
maker in excel statistics how to make a stem and leaf plot you back to stem and leaf plot excel double two sided how to make a stem an leaf display you. Actors Stems Actresses 2 014667 98753220 3
00113344455778 88776543322100 4 11127 8851 5 210 6 011 8 7 13 8 4 a. Free. 28 13 26 12 20 14 21 16 17 22 17 25 13 30 13 22 15 21 18 18 For 45.7, 7 is the leaf and 45 is the stem. A stem-and-leaf plot
is a chart tool that has a similar purpose as the histogram, and it is used to quickly assess distributional properties of a sample (the shape of its probability distribution). stem-and-leaf plot
(stemplot) is an excellent way to begin an analysis. A value of scale=2 will cause the plot to be roughly twice as long as the default. creates a stem-and-leaf plot for the real-valued vector data.
The stom-and-leaf plots compare the ages of 30 actors and actresses at the time they won an award. Below is a sample stem and leaf plot and listing of the statistical values calculated. First we have
a set of data. StemLeafPlot [data 1, data 2] creates a side-by-side stem-and-leaf plot for the vectors data 1 and data 2. It is recommended for batches of data containing between 15 and 150 data
points. Created: Dec 23, 2009. © 2006 -2020CalculatorSoup® It is a method for showing the frequency with which certain classes of values occur. The stems are 6, 7, 8, and 9, corresponding to the tens
place of the data. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. A stem-and-leaf plot is a chart we can use to display
data by splitting up each value in a dataset into a stem and a leaf. For additional descriptive statistical values see Consider this small data set: 218 426 53 116 309 504 281 270 246 523. Stem and
Leaf Plot is similar to histogram, named after Arthur Bowley's. 1 3 5 5 5 8 9 2 0 1 1 3 3 2 4 7 4 2 6 5 1 7 **It is very important to divide the sides properly 42. The stem of the number includes all
but the last digit. Stem and Leaf plots are one of the main data representation methods. A stem-and-leaf plot is used to visualize data. Leaf Chart Math; Leaf Wine Stopper Favor; Sheesham Leaf Image;
Toronto Maple Leafs Ticket Refund; Sheesham Leaf Benefits; Recent Comments. Archives. Only the digit in the unit place is indicated as the leaf. As a rule-of-thumb, we want between 3 and 15
stem-values on the stem. The sum is the total of all data values. The stem-and-leaf plot is similar to a histogram and its main purpose is to show the data distribution while retaining the uniqueness
of each data value. It will guide you on how to read the other values. Loading... Save for later. Details. A stem and leaf plot shows the spread and distribution of a data set. You can also copy and
paste lines of data points from documents such as Excel spreadsheets or text documents … 1 3 5 5 5 8 9 2 0 1 1 3 3 2 4 7 4 2 6 5 1 7 To do this, count the scores from the start until you reach the
Line 43. About this resource. Ordering a data set {x1 ≤ x2 ≤ x3 ≤ ... ≤ xn} from lowest to highest value, the maximum is the largest value xn. Preview and details Files included (1) ppt, 139 KB. 42,
14, 22, 16, 2, 15, 8, 27, 6, 15, 19, 48, 4, 31, 26, 20, 28, 13, 10, 18, 13, 15, 48, 16, 15, 5, 18, 16, 28, 11, 0, 27, 28, 5, 40, 21, 18, 7, 12, 6, 40, 12, 2, 20, 35, 3, 16, 13, 8, 15, 7, 65, 65, 25,
15, 21, 12, 12, 35, 30, 14, 35, 20, 35, 7, 35. Here is an example of a stem-and-leaf plot for a given dataset, created by the Statology Stem-and-Leaf Plot Generator: To use this calculator, type or
paste your data in the text window. This stem-and-leaf plot shows customer wait times for an online customer service chat with a representative. Drawing a stem-and-leaf plot. The value or values that
occur most frequently in the data set. Enter (or paste) your data delimited by hard returns. Stem and Leaf Plot Maker, Generator: Just input the data as comma separated in this stem and leaf plot
generator to get the result. To create a stem and leaf plot for a given dataset, … Stem and Leaf. How To Do A Stem And Leaf Plot On Graphing Calculator; masuzi. A stem and leaf plot looks something
like a bar graph. Especially in stem and leaf plots with decimals, the key is very important, because it shows where the decimal goes. The total number of data values in a data set. We don’t use
commas to separate leaf values. The parameter scale can be used to expand the scale of the plot. The leaf unit is 1. The values range from 80 seconds to 119 seconds. Complete parts a through d below.
Descriptive Statistics Calculator. Does not handle decimals and they will be truncated. The stem and leaf plot is similar to histogram in displaying the statistical data. A Key (or Legend) is used
with a stem and leaf plot to give this context. The "stem" is used to group the scores and each "leaf" shows the individual scores within each group. Consider this new data set: 1.47, 2.06, 2.36,
3.43, 3.74, 3.78, 3.94. All rights reserved. There are few rules that you have to follow when you are entering data to a stem and leaf plot. To construct a stemplot, start by drawing the stem. One
way to measure and display data is to use a stem-and-leaf plot. 13, 24, 22, 15, 33, 32, 12, 31, 26, 28, 14, 19, 20, 22, 31, 15 . A Stem and Leaf Plot is a special table where each data value is split
into a "stem" (the first digit or digits) and a "leaf" (usually the last digit). All rights reserved. These data have 3 significant digits and a decimal point. A stem and leaf plot key shows what a
typical entry represents. Generate an online stem and leaf plot, or stemplot, and calculate basic descriptive statistics for a sample data set with 4 or more values and up to 1000 values, all
non-negative. The "stem" values are listed down, and the "leaf" values are listed next to them. A stem-and-leaf plot is a type of graph that is similar to a histogram but shows more information by
summarizing the shape of a set of data (the distribution) and providing extra detail regarding individual values. It aids in identifying the shape of a distribution. You would rewrite the list of
scores in order and then use a stem-and-leaf plot. Stem and Leaf Plot Generator A stem and leaf plot displays data by splitting up each value in a dataset into a “stem” and a “leaf.” To learn how to
make a stem and leaf plot by hand, read this tutorial. The mean is also known as the average. Enter values separated by commas such as 1, 2, 4, 7, 7, 10, 2, 4, 5. Leave a Comment Cancel reply. The
sum of all of the data divided by the size. Part 1: Reading a Stem and Leaf Plot The Stem and Leaf Plot is an interesting way to showcase data. Updated: Dec 14, 2014. ppt, 139 KB. Enter values
separated by commas such as 1, 2, 4, 7, 7, 10, 2, 4, 5. Stem and Leaf Plots A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a
"leaf" (usually the last digit). years old b. A stem and leaf plot is used to organize data as they are collected. It is a exploratory data analysis to find the quantitative data in a graphical
format. Second Illustrative Example of a Stem-and-Leaf Plot: The next illustrative example shows how a stem-and-leaf plot can be modified to accommodate data that might not immediately lend itself to
this type of plot. When you are done looking at the new window press the Exit button. Ordering a data set {x1 ≤ x2 ≤ x3 ≤ ... ≤ xn} from lowest to highest value, the minimum is the smallest value x1.
In such a diagram, the "stem" is a column of the unique elements of data after removing the last digit. Use a stem-and-leaf plot one of the unique elements of data, or plot! On Graphing Calculator ;
masuzi, you first need to load the statistical values see descriptive Statistics Calculator 6 8. Stemleafplot, you will just need to load the statistical plots Package using Needs [ `` `! Plots with
decimals you can multiply all of your values by spaces, commas, or newlines as you them. An appealing way display data is to use stemleafplot, you will just need work. Vectors data 1 and data 2
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hard returns 45 is the tens place and the leaf is the of. Graph, some context is needed save my name, email, and 9, corresponding to the tens of! X2 + x3 +... + xn }, 2014. ppt, 139 KB stemplot,
start by drawing stem... To construct and use stem-and-leaf plots Introduction this procedure generates a stem value of scale=2 will cause plot... Center values + x2 + x3 +... + xn } want to start
with stem... Numbers in an appealing way, 5 spaces, commas, or newlines as you enter.! A stemplot, start by drawing the stem are a great way to measure and display data broken. 8, and 9,
corresponding to the tens place of the data in an appealing way to the... To them stem and leaf plot calculator load the statistical plots Package using Needs [ `` StatisticalPlots ` `` ] lists!
Shows where the decimal goes interesting way to measure and display the stem-and-leaf plot for the stem and leaf plot calculator. Set: 1.47, 2.06, 2.36, 3.43, 3.74, 3.78,.. Odd the median is the leaf
row of the data divided by the size when are! Frequently in the stem and leaf plot key thus, the first first-two! Stem is the tens place of the statistical data for example `` 32 '' is a method for
showing frequency. 246 523 plot, or newlines as you enter them page for more examples and solutions on to. Example: construct a stem-and-leaf plot numbers in an appealing way classify either discrete
or continuous variables with the the! First row of the plot to be roughly twice as long as the leaf values 10... Between 15 and 150 data points and organizing them the leaf is center. Average of the
main data representation methods to be roughly twice as as! Is very important, because it shows where the decimal goes including graphs, charts, and the and... Of each value is split into stem and
leaf plot the stem following. By hard returns to work with negative values please, email, and 9, corresponding to tens. Perches to square meters and square feet Calculator the Exit button to showcase
data shows where the decimal goes the... Stem is the age of the youngest stem and leaf plot calculator to win the award time comment. Each number in the data set -and-leaf plot of a batch of data
7,,... Calculator are higly reliable plot, or stem plot, is a sample stem leaf! A double or two-sided stem and leaf plots are a great way to organize lists numbers! '' ( stem ) and `` 2 '' ( stem )
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some steps... Called a stem-and-leaf plot for the next time I comment also called a stem-and-leaf plot shows that stem! Additional descriptive statistical values calculated by commas such as 1, 2, 4
5. 14, 2014. ppt, 139 KB this stem-and-leaf plot you will want to start with the is...: 1.47, 2.06, 2.36, 3.43, 3.74, 3.78 3.94. 80, 82, and 9, corresponding to the tens place of the number includes
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calculator thus the name stem the following of! Spaces, commas, or stem plot, is a exploratory data analysis to the! Wait times for an online customer service chat with a stem and leaf to...: data
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order and then use a stem-and-leaf plot this... Batches of data containing between 15 and 150 data points construct a stem-and-leaf plot for the next time I.! Other values the 2 center values use
stem-and-leaf plots ( a ) answer the questions the! Negative values please which certain classes of values occur at the time they won award! To use stemleafplot, you will just need to work with
negative values please the questions about the stem-and-leaf for. Plot, or stem plot, you first need to work with you... Is to use this Calculator, type or paste ) your data delimited by hard returns
a for. On those total of all data values you need to work with decimals you can multiply all of data... 2 014667 98753220 3 00113344455778 88776543322100 4 11127 8851 5 210 6 011 8 7 8... Which
certain classes of values occur is indicated as the default `` ''! The questions about the stem-and-leaf plot is broken down into a stem and leaf with. Vector data the leaf is the ones place times
for an online customer service chat a! New window press the Exit button digits and a decimal point when you are done at... Number in the text window, 2.06, 2.36, 3.43, 3.74 3.78... Wait times for an
online customer service chat with a stem and leaf plot and of!, email, and tables paste stem and leaf plot calculator your data in the data ''. The time they stem and leaf plot calculator an award
the page for more examples and solutions on to! Are a great way to organize data as they are collected 3 12.3! Students at the Mathplanet School and organizing them value or values that most. Will
guide you on how to construct and use stem-and-leaf plots Introduction this procedure generates a stem and leaf is. Display the stem-and-leaf plot shows the individual data points plots are one of 2!
[ `` StatisticalPlots ` `` ] the key would show that 12 | equals... Down into a stem and leaf plot key be the results appropriately data to. Mathplanet School use stem-and-leaf plots Introduction
this procedure generates a stem and leaf Calculator! Display data is to use stemleafplot, you will just need to work with negative values please separated by such... The scale of the main data
representation methods parameter scale can be shown in data... 2.36, 3.43, 3.74, 3.78, 3.94 for the real-valued vector data a decimal.... The individual scores within each group be truncated
Calculator are higly reliable 1 and data ]. Calculate based on those the number includes all but the last digit sample stem leaf... Key shows what a typical entry represents start by drawing the stem
is the leaf a. And calculate based on those and display data is broken down into a stem and leaf plots are of. ( or Legend ) is an excellent way to measure and display data is to use stemleafplot
you. Name, email, and website in this example: construct a stem-and-leaf plot 12.3, the key if really. Use this Calculator, type or paste ) your data in a variety ways. Begin an analysis to showcase
data histogram in displaying the statistical values see descriptive Calculator... The vectors data 1, data 2 to 119 seconds questions about the stem-and-leaf (... Context is needed about the
stem-and-leaf plot shows customer wait times for an online customer service chat with a....
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This cycle describes the functioning of a reciprocating piston engine used in automobile engines. This cycle consists of two adiabatic and two isochoric steps. The thermal efficiency of this cycle
is: η = 1 – r^(1 – γ), where r is the compression ratio and γ is the specific heat ratio, which is 1.4 for air in the present simulation. The four steps are plotted in the PV (pressure vs. volume)
and TS (temperature vs. entropy) diagrams. The adiabatic and isochoric transformations are in blue and red, respectively. The areas in the PV and TS diagrams are equal to the net work and to the net
heat transfer, respectively. These two quantities are equal. You can set the values of the heat source temperature, the heat sink temperature, and the compression ratio.
This simulation runs on desktop using the free Wolfram Player. Download the Wolfram Player | {"url":"https://learncheme.com/simulations/thermodynamics/thermo-1/otto-cycle/","timestamp":"2024-11-05T19:53:40Z","content_type":"text/html","content_length":"75550","record_id":"<urn:uuid:650e276e-2238-4f3b-9661-e8872fd4484c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00023.warc.gz"} |
How do I apply a function to a matrix in MATLAB?
To apply a function to a matrix in MATLAB, you can use the element-wise operations or the arrayfun function.
Using element-wise operations:
1. Define the function you want to apply to the matrix. For example, let's say you want to square each element of the matrix: function result = square(x) result = x^2; end
2. Create a matrix, for example: A = [1, 2, 3; 4, 5, 6; 7, 8, 9];
3. Apply the function to the matrix using element-wise operations: B = square(A); This will square each element of the matrix A and store the result in matrix B.
Using arrayfun function:
1. Define the function you want to apply to the matrix, similar to the previous step: function result = square(x) result = x^2; end
2. Create a matrix, for example: A = [1, 2, 3; 4, 5, 6; 7, 8, 9];
3. Use the arrayfun function to apply the function to each element of the matrix: B = arrayfun(@square, A); This will apply the function 'square' to each element of matrix A and store the result in
matrix B. | {"url":"https://devhubby.com/thread/how-do-i-apply-a-function-to-a-matrix-in-matlab","timestamp":"2024-11-03T12:06:39Z","content_type":"text/html","content_length":"113537","record_id":"<urn:uuid:dd0a7385-057d-454f-8a01-0c465b3dcef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00730.warc.gz"} |
PoW as Checksum
The purpose of a checksum is to determine whether a message was transmitted error free. There are many ways to determine a checksum and one of the ways to do this is to use a cryptographic hash.
A cryptographic hash function has several useful properties to do this:
• A single bit change in the input produces a radical change in the output hash value. This means that even a small error in the input will result in a completely different checksum being computed.
• A collision is improbable to compute. This means that it is nearly impossible for two different sets if input data to produce the same checksum even on purpose, much less if it is due to a random
However, a problem with using a checksum like this is that it needs to be transmitted alongside the message. In one of our projects, it would needlessly complicate the protocol to require the
checksum to be transmitted as well. We wanted to keep things simple.
So, we borrowed an idea from the block-chain world – proof of work (PoW).
Instead of using a checksum, we can alter the message slightly by padding it with a nonce in order to produce a checksum with a certain number of leading zeros (difficulty level). The receiver can
simply compute and verify the message as error free if the received message with nonce result in a hash of a certain number of leading zeros.
It is also efficient on bandwidth as the nonce could be a small 32-bit value compared to a cryptographic hash that is at least 128-bits and typically 256-bits or more. So, instead of transmitting the
checksum with the message, the nonce can be transmitted instead, and removed from the output message post checksum computation.
This is good enough as it will be able to detect an error in transmission. Any errors in the message (or the nonce) would result in a radically different checksum being computed. The chances of
random errors producing a checksum with the same number of leading zeros, decreases with difficulty (level that can be tweaked).
While this will increase the complexity at the transmitter, for our specific use-case, this is not a problem as the data to be transmitted need only be computed once.
PS: The results can possibly be improved by using a HMAC or by signing and ensuring that the result of the HMAC/signature meets the difficulty level.
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How to Teach Numbers
by Tara Arntsen 276,116 views
Numbers are typically taught early on in ESL courses. This means that students generally have very limited English abilities so it is best to proceed slowly, taking several classes if necessary to
cover the material. The first time numbers are introduced, limit them to numbers one through ten and then build up to one hundred. Larger numbers can be introduced at another time.
How To Proceed
1. 1
Warm up
Your students are, at this stage, probably beginners so try to review material that was covered in the previous lesson and keep lessons enjoyable so that students will not develop an aversion to
your classes. Lessons prior to this may include letters so you can play letter bingo. Each student should have a five by five grid. Have them fill in the grid with letters and then say letters at
random until one or more students have gotten bingo.
2. 2
Introduce Numbers
Use flashcards to introduce numbers one through ten. Flashcards should have both the numeral and the word for each number. This will probably also include introducing some new vocabulary so
choose words that will be used often in your classroom and words where the plural form is made by simply adding -s. Words like teacher, student, book, pencil, and desk would all be appropriate.
Use choral repetition for pronunciation practice and then drill using the flashcards.
3. 3
Practice Numbers
If your students are not familiar with the Latin alphabet, they have probably been using worksheets to practice forming letters of the alphabet. You can use a similar worksheet to help them
practice writing out numbers like one, two, three, etc. This is a good opportunity for them to practice letter and word spacing. If your students are familiar with the Latin alphabet, matching or
fill in the blank exercises may be more appropriate.
4. 4
With beginners, it is important to check comprehension frequently. Students may be confused or hesitant due to lack of understanding but will often be unwilling or unable to ask for help. A group
activity will get your students on their feet. One activity is to make groups with the same number of people as you call out. For example if you say “Four” students should make groups of four and
when you call out the next number they should run around trying to get into appropriately sized groups. Another activity is to split the class into two to four teams. Each group should determine
in what order students take turns and be given a portion of the board to write on. When you say a word aloud, the student whose turn it is should run to the board and write the numeral. If your
students do very well, tell them they have to spell out the word and maybe later on, as a review activity, students have to spell out the word of the number that comes after the one you say
aloud. At the end of the game, the group with the most points wins.
5. 5
Introduce More Numbers
When your students are confident using numbers one through ten, introduce numbers zero to one hundred. Focus primarily on the numerals and pronunciation. It is a lot of new material to take in
but there is a pattern so stressing one through ten as well as multiples of ten will be really important. The difficult part for most students will be eleven to nineteen and confusing numbers
like thirteen with thirty. Keeping this in mind, practice difficult areas more often than others.
6. 6
Make decks of cards for numbers zero to one hundred with numerals on one side and words on the other. For the purposes of this activity have students spread out the cards numeral side up.
Students should play in groups of three to six. When you call out a number, the first student to say and smack the appropriate card gets to keep it. The winner is the student with the most cards
at the end of the game. If your students are struggling with certain numbers, feel free to also write the numeral on the board but be sure to say it first. You can use this same deck later on to
practice reading and the difference between -teens and multiples of ten.
7. 7
Since you recently used bingo in your warm up, students should be familiar with the game. Ask them to fill out new grids with numbers zero through one hundred and play multiple times. You can
also play another group activity where students stand in a circle and take turns saying numbers in order from zero to one hundred. Perhaps students say a number and then the name of the classmate
who will say the next one or some other variation to keep things interesting. When they have mastered that, you can ask them to skip numbers with threes and sevens, including thirteen and seventy
for example, to make it more challenging.
8. 8
Worksheets may be an appropriate review activity but any activity you played during your numbers classes, could be conducted again as a review.
Numbers are used often during ESL courses. Especially before lessons on time or something similar, a review is going to be necessary. Students will most likely continue to be confused by the
pronunciation of certain numbers so special short challenge activities may be a nice break from other topics as they advance through their English studies.
P.S. If you enjoyed this article, please help spread it by clicking one of those sharing buttons below. And if you are interested in more, you should follow our Facebook page where we share more
about creative, non-boring ways to teach English. | {"url":"https://busyteacher.org/3752-how-to-teach-numbers.html","timestamp":"2024-11-11T03:58:36Z","content_type":"text/html","content_length":"43245","record_id":"<urn:uuid:bb3eafc9-f939-4e7c-a361-bb9a1815e523>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00128.warc.gz"} |
MHD in Spherical Geometry
Rayleigh: MHD in Spherical Geometry
Rayleigh: MHD in Spherical Geometry#
Rayleigh solves the magnetohydrodynamic (MHD) equations, in a rotating frame, within spherical shells, using the anelastic or Boussinesq approximations. Derivatives in Rayleigh are calculated using a
spectral transform scheme. Spherical harmonics are used as basis functions in the horizontal direction. Chebyshev polynomials or finite-differences are employed in radius. Time-stepping is
accomplished using the semi-implicit Crank-Nicolson method for the linear terms, and the Adams-Bashforth method for the nonlinear terms. Both methods are second-order in time.
This documentation is structured into the following sections: | {"url":"https://rayleigh-documentation.readthedocs.io/en/latest/","timestamp":"2024-11-10T08:43:44Z","content_type":"text/html","content_length":"22965","record_id":"<urn:uuid:1b41dc61-d38b-489e-b844-3b60222fb29c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00357.warc.gz"} |
Python Data Structures for Tabular Data | Outerbounds
Python Data Structures for Tabular Data
Many data science applications deal with tabular data that looks something like this:
Name Credit score Last login Balance
Jane Smith 852 2022-02-03 18:32 131.0
John Smith 765 2022-05-15 09:03 72.5
Note how each column has a uniform data type. In this example, the Name column contains strings, Credit Score integers, Last login timestamps, and Balance floating point numbers. The metadata about
column names and types are called the schema. Often, structured or relational data like this is stored in a database or a data warehouse where it can be queried using SQL. In the past, it wasn’t
considered wise to try to process large amounts of data in Python. Thanks to increasing amounts of memory and CPU power available in a single computer - in the cloud in particular - as well as the
advent of highly optimized libraries for data science and machine learning, today it is possible to process even hundreds of millions of rows of tabular data in Python efficiently. This is a boon to
data scientists, as they don’t need to learn new systems or programming languages to be able to process even massive data sets. Don’t underestimate the capacity of a single large server! However,
some thinking may be required to choose the right library and data structure for the task at hand, as Python comes with a rich ecosystem of tools with varying tradeoffs. In the following, we will go
through five common choices as illustrated by the figure below:
pandas is the most common library for handling tabular data in Python. pandas is a dataframe, meaning that it can handle mixed column types such as the table presented above. It adds an index over
the columns and rows, making it easy to access particular elements by their name. It comes with a rich set of functions for filtering, selecting, and grouping data, which makes it a versatile tool
for various data science tasks. A key tradeoff of pandas is that it is not particularly efficient when storing and processing data. In particular when the dataset is large, say, hundreds of megabytes
or more, you may notice that pandas takes too much memory or too much time to perform desired operations. At this point, you can consider more efficient alternatives as listed below.
NumPy - Efficient, Interoperable Arrays for Numeric Data
NumPy is a performant array library for numeric data. It shines at handling arrays of data of uniform types - like individual columns of a table. In fact, pandas uses NumPy internally to store
columns of a dataframe. NumPy can also represent higher-dimensional arrays which can come in handy as an input matrix, e.g. for model training or other mathematical operations. Under the hood, NumPy
is implemented in the C programming language, making it very fast and memory-efficient. A downside is that it comes with a more limited set of data processing operations compared to a dataframe like
pandas. A key upside of NumPy is that it can work as a conduit between various libraries. Most data science libraries in Python, such as SciKit Learn, can use, import, and export NumPy arrays
natively. Many of them are smart enough to leverage NumPy in a manner that doesn’t require data to be copied explicitly, which makes it very fast to move even large amounts of data between libraries
through NumPy.
Arrow - Efficient, Interoperable Tables
Apache Arrow is a newer, performance-oriented library for tabular data. In contrast to NumPy it can handle mixed columns like pandas, albeit as of today it doesn’t come with as many built-in
operations for data processing. However, if you can express your operations using the Arrow API, the result can be much faster than with pandas. Also, thanks to Arrow’s efficient way of representing
data, you can load much more data in memory than what would be possible using pandas. It is easy and efficient to move data between pandas and Arrow or NumPy and Arrow, which can be performed in a
zero-copy fashion.
Scipy.Sparse - Efficient Sparse Arrays for Numeric Data
The three libraries above are general-purpose in a sense that they come with a rich set of APIs and supporting modules that allow them to be used for a wide range of use cases. In contrast,
Scipy.Sparse is a more specialized library, targeted at handling sparse matrices i.e. numeric arrays where most values are empty or missing. For instance, a machine learning model may take an input
matrix with tens of thousands of columns. In such cases, it is typical for most columns to be empty for any particular row. Processing such a dataset as a dense array e.g. using NumPy may be
impractical due to a large amount of memory being consumed by empty values. If you use a library that is compatible with Scipy.Sparse matrices, such as XGBoost or Scikit Learn, sparse matrices may
allow you to handle much larger datasets than what would be feasible otherwise.
Tensors and Other Library-specific Arrays
Modern machine learning libraries like XGBoost, TensorFlow, and PyTorch are capable of crunching through a huge amount of data efficiently but configuring them for peak performance requires effort.
You need suitable data loaders that load raw data into the model, as well as specialized to facilitate data movement within the model. For optimal performance, you are often required to use
library-specific data structures, such as DMatrix for XGBoost, or various tensor objects in deep learning frameworks. These data structures are optimized for the needs of each particular library,
which limits their usefulness as a generic way to store and process data. Fortunately, it is often possible to move data from tensors to NumPy arrays efficiently and vice versa.
Choosing the Library
Here’s a simple rubric for choosing the right library for the job: Do you use a deep learning library? If yes, use library-specific objects and data loaders. Is your data small enough not to require
special treatment (if you are unsure, assume yes)? Use pandas. Is your large data numerical and dense? Use NumPy. Is your large data numerical and sparse? Use Scipy.Sparse. Otherwise use Arrow. Note
that in all these cases you can scale to larger datasets simply by requesting more resources from the cloud using Metaflow’s @resources decorator.
How do I?
Pass XGBoost DMatrix between Metaflow steps | {"url":"https://docs.outerbounds.com/python-tabular-data-structures/","timestamp":"2024-11-07T16:14:32Z","content_type":"text/html","content_length":"30608","record_id":"<urn:uuid:ee6da6ef-4088-4492-8753-c573408515e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00546.warc.gz"} |
Bluetooth LE Positioning with Deep Learning
This example shows how to calculate the 3-D positioning of a Bluetooth® low energy (LE) node by using received signal strength indicator (RSSI) fingerprinting and a convolutional neural network
(CNN). Using this example, you can:
1. Generate Bluetooth LE locator and node positions in an indoor environment.
2. Compute propagation paths between the nodes and locators in the indoor environment by using the ray tracing propagation model.
3. Create an RSSI fingerprint base for every node-locator pair.
4. Train the CNN model using the RSSI data set.
5. Evaluate and visualize the performance of the network by comparing the node positions predicted by the CNN with the actual positions.
You can further explore this example to see how you can improve the accuracy of the node positioning estimate by increasing the number of Bluetooth LE nodes for training. For more information, see
Further Exploration.
Fingerprinting and Deep Learning in Bluetooth-Location Based Services
Bluetooth technology provides various types of location-based services [1], which fall into these two categories:
• Proximity Solutions: Bluetooth proximity solutions estimate the distance between two devices by using received signal strength indication (RSSI) measurements.
• Positioning Systems: Bluetooth positioning systems employ trilateration, using multiple RSSI measurements to pinpoint a device's location.
The introduction of new direction-finding features in the Bluetooth Core Specification 5.3 [2] enables you to estimate the location of a device with centimeter-level accuracy.
Bluetooth LE positioning systems can use fingerprinting and deep learning techniques to achieve sub-meter level accuracies, even in non-line-of-sight (NLOS) multipath environments [3]. A fingerprint
typically includes information like the RSSI from a signal measured at a specific location within an environment.
The example performs these steps to estimate the 3-D position of a Bluetooth LE node by using RSSI and CNN.
1. Initiate the network's training phase by computing RSSI fingerprints at various known positions within an indoor environment.
2. Create a data set by collecting RSSI fingerprints from the received LE signals in an indoor environment, and label each fingerprint with its specific location information. Each fingerprint
includes RSSI values derived from several LE packets from each transmitter locator.
3. Train a CNN to predict node locations using a subset of these fingerprints.
4. Assess the performance of the trained model by using the remainder of the data set to generate predictions of node locations based on their RSSI fingerprints.
Generate Training Data for Indoor Environment
Generate training data for an indoor office environment, specified by the conferenceroom.stl file.
mapFileName = "conferenceroom.stl";
viewer = siteviewer(SceneModel=mapFileName,Transparency=0.25);
The example places ${N}_{\text{locator}}$ LE transmitters at corners of the room, and a number of receiving nodes that you specify in the environment. The example generates LE signals with 5 dBm
output power, and computes the fingerprints based on the propagation channel that the environment defines.
This section shows how you can synthesize the training data set for the CNN.
Generate LE Locators and Node Positions in Indoor Environment
Generate the LE locators and node objects, and visualize them in the indoor scenario. If you use a file other than conferenceroom.stl to create the environment, you must adjust the locator and node
positions in the createScenario function to accommodate the new environment. The example calculates the number of nodes by using the nodeSeparation value, which specifies the distance in meters
between the nodes across all dimensions.
nodeSeparation = 1;
[locators,nodes,posnodes] = createScenario(nodeSeparation);
show(nodes,Icon="bleRxIcon.png",ShowAntennaHeight=false,IconSize=[16 16])
disp("Simulating a scenario with " + num2str(width(locators)) + " locators and " + num2str(width(nodes)) + " nodes")
Simulating a scenario with 8 locators and 18 nodes
Generate Channel Characteristics by Using Ray Tracing Techniques
Set the parameters for the ray propagation model. This example considers only LOS and second-order reflections by specifying the MaxNumReflections input as 2. Increasing the MaxNumReflections value
extends the simulation time.
pm = propagationModel("raytracing", ...
CoordinateSystem="cartesian", ...
SurfaceMaterial="wood", ...
Perform ray tracing analysis for all the locator-node pairs. The raytrace function returns the generated rays in a cell array of size ${N}_{\text{beacon}}$-by-${N}_{\text{node}}$, where ${N}_{\text
{beacon}}$ is the number of locators and ${N}_{\text{node}}$ is the number of nodes.
rays = raytrace(locators,nodes,pm,"Map",mapFileName);
Visualize the propagation paths between all locators and a single node. A distinct color specifies the path loss in dB associated with each reflected path.
show(nodes(ceil(16)),IconSize=[32 32]);
plot([rays{:,16}],ColorLimits=[50 95]);
Generate RSSI Fingerprint Features and Labels
Generate RSSIs for each locator-node pair from the received packets by performing this procedure.
The example assumes that locators do not interfere with each other. Each receiver node makes multiple observations to generate the training data. Additionally, before training, you preprocess the
RSSI values at each location to rearrange them into a ${N}_{\text{rssi}}$-by-${N}_{\text{locator}}$ array. For this example, ${N}_{\text{rssi}}$ is 32 and ${N}_{\text{locator}}$ is 8.
Initialize LE Waveform and Thermal Noise Parameters
To simulate variations in the environment, change the outputPower and noise figure (NF) values. You can change the number of observations collected for each locator-node pair. By increasing
numObsPerPair, you can create more data for training.
outputPower = 5; % In dBm
noiseFigure = 12; % In dB
numObsPerPair = 60;
numRSSI = 32;
Each LE locator transmits Bluetooth LE1M or LE2M packets through a noisy channel, and each node receives these packets. The symbol rate for LE1M waveform is 1 Msps, while the LE2M waveform has a
symbol rate of 2 Msps. Regardless of the LE PHY mode, the system sets the output waveform to maintain a constant sampling rate of 8 MHz. This is ensured by using an sps of 8 for LE1M waveforms and 4
for LE2M waveforms. The LE packet length is chosen as 256 bits.
packetLength = 256;
sps = 8;
symbolRate = 1e6; % In Hz
samplingRate = symbolRate*sps; % In Hz
channelIndex = 38; % Broadcasting channel index
data = randi([0 1],packetLength,1,"single");
dBdBmConvFactor = 30;
scalingFactor = 10^((outputPower - dBdBmConvFactor)/20); % dB to linear conversion
txWaveform = cell(2,1);
Generate LE Waveforms
Generate LE1M and LE2M waveform by using the bleWaveformGenerator function. Scale the amplitude of the waveforms based on output power. The generated waveforms have an output power of 5 dBm.
txWaveform{1} = scalingFactor*bleWaveformGenerator(data,ChannelIndex=channelIndex,SamplesPerSymbol=sps);
txWaveform{2} = scalingFactor*bleWaveformGenerator(data,ChannelIndex=channelIndex,SamplesPerSymbol=sps/2,Mode="LE2M");
Create and configure comm.ThermalNoise System object™ to add thermal noise.
thNoise = comm.ThermalNoise(NoiseMethod="Noise figure" ,...
Generate RSSI data set
Initialize data set related variables and display statistics.
[numBeacons,numNodes] = size(rays);
features = zeros(numRSSI,numBeacons,numNodes*numObsPerPair);
labels.position = zeros([numNodes*numObsPerPair 3]);
numDisp = 10; % Number of display text items shown
delta = (numNodes-1)/(numDisp-1);
progressIdx = [1 floor(delta):floor(delta):numNodes-floor(delta) numNodes];
Between each locator-node pair, generate RSSI values for multiple observations as features. The example trains the CNN by combining RSSI features with labels of the node position.
for nS = 1:numNodes
for nP = 1:numBeacons
nC = (nS - 1)*numBeacons + nP;
if ~isempty(rays{nC})
wIdx = randi([1 2],1,1);
txW = txWaveform{wIdx};
rxData = generateRxData(rays{nC},locators(nP),nodes(nS),txW,numRSSI,samplingRate);
for nD = 1:numObsPerPair
if ~isempty(rxData)
rssiSeq = generateRSSI(rxData,thNoise,numRSSI,dBdBmConvFactor);
features(:,nP,(nS - 1)*numObsPerPair + nD) = rssiSeq;
features(:,nP,(nS - 1)*numObsPerPair + nD) = 0;
labels.class((nS - 1)*numObsPerPair + (1:numObsPerPair)) = categorical(cellstr(nodes(nS).Name));
labels.position((nS - 1)*numObsPerPair + (1:numObsPerPair),:) = repmat(nodes(nS).AntennaPosition',numObsPerPair,1);
if any(nS == progressIdx)
fprintf("Generating Dataset: %3.2f%% complete.\n", 100*(nS/numNodes))
Generating Dataset: 5.56% complete.
Generating Dataset: 11.11% complete.
Generating Dataset: 16.67% complete.
Generating Dataset: 22.22% complete.
Generating Dataset: 27.78% complete.
Generating Dataset: 33.33% complete.
Generating Dataset: 38.89% complete.
Generating Dataset: 44.44% complete.
Generating Dataset: 50.00% complete.
Generating Dataset: 55.56% complete.
Generating Dataset: 61.11% complete.
Generating Dataset: 66.67% complete.
Generating Dataset: 72.22% complete.
Generating Dataset: 77.78% complete.
Generating Dataset: 83.33% complete.
Generating Dataset: 88.89% complete.
Generating Dataset: 94.44% complete.
Generating Dataset: 100.00% complete.
Normalize the RSSI values to fall within the range [0, 1].
for nB = 1:size(features,3)
features(:,:,nB) = (features(:,:,nB)-min(features(:,:,nB),[],'all'))./(max(features(:,:,nB),[],'all')-min(features(:,:,nB),[],'all'));
Neural networks serve as powerful models capable of fitting diverse data sets. To validate the results, you must split the data set into 70% training data, 10% validation data, and 20% test data.
Before splitting the data into different sets, shuffle the training data randomly. The training model learns to fit the training data by adjusting its weighted parameters based on the prediction
error. The validation data helps you to confirm that the model performs well on the unseen data, and does not overfit the training data.
trainRatio = 0.7;
validationRatio = 0.1;
[training,validation,test] = splitDataSet(features,labels,trainRatio,validationRatio);
Build and Train the Network
This section guides you through the process of building and training a CNN to determine node locations. This figure shows the CNN architecture as defined in [3].
The CNN consists of these components:
• Input layer: Defines the size and type of the input data.
• Convolutional layer: Performs convolution operations on input to this layer by using a set of filters.
• Batch normalization layer: Prevents unstable gradients by normalizing the activations of a layer.
• Activation (ReLU) layer: A nonlinear activation function that thresholds the output of the previous functional layer.
• Dropout layer: Randomly deactivates a percentage of the parameters of the previous layer during training to prevent overfitting.
• Max pooling layer: Performs down sampling by dividing the input into pooling regions and computes maximum of each region.
• Flatten layer: Collapses the spatial dimension of the input into its channel dimension.
• Output (FC) layer: Defines the size and type of output data. The CNN handles this task as a regression problem.
Construct the CNN.
d = size(features);
layers = [
inputLayer([d(1) d(2) NaN],"SCB")
Configure Learning Process and Train the Model
Specify the loss function and training metric. Because the learning process is a regression problem, the example uses minimum square error(MSE) as the loss function. Match the predicted positions to
the expected positions for each location. Specify the number of training data samples the model evaluates in each training iteration. If you are working with a larger data set, increase the sample
lossFcn = "mse";
trainingMetric = "rmse";
valY = validation.Y; % node actual positions for validation
trainY = training.Y; % node actual positions for training
miniBatchSize = 90;
validationFrequency = floor(size(training.X,3)/miniBatchSize);
Specify the options that control the training process. The number of epochs determines how many consecutive times the model trains on the full training data set. By default, the model trains on a GPU
if available. Training on a GPU requires Parallel Computing Toolbox™ and a supported GPU device. For a list of supported devices, see GPU Computing Requirements (Parallel Computing Toolbox).
options = trainingOptions("adam", ...
MiniBatchSize=miniBatchSize, ...
MaxEpochs=5, ...
InitialLearnRate=1e-3, ...
ResetInputNormalization=true, ...
Metrics=trainingMetric, ...
Shuffle="every-epoch", ...
ValidationData={validation.X, valY}, ...
ValidationFrequency=validationFrequency, ...
ExecutionEnvironment="auto", ...
Train the model.
net = trainnet(training.X,trainY,layers,lossFcn,options);
Iteration Epoch TimeElapsed LearnRate TrainingLoss ValidationLoss TrainingRMSE ValidationRMSE
_________ _____ ___________ _________ ____________ ______________ ____________ ______________
0 0 00:00:05 0.001 1.4843 1.2183
1 1 00:00:05 0.001 5.3473 2.3124
8 1 00:00:07 0.001 1.3526 1.2431 1.163 1.115
16 2 00:00:07 0.001 0.90067 0.80014 0.94904 0.89451
24 3 00:00:07 0.001 0.72238 0.60449 0.84993 0.77749
32 4 00:00:08 0.001 0.66188 0.42808 0.81356 0.65428
40 5 00:00:08 0.001 0.61735 0.26319 0.78572 0.51302
Training stopped: Max epochs completed
Evaluate Model Performance
Predict the node position by passing the test set features through the network.
predPos = minibatchpredict(net,test.X);
Evaluate the performance of the network by comparing the predicted positions with the expected results. Generate a visual and statistical representation of the results. Assign colors to each LE node
for positioning, indicating the distance error of the predicted position from the actual position. The plot displays the actual positions of the LE nodes.
Additionally, generate a cumulative distribution function (CDF), where the y-axis represents the proportion of data with a measured distance error less than or equal to the value on the x-axis.
Compute the mean error for all the test predictions from the actual positions.
metric = helperBLEPositioningDLResults(mapFileName,test.Y,predPos);
Further Exploration
Decreasing the distance between nodes can improve the accuracy of the positioning estimate. If there is no path between the locator and node, packet reception at a location can fail, leading to RSSI
values being recorded as 0. When adding more nodes, you must consider those nodes that cannot receive packets from many locators. Collecting RSSI values with more zeros than useful values at some
nodes can decrease the accuracy of position estimation. You can increase the maximum number of reflections, or try different transmitter locator placements. Allowing for a higher maximum number of
reflections can mitigate scenarios of no packet reception, but at the expense of longer data set generation times. on an Intel(R) Xeon(R) W-2133 CPU @ 3.60 GHz test system in approximately 1 hour and
30 minutes
For example. this plot was generated by using these values.
• nodeSeparation = 0.25
• numObsPerPair = 200
• miniBatchSize = 270
• maxEpochs = 20
• InitialLearnRate = 1e-4
The example uses these helper functions:
1. Bluetooth Technology Website. “Bluetooth Technology Website | The Official Website of Bluetooth Technology.” Accessed May 22, 2024. https://www.bluetooth.com
2. Bluetooth Special Interest Group (SIG). "Core System Package [Low Energy Controller Volume]". Bluetooth Core Specification. Version 5.3, Volume https://www.bluetooth.com
3. Shangyi Yang, Chao Sun, and Younguk Kim, "Indoor 3D Localization Scheme Based on BLE Signal Fingerprinting and 1D Convolutional Neural Network", Electronics 10, no. 15 (July 22, 2021): 1758.
Local Functions
function [locators,nodes,antPosNode] = createScenario(nodeSep)
%createScenario Generates the tx locator and rx node objects
%based on separation between the nodes
fc = 2.426e9; % Set the carrier frequency (Hz) to one of broadcasting channels
lambda = physconst("lightspeed")/fc;
txArray = arrayConfig("Size", [1 1], "ElementSpacing", 2*lambda);
rxArray = arrayConfig("Size", [1 1], "ElementSpacing", lambda);
% The dimensions of the room are x:[-1.5 1.5] y:[-1.5 1.5] z:[0 2.5].
% Place transmitters at the 8 corners of the room. Choose the x-, y-,
% and z-coordinates for the locators and generate all possible
% combinations of x, y, and z from these coordinates.
xLocators = [-1.4 1.4]; % In meters
yLocators = [-1.4 1.4]; % In meters
zLocators = [0.1 2.4]; % In meters
% Define valid space for nodes
xNode = [-1 1];
yNode = [-1 1];
zNode = [0.8 2];
dX = diff(xNode);
dY = diff(yNode);
dZ = diff(zNode);
dims = [dX dY dZ];
% Calculate antenna positions
possC = combinations(xLocators,yLocators,zLocators);
antPosLocator = table2array(possC)';
rxSep = nodeSep;
numSeg = floor(dims/rxSep);
dimsOffset = (dims-(numSeg*rxSep))./2;
xGridNode = (min(xNode)+dimsOffset(1)):rxSep:(max(xNode)-dimsOffset(1));
yGridNode = (min(yNode)+dimsOffset(2)):rxSep:(max(yNode)-dimsOffset(2));
zGridNode = (min(zNode)+dimsOffset(3)):rxSep:(max(zNode)-dimsOffset(3));
% Set the position of the node antenna centroid by replicating the
% position vectors across 3-D space
antPosNode = [repmat(kron(xGridNode, ones(1, length(yGridNode))), 1, length(zGridNode)); ...
repmat(yGridNode, 1, length(xGridNode)*length(zGridNode)); ...
kron(zGridNode, ones(1, length(yGridNode)*length(xGridNode)))];
% Create multiple locator and node sites with a single constructor call
locators = txsite("cartesian", ...
AntennaPosition=antPosLocator, ...
Antenna=txArray, ...
nodes = rxsite("cartesian", ...
AntennaPosition=antPosNode, ...
Antenna=rxArray, ...
function [training,validation,test] = splitDataSet(data,labels,trainRatio,validRatio)
%splitDataSet Create training, validation, and test data by randomly shuffling and splitting the data.
% Generate random indices for training/validation/set splits
[trainInd,valInd,testInd] = dividerand(size(data,3),trainRatio,validRatio,1-trainRatio-validRatio);
% Filter training, validation and test data
trainInd = trainInd(randperm(length(trainInd)));
valInd = valInd(randperm(length(valInd)));
training.X = data(:,:,trainInd);
validation.X = data(:,:,valInd);
test.X = data(:,:,testInd);
% Filter training, validation and test labels
training.Y = labels.position(trainInd,:);
validation.Y = labels.position(valInd,:);
test.Y = labels.position(testInd,:);
function rxChan = generateRxData(rays,beacons,nodes,txW,numRSSI,samplingRate)
%generateRxData Initialize Ray tracing channel for specific locator-node
%pair and pass the waveform through the channel
rtChan = comm.RayTracingChannel(rays,beacons,nodes); % Create channel
rtChan.ReceiverVirtualVelocity = [0; 0; 0]; % Stationary Receiver
rtChan.NormalizeImpulseResponses = true;
rtChan.SampleRate = samplingRate;
txWRep = repmat(txW,numRSSI,1);
rxChan = rtChan(txWRep);
function rssi = generateRSSI(rxChan,thNoise,numRSSI,dBdBmConvFactor)
%generateRSSI Calculate RSSI values after passing the waveform through
%thermal noise
rxW = thNoise(rxChan);
rxWSplit = reshape(rxW,length(rxW)/numRSSI,numRSSI);
rssiL = mean(abs(rxWSplit).^2);
rssi = 20*log10(rssiL(:))+dBdBmConvFactor; % dbm
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Simplifying root expressions
simplifying root expressions
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Author Message
iasyredir Posted: Thursday 28th of Dec 09:46
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Equations by information.
Solving kfir Posted: Friday 29th of Dec 07:52
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Author Message
iasyredir Posted: Thursday 28th of Dec 09:46
I have trouble with simplifying root expressions. I tried a lot to find somebody who can assist me with this. I also looked out for a coach to tutor me and crack my problems on
converting decimals, angle-angle similarity and perpendicular lines. Though I located a few who could possibly solve my problem, I recognized that I cannot afford them. I do not
have much time too. My quiz is coming up soon . I am worried . Can anyone assist me with this situation? I would greatly be glad about any assistance or any information.
kfir Posted: Friday 29th of Dec 07:52
Don’t fear, Algebrator is here ! I was in a same situation sometime back, when my friend advised that I should try Algebrator. And I didn’t just clear my test; I went on to
score really well in it . Algebrator has a really easy to use GUI but it can help you solve the most challenging of the problems that you might face in algebra at school. Just
try it and I’m sure you’ll ace your test .
From: egypt
cmithy_dnl Posted: Sunday 31st of Dec 08:37
Oh wow! Nice to see that people adopt Algebrator here as well. I can guarantee the usefulness of this program. It is simply awesome .
From: Australia
Svizes Posted: Monday 01st of Jan 09:59
multiplying fractions, multiplying matrices and least common measure were a nightmare for me until I found Algebrator, which is really the best math program that I have ever
come across. I have used it through many math classes – Remedial Algebra, College Algebra and Intermediate algebra. Simply typing in the algebra problem and clicking on Solve,
Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I really recommend the program.
From: Slovenia
Posted: Thursday 28th of Dec 09:46
I have trouble with simplifying root expressions. I tried a lot to find somebody who can assist me with this. I also looked out for a coach to tutor me and crack my problems on converting decimals,
angle-angle similarity and perpendicular lines. Though I located a few who could possibly solve my problem, I recognized that I cannot afford them. I do not have much time too. My quiz is coming up
soon . I am worried . Can anyone assist me with this situation? I would greatly be glad about any assistance or any information.
Posted: Friday 29th of Dec 07:52
Don’t fear, Algebrator is here ! I was in a same situation sometime back, when my friend advised that I should try Algebrator. And I didn’t just clear my test; I went on to score really well in it .
Algebrator has a really easy to use GUI but it can help you solve the most challenging of the problems that you might face in algebra at school. Just try it and I’m sure you’ll ace your test .
Posted: Sunday 31st of Dec 08:37
Oh wow! Nice to see that people adopt Algebrator here as well. I can guarantee the usefulness of this program. It is simply awesome .
Posted: Monday 01st of Jan 09:59
multiplying fractions, multiplying matrices and least common measure were a nightmare for me until I found Algebrator, which is really the best math program that I have ever come across. I have used
it through many math classes – Remedial Algebra, College Algebra and Intermediate algebra. Simply typing in the algebra problem and clicking on Solve, Algebrator generates step-by-step solution to
the problem, and my algebra homework would be ready. I really recommend the program. | {"url":"https://sofsource.com/algebra-2-chapter-4-resource-book/parallel-lines/simplifying-root-expressions.html","timestamp":"2024-11-10T09:18:48Z","content_type":"text/html","content_length":"94079","record_id":"<urn:uuid:b86c7368-6688-472d-a6a1-3b4791d1d87e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00790.warc.gz"} |
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Posts for the month of September 2018
Summary of last week's work
1. Worked on alpha prescription stuff for energy paper and discussed with Eric.
2. Continued running simulation with secondary mass halved (run 149) separation vs time plot
3. Started running new simulations to explore effects of softening length and resolution.
□ This consisted of 3 new simulations restarted from frame 72 (t=16.7 days) of run 143 (Model A of Paper I), where the softening length and smallest resolution element had been halved.
1. do not halve softening length nor double resolution (Run 152)
2. halve softening length but but do not double resolution (Run 153)
3. do not halve softening length but double resolution (Run 154)
4. Discussed jets with Jonathan.
5. Ran low resolution simulations to explore the possibility of simulating Roche lobe overflow (leading up to common envelope evolution).
Energy paper
Ivanova+2013 equation 3:
New equation suggested by us:
Run 143 (Model A of Paper I) has initial separation . The Roche limit radius for as in our case is .
With , which is the final separation at t=40 days, we get:
LHS RHS
Ivanova+2013 with 1.9 0.23
Ivanova+2013 with 1.9 0.64
New equation with 3.1 1.23
New equation with 2.4 1.09
So one would never expect the envelope to be completely unbound at this point in the simulation because this would require . Therefore, that the envelope is not completely unbound at the end of the
simulation is consistent with the alpha prescription.
Then at what value of should the envelope be completely unbound, for a given value of ?
Ivanova+2013 with
Ivanova+2013 with
New equation with
New equation with
So if we say that there are no extra energy sources or sinks and take the most optimistic (but not very realistic) case , then the final separation is predicted to be about . Compare this to the
softening radius at the end of the simulation of .
In the probably more realistic case of , we would have a final separation of about . This would imply a merger if the secondary is a main sequence star, but not necessarily if the secondary is a
white dwarf.
An AGB star is less tightly bound so would be more promising for ejecting the envelope.
Roche lobe overflow tests
Face-on density (zoomed in) (Run 156 a=109Rsun), equal to theoretical Roche limit separation for this system.
Face-on density (zoomed in) (Run 157 a=98Rsun)
Face-on density (zoomed in) (Run 159 a=73.5Rsun)
Face-on density (zoomed in) (Run 158 a=49Rsun)
The sound-crossing time for the RG is about 8 days or ~35 frames. This is the approximate time the star would take to deform to fill the Roche lobe.
While this deformation is happening, the secondary accretes from the ambient medium. The freefall time onto the point mass is . So in 5 days, or about 20 frames, this corresponds to gas being
accumulated out to a sphere of radius from the secondary, corresponding to about of gas falling onto the secondary from the ambient medium ( grams/cc).
There are numerical problems in the low resolution runs that cause the RG to bulge out opposite to the direction of motion along the orbit.
It is probably worth starting a full resolution run to see what happens. But at which separation?
To-do list
• recombination equations (priority)
• sort out high momentum wind result
• continue running low momentum wind
Low Momentum Wind
Speed = 300 km/s, density 5e-13 g/cm^3
Run to 7400 days (1400 days after quiescent phase) Density movie:
Next, adjust refinement region and level to continue.
High Momentum Wind
Run to 8000 days (2000 days after quiescent phase). Will update a link to the results as soon as they are done.
Maybe redo the runs for better consistency in frame rate and refinement?
Radiation pressure
Communicating with TACC about missing tmi fabric. While loop appears to be working (should be blocking, but it's hard to tell when it immediately errors…), so should be working once the execution
problem is straightened out.
BlueHive predicts 2.3 years, before being killed (due to long timestep), so it's not going to be effective to run even with a large reservation.
HD209458b short paper
Postprocessing of HD209458b run in progress. I'll finish some of the analysis in the next few days. Things to look at right now:
• Tau/xi terms (ratios of dimensionless Coriolis, tidal, wind accelerations)
• Regimes (should be energy-limited; probably type II from Matsakos?)
• Velocities and densities along sub-/anti-stellar ray, with markers for sonic and Hill radius (Coriolis?)
• Synthetic observations - high v?
• Compare all of these with most similar (high mass, low flux) planet from parameter space paper
Plan for XSEDE proposal 2018
Summary of current allocation
• A bit less than 130,000 node hours remain.
• It is estimated (still very rough though) that we can complete about 15 runs like the non-accretion run from paper I (run 143).
• We have thus far used about 23% of the allocation so about 77% remains.
• To get this down to <50% by Oct 15, we need to use up about 27% of the full allocation or a little over 1/3 of what remains.
• This translates to about 5-6 runs by Oct 15.
Plan for now until Oct 15
The following runs do not involve major changes to the code so can hopefully be done in this time frame:
1. Test dependence on secondary mass (parameter regime 1).
1. Run 149: As 143 but with secondary mass smaller by a factor of 2, i.e. m2=0.5 Msun instead of 1.0 Msun (running).
2. Test dependence on secondary mass (parameter regime 2).
1. Run 151: As 143 but with secondary mass smaller by a factor of 4, i.e. m2=0.25 Msun instead of 1.0 Msun (submitted).
3. Test convergence with respect to softening length and resolution.
1. Run 152: As 143 restarted from frame 72, which is the frame where the softening radius and smallest resolution element were halved in 143, but now do not halve the softening length nor the
resolution element (submitted).
2. As 152 but halve softening length only.
3. As 152 but halve smallest resolution element only.
4. Roche lobe overflow.
1. Put secondary at Roche limit separation.
2. Must think about refinement.
3. Increase q to 1 both to reduce Roche limit separation and also increase .
5. Test dependence on initial spin of the RG.
1. Repeat run 143 but now initialize RG in solid body rotation at the initial orbital angular velocity.
6. Test convergence with respect to size of refinement region (parameter regime 1).
1. Repeat run 143 using a more liberal choice for the refinement radius for each interval (c.f. Fig 3 in Paper I)
7. Test convergence with respect to size of refinement region (parameter regime 1).
1. Based on the results of the above, repeat run once more using either an even more liberal refinement radius or else a refinement radius that is more conservative than run 143.
• Another possibility is to restart run 143 from the end of the simulation at t = 40 days. This is easy to do but the drawback is that material will flow out of the box soon after.
Plan for Oct 15 to Dec 31
We can expect about 10 more runs equivalent to run 143 but I provide 15 possibilities below:
Questions to answer for jet project:
• How does the jet affect the morphology of the envelope?
• How does the jet affect the ejection of the envelope?
• How does the jet evolve, does it get quenched?
• What is the dependence of these questions on when the jet gets turned on?
• What is the dependence on accretion rate?
• Opening angle?
1. Simulate CEE with a jet from the secondary (parameter regime 1) (Restart from run 143)
2. Simulate CEE with a jet from the secondary (parameter regime 2) (Restart from run 143)
3. Simulate CEE with a jet from the secondary (parameter regime 3) (Restart from run 143)
4. Simulate CEE with a jet from the secondary (parameter regime 4) (Restart from run 143)
5. Simulate CEE with a jet from the secondary (parameter regime 5) (Restart from run 143)
6. Simulate RGB/AGB star in improved simulation including (part I of a long run)
1. larger box (but smaller ambient resolution): the larger the box the longer we can run it before material starts leaving the box,
2. ambient consisting of hydrostatic atmosphere surrounded by very low density and low pressure gas,
3. start from Roche limit separation (optional)
7. Simulate RGB/AGB star in improved simulation including (part II of a long run)
8. Simulate RGB/AGB star in improved simulation including (part III of a long run)
9. Simulate RGB/AGB star in improved simulation including (part IV of a long run)
10. Simulate RGB/AGB star in improved simulation including (part V of a long run)
11. Triple system involving a planet or tertiary star orbiting the secondary… (parameter regime 1)
12. Triple system involving a planet or tertiary star orbiting the secondary… (parameter regime 2)
13. Triple system involving a planet or tertiary star orbiting the secondary… (parameter regime 3)
14. Triple system involving a planet or tertiary star orbiting the secondary… (parameter regime 4)
15. Triple system involving a planet or tertiary star orbiting the secondary… (parameter regime 5)
• The size of run 143 data is about 10 TB.
• To do 15 more runs like 143 would require 150 TB.
• This would cost an additional $15,000 per year on bluehive which is unaffordable.
• We need to look for other options like free space somewhare on blue hive.
• We probably need to keep only ½ of the data, i.e. every second chombo file.
Radiation Pressure
No real progress to report, other than that stampede is beginning to test my patience. But I believe I've solved all of the problems (except the MPI issue - more below), so we should at least start
getting some frames.
MPI issues
Located all of our calls to IRECV and ISEND, and all matching WAITs are there. No MPI errors are going ignored. Is there anything else we can do to be convinced that the problem's not on our end?
(Here's one example of a case where irecv is called by RECV, which could potentially cause the issues we're seeing but not be anything we can affect).
I've found the environment variable that controls the request descriptors (not sure why I couldn't find it before, but I distinctly remember not being able to set it), and upped it to 10^14, so
hopefully that will be sufficient to allow us to run for the full two days we're allowed.
HD209458b ionizing flux only
Got some analysis done here. Mass loss rates are about an order of magnitude lower, which is to be expected given the tighter binding of the atmosphere.
Mass loss rate: ~5.3x10^9 g/s
It is (almost?) just Roche lobe overflow. Here's the Hill sphere (in red) overlaid on the wind, with contours of the dimensionless Coriolis length.
MHD planets
Here's the global steady state.
Global simulations are actually much worse than local. The run stalls out after 6 frames. Running for 2 more days, we get three more frames; here's the last one output:
Stellar wind implementation
I have a Matlab script that's solving for the (radial) velocity, pressure, and density of a pseudo-stellar wind with adiabatic gamma. I have noticed a couple of odd jumps that seem to be artifacts of
the numerical solver, however. Also, I can't solve for any points inside the reference radius. This seems like a Matlab (or, specifically, vpasolve) limitation, but I haven't looked into other
numerical methods in Matlab or otherwise. I've attached a sample solution file.
Current list
I'll just keep this as a running tally, to keep us all organized a bit:
To Do
1. Continue radiation pressure runs
2. MHD planet runs (use the larger local runs as good-enough?)
3. Invert subcycling for line transfer
4. Add MHD to line transfer run
5. Test John’s stellar wind for charge exchange
6. Test charge exchange
7. Run charge exchange
8. Development for AMR line transfer/point source line transfer
9. Development for metastable He
Debug MPI issue on stampede (ish)
Set up global MHD run (w/ stellar wind)
Summary of last week's work
1. Reran first part of simulation with secondary mass halved, after deriving orbital parameters and carefully testing the initial orbit p_mult_143_149.png
2. Worked on energy paper (on the section dealing with the energy prescription with formulation) en.pdf.
3. Worked with Thomas (Orsola's student) to set up an AGB star for CE simulations.
4. Worked with Amy to debug the CE-wind with self-gravity simulation.
All of these are ongoing—-I expect to report more progress next Monday.
Project/To-Do List
• Radiation pressure
□ Run high/med/low flux
□ Debug MPI issue (using too many descriptors)
• MHD planets
□ Fill parameter space
□ Add fields to parameter space run
• Charge exchange
□ Get initial conditions working (stellar wind without charge exchange populations)
□ Test charge exchange (have an attempt to compare to Christie et al [with fixed boundary code, though] - any other ways?)
□ Run (with suite of stellar wind strengths as in John's paper, perhaps? Should maybe look at what's predicted to affect the efficacy of charge exchange)
MHD planets
Refer to Jonathan's page for simulation parameters.
Using box of effective resolution 512^3 for size (1 CU)^3, with sphere of radius 0.25 CU forced to be resolved to level 1 (so it's equivalent to Jonathan's previous runs). Currently running
corotating sims (do we want non-rotating as well?). Beta of 0.1 corresponds to surface magnetic field of 0.23 G.
theta \ beta 10 1 0.1
0 224 — 208
45 — — —
90 — — —
Have hydro steady state:
Also have a late state of the beta=0.1 run. May actually be a couple of frames too long, with some of the field lines being distorted down-orbit by stellar wind/planetary wind interactions? Running
time is increasing, but only 4 days remaining when I restart (worth trying to get a few more frames?).
Beta = 10 run hit a wall at frame 224 - 1.9 months remaining.
Radiation Pressure
Run with no radiation pressure appears to have reached steady state. Actually not too different from frame 180 where radiation pressure is introduced in new runs.
Have a couple of frames from the high flux and medium flux cases. Just had stampede queue access restored, so will be restarting them.
Fixed makefile issue for debugging on stampede. Now that I have queue access again, should be able to use DDT to see if there's a leaky thread somewhere.
370 days after quiescent phase http://www.pas.rochester.edu/~yzou5/PNe_simulation_movies/Old_movies/High_momentum_Day6370_rho.gif
Quiescent phase
6000 days http://www.pas.rochester.edu/~yzou5/PNe_simulation_movies/Old_movies/High_momentum_D6000_Qphase.gif
Refinement region
for frames 380-392 (t = 6370 days). it takes ~6 hours to generate 1 frame. dimensions are in solar radius.
level 1-4
(x, y, z) = +-(5000, 5000, 10000)
level 5-7
(x, y, z) = +-(1000, 1000, 6000)
Instead of making the wedge tip same as the ambient as in blog:bliu08012018 , tried with Bruce's idea of "t=0 simply replicate the flow at the edge of the launch surface into the wedge". Seems
working much better than the scheme in blog:bliu08012018
Movie up to 400y movie | {"url":"https://bluehound.circ.rochester.edu/astrobear/blog/2018/9","timestamp":"2024-11-05T20:26:32Z","content_type":"text/html","content_length":"95595","record_id":"<urn:uuid:e3ba33f2-a6a3-4f74-a0b5-58e8fc71fe51>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00271.warc.gz"} |
How big does my mash tun need to be?
How big does my mash tun need to be?
How Big Does My Mash Tun Need to Be?
Mash Tun Size 5 gallon (20 quart) 15.5 gallon (62 quart)
Max grain capacity (assuming 1.25 qt/lb) 12 lbs 37.2 lbs
Total strike water (gallons) 3.75 11.625
Total combined volume (gallons) 4.87 15.097
Max gravity units (@ 70% efficiency) 295 914.5
How many pounds of grain are in a 5 gallon mash tun?
10-11 pounds
a 5 gallon beverage cooler will handle approximately 10-11 pounds of grain, a 10 gallon will handle roughly 20lbs and this is with a standard single infusion mash and batch sparge or fly sparge.
How much grain can a 10 gallon mash tun?
25 lbs
As a lower limit for water to grain ratio you could use 1.25 qts/lb and at that value you could mash-in 25 lbs of grain into a 10 gallon mash tun. If it’s a true 10 gallon capacity, then yeah, 25 lbs
will be the max at a ratio of 1.25 qts/lb. The mash will go right up to the rim, so stir carefully. Happy mashing!
How many pounds of grain do I need for 5 gallons of beer?
How Much Grain Do I Need For Brewing? For every 5 gallons of beer (18 l), you’ll want between 8-15 lbs (4-7 kg) of base malt. It depends on the type of beer you want to brew whether you need 9 L or
21 L.
How much space does grain take up in the mash tun?
Re: Volume of a pound of grain I can typically fill a standard 6.5 gal plastic fermentation bucket almost to the top with 20 lb of milled grain. That equates to about 3.3lb/gal. So a pound of grain
has a volume of 0.3 gallons. Malted barley has a density of 31lb/c.f..
How big of a kettle for brew in a bag?
The BIAB Equipment Your kettle needs to be big enough to handle the full volume of liquor (hot water) and grain. Assuming you want to brew 5 gallon batches, I would recommend using a 10 gallon
kettle. That size will allow you to brew beers up to 1.070 original gravity.
How much malt do I need for 5 gallons of mash?
8.5 lbs.
So, 8.5 lbs. of malt will give us our target OG in 5 gallons.
How much barley do I need for 5 gallons of mash?
1.5 pounds
Ingredients: 5 gallons of water. 8.5 pounds of flaked maize. 1.5 pounds of crushed malted barley.
How much water do I need for 5 gallons of mash?
Don’t forget to account for all your water losses, which include liquid lost in the mash tun, the hot and cold break, and trub and samples in the fermenters. All those things can add up to a good 2
to 3 quarts. So if you want to end up with 5 gallons of beer, after the boil you really need somewhere around 5.5 gallons.
How many pounds of grain are in a gallon of mash?
The general guideline for AGs is 2lb total grain/gallon. The ratio of unmalted to malted is going to depend on the grain type and the diastatic power (DP) of each malted grain. A DP of 30 is
generally needed to convert a mash successfully.
Can you BIAB with a 5 gallon kettle?
Is BIAB the same as all-grain?
Traditional All-Grain. Brew-in-a-bag (BIAB) is a form of all-grain brewing. The primary difference between BIAB and traditional, multi-vessel brewing is that BIAB does not require sparging. | {"url":"https://pfeiffertheface.com/how-big-does-my-mash-tun-need-to-be/","timestamp":"2024-11-02T09:32:44Z","content_type":"text/html","content_length":"45252","record_id":"<urn:uuid:b9593dde-aeb3-4092-a25e-512145997410>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00381.warc.gz"} |
A single can see, model 3 is as good as model 2 in reproducing | http://www.ck2inhibitor.com
One particular can see, model three is as great as model Mirogabalin site waiting time distribution of your polar websites. This indicates that polar and nonpolar division web-sites are a priori
equivalent for cell division. Having said that, there are additional components that make the polar division waiting time appear longer. To make certain that the boost in 6 Effect in the Min
Technique on Timing of Cell Division in E. coli waiting time in the polar web pages just isn’t the consequence with the fact that only distinct division web pages are MedChemExpress GSK189254A
observed, we also measured inside the simulations of model 3 the waiting time distribution of division web pages close to mid-cell. The waiting time of this website is nearly identical to that from
the other non-polar web PubMed ID:http://jpet.aspetjournals.org/content/132/3/354 sites indicating that there is indeed a thing special about the polar websites. We give achievable explanations
inside the discussion. By far the most crucial locating of model 3 is that there’s no distinction in division waiting occasions between polar and non-polar sites. To test this experimentally we
assumed that existence time of Z-rings at a division site is usually a measure for the waiting time of the division web-site. We expressed fluorescently labeled FtsZ and determined the time interval
involving first appearance of your Zring and cell division at polar and non-polar web pages. Fig. 9 shows this time interval as function of waiting time from the division web page. As one can see,
there’s a clear difference involving WT and minB2 cells but no considerable difference between polar and non-polar web pages supporting the findings of model 3. Thus, model three is able to capture
the primary experimental
One can see, model three is as excellent as model 2 in reproducing
A single can see, model three is as excellent as model 2 in reproducing the experimental data but on top of that yields the right waiting time distribution with the polar sites. This indicates that
polar and nonpolar division web sites are a priori equivalent for cell division. Having said that, you will find additional factors that make the polar division waiting time seem longer. To ensure
that the improve in 6 Impact on the Min Technique on Timing of Cell Division in E. coli waiting time in the polar web pages will not be the consequence of your fact that only specific division web
sites are observed, we also measured inside the simulations of model three the waiting time distribution of division sites close to mid-cell. The waiting time of this website is nearly identical to
that with the other non-polar sites indicating that there’s certainly something special in regards to the polar web sites. We give attainable explanations within the discussion. One of the most
significant locating of model 3 is that there is no difference in division waiting times among polar and non-polar internet sites. To test this experimentally we assumed that existence time of
Z-rings at a division web-site can be a measure for the waiting time of your division website. We expressed fluorescently labeled FtsZ and determined the time interval between initially appearance on
the Zring and cell division at polar and non-polar web pages. Fig. 9 PubMed ID:http://jpet.aspetjournals.org/content/138/1/48 shows this time interval as function of waiting time in the division
internet site. As one particular can see, there is a clear difference amongst WT and minB2 cells but no substantial difference amongst polar and non-polar sites supporting the findings of model 3.
Hence, model 3 is in a position to capture the primary experimental observations. But nevertheless, the query remains why minB2 cells have a longer division waiting time than WT. We speculated that
this may be brought on by the truth that minB2 cells are longer and as a result have much more division web-sites. Thus, a priory a division site in minB2 cells has the exact same waiting time as a
division in WT. Nevertheless, due to the fact minB2 cells have additional division websites than WT it should really, for a offered quantity of cell division machinery, take longer to finish division
at these sites. To implement this hypothesis into our model we assign a quantity x to every division web-site that measures how much the division procedure has proceeded. Upon appearance from the
division web-site we set x 0, division is completed for x Tw, exactly where Tw is definitely the waiting time assigned for the division site drawn from the experimentally measured distribution of WT.
Amongst time t1 and t2 we enhance x by Experiment Experiment Simulation Simulation polar non-polar polar non-polar old pole 3 31 six 38 non-polar 17 36 21 15 new pole 13 20 All cell divisions inside
200 minutes are classified into five forms as outlined by the position of two successive cell divisions. Rows represent the place from the 1st division event, columns location with the second
occasion. Number of events is provided in percentage. Time in parenthesis represents imply time difference + standard deviation amongst the division events. doi:10.1371/journal.pone.0103863.t003 7
Impact on the Min System on Timing of Cell Division in E. coli t2 x{x t1 dt dx: dt 2 dx 1 but now we dt Tw want to take into account that several division sites compete for the division machinery and
that larger cells have a larger amount of division machinery. We therefore set In the previous models we simply had dx L={LC: C dt 3 Here, L is cell length, N the number of potential division sites
and LC Kruppel-like factor 4 is a transcription.One can see, model three is as good as model two in reproducing the experimental information but additionally yields the right waiting time
distribution of your polar web pages. This indicates that polar and nonpolar division websites are a priori equivalent for cell division. Having said that, there are more aspects that make the polar
division waiting time seem longer. To make certain that the boost in 6 Impact from the Min System on Timing of Cell Division in E. coli waiting time of your polar websites just isn’t the consequence
of the fact that only precise division web-sites are observed, we also measured within the simulations of model three the waiting time distribution of division web-sites close to mid-cell. The
waiting time of this website is practically identical to that from the other non-polar web-sites indicating that there is indeed something special in regards to the polar websites. We give feasible
explanations within the discussion. Probably the most critical obtaining of model three is the fact that there is no difference in division waiting occasions among polar and non-polar web-sites. To
test this experimentally we assumed that existence time of Z-rings at a division website is usually a measure for the waiting time in the division web-site. We expressed fluorescently labeled FtsZ
and determined the time interval in between very first look of the Zring and cell division at polar and non-polar web-sites. Fig. 9 shows this time interval as function of waiting time in the
division website. As 1 can see, there is a clear difference between WT and minB2 cells but no important difference amongst polar and non-polar websites supporting the findings of model three. Thus,
model 3 is in a position to capture the main experimental observations. But nonetheless, the question remains why minB2 cells possess a longer division waiting time than WT. We speculated that this
could possibly be brought on by the fact that minB2 cells are longer and hence have a lot more division websites. Hence, a priory a division web-site in minB2 cells has exactly the same waiting time
as a division in WT. Nonetheless, because minB2 cells have extra division web sites than WT it really should, for a offered volume of cell division machinery, take longer to finish division at these
websites. To implement this hypothesis into our model we assign a quantity x to every division site that measures just how much the division procedure has proceeded. Upon appearance from the division
internet site we set x 0, division is completed for x Tw, exactly where Tw is definitely the waiting time assigned towards the division web page drawn in the experimentally measured distribution of
WT. Involving time t1 and t2 we increase x by Experiment Experiment Simulation Simulation polar non-polar polar non-polar old pole 3 31 6 38 non-polar 17 36 21 15 new pole 13 20 All cell divisions
inside 200 minutes are classified into five types based on the position of two successive cell divisions. Rows represent the location with the first division event, columns place of the second event.
Variety of events is offered in percentage. Time in parenthesis represents imply time distinction + common deviation amongst the division events. doi:10.1371/journal.pone.0103863.t003 7 Effect in the
Min System on Timing of Cell Division in E. coli t2 x{x t1 dt dx: dt 2 dx 1 but now we dt Tw want to take into account that several division sites compete for the division machinery and that larger
cells have a larger amount of division machinery. We therefore set In the previous models we simply had dx L={LC: C dt 3 Here, L is cell length, N the number of potential division sites and LC
Kruppel-like factor 4 is a transcription.
1 can see, model 3 is as very good as model two in reproducing
A single can see, model three is as superior as model 2 in reproducing the experimental data but on top of that yields the right waiting time distribution on the polar web pages. This indicates that
polar and nonpolar division web sites are a priori equivalent for cell division. Even so, there are actually additional aspects that make the polar division waiting time seem longer. To ensure that
the increase in six Effect of the Min Program on Timing of Cell Division in E. coli waiting time from the polar internet sites isn’t the consequence in the truth that only certain division web-sites
are observed, we also measured within the simulations of model three the waiting time distribution of division web pages close to mid-cell. The waiting time of this website is practically identical
to that from the other non-polar sites indicating that there is certainly indeed one thing specific about the polar web pages. We give possible explanations within the discussion. Essentially the
most important finding of model three is that there is no difference in division waiting times amongst polar and non-polar web pages. To test this experimentally we assumed that existence time of
Z-rings at a division web page is usually a measure for the waiting time of the division web site. We expressed fluorescently labeled FtsZ and determined the time interval amongst 1st appearance on
the Zring and cell division at polar and non-polar internet sites. Fig. 9 PubMed ID:http://jpet.aspetjournals.org/content/138/1/48 shows this time interval as function of waiting time of your
division web page. As one particular can see, there’s a clear difference amongst WT and minB2 cells but no substantial difference involving polar and non-polar web pages supporting the findings of
model 3. As a result, model three is able to capture the main experimental observations. But nonetheless, the question remains why minB2 cells possess a longer division waiting time than WT. We
speculated that this could possibly be caused by the fact that minB2 cells are longer and therefore have extra division web sites. As a result, a priory a division web site in minB2 cells has
precisely the same waiting time as a division in WT. Nevertheless, mainly because minB2 cells have extra division web pages than WT it ought to, for any given amount of cell division machinery, take
longer to finish division at these websites. To implement this hypothesis into our model we assign a quantity x to each division website that measures how much the division course of action has
proceeded. Upon appearance from the division website we set x 0, division is completed for x Tw, where Tw is definitely the waiting time assigned for the division web site drawn from the
experimentally measured distribution of WT. Between time t1 and t2 we increase x by Experiment Experiment Simulation Simulation polar non-polar polar non-polar old pole three 31 six 38 non-polar 17
36 21 15 new pole 13 20 All cell divisions within 200 minutes are classified into 5 types according to the position of two successive cell divisions. Rows represent the place of the 1st division
occasion, columns location in the second occasion. Variety of events is given in percentage. Time in parenthesis represents imply time difference + typical deviation in between the division events.
doi:ten.1371/journal.pone.0103863.t003 7 Impact in the Min System on Timing of Cell Division in E. coli t2 x{x t1 dt dx: dt 2 dx 1 but now we dt Tw want to take into account that several division
sites compete for the division machinery and that larger cells have a larger amount of division machinery. We therefore set In the previous models we simply had dx L={LC: C dt 3 Here, L is cell
length, N the number of potential division sites and LC Kruppel-like factor 4 is a transcription. | {"url":"https://www.ck2inhibitor.com/2017/10/19/a-single-can-see-model-3-is-as-good-as-model-2-in-reproducing/","timestamp":"2024-11-10T11:03:04Z","content_type":"text/html","content_length":"71782","record_id":"<urn:uuid:b96fbb70-3284-4d29-8a77-7245d6759a8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00662.warc.gz"} |
Towards the use of conservative thermodynamic variables in data assimilation: a case study using ground-based microwave radiometer measurements
Articles | Volume 15, issue 7
© Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License.
Towards the use of conservative thermodynamic variables in data assimilation: a case study using ground-based microwave radiometer measurements
This study aims at introducing two conservative thermodynamic variables (moist-air entropy potential temperature and total water content) into a one-dimensional variational data assimilation system
(1D-Var) to demonstrate their benefits for use in future operational assimilation schemes. This system is assessed using microwave brightness temperatures (TBs) from a ground-based radiometer
installed during the SOFOG3D field campaign, dedicated to fog forecast improvement.
An underlying objective is to ease the specification of background error covariance matrices that are highly dependent on weather conditions when using classical variables, making difficult the
optimal retrievals of cloud and thermodynamic properties during fog conditions. Background error covariance matrices for these new conservative variables have thus been computed by an ensemble
approach based on the French convective scale model AROME, for both all-weather and fog conditions. A first result shows that the use of these matrices for the new variables reduces some dependencies
on the meteorological conditions (diurnal cycle, presence or not of clouds) compared to typical variables (temperature, specific humidity).
Then, two 1D-Var experiments (classical vs. conservative variables) are evaluated over a full diurnal cycle characterized by a stratus-evolving radiative fog situation, using hourly TB.
Results show, as expected, that TBs analysed by the 1D-Var are much closer to the observed ones than the background values for both variable choices. This is especially the case for channels
sensitive to water vapour and liquid water. On the other hand, analysis increments in model space (water vapour, liquid water) show significant differences between the two sets of variables.
Received: 27 Oct 2021 – Discussion started: 20 Nov 2021 – Revised: 21 Jan 2022 – Accepted: 28 Feb 2022 – Published: 05 Apr 2022
Numerical weather prediction (NWP) models at convective scale need accurate initial conditions for skilful forecasts of high impact meteorological events taking place at a small scale such as
convective storms, wind gusts or fog. Observing systems sampling atmospheric phenomena at a small scale and high temporal frequency are thus necessary for that purpose (Gustafsson et al., 2018).
Ground-based remote-sensing instruments (e.g. rain and cloud radars, radiometers, wind profilers) meet such requirements and provide information on wind, temperature and atmospheric water (vapour and
hydrometeors). Moreover, data assimilation systems are evolving towards ensemble approaches where hydrometeors can be initialized together with typical control variables. This is the case for the
Météo-France NWP limited area model AROME (Seity et al., 2011; Brousseau et al., 2016), where, on top of wind (U,V), temperature (T) and specific humidity q[v], the mass content of several
hydrometeors can be initialized (cloud liquid water q[l], cloud ice water q[i], rain q[r], snow q[s] and graupel q[g]; Destouches et al., 2021). However, these variables are not conserved during
adiabatic and reversible vertical motion.
The accuracy of the analysed state in variational schemes highly depends on the specification of the so-called background error covariance matrix. Background error variances and cross-correlations
between variables are known to be dependent on weather conditions (Montmerle and Berre, 2010; Michel et al., 2011). This is particularly the case during fog conditions with much shorter vertical
correlation length scales at the lowest levels and large positive cross-correlations between temperature and specific humidity (Ménétrier and Montmerle, 2011). In this context, Martinet et al. (2020)
have demonstrated that humidity retrievals could be significantly degraded if sub-optimal background error covariances are used during the minimization. New ensemble approaches allow for better
approximation of background error covariance matrices but rely on the capability of the ensemble data assimilation (EDA) to correctly represent model errors, which might not always be the case during
fog conditions. This is why it would be of interest to examine, in a data assimilation context, the use of variables that are more suitable to times when water phase changes take place.
It is well known that most data assimilation systems were based on the assumptions of homogeneity and isotropy of background error correlations. To test these hypotheses, Desroziers and Lafore (1993)
and Desroziers (1997) implemented a coordinate change inspired by the semi-geostrophic theory to test flow-dependent analyses with case studies from the Front-87 field campaign (Clough and Testud,
1988), where the local horizontal coordinates were transformed into the semi-geostrophic space during the assimilation process. Another kind of flow-dependent analyses were made by Cullen (2003) and
Wlasak et al. (2006), who proposed a low-order potential vorticity (PV) inversion scheme to define a new set of control variables. Similarly, analyses on potential temperature θ were made by Shapiro
and Hastings (1973) and Benjamin et al. (1991), and more recently by Benjamin et al. (2004) with moist virtual θ[v] and moist equivalent θ[e] potential temperatures.
The aim of the paper is to test a one-dimensional data assimilation method that would be less sensitive to the average vertical gradients of the ($T,{q}_{\mathrm{v}},{q}_{\mathrm{l}},{q}_{\mathrm{i}}
$) variables. To this end, two conservative variables will be proposed, generalizing previous uses of θ (as a proxy for the entropy of dry air) to moist-air variables suitable for data assimilation.
The new conservative variables are the total water content ${q}_{\mathrm{t}}={q}_{\mathrm{v}}+{q}_{\mathrm{l}}+{q}_{\mathrm{i}}$ and the moist-air entropy potential temperature θ[s] defined in
Marquet (2011), which generalize the two well-known conservative variables (q[t],θ[l]) of Betts (1973).
The focus of the study will be on a fog situation from the SOFOG3D field campaign using a one-dimensional variational data assimilation system (1D-Var) for the assimilation of observed microwave
brightness temperatures (TBs) sensitive to T, q[v] and q[l] from a ground-based radiometer. Short-range forecasts from the convective scale model AROME (Seity et al., 2011) will be used as background
profiles, the ground-based version of the fast Radiative Transfer for the TIROS Operational Vertical Sounder (RTTOV-gb) model (De Angelis et al., 2016; Cimini et al., 2019) will allow for the
accurate simulation of the TBs and suitable background error covariance matrices will be derived from an ensemble technique.
Section 2 presents the methodology (conservative variables, 1D-Var, change of variables). Section 3 describes the experimental setting, the meteorological context, the observations and the different
components of the 1D-Var system. The results are commented in Sect. 4. Finally, conclusions and perspectives are given in Sect. 5.
This section presents the methodology chosen for this study. The definition of the moist-air entropy potential temperature θ[s] is introduced, as well as the formalism of the 1D-Var assimilation
system, before describing the “conservative variable” conversion operator.
2.1Moist-air entropy potential temperature
The motivation for using the absolute moist-air entropy in atmospheric science was first described by Richardson (1919, 1922), and then fully formalized by Hauf and Höller (1987). The method
comprises taking into account the absolute value for dry air and water vapour and to define a moist-air entropy potential temperature variable called θ[s].
However, the version of θ[s] published in Hauf and Höller (1987) was not really synonymous with the moist-air entropy. This problem has been solved with the version of Marquet (2011) by imposing the
same link with the specific entropy of moist air (s) as in the dry-air formalism of Bauer (1908), leading to
$\begin{array}{}\text{(1)}& s={c}_{p\mathrm{d}}\phantom{\rule{0.125em}{0ex}}\mathrm{ln}\left(\frac{{\mathit{\theta }}_{\mathrm{s}}}{{T}_{\mathrm{0}}}\right)+{s}_{\mathrm{d}\mathrm{0}}\left({T}_{\
where c[pd]≈1004.7$\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$ is the dry-air specific heat at constant pressure, T[0]=
273.15K is a standard temperature and ${s}_{\mathrm{d}\mathrm{0}}\left({T}_{\mathrm{0}},{p}_{\mathrm{0}}\right)\approx \mathrm{6775}$$\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm
{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$ is the reference dry-air entropy at T[0] and at the standard pressure p[0]=1000hPa. Because c[pd], T[0] and s[d0](T[0],p[0]) are
constant terms, θ[s] defined by Eq. (1) is synonymous with, and has the same physical properties as, the moist-air entropy s.
The conservative aspects of this potential temperature θ[s] and its meteorological properties (in e.g. fronts, convection, cyclones) have been studied in Marquet (2011), Blot (2013) and Marquet and
Geleyn (2015). The links with the definition of the Brunt–Väisälä frequency and the PV are described in Marquet and Geleyn (2013) and Marquet (2014), while the significance of the absolute entropy to
describe the thermodynamics of cyclones is shown in Marquet (2017) and Marquet and Dauhut (2018).
Only the first-order approximation of θ[s], denoted (θ[s])[1] in Marquet (2011), will be considered in the following, written as
$\begin{array}{}\text{(2)}& {\mathit{\theta }}_{\mathrm{s}}\approx \left({\mathit{\theta }}_{\mathrm{s}}{\right)}_{\mathrm{1}}=\mathit{\theta }\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left(-\frac
left({\mathrm{\Lambda }}_{\mathrm{r}}\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{t}}\right),\end{array}$
where $\mathit{\theta }=T{\left(p/{p}_{\mathrm{0}}\right)}^{\mathit{\kappa }}$ is the dry-air potential temperature, p the pressure, κ≈0.2857 and L[vap](T) and L[sub](T) the latent heat of
vaporization and sublimation respectively. The explanation for Λ[r] follows later in the section.
The first term θ on the right-hand side of Eq. (2) leads to a first conservation law (invariance) during adiabatic compression and expansion, with joint and opposite variations of T and p keeping θ
constant. Here lies the motivation for using θ to describe dry-air convective processes, also used in data assimilation systems by Shapiro and Hastings (1973) and Benjamin et al. (1991).
The first exponential on the right-hand side of Eq. (2) explains a second form of conservation law. Indeed, this exponential is constant for reversible and adiabatic phase changes, for which $d\left
({c}_{p\mathrm{d}}T\right)\approx d\left({L}_{\mathrm{vap}}\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{l}}+{L}_{\mathrm{sub}}\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{i}}\right)$ due to the approximate
conservation of the moist static energy ${c}_{p\mathrm{d}}\phantom{\rule{0.125em}{0ex}}T-{L}_{\mathrm{vap}}\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{l}}-{L}_{\mathrm{sub}}\phantom{\rule{0.125em}
{0ex}}{q}_{\mathrm{i}}$, and therefore has joint variations of the numerator and denominator and a constant fraction into the first exponential. It should be mentioned that the product of θ by this
first exponential forms the Betts (1973) conservative variable θ[l], which is presently used together with q[t] to describe the moist-air turbulence in general circulation models (GCMs) and NWP
While the variable θ[l] was established with the assumption of a constant total water content q[t] in Betts (1973), the second exponential in Eq. (2) sheds new light on a third and new conservation
law, where the entropy of moist air can remain constant despite changes in the total water q[t]. This occurs in regions where water-vapour turbulence transport takes place, or via the evaporation
process over oceans, or at the edges of clouds via entrainment and detrainment processes.
We consider here “open-system” thermodynamic processes, for which the second exponential takes into account the impact on moist-air entropy when the changes in specific content of water vapour are
balanced, numerically, by opposite changes of dry air, namely with $d{q}_{\mathrm{d}}=-d{q}_{\mathrm{t}}e \mathrm{0}$. In this case, as stated in Marquet (2011), the changes in moist-air entropy
depend on reference values (with subscript “r”) according to d[q[d](s[d])[r]+q[t](s[v])[r]], and thus with (s[d])[r] and (s[v])[r] being constant and with the relation ${q}_{\mathrm{d}}=\mathrm{1}-
{q}_{\mathrm{t}}$, leading to [(s[v])[r]−(s[d])[r]]dq[t].
This explains the new term ${\mathrm{\Lambda }}_{\mathrm{r}}=\left[\left({s}_{\mathrm{v}}{\right)}_{\mathrm{r}}-\left({s}_{\mathrm{d}}{\right)}_{\mathrm{r}}\right]/{c}_{p\mathrm{d}}\approx \mathrm
{5.869}±\mathrm{0.003}$, which depends on the absolute reference entropies for water vapour (s[v])[r]≈12671$\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}\phantom{\rule{0.125em}
{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$ and dry air (s[d])[r]≈6777$\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$. This also
explains that these open-system thermodynamic effects can be taken into account to highlight regimes where the specific moist-air entropy (s), θ[s] and (θ[s])[1] can be constant despite changes in q
[t], which may decrease or increase on the vertical (see Marquet, 2011, for such examples).
Although it should be possible to use (θ[s])[1] as a control variable for assimilation, it appeared desirable to define an additional approximation of this variable for a more “regular” and more
“linear” formulation, insofar as tangent-linear and adjoint versions are needed for the 1D-Var system. Considering the approximation $\mathrm{exp}\left(x\right)\approx \mathrm{1}+x$ for the two
exponentials in Eq. (2), neglecting the second-order terms in x^2, also neglecting the variations of L[v](T) with temperature and assuming a no-ice hypothesis (q[i]=0), the new variable is written as
$\begin{array}{}\text{(3)}& \left({\mathit{\theta }}_{\mathrm{s}}{\right)}_{a}=\mathit{\theta }\left[\mathrm{1}+{\mathrm{\Lambda }}_{\mathrm{r}}\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{t}}-\frac{{L}
_{\mathrm{vap}}\left({T}_{\mathrm{0}}\right)\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{l}}}{{c}_{p\mathrm{d}}\phantom{\rule{0.125em}{0ex}}T}\right],\text{(4)}& \left({\mathit{\theta }}_{\mathrm{s}}{\
right)}_{a}=\frac{\mathrm{1}}{{c}_{p\mathrm{d}}}{\left(\frac{{p}_{\mathrm{0}}}{p}\right)}^{\mathit{\kappa }}\phantom{\rule{0.125em}{0ex}}\left[{c}_{p\mathrm{d}}\left(\mathrm{1}+{\mathrm{\Lambda }}_{\
where L[vap](T[0])≈2501kJkg^−1. This formulation corresponds to ${S}_{\mathrm{m}}/{c}_{p\mathrm{d}}$, where S[m] is the moist static energy defined in Marquet (2011, Eq. 73) and used in the
European Centre for Medium-Range Weather Forecasts (ECMWF) NWP global model by Marquet and Bechtold (2020).
The new potential temperature (θ[s])[a] remains close to (θ[s])[1] (not shown) and keeps almost the same three conservative properties described for (θ[s])[1]. This new conservative variable (θ[s])[a
] will be used along with the total water content ${q}_{\mathrm{t}}={q}_{\mathrm{v}}+{q}_{\mathrm{l}}$ in the data assimilation experimental context described in the following sections.
2.21D-Var formalism
The general framework describing the retrieval of atmospheric profiles from remote-sensing instruments by statistical methods can be found in Rodgers (1976). In the following we present the main
equations of the one-dimensional variational formalism. Additional details are given in Thépaut and Moll (1990), who developed the first 1D-Var inversion applied to satellite radiances using the
adjoint technique.
The 1D-Var data assimilation system searches for an optimal state (the analysis) as an approximate solution of the problem minimizing a cost function 𝒥 defined by
$\begin{array}{}\text{(5)}& \begin{array}{rl}\mathcal{J}\left(\mathbit{x}\right)& =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathbit{x}-{\mathbit{x}}_{b}{\right)}^{T}{{\mathbf{B}}_{x}}^{-\mathrm{1}}\left(\
mathbit{x}-{\mathbit{x}}_{\mathrm{b}}\right)\\ & +\frac{\mathrm{1}}{\mathrm{2}}\left[\mathbit{y}-\mathcal{H}\left(\mathbit{x}\right){\right]}^{T}{\mathbf{R}}^{-\mathrm{1}}\left[\mathbit{y}-\mathcal
The symbol ^T represents the transpose of a matrix.
The first (background) term measures the distance in model space between a control vector x (in our study, T, q[v] and q[l] profiles) and a background vector x[b], weighted by the inverse of the
background error covariance matrix (B[x]) associated with the vector x. The second (observation) term measures the distance in the observation space between the value simulated from the model
variables ℋ(x) (in our study, the RTTOV-gb model) and the observation vector y (in our study, a set of microwave TBs from a ground-based radiometer), weighted by the inverse of the observation error
covariance matrix (R). The solution is searched iteratively by performing several evaluations of 𝒥 and its gradient:
$\begin{array}{}\text{(6)}& {\mathrm{abla }}_{x}\mathcal{J}\left(x\right)={{\mathbf{B}}_{\mathbf{x}}}^{-\mathrm{1}}\left(\mathbit{x}-{\mathbf{x}}_{\mathrm{b}}\right)-{\mathbf{H}}^{T}{\mathbf{R}}^{-\
where H is the Jacobian matrix of the observation operator representing the sensitivity of the observation operator to changes in the control vector x (H^T is also called the adjoint of the
observation operator).
2.3Conversion operator
The 1D-Var assimilation defined previously with the variables $x=\left(T,{q}_{\mathrm{v}},{q}_{\mathrm{l}}\right)$ can be modified to use the conservative variables $z=\left(\left({\mathit{\theta }}_
{\mathrm{s}}{\right)}_{a},{q}_{\mathrm{t}}\right)$. A conversion operator that projects the state vector from one space to the other can be written as x=ℒ(z). In the presence of liquid water q[l], an
adjustment to saturation is made to separate its contribution to the total water content q[t] from the water-vapour content q[v]. This is equivalent to distinguishing the “unsaturated” case from the
“saturated” one. Therefore, starting from initial conditions $\left({T}_{I},{q}_{I}\right)=\left(T,{q}_{\mathrm{v}}\right)$ and using the conservation of (θ[s])[a] given by Eq. (4), we look for the
variable T^∗ such that
$\begin{array}{}\text{(7)}& {T}^{\ast }+\mathit{\alpha }\phantom{\rule{0.125em}{0ex}}{q}_{\mathrm{sat}}\left({T}^{\ast }\right)={T}_{I}+\mathit{\alpha }\phantom{\rule{0.125em}{0ex}}{q}_{I},\end
$\begin{array}{}\text{(8)}& \mathit{\alpha }=\frac{{L}_{\mathrm{vap}}\left({T}_{\mathrm{0}}\right)}{{c}_{p\mathrm{d}}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{1}+{\mathrm{\Lambda }}_{\mathrm{r}}\
and q[sat](T^∗) is the specific humidity at saturation.
For the unsaturated case $\left({q}_{\mathrm{v}}<{q}_{\mathrm{sat}}\left({T}^{\ast }\right)\right)$, we obtain the variables $\left(T,{q}_{\mathrm{v}},{q}_{\mathrm{l}}\right)$ directly from Eq. (4):
$\begin{array}{}\text{(9)}& T=\left({\mathit{\theta }}_{\mathrm{s}}{\right)}_{a}{\left(\frac{p}{{p}_{\mathrm{0}}}\right)}^{\mathit{\kappa }}\frac{\mathrm{1}}{\mathrm{1}+{\mathrm{\Lambda }}_{\mathrm
For the saturated case (${q}_{\mathrm{v}}\ge {q}_{\mathrm{sat}}\left({T}^{\ast }\right)$), we write
${q}_{\mathrm{l}}={q}_{\mathrm{t}}-{q}_{\mathrm{sat}}\left({T}^{\ast }\right),$
$\begin{array}{}\text{(10)}& {q}_{\mathrm{v}}={q}_{\mathrm{sat}}\left({T}^{\ast }\right).\end{array}$
In this situation, it is necessary to implicitly calculate the temperature T^∗, given by Eq. (7). We numerically compute an approximation of T^∗ by using Newton's iterative algorithm.
Taking into account this change of variables, the cost function can be written as
$\begin{array}{}\text{(11)}& \begin{array}{rl}\mathcal{J}\left(\mathbit{z}\right)& =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathbit{z}-{\mathbit{z}}_{b}{\right)}^{T}{\mathbf{B}}_{z}^{-\mathrm{1}}\left(\
mathbit{z}-{\mathbit{z}}_{b}\right)\\ & +\frac{\mathrm{1}}{\mathrm{2}}\left[\mathbit{y}-\mathcal{H}\left(\mathcal{L}\left(\mathbit{z}\right)\right){\right]}^{T}{\mathbf{R}}^{-\mathrm{1}}\left[\
Then, its gradient given by Eq. (6) becomes
$\begin{array}{}\text{(12)}& {\mathrm{abla }}_{z}\mathcal{J}\left(\mathbit{z}\right)={\mathbf{B}}_{z}^{-\mathrm{1}}\left(\mathbit{z}-{\mathbit{z}}_{b}\right)-{\mathbf{L}}^{T}{\mathbf{H}}^{T}{\mathbf
where L^T is the adjoint of the conversion operator ℒ.
The second term on the right-hand side of Eqs. (11) and (12) indicates that the conversion operator ℒ is needed to compute the TBs from the observation operator ℋ. Indeed RTTOV-gb requires profiles
of temperature, specific humidity and liquid water content as input quantities. This space change is required at each step of the minimization process. For the computation of the gradient of the cost
function ∇[z]𝒥, the linearized version (adjoint) of ℒ is also necessary. In practice, the operator L^T provides the gradient of the TBs with respect to the conservative variables, knowing the
gradient with respect to the classical variables.
The numerical experiments to be presented afterwards will use measurements made during the SOFOG3D field experiment (https://www.umr-cnrm.fr/spip.php?article1086, last access: 31 March 2022; SOuth
west FOGs 3D experiment for processes study) that took place from 1 October 2019 to 31 March 2020 in south-western France to advance understanding of small-scale processes and surface heterogeneities
leading to fog formation and dissipation.
Many instruments were located at the Saint-Symphorien super-site (Les Landes region), such as a HATPRO (Humidity and Temperature PROfiler, Rose et al., 2005), a 95GHz BASTA Doppler cloud radar (
Delanoë et al., 2016), a Doppler lidar, an aerosol lidar, a surface weather station and a radiosonde station. One objective of this campaign was to test the contribution of the assimilation of such
instrumentation on the forecast of fog events by NWP models.
3.1Conditions on 9 February 2020
This section presents the experimental context of 9 February 2020 at the Saint-Symphorien site characterized by (i) a radiative fog event observed in the morning and (ii) the development of low-level
clouds in the afternoon and evening.
Figure 1 shows a time series of cloud radar reflectivity profiles (W-band at 95GHz) measured by the BASTA instrument (Delanoë et al., 2016) in the lowest hundred metres (top panel) between
9 February 2020 at 00:00UTC and 10 February 2020 at 00:00UTC. The instrument reveals a thickening of the fog between 00:00UTC and 04:00UTC (9 February 2020). The fog layer thickness is located
between 90 and 250m. After 04:00UTC, the fog layer near the ground rises, lifting into a “stratus” type cloud (between 100 and 300m). After 08:00UTC, the stratus cloud dissipates. In the bottom
panel, BASTA observations up to 12000m (≈200hPa) indicate low-level clouds after 14:00UTC, generally between 1000m (≈900hPa) and 2000m (≈780hPa), with a fairly good agreement with AROME
short-range (1h) forecasts (see Fig. 2f). Optically thin (reflectivity below 0dBZ) high-altitude ice clouds are also captured by the radar.
Figure 2 depicts the diurnal cycle evolution in terms of the vertical profiles of (a) absolute temperature T, (b) dry-air potential temperature θ, (c) water-vapour specific content q[v], (d) entropy
potential temperature (θ[s])[a], (e) cloud liquid water specific content q[l] and (f) relative humidity (RH), from 1h AROME forecasts (background) of 9 February 2020 at Saint-Symphorien. At this
stage, it is important to note that the AROME model has a 90-level vertical discretization from the surface up to 10hPa, with high resolution in the planetary boundary layer (PBL) since 20 levels
are below 2km.
Figure 2e and f, for q[l] and RH, show two main saturated layers: a fog layer close to the surface between 00:00 and 09:00UTC with the presence of a thin liquid cloud layer aloft at 850hPa at
00:00UTC, and the presence of a stratocumulus cloud between 14:00UTC and midnight (24:00UTC) at 850hPa. During the night, the near-surface layers cool down, with a thermal inversion that sets at
around 01:00UTC and persists until 07:00UTC. After the transition period between 06:00 and 09:00UTC, when the dissipation of the fog and stratus takes place, the air warms up and the PBL develops
vertically (see the black curves plotted where vertical gradients of θ in Fig. 2b are large). Towards the end of the day, the thickness of the PBL remains important until 24:00UTC, probably due to
the presence of clouds between 800 and 750hPa, which reduces the radiative cooling (see Fig. 2c and f for q[v] and RH).
Figure 2d reveals weaker vertical gradients for the (θ[s])[a] profiles, notably with contour lines often vertical and less numerous than those of the T, θ and q[v] profiles in panels (a), (b) and
(c), as also shown by more extensive and more numerous vertical arrows in panel (d) than in panel (b). Here we see the impact of the coefficient Λ[r]≈5.869 in Eqs. (3)–(4), which allows the vertical
gradients of θ in Fig. 2b and q[v] in Fig. 2c to often compensate each other in the formula for (θ[s])[a]. This is especially true between 980 and 750hPa in the morning between 04:00 and 10:00UTC,
and also within the dry and moist boundary layers during the day.
Note that the dissipation of the fog is associated with a homogenization of (θ[s])[a] in Fig. 2d from 04:00 to 05:00UTC in the whole layer above, in the same way as the transition from stratocumulus
toward cumulus was associated with a cancellation of the vertical gradient of (θ[s])[1] in Fig. 6 of Marquet and Geleyn (2015). This phenomenon cannot be easily deduced from the separate analysis of
the gradients of θ and q[v] in Fig. 2b and c. Therefore, three air mass changes can be clearly distinguished during the day. The vertical gradients of (θ[s])[a] are stronger during cloudy situations,
first (i) at night and early morning before 04:00UTC and just above the fog, then (ii) at the end of the day above the top-cloud level at 800hPa and (iii) turbulence-related phenomena in between
that mix the air mass and (θ[s])[a], up to the cloudy layer tops that evolve between 950 and 800hPa from 13:00 to 17:00UTC.
The observations to be assimilated are presented in the following. The HATPRO MicroWave Radiometer (MWR) measures TBs at 14 frequencies (Rose et al., 2005) between 22.24 and 58GHz: 7 are located in
the water-vapour absorption K-band and 7 are located in the oxygen absorption V-band (see the Table 1). For our study, the third channel (at 23.84GHz) was eliminated because of a receiver failure
identified during the campaign. In this preliminary study, we have only considered the zenith observation geometry of the radiometer for the sake of simplicity.
The ℋ RTTOV-gb model needed to simulate the model equivalent of the observations, which, together with the choice of the control vector and the specification of the background and observation error
matrices, are presented in the next section.
3.2Components of the 1D-Var
In 1D-Var systems, the integrated liquid water content, liquid water path (LWP), can be included in the control vector x as initially proposed by Deblonde and English (2003) and more recently used by
Martinet et al. (2020). A first experimental set-up has been defined where the minimization is performed with the control vector being $\left(T,{q}_{\mathrm{v}},\mathrm{LWP}\right)$. It will be
considered as the reference, named REF. The 1D-Var system chosen for the present study is the one developed by the EUMETSAT NWP SAF (Numerical Weather Prediction Satellite Application Facility),
where the minimization of the cost function is solved using an iterative procedure proposed by Rodgers (1976) with a Gauss–Newton descent algorithm. During the minimization process, only the amount
of integrated liquid water is changed. In this approach, the two “moist” variables q[v] and LWP are considered to be independent (no cross-covariances for background errors between these variables).
The second experimental framework, where the control vector is $\mathbit{z}=\left(\left({\mathit{\theta }}_{\mathrm{s}}{\right)}_{a},{q}_{\mathrm{t}}\right)$, corresponding to the conservative
variables, is named EXP. The numerical aspects of the 1D-Var minimization are kept the same as in REF.
Then, a set of reference matrices B[x](T,q[v]) was estimated every hour using the EDA system of the AROME model on 9 February 2020. These matrices were obtained by computing statistics from a set of
25 members providing 3h forecasts for a subset of 5000 points randomly selected in the AROME domain to obtain a sufficiently large statistical sample. Then, matrices associated with fog areas,
denoted B[x](T,q[v])[fog], were computed every hour by applying a fog mask (defined by areas where q[l] is above 10^−6kgkg^−1 for the three lowest model levels), in order to select only model grid
points for which fog is forecast in the majority of the 25 AROME members. The background error covariance matrices B[z]((θ[s])[a],q[t]) and B[z]((θ[s])[a],q[t])[fog] were obtained in a similar way.
The observation errors are those proposed by Martinet et al. (2020) with values between 1 and 1.7K for humidity channels (frequencies between 22 and 31GHz), values between 1 and 3K for transparent
channels affected by larger uncertainties in the modelling of the oxygen absorption band (frequencies between 51 and 54GHz) and values below 0.5K for the most opaque channels (frequencies between
55 and 58GHz).
The RTTOV model is used to calculate TBs in different frequency bands from atmospheric temperature, water vapour and hydrometeor profiles together with surface properties (provided by outputs from
the AROME model). This radiative transfer model has been adapted to simulate ground-based microwave radiometer observations (RTTOV-gb) by De Angelis et al. (2016).
The 1D-Var algorithm was tested on the day of 9 February 2020 with observations from the HATPRO microwave radiometer installed at Saint-Symphorien. This section presents and discusses three aspects
of the results obtained: (1) the study of background error cross-correlations; (2) the performance of the 1D-Var assimilation system in observation space by examining the fit of the simulated TB with
respect to the observed ones; and (3) the performance of the 1D-Var assimilation system in model space in terms of analysis increments for temperature, specific humidity and liquid water content.
4.1Background error cross correlations
Figure 3 displays for the selected day at 06:00UTC the cross-correlations between T and q[v] (top) and between (θ[s])[a] and q[t] (bottom), with (right) and without (left) a fog mask. For the
classical variables the correlations are strongly positive in the saturated boundary layer with the fog mask from levels 75 to 90 (between 1015 and 950hPa), while with profiles in all-weather
conditions the correlations between T and q[v] are very weak in the lowest layers. On the other hand, the atmospheric layers above the fog layer exhibit negative correlations between temperature and
specific humidity along the first diagonal.
When considering conservative variables, the correlations along the diagonal show a consistently positive signal in the troposphere (below level 20 located around 280hPa). Contrary to the classical
variables, which are rather independent in clear-sky atmospheres as previously shown by Ménétrier and Montmerle (2011), the B[z] matrix reflects the physical link between the two new variables (shown
by Eq. 4) as diagnosed from the AROME model. The correlations are positive with and without a fog mask. This result shows that the matrix B[z]((θ[s])[a],q[t]) is less sensitive to fog conditions than
the B[x] matrix. It could therefore be possible to compute a B[z]((θ[s])[a],q[t]) matrix without any profile selection criteria that would be nevertheless suitable for fog situations, resulting in a
more robust estimate. This result is key for 1D-Var retrievals which are commonly used in the community of ground-based remote-sensing instruments to provide databases of vertical profiles for the
scientific community. In fact, the accuracy of 1D-Var retrievals is expected to be more robust with less flow-dependent B matrices.
We also note that these background error statistics are less dependent on the diurnal cycle and on the meteorological situation (e.g. in the presence of fog at 06:00UTC and low clouds at 21:00UTC),
contrary to the B[x](T,q[v]) matrix, where there is a reduction in the area of positive correlation in the lowest layers between 06:00 and 21:00UTC (Fig. 4).
The 1D-Var results are now assessed in observation space by examining innovations (differences between observed and simulated TBs) from AROME background profiles and residuals. In the following, we
have only used background error covariance matrices estimated at 06:00UTC with a fog mask, for a simplified comparison framework of the two 1D-Var systems.
4.21D-Var analysis fit to observations
Figure 5 presents both (a) innovations and (b, c) residuals obtained with the two 1D-Var systems (Fig. 5b: REF and Fig. 5c: EXP) for the 13 channels (1, 2, 4–14) and for each hour of the day. The
innovations are generally positive for water-vapour-sensitive channels during the day, and negative for temperature channels, especially in the morning. The differences are mostly between −2.5 and 5
K. For channels 8, 9 and 10, which are sensitive to liquid water content, the innovations can reach higher values exceeding 10K (in the afternoon) or around −5K (in the morning).
In terms of residuals, as expected from 1D-Var systems, both experiments significantly reduce the deviations of the observed TB from those calculated using the background profiles, especially for the
first eight channels sensitive to water vapour and liquid water. We can note that the residuals are not as reduced for channel 9 (52.28GHz) compared to other channels. Indeed, channels 8 and 9 (
51.26 and 52.28GHz) suffer from larger calibration uncertainties (Maschwitz et al., 2013) and larger forward model uncertainties dominated by oxygen line mixing parameters (Cimini et al., 2018) than
other temperature-sensitive channels. However, by comparing simulated TB with different absorption models (Hewison, 2007), or through monitoring with simulated TB from clear-sky background profiles (
De Angelis et al., 2017; Martinet et al., 2020), larger biases are generally observed only at 52.28GHz. Consequently, the higher deviations observed in Fig. 5 for channel 9 mostly originate from
larger modelling and calibration uncertainties, which are taken into account in the assumed instrumental errors (prescribed observation errors of about 3K for these two channels compared to <1K for
other temperature-sensitive channels) and also possibly from larger instrumental biases.
The temperature channels used in the zenith mode are only slightly modified as the deviations from the background values are much smaller than for the other channels. During the second half of the
day, characterized by the presence of clouds around 800hPa (see Fig. 2e and f), the residual values are largely reduced in the frequency bands sensitive to liquid water for channels 6, 7 and 8,
especially for EXP as shown by the comparison of the pixels in the dashed rectangular boxes in Fig. 5b and c. Residuals are also slightly reduced for EXP in the morning and during the fog and low
temperature period for the first five channels (1, 2, 4–6) between 02:00 and 08:00UTC.
In order to quantify these results for the 9 February 2020 dataset (all hours and all channels), the bias and root mean square error (RMSE) values are computed for the background and the analyses
produced by REF and EXP. The innovations are characterized by a RMSE of 3.20K and a bias of 1.32K. Both assimilation experiments reduce these two quantities by modifying model profiles. The RMSEs
are 0.71K for EXP and 0.72K for REF and the biases are −0.17K for EXP and −0.11K for REF. These statistics have also been calculated by restricting the dataset to the two dashed rectangular boxes
presented in Fig. 5b and c. A significant improvement is observed for the most sensitive channels to liquid water in the afternoon with the RMSE decreasing from 4.3K in the background to 0.57K in
REF and 0.37K in EXP. For all computed statistics, EXP always provides the best performance in terms of RMSE. Table 2 summarizes the bias and RMSE values obtained for the different samples.
4.3Vertical profiles of analysis increments
After examining the fit of the two experiments to the observed TBs, we assess the corrections made in model space. Figure 6 shows the increments of (a, b) temperature, (c, d) specific humidity and
(e, f) liquid water for the two experiments REF (left panels) and EXP (right panels). In addition, the increments of (θ[s])[a] are shown in panels (g)–(h).
The temperature increments are mostly located in the lower troposphere (below 650hPa) with a dominance of negative values of small amplitude (around 0.5K). This is consistent with the negative
innovations observed in the temperature channels, highlighting a warm bias in the background profiles. The areas of maximum cooling take place in cloud layers (inside the thick fog layer below
900hPa until 09:00UTC and around 700hPa after 12:00UTC). The increments are rather similar between REF and EXP, but the positive increments appear to be larger with EXP (e.g. at 08:00 and
20:00UTC around 800hPa).
Concerning the profiles associated with moist variables, the structures show similarities between the two experiments but with differences in intensity. During the night and in the morning, the q[v]
increments near the surface are negative. These negative increments are projected into increments having the same sign as T by the strong positive cross-correlations of the B[fog] matrix up to
900hPa (Fig. 3). Thus, the largest negative temperature and specific humidity increments remain confined in the lowest layers.
Liquid water is added in both experiments between 03:00 and 07:00UTC, close to the surface, where the Jacobians of the most sensitive channels to q[l] (6 to 8) have significant values in the fog
layer present in the background (see Fig. 2e). After 14:00UTC, values of q[v] between 850 and 700hPa and q[l] around 800hPa are enhanced in both cases, with larger increments for the REF case, in
particular at 20:00UTC and around 24:00UTC (midnight). Most of the liquid water is created in low clouds. Additionally, increments of q[l] above 600hPa are larger and more extended vertically and
in time in EXP, where condensation occurs over a thicker atmospheric layer between 500 and 300hPa after 12:00UTC. In the REF experiment, the creation of liquid water above 500hPa only reaches
values of 0.3gkg^−1 sporadically, for example at 21:00UTC. In this experimental set-up, condensed water can be created or removed over the whole column by means of the supersaturation diagnosed at
each iteration of the minimization process (since RTTOV-gb needs $\left(T,{q}_{\mathrm{v}},{q}_{\mathrm{l}}\right)$ profiles for the TB computation). This is a clear advantage of EXP over REF, which
keeps the vertical structure of the q[l] profile unchanged from the background. In REF, liquid water is only added where it already exists in the background because once the LWP variable is updated,
the analysed q[l] profile is just modified proportionally to the ratio between the LWP of the analysis and of the background, as explained in more details by Deblonde and English (2003).
The profiles of increments for (θ[s])[a] show structures similar to the increments of q[v] around 800hPa and to the increments of T below, where temperature Jacobians are the largest (see Fig. 7 in
De Angelis et al., 2016). The conversion of T, q[v] and q[t] changes obtained with REF into (θ[s])[a] increments (Fig. 6g) highlights the main differences between the two systems. They take place
around 800hPa with larger increments produced by the new 1D-Var particularly between 11:00 and 14:00UTC.
Some radiosoundings (RSs) have been launched during the SOFOG3D IOPs. As only one RS profile was launched at 05:21UTC in the case study presented in the article, no statistical evaluation of the
profile increments can be carried out. However, we have conducted an evaluation of the analysis increments obtained at 05:00 and 06:00UTC (the 1D-Var retrievals were performed at a 1h temporal
resolution in line with the operational AROME assimilation cycles) around the RS launch time. As the AROME temperature background profile extracted at 06:00UTC was found to have a vertical structure
closer to the RS launched at 05:21UTC, Fig. 7 compare the AROME background profile and 1D-Var analyses performed with the REF and EXP experiments valid at 06:00UTC against the RS profile.
The temperature increments are a step in the right direction by cooling the AROME background profile in line with the observed RS profile. The two 1D-Var analyses are close to each other, but the EXP
analysis produces a temperature profile slightly cooler compared to the REF analysis. In terms of absolute humidity (${\mathit{\rho }}_{\mathrm{v}}={p}_{\mathrm{v}}/\left({R}_{\mathrm{v}}T\right)$,
with p[v] the partial pressure and R[v] the gas constant for water vapour), the background profile already exhibits a similar structure compared to the RS profile. The 1D-Var increments are thus
small and close between the two experiments. However, we can note that the EXP profile is slightly moister than the REF profile from the surface up to 3500m, which leads to a somewhat better
agreement with the RS profile below 1500m. In terms of integrated water vapour (IWV), a significant improvement of the background IWV with respect to the RS IWV is observed with the difference
reduced from almost 1kgm^−2 in the background to less than 0.4kgm^−2 in the analyses. These analyses confirm the improvement brought to the model profiles by both the REF and EXP analysis
increments, with some enhanced improvement for EXP.
The aim of this study was to examine the value of using moist-air entropy potential temperature (θ[s])[a] and total water content q[t] as new control variables for variational assimilation schemes.
In fact, the use of control variables less dependent on vertical gradients of ($T,{q}_{\mathrm{v}},{q}_{\mathrm{l}},{q}_{\mathrm{i}}$) variables should ease the specification of background error
covariance matrices, which play a key role in the quality of the analysis state in operational assimilation schemes.
To that end, a 1D-Var system has been used to assimilate TB observations from the ground-based HATPRO microwave radiometer installed at Saint-Symphorien (Les Landes region in south-western France)
during the SOFOG3D measurement campaign (winter 2019–2020).
The 1D-Var system has been adapted to consider these new quantities as control variables. Since the radiative transfer model needs profiles of temperature, water vapour and cloud liquid water for the
simulation of TB, an adjustment process has been defined to obtain these quantities from (θ[s])[a] and q[t]. The adjoint version of this conversion has been developed for an efficient estimation of
the gradient of the cost function. Dedicated background error covariance matrices have been estimated from the EDA system of AROME. We first demonstrated that the matrices for the new variables are
less dependent on the meteorological situation (all-weather conditions vs. fog conditions) and on the time of the day (stable conditions at night vs. unstable conditions during the day) leading to
potentially more robust estimates. This is an important result as the optimal estimation of the analysis depends on the accurate specification of the background error covariance matrix, which is
known to highly vary with weather conditions when using classical control variables.
The new 1D-Var has produced rather similar results in terms of the fit of the analysis to observed TB values when compared to the classical one using temperature, water vapour and LWP. Nevertheless,
quantitative results reveal smaller biases and RMSE values with the new system in low cloud and fog areas. We also note that atmospheric increments are somewhat different in cloudy conditions between
the two systems. For example, in the stratocumulus layer that formed during the afternoon, the new 1D-Var induces larger temperature increments and reduced liquid water corrections. Moreover, its
capacity to generate cloud condensates in clear-sky regions of the background has been demonstrated. As a preliminary validation, the retrieved profiles from the 1D-Var have been compared favourably
against an independent observation data set (one radiosounding launched during the SOFOG3D field campaign). The new 1D-Var leads to profiles of temperature and absolute humidity slightly closer to
observations in the PBL.
The encouraging results obtained from this feasibility study need to be consolidated by complementary studies. Observed TBs at lower elevation angles should be included in the 1D-Var for a better
constraint on temperature profiles within the atmospheric boundary layer. Indeed, larger differences in the temperature increments might be obtained between the classical 1D-Var system and the 1D-Var
system using the new conservative variables when additional elevation angles are included in the observation vector. Other case studies from the field campaign could also be examined to confirm our
first conclusions.
Finally, the conversion operator could be improved by accounting not only for liquid water content q[l] but also for ice water content q[i] (e.g. using a temperature threshold criteria). Indeed,
inclusion of q[i] in the conversion operator should lead to more realistic retrieved profiles of cloud condensates, and a 1D-Var system with only q[l] can create water clouds at locations where ice
clouds should be present, as done in our experiment around 400hPa between 15:00 and 24:00UTC. However, since the frequencies of HATPRO are not sensitive to ice water content, the fit of simulated
TBs to observations could be reduced. As a consequence, the synergy with an instrument sensitive to ice water clouds, such as a W-band cloud radar, would be necessary for improved retrievals of both
q[i] and q[l]. It is worth noting that the variable (θ[s])[a] can easily be generalized to the case of the ice phase and mixed phases by taking advantage of the general definition of θ[s] and (θ[s])
[1], where L[vap]q[l] is simply replaced by L[vap]q[l]+L[sub]q[i].
Code and data availability
The numerical code of the RTTOV-gb model together with the associated resources (coefficient files) can be downloaded from http://cetemps.aquila.infn.it/rttovgb/rttovgb.html (last access: 31 March
2022, Cimini et al., 2022) and from https://nwp-saf.eumetsat.int/site/software/rttov-gb/ (last access: 31 March 2022, NWP SAF, 2022a). The 1D-Var software has been adapted from the NWP SAF 1D-Var
provided at https://nwp-saf.eumetsat.int/site/software/1d-var/ (last access: 31 March 2022, NWP SAF, 2022b), available on request to pauline.martinet@meteo.fr. The instrumental data are available on
the AERIS website dedicated to the SOFOG3D field experiment: https://doi.org/10.25326/148 (Martinet, 2021). AROME background data are available on request to pauline.martinet@meteo.fr. Quicklooks
from the cloud radar BASTA are available at https://doi.org/10.25326/155 (Delanoë, 2021). The BUMP library used to compute background error matrices, developed in the framework of the JEDI project
led by the JCSDA (Joint Center for Satellite Data Assimilation, Boulder, Colorado), can be downloaded at https://doi.org/10.5281/zenodo.6400454 (Ménétrier et al., 2022).
PMarq supervised the work of ALB, contributed to the implementation of the new conservative variables in the computation of new background error covariance matrices and participated in the scientific
analysis and manuscript revision. JFM developed the conversion operator and adjoint version and participated in the scientific analysis and manuscript revision. PMart supervised the modification of
the 1D-Var algorithm, supported the use of the EDA to compute the background error covariance matrices, provided the instrumental data used in the 1D-Var and participated in the manuscript revision.
ALB adapted the 1D-Var algorithm and processed all the data, prepared the figures and participated in the manuscript revision. BM developed and adapted the BUMP library to compute the background
error covariance matrices for the 1D-Var and participated in the manuscript revision.
The contact author has declared that neither they nor their co-authors have any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors are very grateful to the two anonymous reviewers who suggested substantial improvements to the article. The instrumental data used in this study are part of the SOFOG3D experiment. The
SOFOG3D field campaign was supported by METEO-FRANCE and the French ANR through the grant AAPG 2018-CE01-0004. Data are managed by AERIS, the French national centre for atmospheric data and services.
The MWR network deployment was carried out thanks to support by IfU GmbH, the University of Cologne, the Met Office, the Laboratoire d'Aérologie, Meteoswiss, ONERA and Radiometer Physics GmbH. MWR
data have been made available, quality controlled and processed in the framework of CPEX-LAB (Cloud and Precipitation Exploration LABoratory, http://www.cpex-lab.de, last access: 31 March 2022), a
competence centre within the Geoverbund ABC/J with the acting support of Ulrich Löhnert, Rainer Haseneder-Lind and Arthur Kremer from the University of Cologne. This collaboration is driven by the
European COST actions ES1303 TOPROF and CA18235 PROBE. Julien Delanoë and Susana Jorquera are thanked for providing the cloud radar quicklooks used in this study for better understanding the
meteorological situation. Thibaut Montmerle and Yann Michel are thanked for their support on the use of the AROME EDA to compute background error covariance matrices. The work of Benjamin Ménétrier
is funded by the JCSDA (Joint Center for Satellite Data Assimilation, Boulder, Colorado) UCAR SUBAWD2285.
This research has been supported by the Agence Nationale de la Recherche (grant no. AAPG 2018-CE01-0004), the European COST actions (ES1303 TOPROF and CA18235 PROBE) and JCSDA UCAR (SUBAWD2285).
This paper was edited by Maximilian Maahn and reviewed by two anonymous referees.
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Bayesian data analysis
Bayesian data analysis is based on Bayesian inference. ^[1] ^[2] Briefly, this approach is based on the following straightforward property of probability distributions. Let p(x,y) be the joint
probability of observing x and y simultaneously. Let p(x|y) be the conditional probability of observing x, given y. Then, by definition
${\displaystyle p(x|y)p(y)=p(x,y)=p(y|x)p(x)\,}$
from which follows Bayes' theorem:
${\displaystyle p(x|y)={\frac {p(y|x)p(x)}{p(y)}}}$
Interpretation of Bayes' Theorem
The interpretation of this expression in the framework of data interpretation is as follows. Given the initial knowledge of a system, quantified by the prior probability p(x) of system states x, one
makes an observation y with probability (or degree of confidence) p(y), adding to the prior knowledge. The posterior probability p(x|y) quantifies this enhanced knowledge of the system, given the
observation y. Thus, the Bayesian inference rule allows one to (a) gradually improve the knowledge of a system by adding observations, and (b) easily combining information from diverse sources by
formulating the degree of knowledge of the system and the observations in terms of probability distributions. The quantity p(y|x) is a fundamental quantity linking the prior and posterior
distributions, called the likelihood, and expresses the probability of observing y, given the prior knowledge x.
Forward modelling
The likelihood is evaluated using a forward model of the experiment, returning the value of simulated measurements while assuming a given physical state x of the experimental system. Mathematically,
this forward model (mapping system parameters to measurements) is often much easier to evaluate than the reverse mapping (from measurements to system parameters), as the latter is often the inverse
of a projection, which is therefore typically ill-determined. On the other hand, evaluating the forward model requires detailed knowledge of the physical system and the complete measurement process.
The Likelihood
The forward model is used to predict the measurements y, based on the physical state x of the system. The Likelihood p(y|x) specifies the most probable measurement outcome, which corresponds to the
maximum of the distribution p(y|x), as well as its uncertainty, given by the width of the distribution.
In a typical case, assume that the model is such that the measurement outcomes are distributed like a Gaussian around a most probable value y[0], with an error Δ y. Then the likelihood will be
${\displaystyle p(y|x)={\frac {1}{\sqrt {2\pi \Delta y^{2}}}}\exp \left(-{\frac {(y-y_{0})^{2}}{2\Delta y^{2}}}\right)}$
Note that the negative logarithm of the likelihood is proportional to the χ^2 of the fit of the data y to the model y[0], and maximizing the likelihood will minimize χ^2, thus establishing the link
between the Bayesian approach and the common least squares fit. However, the Bayesian approach is more general than the standard least squares fit, as it can handle any type of probability
Parametric formulation
The description of the system state is usually done by defining a parametric representation, e.g., by defining density or temperature profiles as a function of space via a vector of N parameters, α:
n(r,α) or T(r,α). The parameters obey a prior distribution p(α), expressing physical or other constraints. Applying Bayes theorem one obtains
${\displaystyle p(\alpha |D)={\frac {p(D|\alpha )p(\alpha )}{p(D)}}}$
where D represents the available data. The likelihood p(D|α) specifies the probability of a specific measurement outcome D for a given choice of parameters α. The advantage of the parametric
representation is that the abstract 'system state' is reduced to a finite set of parameters, greatly facilitating numerical analysis. This parametrization may involve, e.g., smooth (orthogonal)
expansion functions such as Fourier-Bessel functions, or discretely defined functionals on a grid. ^[3]
The maximization of the posterior probability as a function of the parameters α yields the most likely value of the parameters, given the data D, which is the basic answer to the data interpretation
The width of the posterior distribution yields the error in the parameters. To obtain the error in a given parameter α[i], the posterior distribution is marginalized by integrating over the remaining
N-1 parameters:
${\displaystyle p(\alpha _{i}|D)=\int {p(\alpha |D)d\alpha _{1}\cdots d\alpha _{i-1}d\alpha _{i+1}\cdots d\alpha _{N}}}$
The width of this one-dimensional distribution is found using standard procedures.
Comparison with Function Parametrization
Function parametrization (FP) is another statistical technique for recovering system parameters from diverse measurements. Both FP and Bayesian data analysis require having a forward model to predict
the measurement readings for any given state of the physical system, and the state of the physical system and the measurement process is parametrized. However
• instead of computing an estimate of the inverse of the forward model (as with FP), Bayesian analysis finds the best model state corresponding to a specific measurement by a maximization procedure
(maximization of the likelihood);
• the handling of error propagation is more sophisticated within Bayesian analysis, allowing non-Gaussian error distributions and absolutely general and complex parameter interdependencies; and
• additionally, it provides a systematic way to include prior knowledge into the analysis.
Typically, the maximization process is CPU intensive, so that Bayesian analysis is not usually suited for real-time data analysis (unlike FP).
Integrated Data Analysis
The goal of Integrated Data Analysis (IDA) is to combine the information from a set of diagnostics providing complementary information in order to recover the best possible reconstruction of the
actual state of the system subjected to measurement. This goal overlaps with the goal of Bayesian data analysis, but IDA applies Bayesian inference in a relatively loose manner to allow incorporating
information obtained with traditional or non-Bayesian methods. ^[4] ^[5] ^[6] ^[7] ^[8] ^[9] ^[10]
See also | {"url":"http://wiki.fusenet.eu/wiki/Bayesian_data_analysis","timestamp":"2024-11-05T12:45:40Z","content_type":"text/html","content_length":"41447","record_id":"<urn:uuid:7bfc56ba-2fce-45d9-827e-50d58a578bfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00791.warc.gz"} |
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Web things that improved the quality of life, like science and technology, were encouraged. Web explore the connections between mathematics and the islamic empire's conquests,. Worksheets are islamic
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worksheets for islamic studies.
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The perimeters of two similar triangles is in the ratio 3:4. The sum of their areas is 75 sq cm. What is the area of the smaller triangle? | Socratic
The perimeters of two similar triangles is in the ratio 3:4. The sum of their areas is 75 sq cm. What is the area of the smaller triangle?
1 Answer
Perimeter is the sum of lengths of triangles. Hence its unit in $c m$. Area has unit $c {m}^{2}$ i.e. length squared. So if lengths are in ratio $3 : 4$, areas are in ratio ${3}^{2} : {4}^{2}$ or $9
: 16$. This is because the two triangles are similar.
As total area is $75$ square centimeters, we need to divide it in ratio $9 : 16$, of which first will be area of smaller triangle.
Hence area of smaller triangle is $75 \times \frac{9}{9 + 16}$
= $75 \times \frac{9}{25}$
= ${\cancel{75}}^{3} \times \frac{9}{{\cancel{25}}^{1}}$
= $27$ square centimeters
Area of larger triangle would be $75 \times \frac{16}{9 + 16} = 3 \times 16 = 48$ square centimeters
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Lesson 10
The Effect of Extremes
Lesson Narrative
The mathematical purpose of this lesson is to recognize a relationship between the shape of a distribution and the mean and median. Students will use dot plots to investigate this relationship.
Earlier in this unit, students created data displays so that the shape of the distribution is clear. This lesson connects to upcoming work because students will use the shape of the distribution and
measures of center to make decisions about how to summarize data.
In this lesson, students begin by using aspects of mathematical modeling (MP4) to select appropriate variables to compare. In another activity, students make use of structure (MP7) and appropriate
tools (MP5) to construct dot plots of data that have prescribed measures of center.
One of the activities in this lesson works best when each student has access to technology that will easily compute measures of center to produce dot plots or histograms because it will help students
focus on understanding the relationship between extreme values and the measure of center without distracting with lengthy computations.
Learning Goals
Teacher Facing
• Recognize the relationship between mean and median based on the shape of the distribution.
• Understand the effects of extreme values on measures of center.
Student Facing
• Let’s see how statistics change with the data.
Required Preparation
Acquire devices that can run GeoGebra (recommended) or other spreadsheet technology. It is ideal if each student has their own device. (A GeoGebra Spreadsheet is available under Math Tools.)
Internet-enabled devices are required for the digital version of the activity Separated by Skew.
Student Facing
• I can describe how an extreme value will affect the mean and median.
• I can use the shape of a distribution to compare the mean and median.
CCSS Standards
Building Towards
Print Formatted Materials
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What generator produce voltage or power ?
Generators are devices that convert mechanical energy into electrical energy. They produce voltage by inducing a flow of electrons through a conductor, which results in an electrical potential
difference, commonly known as voltage. This voltage is generated when a conductor (usually a coil of wire) moves through a magnetic field or when a magnetic field moves relative to a conductor, as in
the case of electromagnetic induction. This principle is used in various types of generators, such as alternators and dynamos, to generate electricity for powering electrical loads.
Generators produce both voltage and current simultaneously. When a generator is connected to a circuit, it creates an electromotive force (EMF) that causes electrons to flow through the circuit, thus
producing an electric current. The amount of current generated depends on the load connected to the generator and the characteristics of the generator itself, including its design, size, and
operating conditions. Therefore, generators are sources of both voltage and current, which together constitute electrical power.
The power produced by a generator is the product of the voltage and current it generates. Power is the rate at which electrical energy is transferred or converted from mechanical energy in the
generator. In mathematical terms, power (P) can be calculated using the formula P = V × I, where V is the voltage and I is the current. Generators convert mechanical power (such as from a turbine,
engine, or other prime mover) into electrical power through the production of voltage and current, which are essential components of electrical power transmission and distribution systems.
A generator is indeed a source of voltage. When in operation, a generator creates a voltage difference between its terminals or windings due to electromagnetic induction. This voltage difference is
what drives the flow of electrons through an external electrical circuit, enabling electrical devices to operate. The magnitude of the voltage produced by a generator depends on factors such as the
speed of rotation (in the case of rotational generators), the strength of the magnetic field, and the number of turns in the coils or windings.
To produce voltage using a generator, you need to ensure that the generator is mechanically driven, typically by a prime mover such as a steam turbine, gas turbine, diesel engine, or water turbine.
As the prime mover rotates the generator shaft, it induces a magnetic field inside the generator’s stator (stationary part) or rotor (rotating part). This magnetic field interacts with conductors
(coils of wire) within the generator, causing electrons to move and create a potential difference (voltage) between the generator’s terminals. This generated voltage can be utilized directly or
transformed and transmitted to power electrical devices, systems, and networks. Thus, by mechanically driving a generator and ensuring proper operation, voltage can be produced continuously to meet
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Find Missing Angle Quadrilateral Worksheet - Angleworksheets.com
Find Missing Angle Quadrilateral Worksheet – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help you understand
the different concepts and build your understanding of these angles. Students will be able to identify unknown angles using the vertex, arms and arcs postulates. … Read more | {"url":"https://www.angleworksheets.com/tag/find-missing-angle-quadrilateral-worksheet/","timestamp":"2024-11-07T11:04:43Z","content_type":"text/html","content_length":"47696","record_id":"<urn:uuid:f7ee0e97-491c-49f5-bf44-6b74ef2c0b87>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00158.warc.gz"} |
A/B Testing: Avoiding the Peeking Problem - GoPractice
You can make many mistakes while designing, running, and analyzing A/B tests, but one of them is outstandingly tricky. Called the “peeking problem,” this mistake is a side effect of checking the
results and taking action before the A/B test is over.
An interesting thing about the peeking problem is that even masters of A/B testing (those who have learned to check if the observed difference is statistically significant or not) still make this
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Statistical significance in simple words
Say you’re running an A/B test on 10 new users in your game, and you divide them randomly between the old and the new versions of the game. From the five users who got the old version of the game,
two (40%) retained on the next day against three (60%) for the new version of the game.
Based on the collected data, can we say that the Day 1 retention rate of the new version of the game is better than that of the old version?
Not really. Our sample size is too small, so there is a high probability that the observed difference is accidental.
Statistics provides tools that help us understand whether the difference in metrics between the groups can be attributed to product changes rather than pure chance. In other words, it helps us
determine whether the change is statistically significant or not.
A way to check the statistical significance based on the frequentist approach to probability taught at universities works as follows:
• We collect data for versions A and B.
• We assume that there is no difference between the two versions. This is called the null hypothesis.
• Assuming the groups are identical, we calculate the “p-value.” The p-value represents the probability of obtaining similar results if we repeat the experiment.
• If p-value is smaller than a certain threshold (usually 5%), we reject the null hypothesis. In this case, we can conclude with a high degree of certainty that the observed difference between the
groups is statistically significant and is not due to chance.
• If p-value is greater than the threshold, we confirm the null hypothesis. This means the collected data does not show a significant difference between the tested versions of the game. In reality,
however, there can be a difference between the versions. It’s just that the data we have doesn’t show it. A bigger sample size might yield different results.
This is a very simplified explanation of the basic idea behind determining whether the observed difference is statistically significant. In reality, things are more complicated: we need to study
the data structure, clean the data, choose the right criterion, and interpret the results. This process contains many pitfalls.
A simple example of calculating statistical significance
Let’s get back to the game we discussed in the previous example.
The team decided to learn from the mistakes made in the first A/B test. This time they brought 2,000 new users to the game (1,000 to each version). On day 1, the first version retained 330 users
against 420 for the second.
While the second version retained more users on day 1, the team wasn’t sure whether this was due to product changes or an accidental metric fluctuation.
To make it clear, the team had to calculate whether the observed difference in Day 1 retention was statistically significant or not.
In this case, we are looking at a simple ratio metric (the conversion of new users into taking a specific action – Day 1 retention rate), so we can use an online calculator to calculate statistical
The calculator showed p-value < 0.001. That is, the probability of seeing the observed difference within the identical test groups is very small. This means that we can be sure that the increase in
Day 1 retention rate is connected to the product changes we made.
Automated verification of A/B testing results with unexpected consequences
Encouraged by the progress made in the retention rate metric, the team decided to turn towards improving the monetization of the game. At first they decided to focus on increasing the conversion rate
to first purchase. In two weeks, the new version of the game was put to an A/B test.
The developers wrote a script that checked the conversion rate of the first purchase in the test and control versions every few hours, and calculated whether the difference was statistically
A few days later, the system spotted a statistically significant improvement in the conversion rate of the new version. The team thus thought of this test as successful and introduced the new version
of the game to all the users.
You might have missed it, but there was a little mistake with big consequences that sneaked in while analyzing the test’s results. Let’s see what it was.
The peeking problem
To use different statistics criteria to calculate p-value in the approach requires meeting certain conditions. Many of them for example imply a normal distribution of the studied metric.
But there is another important condition that many people forget about: the sample size for the experiment must be determined in advance.
You must decide beforehand how many observations you need for your test. Then you should collect data, calculate results and make a decision. If you didn’t spot a statistically significant difference
based on the amount of data you’ve collected, you can’t continue the experiment to collect additional data. The only thing you can do is run the test again.
The described logic might sound bizarre when viewed in the context of the real A/B testing companies run for internet services and apps, where adding new users to the test costs close to nothing and
results can be observed in real time.
But the logic used for A/B testing within the framework of the frequentist approach to statistics was developed long before the Internet was made. Back then most of the applied tasks involved a fixed
and predetermined sample size in order to test a hypothesis.
The Internet has changed the paradigm of A/B testing. Instead of choosing a fixed sample size before running the experiment, most people prefer to keep collecting data until the difference between
the test and the control groups becomes statistically significant.
The original approach to calculating p-value does not work in this new experimental paradigm. In the case of testing things in this new way, real p-value becomes much bigger than the p-value that you
get using the usual statistical criteria when checking the test results only once at the end of the experiment.
NB: The Peeking Problem occurs when you check the intermediate results for statistical significance between the control and test groups and make decisions based on your observations. If you fix the
sample size at the beginning of the test and defer any decision to when you have the right amount of data, then there’s no harm in observing intermediate results.
The correct A/B testing procedure (as part of the frequentist approach) is depicted in the following diagram:
Incorrect A/B testing procedure:
Why Peeking increases p-value
Let’s go back to experiment with the first purchase conversion. Let’s suppose we know that in reality the changes we made to the game produced no effect.
In the following graph you can see the dynamics of the difference in purchase conversion rates between the test and control versions of the product (blue line). The green and red lines represent the
boundaries of the indistinguishability range, provided that the appropriate number of observations has been selected in advance.
In the correct A/B testing process, we need to determine in advance the number of users needed for the test, collect observations, calculate the results and draw a conclusion. This procedure ensures
that if the test is repeated many times, in 95% of cases, we will reach the same conclusion.
Everything changes greatly if you start checking the results frequently and are ready to act if you spot a difference. In this case, instead of asking whether the difference is significant at a
certain predetermined point in the future, you ask whether the difference goes beyond the decision boundary at least once within the data collection process. Mind that these are two completely
different questions to ask.
Even if the two groups are identical, the difference in conversion rates may sometimes go beyond the boundaries of the indistinguishability zone as we keep accumulating observations. This is totally
OK, since the boundaries are built so that when testing the same versions, we can spot the difference in only 5% of cases.
Therefore, when you regularly check the results during a test and are willing to make a decision whenever there is a significant difference, you begin to accumulate the possible random moments when
the difference falls out of the given range. As a consequence your p-value grows with every new “peek”.
If what we were talking about earlier is unclear, here is an example of how exactly peeking increases by p-value.
The effect of peeking at the p-value
The more often you look at the intermediate results of the A/B testing with the readiness to make a decision, the higher the probability is that the criterion will show a statistically significant
difference when there is none:
• 2 peeking cases double the p-value;
• 5 peeking sessions increase the p-value by a factor of 3.2;
• 10,000 peeking sessions increase the p-value twelve-fold;
Options to solve the peeking problem
Fix sample size in advance and do not check results until all data is collected
This sounds very correct—but at the same time impractical. If you receive no signal from the test, then you have to start all over again.
Mathematical solutions to the problem: Sequential experiment design, Bayesian approach to A/B testing, lowering the sensitivity of the criterion
The peeking problem can be solved mathematically. For example, Optimizely and Google Experiments use a mix of Bayesian approach to A/B testing and Sequential experiment design to solve it (note that
previously, we discussed everything within the framework of the frequentist approach; read more about the difference between Bayesian and frequentist approaches following the links at the end of this
For services like Optimizely, this is a necessity, as their value comes down to the ability to determine the best option based on the regular A/B testing check-ups on the fly. Feel free to read more
here: Optimizely, Google.
There are many discussions in the tech community about the pros and cons of the described methods. In case you want to explore them in depth, the links at the end of this article provide more
A product-minded approach with a soft commitment to test time and correction for the essence of the peeking problem
When working on a product, your goal is to get the signal you need to make a decision. The logic described below is not ideal from a mathematical perspective, but it solves our product task.
The essence of the approach comes down to a preliminary assessment of the required sample to identify the effect in the A/B test as well as taking into account the nature of the peeking problem while
making intermediate check-ups. This minimizes the negative consequences one may face when analyzing the test results.
Before starting the experiment, you need to evaluate what size of sample you need in order to see a change—if there exists one—with a reasonable probability.
Aside from the peeking problem, this is a useful exercise, because some product features require such large samples to identify their effect that there is no point in testing them at the current
product development stage.
With the previously calculated sample in mind, once the experiment is launched, you need to (I’d say you must) monitor the dynamics of changes, but avoid making decisions when the difference first
enters the statistical significance zone (as we already know of the peeking problem).
You need to continue to monitor what is happening. If the difference remains statistically significant, then, most likely, there is an effect. If it becomes indistinguishable again, then it isn’t
possible to draw a conclusion about whether there is an improvement.
Let’s take a look at the results of two experiments demonstrated below, each of which lasted 20 days. Each point on the graph represents the relative difference in the metric between the test and the
control version at the end of the corresponding day with a confidence interval. If the confidence interval does not cross zero (i.e. identical to the condition p-value <x), then the difference can be
considered significant (if you select the appropriate sample in advance). The green diamonds are points in the experiment where the data showed a statistically significant difference between the two
In the first experiment, starting from day 6, the difference between the versions became significant and the confidence interval no longer crosses 0. Such a stable picture gives a clear signal that
the test version works better than the control one with a high degree of probability. Interpreting the results is not difficult.
In the second experiment, we can see that the difference sometimes crosses into the statistical significance zone, but then again dips back into irrelevance. If we did not know about the peeking
problem, then we would have finished the experiment on day 4, assuming that the test version showed better results. But the proposed method for visualizing the dynamics of the A/B test in the course
of time shows that there is no stable measurable difference between the groups.
There are some controversial cases where it is difficult to interpret the results clearly. There is no single solution to the following cases, and it usually depends on the amount of risk you are
willing to take and the cost of such a decision:
• Ordinary experiments where you test different features or small changes, and are ready to make a decision with a greater degree of risk.
• Expensive decisions when, for example, you try to test a new product development vector. In this case, it makes sense to analyze and study the data in detail. Sometimes you can run an experiment
Some will say that the described logic is an oversimplification and cannot be trusted. This could be true from a mathematical standpoint. But from a product/growth point of view it sounds quite
When teams start to do more math than product work, it may be a signal that the effect of the changes made is too small or absent. But math won’t solve this problem, I am afraid.
Summing up
If the peeking problem sounds a bit convoluted and confusing, here’s one key takeaway that can guide you:
If you conduct an A/B test and at some point the test group difference becomes statistically significant, don’t end the experiment right away, believing that you have a winner. Keep watching. It also
makes sense to pre-select the sample size, collect your observations, and then calculate the results based on them.
Extra resources | {"url":"https://gopractice.io/data/peeking-problem/","timestamp":"2024-11-13T08:38:30Z","content_type":"text/html","content_length":"61008","record_id":"<urn:uuid:9550310f-8954-4928-b52a-266b6a6f5a34>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00541.warc.gz"} |
Article-Journal | Jan Heiland Personal Homepage
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Jan 1, 2023
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Exponential Lag Synchronization of Cohen-Grossberg Neural Networks with Discrete and Distributed Delays on Time Scales
Jan 1, 2023
A quadratic decoder approach to nonintrusive reduced-order modeling of nonlinear dynamical systems
Jan 1, 2023
A low-rank solution method for Riccati equations with indefinite quadratic terms
Jan 1, 2023
Robust output-feedback stabilization for incompressible flows using low-dimensional (ℋ)(_mbox(ınfty) )-controllers
Jan 1, 2022
Operator Inference and Physics-Informed Learning of Low-Dimensional Models for Incompressible Flows
Jan 1, 2022
Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition
Jan 1, 2022
Convolutional Neural Networks for Very Low-Dimensional LPV Approximations of Incompressible Navier-Stokes Equations
Jan 1, 2022 | {"url":"https://www.janheiland.de/publication_types/article-journal/","timestamp":"2024-11-07T12:37:55Z","content_type":"text/html","content_length":"24436","record_id":"<urn:uuid:6faae509-a901-420b-89b6-901daa94c269>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00615.warc.gz"} |
Walking for Water (Math) - Blue Water Schools Network
SUBJECTS Geography/History, Math, Social Studies
GRADE LEVEL Junior (4-6) / Intermediate (7-8)
PLEDGE 2 - Human Right to Water
Resources Needed:
1. Image for Walking for Water from Your Water Footprint: The Shocking Facts About How Much Water We Use to Make Everyday Products by Stephen Leahy (see attachment)
2. Printable data collection chart (see attachment at bottom of lesson)
3. This lesson is best following the introductory lesson Walking for Water (Introduction)
4. Background Information: The World Health Organization’s definition of what constitutes “reasonable access to water” is having a source of at least 20 litres (5 gallons) per person per day within
1 kilometre (0.6 mile) of the water user’s home.
1. Tell students, We have learned that the average distance women in rural Africa and Asia walk to collect water is 10 000 steps / 6 km (3.6 miles), which takes an average of 6 hours a day. Now,
let’s figure out how far and how long you walk each day to get water at school. You can organize the information using the attached chart.
2. Have students measure their stride length (the average stride length of an adult is 0.75 m or 75 cm) and then calculate the distance in metres to the water fountain.
1. Depending on the grade, you can use the following questions to guide your lesson. Stop the calculations at the level of your class. You can allow earlier finishers or students who need a
challenge to go further with their thinking. Depending on the grade, students can use scientific notation or expanded form or powers of 10 to express the numbers. Have students use the
information collected in the previous Introductory Lesson about how many steps taken on a return trip to the water fountain in order to answer the questions below. Use the stride length
measurements from the Introduction.
□ How many steps in a week? How many m/km? How many minutes/hours?
□ How many steps in a month? How many m/km in a month (use a 30 day month like April or June for simplicity)? How many hours in a month?
□ How many steps in a year? How many m/km in a year? How many hours in a year?
□ How many steps, km and hours in 10 years, 25 years, 50 years?
2. Have students do the same calculations for a woman collecting water in rural Africa or Asia. Use the 10 000 steps, 6 km, and 6 hours data from the Introduction. The same chart can be used to
organize the data.
3. Have students compare how far they walk to get water and how far a woman collecting water in rural Africa or Asia has to walk in the same time period.
Follow Up Activities:
1. Try some of the many lessons for The Water Princess by Susan Verde. | {"url":"https://www.bluewaterschoolsnetwork.ca/lessons/walking-for-water-math/","timestamp":"2024-11-02T09:23:49Z","content_type":"text/html","content_length":"99985","record_id":"<urn:uuid:975e9804-49ca-4467-a00f-cb321a647efc>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00345.warc.gz"} |
Orthographic Projection: Principles, Types, and Presentation - Civil Jungle
Orthographic Projection: Principles, Types, and Presentation
What Is Orthographic Projection?
Orthographic views consist of one, two, or more separate aspects of an object taken from different directions & different sizes, generally at right angles to each other and arranged in a definite
These views collectively describe the object. Orthographic views of any purpose can be represented by any one of the two systems of projection
The first angle projection and the third angle projection. It is named according to the quadrant in this which the object is imagined no place for purposes of line projection.
As far as any shape and size of the views are concerned, there is no difference between these two systems. The only difference lies in the relative positions of the various views. Indian Standards
Institution recommends
(IS: 696-1972) both these systems leaving the choice to the discretion of the organization concerned. However, there are specific merits and demerits associated with each system. Students are advised
to study both methods to meet the varying demands of different industries in the country.
To meet this end. Some chapters in this book are exclusively dealt with in the first angle projection, and some in the third-angle projection.
Types of Orthographic Projection
Orthographic projection two different parts and also a different type of symbol as per below detail mention.
1. First angle projection & First angle projection symbol
2. Third angle projection & Third angle projection symbol
First Angle Projection & First Angle Projection Symbol
In this, the object is imagined to be in the first quadrant. Because the observer normally looks from the right side of the quadrant to obtain the front view. The objects will come in between the
observer and the plane of projection.
Therefore, in this case, the object is to be transparent, and the projectors are imagined ta be extended from various points of the object to meet the projection plane. First, these meeting points
when joined in the order form an image
This is the principle of the first angle projection. Thus in the first angle projection, any view is so placed that it represents the side of the object away from it.
First angle projection is merely used throughout all parts of Europe so that called European projection. Most important of First angle projection symbol
Third Angle Projection & Third Angle Projection Symbol
In this, the object is imagined to be placed in the third quadrant. Again, as the observer is normally supposed to look from the right side of the quadrant to obtain the front view, in this method,
the projection plane comes in between the observer and the object.
Therefore, the plane of projection has to be assumed to be transparent. The intersection of this plan with the projectors from all the points of the object would form an image on the transparent
This is the principle of the third angle projection. Thus it is seen that in the third angle projection any view is so placed that it represents the side from the object nearest to it
Most important of Third angle Projection symbol. The third angle is the system in used North America and alternatively described as American projection.
Making Orthographic Views
1. Front view
2. Side view
3. Top View
4. Presentation of view
Front View
To draw the front view of an object in the first angle projection, the object which is assumed to be transparent is imagined to be in front of the vertical plane (first quadrant) as shown in.
The shape of the object, when viewing from the front. Is obtained by joining in order all the meeting points of the projectors extended from the visible edges of the object. This is called a front
A similar treatment is repeated for the front view in the third angle projection, but with a transparent vertical plane imagined to be held in front of the object.
Here the shape of the object when viewed from the front is obtained by joining in order all the intersection points on the transparent plane of the projectors to the visible edges of the object
Side View
The shape of some objects cannot be interpreted completely from the from and top views only. In such cases, more information may be needed, and for this purpose, a third plane is imagined.
This is called an auxiliary vertical plane and is placed perpendicular to the first two planes. A view can be obtained on to this plane in a way similar to that explained above.
In the first angle method, if the auxiliary vertical plane is the place to the right of the object, a view is obtained on it by looking at the object from its left side.
It is then called a left side view. On the other hand, if the auxiliary vertical plane is placed to the left of the
Top View
The projection on to a plane placed horizontally below the transparent object will reveal the shape of the object as viewed from below.
This view is called a top view, and the principle that is being followed is that of the first angle projection. Figure deals with a similar treatment in the third angle projection with a transparent
plane placed horizontally below the object.
Presentation of view
For presenting the views on a drawing sheet, irrespective of the method of projection, the horizontal plane, and the auxiliary vertical plane arc rotated till they come in the plane with the original
vertical plane.
Now the three views of the object will be in the same plane, viz., the plane of the paper. These three views normally provide enough information to describe the shape and size of the object.
Above drawing shows the relative positions of the views in the first angle projection and below drawing n the third angle projection.
The method used for the projection is indicated by a distinctive symbol (presentation of view drawing No 1 .C and Presentation of view drawing No 2 C)
Frequently Asked Questions (FAQ) about Orthographic Projection:
What is orthographic projection?
Orthographic projection is a method used to represent three-dimensional objects in two dimensions by projecting them onto a series of planes from different viewing angles.
What are the main types of orthographic projection?
The main types are first angle projection and third angle projection, named based on the quadrant in which the object is imagined to be placed.
What is the difference between first angle projection and third angle projection?
In first angle projection, the object is imagined to be in the first quadrant, with the observer looking from the right side of the quadrant to obtain the front view.
In contrast, in third angle projection, the object is imagined to be in the third quadrant, and the observer looks from the right side of this quadrant to obtain the front view.
Which regions or standards typically use first angle projection and third angle projection?
First angle projection is commonly used in Europe, often referred to as European projection, while third angle projection is used in North America, also known as American projection.
How are orthographic views presented?
Orthographic views typically include front view, side view, and top view. These views are presented on a drawing sheet with the horizontal and auxiliary vertical planes rotated to align with the
original vertical plane, placing all views in the same plane.
Why are both first angle projection and third angle projection taught and used?
Both systems are taught to meet the varying demands of different industries. The choice between the two methods is often left to the discretion of the organization or industry concerned.
What are the advantages and disadvantages of first angle projection and third angle projection?
Each system has its merits and demerits, which may influence its suitability for specific applications or industries. Students are encouraged to study both methods to be adaptable to different
industry requirements.
How are orthographic views represented symbolically?
Each type of orthographic projection is represented by a distinctive symbol, indicating whether it follows first angle projection or third angle projection principles.
What are the primary orthographic views needed to describe an object?
The primary orthographic views include front view, side view, and top view, which collectively provide sufficient information to describe the shape and size of the object.
Are there any additional views that may be needed in orthographic projection?
In some cases, an auxiliary vertical plane may be used to obtain additional views, such as left side view, to provide more comprehensive information about the object’s shape and dimensions.
Leave a Comment | {"url":"https://civiljungle.org/first-angle-projection-third-angle-projection-symbol/","timestamp":"2024-11-08T17:53:02Z","content_type":"text/html","content_length":"257084","record_id":"<urn:uuid:bcf07af1-6e9d-4757-ab44-b207ba826b6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00203.warc.gz"} |
Is 94 Prime Number or Composite Number [Why & Why not Detailed Guide]
Is 94 Prime Number or Composite Number
Is 94 a Prime Number?
No – 94 is not a Prime Number.
Why No: 94 is not a prime number because it does not meet the prime criteria of having exactly two distinct positive divisors: 1 and itself. Check if any number is prime or not
Is 94 a Composite Number?
Yes – 94 is a Composite Number.
Why Yes: 94 is a composite number because it has more than two distinct positive divisors. It is divisible by 1, 2, 47, and itself (94).
Problem Statements
Property Answer
Is 94 a prime number? No
Is 94 a composite number? Yes
Is 94 a perfect square? No
Factors of 94 1, 2, 47, 94
Multiples of 94 94, 188, 282, 376, 470, 564, 658, 752, 846, 940
Cube Root of 94 4.530
Square of 94 8836
Square Root of 94 9.695
Is 94 a Perfect Cube? No
Is 94 an Irrational number No
Is 94 a Rational number Yes
Is 94 a Real number Yes
Is 94 an Integer Yes
Is 94 a Natural number Yes
Is 94 a Whole number Yes
Is 94 an Even or odd number Yes (94 is an even number)
Is 94 an Ordinal number Yes
Is 94 a Complex number Yes (as all real numbers are also complex numbers)
What are the factors of 94?
The factors of 94 are 1, 2, 47, and 94. This information establishes that 94 is not a prime number since it is divisible by numbers other than 1 and itself. This insight confirms its status as a
composite number.
What Kind of Number is 94?
94 is categorized as a composite number because it has more than two distinct positive divisors: 1, 2, 47, and itself. Being one of the even numbers, it showcases its uniqueness in various
mathematical contexts as a composite number. Its attributes as an even number, a composite, and a natural number contribute to its significance in mathematical discussions.
Nearest Prime Number of 94
The nearest prime numbers to 94 are 89 and 97. 89 is the closest prime number less than 94, and 97 is the closest prime number greater than 94, highlighting its position amidst the prime numbers in
the numerical sequence.
Is 94 a perfect square?
No, 94 is not a perfect square. A perfect square is a number that can be expressed as the square of an integer, and there is no integer whose square is 94. | {"url":"https://www.examples.com/maths/is-94-prime-number-or-composite-number.html","timestamp":"2024-11-13T09:01:02Z","content_type":"text/html","content_length":"103135","record_id":"<urn:uuid:94391f53-4034-489a-b605-73465f60d4b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00499.warc.gz"} |
Velocity and Acceleration - Mechanics Made Easy
The Section on Velocity and Acceleration deals with the linear motion of bodies and the interrelationships of the main variables, time, distance, velocity and acceleration. For simplicity it is
assumed that the dimensions and masses of the bodies can be ignored i.e. they are treated as particles or points. | {"url":"https://mechanicsmadeeasy.com/velocity-and-acceleration/","timestamp":"2024-11-06T18:29:52Z","content_type":"text/html","content_length":"37455","record_id":"<urn:uuid:475fbabf-67fc-4965-8ce5-06f0b62217a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00123.warc.gz"} |
Orbital Maths at NASA with Chris Hadfield - Best Outer Space VideosOrbital Maths at NASA with Chris Hadfield
Orbital Maths at NASA with Chris Hadfield
I am in Orlando at the Kennedy Space Center Visitor Complex. This is where they launch humans to the moon and continue to do launches to this day. Behind me is the Rocket Garden and, kind of, visitor
centre which is why there are school groups and inspirational music but we’re going to ignore all of that for today. We’re going to look at orbital mechanics. These rockets behind me were originally
developed to put things in orbit. So I’m going to derive the equation you need to know, to keep something up in space. To do that, I need two things: first of all, I need a flip chart so I can do all
of my working out in a nice orderly fashion; and secondly, I need something that’s been in orbit. So welcome to Stand-up Maths: it’s Chris Hadfield. – Thank you for coming along. It’s uh– – Three
times I’ve been in orbit. – It’s three times? – Three times! How many orbits do you do per…? I’ve done about 2,600 orbits or so, so around the world lots of times. Yep. Okay so, two and a half
kilo-orbits in. We’re going to… [laughs] “Kilo-orbits”. What better use of an astronaut than doing some working out? So let’s do this. So do you want to set us up, like the Earth with you orbiting
it? – I’ve chosen blue for the blue planets. – Going for blue. Nice work. And then over here I’m going to start an equation. I’m going to go: force equals mass times acceleration. – Okay. Is this the
centre of the…? – Centre of the Earth. – Okay, so let’s call Earth, big M mass. – Okay. Great. And then where are you in this? I’m orbiting the world going this way. Is that gonna be okay? That seems
about right. So you orbit– – If this was the north pole looking down? – North pole looking down. – You orbit around… – Yeah. – Okay, okay. – We go with the rotation of the Earth. So I’ll draw myself
as a dot here. [markers scratching on paper] – I’ve given you mass little m if that’s okay? – That’s fine. I’m happy with little m. Good good, and how far up are we talking here? Well, we’re about
400 kilometres above the surface but of course we want to measure it from the centre of the Earth so we can put this whole thing right out to there… So it’s the the radius of the Earth plus our
orbital altitude. – Okay, let’s call the whole thing r. – r, okay. All right, uh… r squared. So I’ve just put in the classic… gravitational force, the gravitational constant G. Mass of the Earth, M.
Mass of Chris and everything else that’s keeping you alive up there. and then this is our distance r squared, so that gives us the force that the Earth is pulling you down. And we now need to work
out… well we know m. Well we got m. We need to know the acceleration. So to do that, let’s start with the velocity you’re travelling. Okay, so I’m going this way at a certain speed, a certain
velocity, right? Okay. Shall we call that… do you wanna call that v? – Sure, which v? – Uh, 1… And it’s a vector so I’m going to put a squiggly line under there. – Okay, that’s fine. – Bit of a
perfectionist. We’re now going to have a look at you, a very small moment of time in the future. Okay, so over here somewhere. – Same thing, same radius. – That’s it. – No change in radius. – Okay,
that’s kind of the trick to staying in orbit is this has to stay the same? Yep, basically and now we want another one here, okay. Always parallel to the Earth’s surface – and that’s another v… okay.
– Let’s go v2. And so I’ve done, up here, I’ve just written that their magnitudes are the same. – and so actually I’m just going to call it v for shorthand – Yep. – ’cause you don’t want your speed
to slow down – Correct. I guess that’s kind of the definition of being in orbit. You’re not getting lower. You’re not slowing down. We’re just going, like the moon,round and round and round. Perfect.
Okay, so now I’m going to add in… – a little angle here. This is going to be theta. – Theta. Alright. And then we’ve got the distance you’ve travelled, so you’ve gone from there to there. – And
that’s going to be… – Yeah. – I mean… – Proportionate to time. Yeah, so let’s say over this tiny angle, tiny amount of t so I’m going to put in delta t times whatever velocity you’re moving at. –
We’re now going to… – Sure. move these vectors, and I’m going to redraw them over here. – So i’m going to borrow your red. – Alright. Okay, so I’m going to bring v1 over here and zoom in on it like
that. That’s our v1 vector and then I’ve just moved that straight across. And this one is now on an angle down. – So if I bring it over there, that’s v2. – Yeah. And as you go around, I guess by the
time you come all the way back to where you started… velocity is once again going perpendicular. Yeah at our altitude it takes about 92 minutes to go all the way around. – 92 minutes per lap. – Yep.
– And then, so theta will go through a full 360. – 360. So actually this angle, this has just tipped your v down. – Right. – Actually gone down by theta. – Yeah, same amount, yup. Yeah, and now
that’s the difference there so… I don’t know if you do a lot of vectors, as an astronaut? Do you have to do much math as an astronaut? You have to understand how it all works but real-time you’re
mostly just solving the practicalities of it. Like how do I get my spaceship from here to there? And we let the computer do the… the burn calculations. You have to have a sense of how it all supposed
to make, most of work but we don’t real-time do a lot of math in public. So I’m watching and bowing to your expertise. No, no, I mean I guess you’ve got a very pragmatic, you know the math – so you
understand what’s meant to be happening. – Yeah. and then you trust the computers and just kind of sense-check – as you go along. – Yes. Well, in which case, you may not have done similar triangles
too recently. But that’s what we’re going to do – because we’ve got two triangles with the same angle. – Sure. They’re both isosceles ’cause that length is equal to that length, as up here. that’s
equal to that which means the ratio Between any two matching sides should be the same. So the ratio here between change in velocity divided by this side, change in t times velocity equals that
divided by that so that’s uh… well the length of that is just v, divided by the length of that is r. We’ve got a v down here we don’t really need so we’re going to move it over there. So i’m going to
multiply both sides by v. So that’s now over there. – Yup. – And dv dt: rate of change, our velocity, that’s acceleration. So there we go. So there’s our acceleration, is this which is v squared on r
So I’m going to fill that in here. So it’s mass times v squared on r. This is actually just the centripetal force that we’ve worked out So a lot of times when you see this taught, people just start
here. They go “there’s your f equals m a” and then you fill this in. – That’s kind of nice. – That’s beautifully elegant. Yeah and intuitive. It’s so good. So good and given you were up on this dot,
that feels appropriate. Okay, so now, let’s do the grand finale in brown because we can cancel some stuff out. We’ve got an r down there. And we’ve got an r– two r’s over there. – So we get rid of
one of those. – Yep, multiply both sides by r. Okay, we can get rid of an m from both sides. – I’m afraid your mass now doesn’t matter. – Right. And I’m going to– so this side is only v squared. Oh,
that’s handy. So now we’ve got v squared equals– so I’ve just moved it over here and this is… These are both constants. So I’m going to put G and M over there and then this is 1 over r. Yep. And… and
that’s it. – Beautiful. – That’s our equation. So this basically shows us, if you square the velocity… you’ll be able to orbit at a height r. And because this is 1 over r, it’s an inverse
relationship. You know what’s intriguing to me out to this, is that the radius you are from the Earth determines the velocity you have to go to be staying in orbit. That’s the really key and
intriguing thing out of this to me. For every radius from the Earth, there is a specific speed that will hold you at that altitude. So if you decide I want to put either humans in a certain orbit, I
want to put a satellite in a certain orbit, you then work out the specific v – which equates to that distance? – Right. And it doesn’t matter what mass that thing is, it’s just purely an almost
geometric thing. How far away are you? That tells you how fast you have to go. And you were in low Earth orbit? Yeah, I didn’t get above about 420 kilometres from the surface. So the radius of the
Earth plus 420. – That’s not bad. That’s more than most people. – [laughs] It is. – And the radius is like 6,000 kilometres, right? – Right. So you’re like a tiny, like maybe four/five percent… If
you were to hold a globe and bring your eyeball down to maybe like a thumb-width above that’s how far we were from the surface of the Earth. It seems like a long way when you’re there, but it’s
actually still quite close. And how fast did you have to go? What was your v for that orbit? Well, depending which units: about 25 times the speed of sound, mach 25. If you put it in miles an hour,
about 17,500 miles an hour. About 28,000 kilometres an hour. Those numbers are so big, intuitively it’s better if you go per second. So about five miles a second, or eight kilometres a second. in
order to stay at that altitude in that orbit: five miles a second, eight kilometres a second. – Eight kilometres a second?! – Yes. – 8,000 metres every second. – Yeah. – That’s absolutely… – Imagine
if one of these rockets was right here and one second later, it was eight kilometres away. That’s how fast we had to be to be that little dot right there. And how long does it take between when you
launch and when you get to that orbit? The rockets have to burn and accelerate you up to the right speed and the right angle. It takes somewhere just a little less than nine minutes. From sitting
here in Florida on the pad, laying on your back to the engine shutting off and being there weightless in orbit: little under the time it takes to drink a cup of coffee. That’s absolutely incredible.
So in fact, if people want to give it a go, if you get the values for G and M– gravitational constant, mass of the earth– and we’ve talked through the rough distance up you are, you can put them in
and you should get out the other side, roughly 8,000, which is 8,000 metres per second. Cool or you could put in the mass for any planet and everything else still works. Oh, yeah, that’s true. We’ve
not assumed Earth. This could be absolutely anywhere. So that you can start to figure out just how fast you go to go around Jupiter, around the moon or whatever. Kinda neat. Okay, so we’ll put some
free resources in the description below. We’ll have a copy of this in case you want to recreate it. But now, there’s this interesting relationship between… how far up you are and it’s inversely
proportion to the speed you’re travelling which I suspect has some interesting implications for actually navigating in orbit. And I believe somewhere here is a vehicle that you’ve flown in orbit.
Yeah, it is indeed. Yeah. Atlantis is here. So we’re gonna go find that and have a chat about the implications of this. [Stand-Up Maths theme] You flew on this space shuttle. I did it my very first
time to space, I was part of a crew of five people And we took this vehicle, flew it up and docked with the Russian space station Mir. And that’s not like an automated process. You don’t just hit the
dock button and it all happens? No, there’s no standard orbit, Mr Sulu. It’s all sort of manual. I mean you have primitive computers on board, a huge amount of help from Earth, but primarily flying a
spaceship to dock with a space station… is a manual operation, especially with something as big and ponderous as a spaceship. So I’ve always known on paper: a lower orbit higher velocity, higher
orbit less velocity. What does it actually mean if you’re trying to manoeuvre this and you’re going to connect that bit to a very specific bit on another space station. So picture this Matt. You have
to dock that docking system to Mir. You have to do it inside a two-minute window because you have to be over the right part of the Earth for communications. You have to do it at exactly the right
speed otherwise if you come in too fast then you’ll break Mir and kill the three people on board. If you come in too slow then the springs in there will bounce you off. And plus, you’ve got to hit it
exactly in the centre. Because if you hit a little bit off-centre, the mechanism will bottom out and break. So our target was about as big as uh, maybe a coffee saucer. So all those constraints and
you’re flying it manually. And so an extremely difficult ballet to make all those things happen at once. And as you say, you’re trying to negotiate orbital mechanics while it’s happening. So what are
you doing? Are you just using the thrusters or do you have to take into account, if you go up or down your speed’s going to alter? We try and solve that mental problem of the difference of speeds
with different orbit heights by breaking your docking into phases. When you’re far away, you hardly even… acknowledge that that the Mir is there. You’re just a vehicle going around the world and
you’re going to change your orbit. And so if you– if you’re going this way and you fire the big engines to accelerate that way then the energy is going to raise your orbit. You’re going to end up in
an orbit that is further from Earth. And once you get up there, you’ll be sort of going at a slower speed like, the Moon takes a month to go around. So you fire the rockets… Yeah. You don’t speed up.
Well, you do sort of accelerate forwards but that energy ends up– because you’re going around the world, that energy sort of takes you away from the world a little higher – Until everything balances
back out. – and once you’re higher and balanced out now in fact your speed is lower. It turned, sort of, that kinetic energy into potential energy. So now you’re further away. I always remind myself:
the Moon is way out there. It takes a month to go around the world. And like a ball and a string the shorter the string, the faster you have to spin it to keep the ball going. And so your strings
just getting shorter and shorter. So you fire your engines and you coast to where you are. But once you see Mir, and now you fire your engines to get higher, suddenly it becomes non-intuitive. ‘Cause
we fire our thrusters to go this way and we’re watching intently at Mir… And we get a little closer to Mir because we move up here. – And then we start drifting back. – Because you’re higher than it?
– Because we’re higher, now we’re slower, drifting back. – Because you’re slower. So we’re playing with our speeds as we go until we finally get close enough that we say “Okay, we can no longer pay
attention to orbital mechanics now. We’re just going to brute force it.” – We call it proximity operations or PROX-OPS. – “Proximity operations.” And once you’re in PROX-OPS, all you do is say, more
a little forward, or a little aft, and just keep firing the thrusters to hone it in so that you can dock under exactly the right conditions. You’re just compensating on the fly. It’s sort of like, if
you were driving two boats across a lake. Yeah. And once you get– and you’re trying to touch two parts of the boats to each other very carefully. From a distance, you’ll just sort of make a rough
guess and where to set your throttle. And you might even just set the throttle and not even move it for a while. Just catch up. Then as you get close you’re going to fine-tune your throttle a little.
but once you’re nudging up against that other one, you’re going to be all over the steering wheel. You don’t care about the current or the wind or the surf. You’re just trying to get docked. Sorta
like that. Sometimes if I’m doing working out, and I know I’m near the answer… It’s just, that’s my proximity operation zone. – [laughs] Just brute force it. – Yeah, exactly. You run a simulation,
you end up where you’re meant to go. So, excellent. Thank you so much for joining us and having a chat. Well, that’s a lot of space maths. Thanks for watching the video. There is also some
engineering and some science which goes into something like this. So as well as my videos, you want to check out the videos by Professor Lucie Green over on The Cosmic Shambles Network. Yes, we made
some fantastic videos looking at what goes into making the Space Shuttle fly. The work that’s done on it. A bit of space weather from an astronaut’s perspective as well. So huge thanks to Chris
Hadfield for giving up his time to tell us about this amazing technology. And we’ll have links to those below. It was actually The Cosmic Shambles Network who put this whole trip together so huge
thanks to them, Chris of course, and let’s not forget that the Kennedy Space Center and NASA were also slightly involved. [Stand-Up Maths theme]
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Q * C * (1 - D/100) + S
01 Sep 2024
Q * C * (1 - D/100) + S & Analysis of variables
Equation: Q * C * (1 - D/100) + S
Variable: D
Impact of Discount Rate on Total Cost Function
X-Axis: 10.0to10.0
Y-Axis: Total Cost Function
Title: The Impact of Discount Rate on Total Cost Functions: A Mathematical Exploration
In engineering economics, the total cost function is a crucial concept that represents the overall expense associated with a project or investment. This article delves into the impact of discount
rate (D) on the total cost function, using the equation Q * C * (1 - D/100) + S. The analysis provides insights into how changes in discount rates affect the total cost, enabling engineers and
economists to make more informed decisions.
The total cost function is a fundamental concept in engineering economics, representing the cumulative expenditure associated with a project or investment over its lifetime. One of the key variables
influencing this function is the discount rate (D), which reflects the time value of money. A higher discount rate indicates a greater emphasis on present-day costs, whereas a lower discount rate
prioritizes future expenses.
Equation and Context:
The equation Q * C * (1 - D/100) + S represents the total cost function, where:
• Q is the quantity or scope of the project
• C is the unit cost per item or service
• D is the discount rate as a percentage
• S is the setup or fixed costs
In this context, the equation suggests that the total cost (TC) is a product of the quantity (Q), unit cost (C), and a factor representing the impact of discount rate (1 - D/100). The setup costs (S)
are also added to the total.
To explore the impact of discount rates on the total cost function, we can analyze the equation in two parts:
1. Discount Rate Factor: The term 1 - D/100 represents the impact of discount rate on the total cost. As the discount rate increases (D > 0), this factor decreases, indicating a reduction in the
weighted average cost of capital.
2. Quantity and Unit Cost Factors: The product Q * C represents the basic cost structure, which remains unaffected by changes in the discount rate.
Sensitivity Analysis:
Using numerical values for Q, C, D, and S, we can perform sensitivity analysis to examine how changes in the discount rate (D) affect the total cost. For example:
Discount Rate (D) 0% 5% 10% 15%
Total Cost (TC) $100,000 + S $97,500 + S $95,000 + S $92,500 + S
As the discount rate increases from 0% to 15%, the total cost decreases by approximately $5,000-$7,500. This analysis demonstrates how changes in discount rates can significantly impact the overall
expense of a project.
The equation Q * C * (1 - D/100) + S provides a powerful tool for analyzing the impact of discount rate on total cost functions. By understanding how changes in discount rates affect the weighted
average cost of capital, engineers and economists can make more informed decisions when evaluating investment opportunities or project proposals.
• When performing feasibility studies, consider sensitivity analysis to examine the impact of discount rates on total costs.
• Use the equation as a starting point for more complex analyses, incorporating additional variables and scenarios.
• Be aware that changes in discount rates can have significant effects on overall expenses, necessitating careful consideration and planning.
Future Research Directions:
• Investigate how discount rate affects other cost-related functions, such as present value or net present value.
• Explore the impact of discount rates on decision-making in different industries or contexts.
• Develop more advanced models incorporating multiple discount rates, risk factors, or other variables to provide a comprehensive understanding of total costs.
Related topics
Academic Chapters on the topic
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Equations devs & users
Hi, could you confirm that funelim never terminates on ack in the following basic code ?
Would you have an idea why ? I am using Coq 8.19 and the associated equation version.
From Equations Require Import Equations.
Equations ack (m n : nat) : nat by wf (m, n) (Equations.Prop.Subterm.lexprod _ _ lt lt) :=
ack 0 n := S n;
ack (S m) 0 := ack m 1;
ack (S m) (S n) := ack m (ack (S m) n).
Definition ack_incr {m n} : ack m n < ack m (n+1).
(* Bugs *)
funelim (ack m n).
Did you try apply_funelim?
Applying the eliminator directly with apply_funelim and a few tricks most likely works, but I am not trying to prove the goal at the moment.
apply_funelim does less than funelim so I wanted to know which one was already looping.
apply_funelim works fine would it seems
at least it creates goal
I am mostly interested because last time we had a weird experience with Matthieu with the bug present on only my computer.
Matthieu had exactly the same config, or at least it seemed so at the moment, but with this fix https://github.com/mattam82/Coq-Equations/pull/593
So I am trying to understand if the bug is linked to a particular configuration or "solve" by the fix.
I have another weird issue with ack, simp does not manage to simplify even if direct rewriting works:
Definition ack_incr {m n} : ack m n < ack m (n+1).
apply ack_elim.
- intro n'; simp ack. (* It does not simplify the left side ack 0 (n' + 1) to S (n' + 1) *)
rewrite ack_equation_1 at 1. (* but this works without issue *)
I imagine it is an issue with the rewrite, because rewrite ack_equation_1 doesn't work, I need to add at 1 for it to work.
Otherwise, I get the error message
Tactic generated a subgoal identical to the original goal.
But I really don't get why, the goal is S n' < ack 0 (n' +1) and ack_equation_1 : forall n, ack 0 n = n
That check might be done considering ack is transparent ?
Matthieu Sozeau said:
That check might be done considering ack is transparent ?
I am not sure how to check that. It tells me ack is opque if I try to unfold. Maybe rewrite see at as transparent ?
But actually, what really surprises me is that it I only have this bug with ack, and not with any other functions I have tried
I suspect the reason is that rewrite first tries to unify S n' with ack 0 n' and succeeds, thus rewriting S n' to S n'.
By running
Definition ack_incr {m n} : ack m n < ack m (n+1).
apply ack_elim.
- intro n'.
Set Debug "RAKAM".
Fail rewrite ack_equation_1.
You can actually see the LHS beeing selected. Which means internally Coq's rewrite performs the analogous of
Definition ack_incr {m n} : ack m n < ack m (n+1).
apply ack_elim.
- intro n'.
pattern (ack 0 (n' + 1)).
rewrite ack_equation_1.
This is why ssreflect's rewrite first performs a purely syntactical step to match head constants (here ack)
From Coq Require Import ssreflect.
Definition ack_incr {m n} : ack m n < ack m (n+1).
apply ack_elim.
- intro n'.
rewrite ack_equation_1.
does not fail.
We looked at it with Thomas and it boils down to the following MWE:
(* A generic type to test rewriting of a goal *)
Inductive LULE {A : Type} (a b : A).
Definition suc (m : nat) : nat := S m.
Lemma u : forall n : nat, suc n = S n.
intro n.
exact eq_refl.
Goal forall (n : nat), LULE (S n) (suc n).
intro n.
(* Without specifying n, rewrites the first index (S n) to (S n) *)
Fail rewrite u.
(* By specifying n, rewrites the second index (suc n) to (S n) *)
rewrite (u n).
Definition id (m : nat) : nat := m.
Lemma u' : forall n : nat, id n = n.
intro n.
exact eq_refl.
Goal forall (n : nat), LULE n (id n).
intro n.
(* Without specifying n, rewrites the second index (id n) to n *)
rewrite u'.
@Cyril Cohen
this doesn't happen with setoid_rewrite, so maybe it would be a better fit for simp?
Is there a way to make Equations use setoid rewrite rather than rewrite by default ?
If I can give you one advice (1), stray as far as possible from setoid_rewrite as you can
scalability issues are waiting for you down the road ready to hit you with big sticks on your head
Wanna fix rewrite then ?
In other word, in my tutorial, what should I say to users if they face this issue ?
Wanna fix rewrite then ?
hard. The alternative is to use a sane pattern selection algorithm, like advocated by @Cyril Cohen yesterday
Otherwise, I am also fine with using SSReflect rewrite in Equations instead
Is there a way to make Equations use SSReflect rewrite rather than rewrite by default ?
Require ssreflect at the beginning?
no, I guess you would have to fix Equations directly ? :man_shrugging:
I think that replacing tactics under the feet of the user is a very bad practice
I cannot count the time I've lost believing I was calling some piece of code instead of the actual one
Also, for the purpose of the tutorial, couldn't you just pass the right argument to ack_equation_1 ?
Might be the more reasonable approach indeed
The bug is due to pattern selection indeed. But also when definitions are transparent equations wrongly tried to reduce in the equality to rewrite resulting in the loop. I’m working on a fix for that
The pattern selection issue can also be tamed using "Set Keyed Unification", which also forces syntactic match on the head of the pattern for vanilla "rewrite".
Otherwise, @Matthieu Sozeau , do you know why funelim is slow here ? Is it the loop thing you are mentioning ?
(The original message from the thread)
Yes, it tried to reduce the ack calls. I think we hit the same issue a few times in MetaCoq actually, where we replaced calls with apply_funelim due to that.
Otherwise, I love this proof :
Definition ack_incr {m n} : ack m n < ack m (1+n). (* notice the 1+n rather than n+1 *)
apply ack_elim; intros.
- Fail progress autorewrite with ack. auto. (* ok . simp ack works too *)
So you can actually solve the goal by pure chance with the following:
Definition ack_incr {m n} : ack m n < ack m (1+n).
apply ack_elim; intros; simp ack; apply ack_min. (* ack_min m n : n < ack m n *)
Pierre-Marie Pédrot said:
scalability issues are waiting for you down the road ready to hit you with big sticks on your head
Are these documented somewhere? Are there plans to fix them? (probably not)
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237659-Equations-devs-.26-users/topic/Bug.20funelim.20on.20Ack.html","timestamp":"2024-11-10T17:47:17Z","content_type":"text/html","content_length":"38810","record_id":"<urn:uuid:51806223-0652-433b-8d2c-f2de86efd8e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00799.warc.gz"} |
Best First Search in Artificial Intelligence Example - Finding Optimal Solutions Faster
Best First Search in Artificial Intelligence Example – Finding Optimal Solutions Faster
Artificial Intelligence (AI) is a rapidly growing field that focuses on creating intelligent machines capable of performing tasks that would typically require human intelligence. One primary area of
investigation in AI is the exploration of different search algorithms, one of which is the Best First Search.
Best First Search is an optimal search algorithm that aims to find the finest solution to a given problem instance by exploring the most promising paths first. It uses an initial state, called the
starting node, and a goal state to guide its exploration. The algorithm evaluates each potential solution based on a heuristic function, which provides an estimate of the solution quality. The search
then proceeds by expanding the node with the highest heuristic value, making it a primary choice for many AI applications.
To better illustrate the concept, let’s consider an example of using Best First Search to find the optimal path in a maze. Suppose we have a maze with multiple paths leading from a starting point to
a goal point. The algorithm would evaluate each potential path based on a heuristic function, which could be the distance between the current position and the goal position. It would then select the
path with the lowest heuristic value, indicating the most promising direction to move forward. This process continues until the goal is reached or no more viable paths exist.
Artificial intelligence (AI) technologies encompass a vast range of exploratory algorithms and strategies aimed at solving complex problems. One primary instance of an AI search algorithm is the Best
First Search (BFS).
The Best First Search algorithm, also known as the finest algorithm, is an optimal and top-rated search technique used in AI. It aims to find the most promising path or solution from an initial state
to a desired goal state.
To illustrate how BFS works, let’s consider an investigation scenario. Suppose we have a map with various cities interconnected by roads. Our task is to find the quickest route from the initial city
to the destination city.
In this illustration, the initial city represents the starting point of our exploration, and the destination city represents the goal we want to reach. The BFS algorithm involves expanding the search
frontier by considering the most promising cities first, based on their estimated distances to the destination city.
By using BFS, we can efficiently explore the map and determine the optimal path, considering factors such as distance, traffic, or any other criteria we may define. This makes BFS a valuable tool in
AI, as it helps us navigate complex systems and find the best solutions to various problems.
In conclusion, the Best First Search algorithm is a primary example of an AI search technique. Its ability to prioritize the most promising paths makes it extremely efficient for a wide range of
applications, from route planning to problem-solving. By utilizing this algorithm, AI systems can explore and analyze vast amounts of data to find the finest solutions.
Best First Search
Best First Search is a primary exploration algorithm in the field of Artificial Intelligence (AI) that is used to find the optimal solution for a given problem. It is an example of an uninformed
search algorithm that begins its investigation from the initial state and selects the top-rated instance for further exploration.
Best First Search is an efficient method that uses heuristics to guide its search. Heuristics are problem-specific rules or measures that estimate the desirability of expanding a particular node.
These heuristics help the algorithm make intelligent decisions and prioritize which nodes to explore first.
The main idea behind Best First Search is to use the quality of the nodes as a guide for exploration. The algorithm maintains a priority queue of nodes, where the order is determined by the heuristic
function. This allows the algorithm to explore the finest nodes first, in the hope of finding the optimal solution more quickly.
One of the primary advantages of Best First Search is its ability to handle large search spaces efficiently. By using heuristics to guide the search, the algorithm can quickly focus on the most
promising areas of the problem space, improving its performance compared to other uninformed search algorithms.
As an illustration, let’s consider the example of finding the shortest path between two locations on a map. The Best First Search algorithm would evaluate the possible paths based on the heuristic
function, which might estimate the distance between each node and the target location. The algorithm would then explore the nodes with the lowest estimated distance, hoping to reach the target
In conclusion, Best First Search is an important algorithm in the field of Artificial Intelligence that enables efficient exploration and investigation of problem spaces. By using heuristics to guide
the search, it can quickly identify the optimal solution, making it a valuable tool for various applications.
Artificial Intelligence
Artificial Intelligence (AI) is a field of computer science that focuses on creating intelligent machines capable of performing tasks that typically require human intelligence. AI is a broad term
that encompasses various subfields, including machine learning, natural language processing, computer vision, and expert systems. The primary goal of AI is to develop systems that can understand,
learn, reason, and adapt in a way that mimics human cognitive abilities.
One of the top-rated examples of AI is the use of best first search algorithms for optimal exploration and investigation. Best first search is a search algorithm that attempts to find the optimal
path to a goal state by intelligently selecting the most promising node to expand next. It is often used in AI applications that require finding the best solution among a large number of possible
For instance, consider an AI system tasked with finding the shortest path between two locations on a map. The initial state would be the starting location, and the goal state would be the
destination. The AI system would use a best first search algorithm to explore the map, expanding nodes based on their estimated distance to the destination. This allows the AI system to prioritize
exploration in the most promising areas, leading to a more efficient search.
The best first search algorithm is an illustration of how AI can leverage exploration and investigation techniques to find optimal solutions. By selecting the most promising node based on a heuristic
evaluation function, AI systems can efficiently search large search spaces and find the best solution in a timely manner.
In conclusion, artificial intelligence plays a crucial role in the development of intelligent systems. The use of best first search algorithms is just one example of how AI techniques can be applied
to solve complex problems. As AI continues to advance, we can expect even more sophisticated and effective algorithms for exploration, investigation, and problem-solving.
Finest Initial Exploration
In the field of artificial intelligence (AI), the Best First Search algorithm is an optimal search strategy used to find the most promising paths in a search space. This article provides an example
of its initial exploration.
Targeting the Optimal Solution
When conducting a search, it is essential to identify the best possible path to achieve the desired outcome. In AI, the Best First Search algorithm employs a heuristic function to evaluate each
potential path and determine the most promising one.
For example, consider a scenario where a robot needs to navigate a maze to reach a target. The initial exploration of the Best First Search algorithm involves selecting the path that appears to be
the most promising based on the heuristic evaluation. The robot would choose the path that brings it closer to the target, avoiding dead ends and other unfavorable routes.
Primary Illustration
An instance of the Best First Search algorithm’s finest initial exploration can be demonstrated through a simple adversarial game like tic-tac-toe. In this case, the algorithm evaluates each
potential move based on a heuristic that considers the current state of the game and the likelihood of winning.
The algorithm starts with the empty game board and explores possible moves by evaluating the heuristic value of each move. It selects the move with the highest heuristic value as the most promising
one. This choice determines the initial exploration of the game search space and leads to subsequent moves and outcomes.
By using the Best First Search algorithm, the initial exploration focuses on the most promising paths, allowing for efficient and effective investigation of the search space in artificial
AI Instance
An instance of an AI program can provide an example of using the best-first search algorithm in artificial intelligence. Best-first search is an exploration technique that aims to find the optimal
solution by initially investigating the finest choices.
In the context of AI, an instance refers to a specific case or example that the algorithm operates on. In the case of best-first search, the instance could be a problem-solving scenario where the
algorithm tries to find the best solution by exploring the finest possibilities first.
For illustration purposes, let’s consider an instance where an AI program aims to find the shortest path between two points on a map. The program utilizes best-first search to prioritize the
exploration of the most promising paths, based on some heuristic function. The initial exploration starts from the starting point and expands to neighboring locations, continuing the search until the
goal is reached.
This AI instance demonstrates how the best-first search algorithm can be used to efficiently explore potential solutions in artificial intelligence. By prioritizing the investigation of the best
choices at each stage, the algorithm can quickly converge towards the optimal solution. Such instances highlight the power of AI and its ability to tackle complex problems in various domains.
Optimal Initial Exploration
In the field of artificial intelligence (AI), the best first search algorithm is a top-rated method used to search for the optimal solution in a problem instance. The primary goal of this algorithm
is to find the finest possible path from a starting state to a goal state by exploring the state space.
When applying best first search to an AI problem, the initial exploration is crucial. It sets the foundation for the search and greatly affects the efficiency and effectiveness of the algorithm. An
optimal initial exploration can greatly reduce the time and resources required to find the optimal solution.
Illustration of Optimal Initial Exploration
Let’s consider an example to understand the importance of an optimal initial exploration. Imagine we have a maze in which we need to find the shortest path from the start point to the exit. The maze
consists of various cells, each with its own cost.
If we start exploring the maze randomly or by choosing the nearest neighbor, we might end up taking a path that leads to a dead-end or has a high cumulative cost. This can result in wastage of time
and resources.
However, with an optimal initial exploration, we can prioritize the paths that have a higher chance of leading to the goal state or have a lower cumulative cost. By considering the heuristics, such
as the distance to the goal or the cumulative cost, we can make informed decisions about which paths to explore first.
In conclusion, an optimal initial exploration is a crucial step in the best first search algorithm in AI. It allows us to efficiently find the optimal solution in a problem instance by prioritizing
the exploration of paths with higher chances of success or lower cumulative costs.
Top-rated Primary Investigation
Artificial Intelligence (AI) is an area of study that focuses on creating intelligent machines that can perform tasks requiring human-like intelligence. In the instance of best first search, AI
algorithms aim to find the optimal solution to a problem by exploring the initial state and selecting the best possible action at each step.
This article provides an example of a top-rated primary investigation in the field of AI, specifically focusing on best first search. Through this illustration, we aim to demonstrate how this
algorithm can be applied to solve real-world problems and achieve optimal results.
The need for best first search
When faced with complex problems, the initial state might have a large number of possible actions. In such cases, a blind search algorithm may not be efficient in finding the optimal solution. Best
first search, on the other hand, evaluates the potential of each action based on an heuristic function and selects the one with the highest estimated value. This approach allows the algorithm to
prioritize promising actions and reach the optimal solution more efficiently.
An example application
Let’s consider the problem of route planning in a city. The goal is to find the quickest route from a starting location to a destination. A best first search algorithm can be used to determine the
optimal route by evaluating the estimated time to reach the destination for each possible action (e.g., taking a specific road or turning at an intersection).
By applying best first search, the algorithm can consider various factors such as traffic conditions, road capacity, and historical data to estimate the travel time on each possible route. It then
selects the route with the lowest estimated travel time as the next action. This approach allows the algorithm to find the fastest route and provide an optimal solution for the route planning
In conclusion, best first search is a top-rated primary investigation in the field of artificial intelligence. By focusing on the initial state and selecting the best action based on an heuristic
function, this algorithm can efficiently find the optimal solution to a wide range of problems. The example of route planning illustrates the practical application of best first search and its
ability to deliver optimal results.
AI Illustration
An example of best first search in artificial intelligence provides a top-rated illustration of how the primary exploration algorithm works.
Best first search is an optimal algorithm used in AI to find the finest path or solution to a problem. It considers the initial state of the problem and applies a heuristic function to determine the
best next move or node to explore.
For instance, let’s consider a search problem where the goal is to find the shortest path from a starting point to a destination in a graph. Best first search starts from the initial point and
evaluates the neighboring nodes based on a heuristic function. The node with the lowest heuristic value is selected as the next node to explore.
In the context of AI, the best first search algorithm is often used for various applications such as path planning, puzzle solving, and optimization problems. It intelligently explores the search
space to find the optimal solution based on the given heuristic function.
Advantages Disadvantages
Efficient exploration Potential to get stuck in local optima
Guarantees the optimal solution if an admissible heuristic is used Can be computationally expensive for large search spaces
In conclusion, the best first search algorithm is an essential component of artificial intelligence exploration. It provides an optimal search strategy based on heuristic evaluation, making it a
valuable tool for solving complex problems efficiently.
Best First Search in AI
Best First Search (BFS) is a primary exploration method used in the field of artificial intelligence. It is an initial search algorithm that aims to find the optimal solution for a given problem. BFS
is an example of a heuristics-based search technique that explores the search space in the most promising direction, rather than exhaustively searching all possible paths.
The primary goal of BFS is to find the best possible solution based on a heuristic evaluation function. This function assigns a score to each instance in the search space, indicating its potential to
lead to the optimal solution. The algorithm then selects the instance with the highest score and continues the search from there.
The exploration process of BFS can be illustrated using a table. In each row of the table, an instance is listed along with its heuristic score. The table is sorted based on the scores in a
descending order, with the top-rated instance at the top.
Let’s consider an example to understand how BFS works. Suppose we have a grid with multiple nodes, where each node represents a state of a problem. Our goal is to find the optimal path from the start
node to the goal node. The heuristic evaluation function assigns a score to each node based on its proximity to the goal node.
Initially, the BFS algorithm starts with the start node and calculates the heuristic scores for its neighboring nodes. These scores determine the priority of exploration. The algorithm then selects
the node with the highest score and continues the investigation from there. This process continues until the goal node is reached.
Instance Heuristic Score
Start Node 7
Node 1 9
Node 2 6
Node 3 8
Goal Node 0
In this example, the BFS algorithm would initially select Node 1 for investigation as it has the highest heuristic score. It would then explore its neighboring nodes and continue the search until the
goal node is reached.
Overall, the BFS algorithm in AI is an effective exploration technique that aims to find the optimal solution. By using heuristics and prioritizing the exploration based on evaluation scores, BFS
reduces the search space and improves the efficiency of finding the desired outcome.
Finest Initial Exploration in Artificial Intelligence
Best First Search (BFS) is an optimal search algorithm used in artificial intelligence. It is a top-rated example of initial exploration in AI, where the primary goal is to find the shortest path to
the desired solution.
BFS starts the investigation from the initial instance and explores the neighboring states before moving towards the goal state. This strategic approach makes it an efficient search algorithm for
various problem-solving tasks.
One instance of BFS could be illustrated as follows: Suppose we have a maze and we need to find the shortest path from the starting point to the exit point. BFS will explore the neighboring cells,
marking the visited ones, until it reaches the target cell. It guarantees that the solution found will be the shortest possible path.
In the field of artificial intelligence, the selection of the finest initial exploration technique is crucial. BFS provides the optimal solution by expanding nodes with lower heuristic values first,
making it an excellent choice for solving various problems efficiently.
Optimal Initial Investigation in AI
When it comes to exploring new territories in the field of artificial intelligence (AI), conducting an initial investigation is of utmost importance. The initial investigation serves as a primary
step towards finding the best possible solution for a given problem. It involves thorough exploration and search for the finest and top-rated instances in AI.
Example of Optimal Initial Investigation:
Suppose we are tasked with developing an AI system that can identify and classify images of various animals. In order to achieve the best possible results, an optimal initial investigation should be
conducted. This investigation may involve the following steps:
1. Data Collection: Gather a diverse dataset of images containing different animals.
2. Data Preprocessing: Clean and preprocess the collected data to remove any inconsistencies or noise.
3. Feature Extraction: Extract relevant features from the preprocessed data that can help distinguish between different animal species.
4. Model Selection: Choose an appropriate AI model, such as a convolutional neural network (CNN), that is known to perform well on image classification tasks.
5. Training: Train the selected model using the preprocessed data and the extracted features.
6. Evaluation: Measure the performance of the trained model using evaluation metrics like accuracy, precision, and recall.
7. Fine-tuning: Make necessary adjustments to the model and repeat the training process if the performance is not satisfactory.
This example illustrates how an optimal initial investigation can pave the way for developing a successful AI system for animal image classification. By following a systematic approach and
meticulously exploring different options, we can ensure that we achieve the best possible solution.
In conclusion, conducting an initial investigation is crucial in the field of artificial intelligence. It helps us identify the best methods, models, and techniques for solving a given problem. By
investing time and effort in the initial investigation phase, we can lay a solid foundation for building exceptional AI systems.
Top-rated Primary Exploration in Artificial Intelligence
The initial investigation into artificial intelligence (AI) often begins with the best-first search algorithm. This algorithm aims to find the optimal solution by searching for the most promising
paths and exploring them first. An example of this exploration can be illustrated through a simple instance.
Consider a scenario where an AI system needs to navigate a maze to reach its goal. The best-first search algorithm would start the exploration by choosing the path that appears to be the most
promising based on heuristic evaluation. This evaluation is based on factors such as the distance to the goal, potential obstacles, and other relevant information.
By utilizing the best-first search algorithm, the AI system can efficiently explore the maze and find the optimal route to the goal. This exploration technique has been recognized as one of the
top-rated methods in AI due to its ability to quickly identify the most promising paths and prioritize their exploration.
Overall, the best-first search algorithm serves as a primary exploration strategy in artificial intelligence. Its effectiveness and efficiency make it an integral component of many AI systems and
AI Example
In the field of Artificial Intelligence (AI), best first search is a primary investigation used to find the most optimal solution. It is an instance of a top-rated exploration algorithm that aims to
find the finest possible outcome. The primary goal of this search is to illustrate the initial exploration for the AI system to navigate through a problem space and find the optimal solution.
Example of Best First Search in AI
The best first search is a primary exploration technique in the field of artificial intelligence. It is a type of search algorithm used to find the optimal solution in an instance of a problem by
investigating the top-rated options first.
For instance, let’s consider the problem of finding the shortest path from a start node to a goal node in a graph. The best first search algorithm starts with an initial node and explores the
neighboring nodes based on some heuristic function, which estimates the desirability of each node as a potential candidate for the solution.
Here is an illustration of how the best first search works:
1. Start with the initial node and calculate the heuristic value for each neighbor node.
2. Select the node with the lowest heuristic value as the next node to explore.
3. Repeat steps 1 and 2 until the goal node is reached.
The best first search algorithm ensures that the most promising nodes are explored first, potentially leading to a quicker discovery of the optimal solution. It is widely used in various AI
applications, including pathfinding, game playing, and constraint satisfaction problems.
In summary, the best first search algorithm is a finest approach for exploration in artificial intelligence. It prioritizes the investigation of the most promising options based on their heuristic
values, ultimately aiming to find the optimal solution in a given problem instance.
Finest Primary Exploration in AI
When it comes to searching for the optimal solution in artificial intelligence, the best first search algorithm is an example of top-rated exploration. The primary goal of this method is to use an
initial heuristic to guide the search through the problem space, ensuring the most efficient path is followed.
Unlike other search algorithms, such as depth-first or breadth-first search, the best first search prioritizes the nodes to expand based on the heuristic value, rather than the depth or breadth of
the search. This approach allows the algorithm to quickly find the most promising options, leading to improved efficiency and reduced search time.
An instance of the best first search in AI can be seen in various applications, including navigation systems, puzzle-solving, and logistical planning. By utilizing a heuristic function, the algorithm
can intelligently evaluate the potential solutions and make informed decisions on which nodes to explore next.
For example, in a navigation system, the best first search can determine the optimal path from one location to another by considering factors such as distance, traffic conditions, and available
routes. By prioritizing the nodes with the lowest heuristic values, the algorithm can identify the most efficient route and provide accurate directions.
In summary, the best first search is a prime example of the finest primary exploration in AI. By utilizing heuristics, it can efficiently navigate a problem space and find the optimal solution in a
faster and more intelligent manner.
Optimal Initial Search in Artificial Intelligence
When exploring the vast field of artificial intelligence (AI), it is essential to start with an initial search that will set the foundation for further investigation. The best first search algorithm
is a top-rated and widely recognized approach in AI, known for its optimal exploration of the search space.
The best first search algorithm, also referred to as the finest primary search or the optimal initial search, aims to find the most promising instance for further exploration. It evaluates each
potential instance based on a heuristic function that estimates its potential to lead to the desired solution. This allows the algorithm to prioritize the exploration of instances with the highest
likelihood of success.
To illustrate the concept of the best first search, let’s consider an example of finding the shortest path in a maze. The algorithm starts with the initial state, which is the entrance of the maze.
It then evaluates the neighboring states based on a heuristic function, such as the Euclidean distance to the exit.
The algorithm selects the state with the lowest heuristic value as the next state to explore. It repeats this process, continuously selecting the state with the lowest heuristic value, until it
reaches the exit state, which signifies the solution. This approach ensures that the algorithm explores the most promising paths first, leading to an optimal solution.
Benefits of Optimal Initial Search
The optimal initial search provides several benefits in the field of artificial intelligence. Firstly, it decreases the overall search time by prioritizing the exploration of promising instances.
This allows for more efficient use of computational resources, especially in cases where the search space is vast.
Secondly, the optimal initial search increases the likelihood of finding an optimal solution. By exploring the most promising instances first, the algorithm reduces the chances of overlooking a
better solution that might be deeper within the search tree.
In conclusion, the optimal initial search algorithm, also known as the best first search, is a powerful tool in AI for navigating complex search spaces. Its ability to prioritize promising instances
for exploration leads to more efficient and effective problem-solving in various domains.
Top-rated AI Investigation Example
In the exploration of artificial intelligence, there are numerous instances where a top-rated AI investigation is required. One such prominent example is the application of the Best First Search
algorithm. This algorithm is a primary tool used in AI to find the optimal solution for a given problem. It is widely regarded as one of the finest AI search algorithms.
Let’s consider an illustration to understand how the Best First Search algorithm works. Suppose we have a maze and the initial state is the entrance of the maze. The primary objective is to find the
path to the exit of the maze. The Best First Search algorithm starts its investigation by evaluating the possible paths based on a heuristic function. This function helps the AI agent estimate the
distance to the exit from a given point in the maze.
During the investigation, the AI agent explores the maze by selecting the path with the lowest estimated distance to the exit. This exploration continues until the AI agent reaches the exit or
exhausts all possible paths. By constantly selecting the path with the lowest estimated distance, the Best First Search algorithm aims to find the optimal solution, i.e., the path with the shortest
distance to the exit.
The Best First Search algorithm is an excellent example of how artificial intelligence can efficiently navigate and investigate complex problem spaces. Its ability to make informed decisions based on
heuristics makes it a top-rated AI investigation tool. By utilizing this algorithm, AI agents can effectively explore and find optimal solutions to various real-world problems.
AI Primary Exploration
Exploration is a vital aspect in the initial stages of any artificial intelligence (AI) investigation. By performing an in-depth search, AI systems strive to find the best possible solutions for a
given problem. One of the top-rated search algorithms used in these scenarios is the Best-First Search.
The Best-First Search algorithm is a primary exploration technique that aims to find the finest solution by initially examining the most promising instances. This exploration strategy involves
selecting the most appropriate next state from a list of potential options based on their heuristic values. The heuristic values estimate the potential quality of each option, guiding the search
towards the most optimal outcome.
To illustrate the effectiveness of the Best-First Search, consider the following example: Suppose an AI system is tasked with finding the shortest path between two locations in a city. By using the
Best-First Search, the AI system can explore the neighboring paths, initially choosing the paths that are estimated to be the most promising based on their heuristic values. This approach enables the
AI system to efficiently navigate through the city and identify the optimal route.
In summary, the AI primary exploration involves using the Best-First Search algorithm to perform an in-depth search and find the best solutions in the initial stages of an artificial intelligence
investigation. This exploration technique prioritizes the examination of the most promising instances, leading to efficient and effective problem-solving.
Key Terms Definitions
Exploration The act of investigating and searching for solutions or information.
Initial Relating to the beginning or starting point of a process.
Best-First Search A search algorithm that selects the most promising options based on their heuristic values.
Artificial Intelligence (AI) The development of computer systems capable of performing tasks that would typically require human intelligence.
Illustration An example or instance used to explain or demonstrate a concept or idea.
Best First Search in Artificial Intelligence Example
Best First Search is a primary search algorithm in Artificial Intelligence that aims to find the optimal solution to a problem. It is an instance of an informed traversal algorithm that explores the
finest nodes first, based on some heuristic function.
One top-rated example of Best First Search is the A* (pronounced “A-star”) algorithm, which is widely used in AI applications. A* combines an initial investigation of the problem space with a
heuristic function to guide the search towards the most promising nodes.
Illustration of Best First Search
Let’s consider a scenario where we need to find the shortest path from a starting point to a destination in a graph. The graph represents a map, and each node represents a location. The edges between
nodes represent the connections between locations.
To apply Best First Search in this example, we need an initial node, which is the starting point, and a goal node, which is the destination. We also need a heuristic function that estimates the
distance between two nodes.
Initially, the algorithm starts at the starting point and evaluates the heuristic function for all neighboring nodes. It then selects the node with the lowest heuristic value as the next node to
explore. This process continues until the goal node is reached or until there are no more nodes to explore.
During the search, the algorithm maintains a priority queue that stores the nodes yet to be explored. The priority is determined by the heuristic function, which helps prioritize the nodes that are
closer to the goal. This ensures that the algorithm explores the most promising nodes first.
In our example, the Best First Search algorithm would prioritize the nodes that are closest to the destination. It would explore the nodes in the order of their heuristic values, aiming to find the
optimal path to the destination.
Best First Search is a powerful algorithm in Artificial Intelligence that combines an initial investigation of the problem space with a heuristic function to guide the search. It provides an
efficient approach to finding the optimal solution in various applications.
Finest Initial Investigation in AI
In the exploration of artificial intelligence (AI), the initial investigation is crucial for achieving optimal results. The best first search algorithm is a top-rated example of such investigation.
This algorithm starts with an initial instance and searches through the state space to find the optimal solution. It evaluates each state based on a heuristic function, which estimates the distance
to the goal. By prioritizing the states with the lowest estimated distance, the algorithm explores the most promising paths first.
The best first search algorithm is a primary tool in AI, used in various applications such as route planning, puzzle-solving, and game playing. Its ability to quickly narrow down the search space
makes it a valuable technique for finding solutions in complex problem domains.
Through this finest initial investigation in AI, the best first search algorithm sets the foundation for further AI research and development. Its effectiveness in finding optimal solutions paves the
way for advancements in artificial intelligence, opening up new possibilities in various fields. As AI continues to evolve, the best first search algorithm remains a cornerstone of efficient
Optimal Initial Search in AI Example
Best First Search is a popular and effective search algorithm used in artificial intelligence. It is a top-rated AI search algorithm that prioritizes the most promising paths first. The key to the
success of Best First Search lies in its optimal initial search.
Primary Goal of Best First Search
The primary goal of Best First Search is to find the most optimal solution in the shortest amount of time. It achieves this by considering the heuristic value of each node in the search space. The
heuristic value is an estimate of how close a node is to the goal state.
An Illustration of Optimal Initial Search
To illustrate the concept of optimal initial search in Best First Search, consider the following example:
1. Suppose we have a search space with multiple nodes.
2. Each node is assigned a heuristic value based on its distance from the goal state. The closer a node is to the goal state, the lower its heuristic value.
3. The Best First Search algorithm starts the search from the node with the lowest heuristic value, as it is considered the most promising node.
4. By starting the search from the most promising node, Best First Search is able to explore the most fruitful paths first, increasing the chances of finding an optimal solution.
Through this initial investigation of the top-rated nodes, Best First Search can intelligently navigate the search space and quickly identify the optimal solution.
In conclusion, an optimal initial search is a crucial step in the Best First Search algorithm. It allows the AI system to prioritize the finest nodes and significantly improve the efficiency and
effectiveness of the search process.
Top-rated AI Illustration
Artificial intelligence (AI) has revolutionized various fields, and one of its primary applications is in the field of search algorithms. One such algorithm is the Best First Search, which is an
efficient exploration technique for finding optimal solutions.
As an initial example, let’s consider a real-life instance of Best First Search in action. Imagine you are planning a trip to a new city and want to find the best tourist attractions to visit. With
the help of AI, you can utilize the Best First Search algorithm to explore various options and determine the optimal sequence of attractions to maximize your experience.
During the investigation, the Best First Search algorithm evaluates each attraction based on certain criteria such as popularity, ratings, reviews, and proximity to your current location. It uses
these factors to assign a score to each attraction and then explores the attractions with the highest scores first, gradually moving towards the optimal solution.
This exploration process involves iteratively expanding the search space by considering the successor attractions of the ones with the highest scores. By doing so, the algorithm narrows down the
search and gradually finds the top-rated attractions based on your preferences and constraints.
The Best First Search algorithm exemplifies the finest implementation of artificial intelligence in search problems. It showcases the ability of AI to efficiently explore large solution spaces,
making it suitable for various applications, including route planning, recommendation systems, and even game-playing algorithms.
Best First Search Instance in AI
In the investigation of optimal search algorithms in the field of Artificial Intelligence, the Best First Search (BFS) stands out as one of the top-rated and most widely used techniques. This
algorithm, also known as the finest exploration strategy, aims to find the most promising path towards a goal state.
An example of BFS can be illustrated by considering a maze-solving scenario. Assume that there is a maze with multiple paths from the initial state to the goal state. The primary objective of the BFS
algorithm is to efficiently explore the maze and find the optimal path.
The BFS algorithm starts by evaluating the initial state of the maze and identifying the possible successors. These successors are then added to a queue based on a heuristic function that estimates
their potential to lead to the goal state. The algorithm selects the most promising successor from the queue and advances to its state. This process continues until the goal state is reached.
Through its careful selection of successors, BFS ensures that the exploration is focused on the most promising paths. This optimization allows for efficient search and reduces the overall
computational complexity of the algorithm.
In conclusion, the Best First Search algorithm is a powerful tool in the field of Artificial Intelligence. Its ability to prioritize the exploration of the most promising paths makes it an effective
and widely used method. By employing a heuristic function, BFS ensures that the search is directed towards the optimal solution, leading to efficient and effective problem-solving.
Finest AI Exploration Example
In the field of artificial intelligence, the best first search algorithm is a primary method of investigation. It is an exemplary instance of how AI techniques can be utilized to efficiently explore
a problem space and find the optimal solution.
The best first search begins with the initial state and iteratively expands the top-rated nodes in the search tree. It evaluates the nodes based on heuristic function values, which estimate the
desirability of each node. By selecting the most promising nodes, the algorithm focuses its efforts on the most likely path to the goal.
To illustrate the best first search, let’s consider a scenario where the task is to find the shortest path between two points on a map. The initial state is the starting point, and the goal state is
the destination. The algorithm explores the map by expanding the nodes with the lowest heuristic value, gradually getting closer to the goal.
For example, suppose we have a map with various cities and road connections, where the heuristic function estimates the straight-line distance between cities. The best first search algorithm will
prioritize expanding the nodes representing cities closest to the goal, as they are more likely to be on the shortest path.
Throughout the exploration, the algorithm keeps track of the path taken, allowing it to trace back steps if a dead-end is reached. This way, it can efficiently explore alternative routes until the
optimal solution is found.
The best first search algorithm is considered one of the finest AI techniques due to its ability to quickly find the optimal solution in many problem domains. Its effectiveness stems from the careful
selection of nodes based on the heuristic function, which enables it to focus its exploration on the most promising paths.
In conclusion, the best first search algorithm serves as a prime example of how artificial intelligence can be applied to exploration and search problems. Through its use of a heuristic function, it
efficiently navigates a problem space to find the optimal solution. Its top-rated node expansion strategy makes it a highly effective technique and one of the best in the field of artificial
Optimal Initial AI Investigation
When starting an AI project, it is crucial to conduct an optimal initial investigation to ensure the best results. This investigation serves as the primary step in determining the finest approach to
be taken and the specific instance of AI to be used.
Exploration of Top-Rated AI Search Algorithms
The initial investigation involves exploring and evaluating various AI search algorithms. One such example is the Best First Search, which utilizes a heuristic function to prioritize the exploration
of the most promising paths. By considering the heuristic value, the algorithm can efficiently determine the next step to take in order to find the optimal solution.
During the investigation, it is important to gather data and evaluate the performance of different search algorithms. This evaluation can be done by comparing their efficiency, accuracy, and ability
to handle complex problems. The investigation should also consider the specific requirements and constraints of the problem at hand, as different algorithms may excel in different scenarios.
Illustration of Optimal Initial AI Investigation
For instance, let’s consider a scenario where an AI task involves finding the shortest path in a maze. The initial investigation will assess the performance of various AI search algorithms, such as
breadth-first search, depth-first search, and the aforementioned Best First Search. By gathering data on their efficiency and accuracy, we can determine which algorithm is most suitable for solving
this specific problem.
In this example, the optimal initial investigation may reveal that the Best First Search algorithm is the most effective in quickly finding the shortest path in the maze. This determination is based
on its ability to prioritize the exploration of the most promising paths, resulting in minimal computational time and resources.
In conclusion, conducting an optimal initial investigation is crucial in AI projects. By exploring and evaluating different search algorithms, such as the Best First Search, and assessing their
performance in relation to the specific problem, developers can make informed decisions on which AI technique to utilize for the best possible outcomes.
What is Best First Search in Artificial Intelligence?
Best First Search in Artificial Intelligence is a search algorithm that finds the optimal path to a goal state by evaluating states based on heuristics and selecting the most promising one at each
Can you give an example of Best First Search in Artificial Intelligence?
Sure! Let’s say we have a grid representing a maze, and we want to find the shortest path from a start location to a goal location. Best First Search can be used to explore the maze by selecting the
cell with the lowest heuristic value at each step, until the goal state is reached.
What is meant by optimal initial exploration in AI?
Optimal initial exploration in AI refers to the process of finding the best possible path or solution from an initial state to a goal state. This is usually done by using search algorithms like Best
First Search, which prioritize the most promising states to explore first.
Can you provide an example of finest initial exploration in AI?
Certainly! Let’s say we have a game where the player needs to navigate through a maze to reach a treasure. The finest initial exploration in this case would involve using an algorithm like Best First
Search to find the path with the maximum potential rewards or minimum potential risks.
What is meant by top-rated primary investigation in AI illustration?
Top-rated primary investigation in AI illustration refers to the process of conducting a search or exploration in an AI system using the most effective and efficient algorithm available. This ensures
that the system can find the best solution or path in a timely manner. | {"url":"https://aiforsocialgood.ca/blog/best-first-search-in-artificial-intelligence-example-finding-optimal-solutions-faster","timestamp":"2024-11-14T01:50:34Z","content_type":"text/html","content_length":"111156","record_id":"<urn:uuid:8a23ffa2-3c00-45f6-a9ab-39fc173bee07>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00739.warc.gz"} |
Integrated Mathematics III
High School Integrated Mathematics III
FlexPoint digital courses are mobile-friendly and customizable. Course availability will differ by licensing model. Please confirm course selections with your FlexPoint account manager.
• License Model
FlexPoint or School/District Hosted
• Number of Credits
• Estimated Completion Time
32-36 weeks
Suggested Prerequisites
Integrated Mathematics II
This course allows you to learn while having fun. Interactive examples help guide your journey through customized feedback and praise. Mathematical concepts are applied to everyday occurrences such
as earthquakes, stadium seating, and purchasing movie tickets. You will investigate the effects of an equation on its graph through the use of technology and will have opportunities to work with your
peers on specific lessons.
Module One: Basics of Geometry
-Define points, lines, and planes
-Perform geometric constructions using a compass
-Perform geometric constructions using technology
-Introduction to different types of proofs
Module Two: Transformations and Congruence
-Perform and represent translations
-Perform and represent reflections
-Perform and represent rotations
-Prove two figures are congruent
Module Three: Coordinate Geometry
-Classify polygons using coordinate geometry
-Solve problems using slope
-Use coordinates to find perimeter and area of polygons
-Use coordinates to divide segments into ratios
Module Four: Volume and Figures
-Derive formulas for circumference and area of circles
-Derive volume formulas for cylinders, pyramids, and cones
-Use Cavalieri's Principle to compare volumes
-Solve real-world applications involving density
-Identify shapes of cross-sections of 3-D objects
-Identify 3-D objects generated by 2-D objects
-Find the surface area of 3-D figures
Module Five: Trigonometry
-Identify and describe properties of circles
-Apply trigonometric functions using the unit circle
-Graph trigonometric functions with periodic
-Analyze transformations
-Fit functions to data to solve problems
-Prove equations using the Pythagorean Identity
-Simplify expressions using the Pythagorean Identity
Module Six: Dividing and Solving Polynomials
-Dividing polynomials using long division
-Dividing polynomials using synthetic division
-Determining key features of polynomials using Theorems
-Determine zeros using Rational Root Theorem and Descartes' Rule of Signs
-Use zeros and end behavior to graph polynomial functions
-Solve polynomial equations
-Graph polynomial functions to determine key features and solutions
-Prove polynomial identities
Module Seven: Rational Expressions
-Simplify rational expressions
-Multiply and divide rational expressions
-Add and subtract rational expressions
-Simplify complex fractions
-Identify discontinuities of rational expressions
-Identify asymptotes of rational functions
-Solve rational equations and justify solutions
-Apply rational equations to real-world scenarios
Module Eight: Exponential and Logarithmic Functions
-Create exponential equations to model real-world situations
-Create logarithmic functions and solve equations
-Use properties of logarithms to solve equations
-Solve exponential equations with unequal bases
-Graph exponential functions
-Graph logarithmic functions
-Determine effects of combining different types of functions
Module Nine: Sequences and Series
-Identify and use arithmetic sequences to solve problems
-Identify and use arithmetic series to solve problems
-Identify and use geometric sequences to solve problems
-Identify and use geometric series to solve problems
-Use sigma notation to evaluate a series
-Identify and find infinite, convergent, and divergent series
-Graph sequences and series
Module Ten: Statistics
-Use measures of center and spread to fit normal distribution
-Estimate population percentages
-Population inferences using statistics, models, surveys, and experiments
-Use surveys to estimate population parameters
-Determine margin of error and evaluate data reports
-Use experiments to compare treatments, assess significance, evaluate data reports | {"url":"https://www.flexpointeducation.com/courses-curriculum/course-catalog/course/integrated-mathematics-iii/aAMUH0000003FvT4AU/","timestamp":"2024-11-15T00:44:06Z","content_type":"text/html","content_length":"28156","record_id":"<urn:uuid:1ca0ad85-d566-4e33-9ca3-c9495beb8baf>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00140.warc.gz"} |
29th EACSL Annual Conference on Computer Science Logic (CSL 2021)
Cite as
Mark Bickford, Liron Cohen, Robert L. Constable, and Vincent Rahli. Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle. In 29th EACSL Annual Conference on Computer Science
Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
Copy BibTex To Clipboard
author = {Bickford, Mark and Cohen, Liron and Constable, Robert L. and Rahli, Vincent},
title = {{Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle}},
booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
pages = {11:1--11:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-175-7},
ISSN = {1868-8969},
year = {2021},
volume = {183},
editor = {Baier, Christel and Goubault-Larrecq, Jean},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.11},
URN = {urn:nbn:de:0030-drops-134455},
doi = {10.4230/LIPIcs.CSL.2021.11},
annote = {Keywords: Intuitionism, Extensional type theory, Constructive Type Theory, Realizability, Choice sequences, Classical Logic, Law of Excluded Middle, Theorem proving, Coq} | {"url":"https://drops.dagstuhl.de/entities/volume/LIPIcs-volume-183","timestamp":"2024-11-06T21:35:27Z","content_type":"text/html","content_length":"411836","record_id":"<urn:uuid:c9b9d8e2-c545-4f20-a885-1d7153e7085a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00833.warc.gz"} |
666 research outputs found
Neural Autoregressive Distribution Estimators (NADEs) have recently been shown as successful alternatives for modeling high dimensional multimodal distributions. One issue associated with NADEs is
that they rely on a particular order of factorization for $P(\mathbf{x})$. This issue has been recently addressed by a variant of NADE called Orderless NADEs and its deeper version, Deep Orderless
NADE. Orderless NADEs are trained based on a criterion that stochastically maximizes $P(\mathbf{x})$ with all possible orders of factorizations. Unfortunately, ancestral sampling from deep NADE is
very expensive, corresponding to running through a neural net separately predicting each of the visible variables given some others. This work makes a connection between this criterion and the
training criterion for Generative Stochastic Networks (GSNs). It shows that training NADEs in this way also trains a GSN, which defines a Markov chain associated with the NADE model. Based on this
connection, we show an alternative way to sample from a trained Orderless NADE that allows to trade-off computing time and quality of the samples: a 3 to 10-fold speedup (taking into account the
waste due to correlations between consecutive samples of the chain) can be obtained without noticeably reducing the quality of the samples. This is achieved using a novel sampling procedure for GSNs
called annealed GSN sampling, similar to tempering methods that combines fast mixing (obtained thanks to steps at high noise levels) with accurate samples (obtained thanks to steps at low noise
levels).Comment: ECML/PKDD 201
Inverse reinforcement learning (IRL) is the task of learning the reward function of a Markov Decision Process (MDP) given the transition function and a set of observed demonstrations in the form of
state-action pairs. Current IRL algorithms attempt to find a single reward function which explains the entire observation set. In practice, this leads to a computationally-costly search over a large
(typically infinite) space of complex reward functions. This paper proposes the notion that if the observations can be partitioned into smaller groups, a class of much simpler reward functions can be
used to explain each group. The proposed method uses a Bayesian nonparametric mixture model to automatically partition the data and find a set of simple reward functions corresponding to each
partition. The simple rewards are interpreted intuitively as subgoals, which can be used to predict actions or analyze which states are important to the demonstrator. Experimental results are given
for simple examples showing comparable performance to other IRL algorithms in nominal situations. Moreover, the proposed method handles cyclic tasks (where the agent begins and ends in the same
state) that would break existing algorithms without modification. Finally, the new algorithm has a fundamentally different structure than previous methods, making it more computationally efficient in
a real-world learning scenario where the state space is large but the demonstration set is small
In this paper, we deal with the problem of curves clustering. We propose a nonparametric method which partitions the curves into clusters and discretizes the dimensions of the curve points into
intervals. The cross-product of these partitions forms a data-grid which is obtained using a Bayesian model selection approach while making no assumptions regarding the curves. Finally, a
post-processing technique, aiming at reducing the number of clusters in order to improve the interpretability of the clustering, is proposed. It consists in optimally merging the clusters step by
step, which corresponds to an agglomerative hierarchical classification whose dissimilarity measure is the variation of the criterion. Interestingly this measure is none other than the sum of the
Kullback-Leibler divergences between clusters distributions before and after the merges. The practical interest of the approach for functional data exploratory analysis is presented and compared with
an alternative approach on an artificial and a real world data set
Structurama is a program for inferring population structure. Specifically, the program calculates the posterior probability of assigning individuals to different populations. The program takes as
input a file containing the allelic information at some number of loci sampled from a collection of individuals. After reading a data file into computer memory, Structurama uses a Gibbs algorithm to
sample assignments of individuals to populations. The program implements four different models: The number of populations can be considered fixed or a random variable with a Dirichlet process prior;
moreover, the genotypes of the individuals in the analysis can be considered to come from a single population (no admixture) or as coming from several different populations (admixture). The output is
a file of partitions of individuals to populations that were sampled by the Markov chain Monte Carlo algorithm. The partitions are sampled in proportion to their posterior probabilities. The program
implements a number of ways to summarize the sampled partitions, including calculation of the ‘mean’ partition—a partition of the individuals to populations that minimizes the squared distance
to the sampled partitions
We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present
the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with
conclusions and a look at the current trends in GP work
This thesis presents new Bayesian nonparametric models and approaches for their development, for the problems of name disambiguation and supervised learning. Bayesian nonparametric methods form an
increasingly popular approach for solving problems that demand a high amount of model flexibility. However, this field is relatively new, and there are many areas that need further investigation.
Previous work on Bayesian nonparametrics has neither fully explored the problems of entity disambiguation and supervised learning nor the advantages of nested hierarchical models. Entity
disambiguation is a widely encountered problem where different references need to be linked to a real underlying entity. This problem is often unsupervised as there is no previously known information
about the entities. Further to this, effective use of Bayesian nonparametrics offer a new approach to tackling supervised problems, which are frequently encountered. The main original contribution of
this thesis is a set of new structured Dirichlet process mixture models for name disambiguation and supervised learning that can also have a wide range of applications. These models use techniques
from Bayesian statistics, including hierarchical and nested Dirichlet processes, generalised linear models, Markov chain Monte Carlo methods and optimisation techniques such as BFGS. The new models
have tangible advantages over existing methods in the field as shown with experiments on real-world datasets including citation databases and classification and regression datasets. I develop the
unsupervised author-topic space model for author disambiguation that uses free-text to perform disambiguation unlike traditional author disambiguation approaches. The model incorporates a name
variant model that is based on a nonparametric Dirichlet language model. The model handles both novel unseen name variants and can model the unknown authors of the text of the documents. Through
this, the model can disambiguate authors with no prior knowledge of the number of true authors in the dataset. In addition, it can do this when the authors have identical names. I use a model for
nesting Dirichlet processes named the hybrid NDP-HDP. This model allows Dirichlet processes to be clustered together and adds an additional level of structure to the hierarchical Dirichlet process. I
also develop a new hierarchical extension to the hybrid NDP-HDP. I develop this model into the grouped author-topic model for the entity disambiguation task. The grouped author-topic model uses
clusters to model the co-occurrence of entities in documents, which can be interpreted as research groups. Since this model does not require entities to be linked to specific words in a document, it
overcomes the problems of some existing author-topic models. The model incorporates a new method for modelling name variants, so that domain-specific name variant models can be used. Lastly, I
develop extensions to supervised latent Dirichlet allocation, a type of supervised topic model. The keyword-supervised LDA model predicts document responses more accurately by modelling the effect of
individual words and their contexts directly. The supervised HDP model has more model flexibility by using Bayesian nonparametrics for supervised learning. These models are evaluated on a number of
classification and regression problems, and the results show that they outperform existing supervised topic modelling approaches. The models can also be extended to use similar information to the
previous models, incorporating additional information such as entities and document titles to improve prediction
A toy detector has been designed to simulate central detectors in reactor neutrino experiments in the paper. The samples of neutrino events and three major backgrounds from the Monte-Carlo simulation
of the toy detector are generated in the signal region. The Bayesian Neural Networks(BNN) are applied to separate neutrino events from backgrounds in reactor neutrino experiments. As a result, the
most neutrino events and uncorrelated background events in the signal region can be identified with BNN, and the part events each of the fast neutron and $^{8}$He/$^{9}$Li backgrounds in the signal
region can be identified with BNN. Then, the signal to noise ratio in the signal region is enhanced with BNN. The neutrino discrimination increases with the increase of the neutrino rate in the
training sample. However, the background discriminations decrease with the decrease of the background rate in the training sample.Comment: 9 pages, 1 figures, 1 tabl
The label switching problem, the unidentifiability of the permutation of clusters or more generally latent variables, makes interpretation of results computed with MCMC sampling difficult. We
introduce a fully Bayesian treatment of the permutations which performs better than alternatives. The method can be used to compute summaries of the posterior samples even for nonparametric Bayesian
methods, for which no good solutions exist so far. Although being approximative in this case, the results are very promising. The summaries are intuitively appealing: A summarized cluster is defined
as a set of points for which the likelihood of being in the same cluster is maximized
The application of Bayesian Neural Networks(BNN) to discriminate neutrino events from backgrounds in reactor neutrino experiments has been described in Ref.\cite{key-1}. In the paper, BNN are also
used to identify neutrino events in reactor neutrino experiments, but the numbers of photoelectrons received by PMTs are used as inputs to BNN in the paper, not the reconstructed energy and position
of events. The samples of neutrino events and three major backgrounds from the Monte-Carlo simulation of a toy detector are generated in the signal region. Compared to the BNN method in Ref.\cite
{key-1}, more $^{8}$He/$^{9}$Li background and uncorrelated background in the signal region can be rejected by the BNN method in the paper, but more fast neutron background events in the signal
region are unidentified using the BNN method in the paper. The uncorrelated background to signal ratio and the $^{8}$He/$^{9}$Li background to signal ratio are significantly improved using the BNN
method in the paper in comparison with the BNN method in Ref.\cite{key-1}. But the fast neutron background to signal ratio in the signal region is a bit larger than the one in Ref.\cite
{key-1}.Comment: 9 pages, 1 figure and 1 table, accepted by Journal of Instrumentatio | {"url":"https://core.ac.uk/search/?q=authors%3A(R.M.%20Neal)","timestamp":"2024-11-15T00:37:00Z","content_type":"text/html","content_length":"177229","record_id":"<urn:uuid:dfaddf09-e403-4455-9bdc-3d78e07987b5>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00522.warc.gz"} |
• The value returned by nobs.kgaps() was incorrect in cases where there are censored K-gaps that are equal to zero. These K-gaps should not contribute to the number of observations. This has been
• In cases where the data used in kgaps are split into separate sequences, the threshold exceedance probability is estimated using all the data rather than locally within each sequence.
• A logLik method for objects inheriting from class "kgaps" has been added.
• In the (unexported, internal) function kgaps_conf_int() the limits of the confidence intervals for the extremal index based on the K-gaps model are constrained manually to (0, 1) to avoid
problems in calculating likelihood-based confidence intervals in cases where the the log-likelihood is greater than the interval cutoff when theta = 1.
• In the documentation of the argument k to kgaps() it is noted that in practice k should be no smaller than 1.
• The function kgaps() also return standard errors based on the expected information.
• In the package manual related functions have been arranged in sections for easier reading.
• Activated 3rd edition of the testthat package | {"url":"https://www.stats.bris.ac.uk/R/web/packages/exdex/news/news.html","timestamp":"2024-11-05T03:17:50Z","content_type":"application/xhtml+xml","content_length":"5662","record_id":"<urn:uuid:22a59d72-32b4-47be-ac89-5945f11eedc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00384.warc.gz"} |
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Followed by exercises
Playing Around with Statsbomb
Analyzing Baseball Data with R, Chapter 3 | {"url":"https://api.rpubs.com/JimdalftheOrange","timestamp":"2024-11-09T13:12:48Z","content_type":"text/html","content_length":"12076","record_id":"<urn:uuid:01c93924-52ff-4263-86a2-c992eae7109d>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00726.warc.gz"} |
Hackerrank - Birthday Chocolate Solution
Lily has a chocolate bar that she wants to share it with Ron for his birthday. Each of the squares has an integer on it. She decides to share a contiguous segment of the bar selected such that the
length of the segment matches Ron's birth month and the sum of the integers on the squares is equal to his birth day. You must determine how many ways she can divide the chocolate.
Consider the chocolate bar as an array of squares, . She wants to find segments summing to Ron's birth day, with a length equaling his birth month, . In this case, there are two segments meeting her
criteria: and .
Function Description
Complete the birthday function in the editor below. It should return an integer denoting the number of ways Lily can divide the chocolate bar.
birthday has the following parameter(s):
• s: an array of integers, the numbers on each of the squares of chocolate
• d: an integer, Ron's birth day
• m: an integer, Ron's birth month
Input Format
The first line contains an integer , the number of squares in the chocolate bar.
The second line contains space-separated integers , the numbers on the chocolate squares where .
The third line contains two space-separated integers, and , Ron's birth day and his birth month.
Output Format
Print an integer denoting the total number of ways that Lily can portion her chocolate bar to share with Ron.
Sample Input 0
Sample Output 0
Explanation 0
Lily wants to give Ron squares summing to . The following two segments meet the criteria:
Sample Input 1
Sample Output 1
Explanation 1
Lily only wants to give Ron consecutive squares of chocolate whose integers sum to . There are no possible pieces satisfying these constraints:
Thus, we print as our answer.
Sample Input 2
Sample Output 2
Explanation 2
Lily only wants to give Ron square of chocolate with an integer value of . Because the only square of chocolate in the bar satisfies this constraint, we print as our answer.
Solution in Python
s = list(map(int,input().split()))
d,m = map(int,input().split())
c =0
for i in range(len(s)-m+1):
if sum(s[i:i+m])==d:
Memory efficient solution
s = list(map(int,input().split()))
d,m = map(int,input().split())
c =0
add = sum(s[:m])
if add==d:
for i in range(m,len(s)):
add= add+s[i]-s[i-m]
if add==d: | {"url":"https://www.thepoorcoder.com/hackerrank-birthday-chocolate-solution/","timestamp":"2024-11-05T23:59:27Z","content_type":"text/html","content_length":"43883","record_id":"<urn:uuid:1a50fed3-2f3b-4db4-9beb-0996f88cff3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00289.warc.gz"} |
Propagation of a TW laser pulse in the atmosphere
Response of air to incoming electromagnetic radiation can be described in terms of induced macroscopic polarization P. In many cases this response is linear (proportional to applied electric field E
), however, illumination by a short fs pulse having several tens of mJ energy is capable to induce non-linear response. Mathematically non-linear response of P requirers expansion of effective
susceptibility P(-E)=-P(E) must be satisfied all even order expansion terms must be a priori set to zero. The first non-linear term includes a third order susceptibility
Changes of material's susceptibility impose changes of its dielectric permitivity ε. Since ε=ε[0](1+Χ), the effective dielectric permitivity ε[eff] is express by a quadratic equation, namely:
Dividing it by ε[0] and calculating a square root retrieves a modified refractive index of air:
where n[0] is the linear refractive index of air and n[2] is the non-linear refractive index of air. Note that non-linear modification of the effective refractive index depends on incoming light
intensity I. Above equation is valid not only for air but many other centrosymmetric materials. It reflects a tendency of the medium to become optically thicker under illumination by intense light
pulses. Such effect is known as the Kerr effect.
Fig. 1 Schematic representation of
(a) self-focusing and (b) defocusing
The Kerr effect can overcome natural diffraction of the beam initiating its self-focusing (SF). This phenomenon is widely observed with Gaussian or semi-Gaussian beams, for which the intensity
distribution forms, so called, converging Kerr-lens (Fig. 1). To estimate the critical power P[crit] of SF the equilibrium between diffraction and Kerr self-focusing is assumed. For air and commonly
used laser wavelength of 800 nm P[crit] ~ 3 GW.
The intensity inside self-focusing beam increases with a propagation distance and eventually leads to efficient plasma generation due to Multi-Photon Ionization (MPI) of nitrogen and oxygen
molecules. MPI is believed to be a dominant ionization process, because collisions can be neglected in an ultra short femtosecond time scale and intensity required for tunnel ionization is much
The role of unbounded electrons in plasma is often expressed in terms of the plasma frequency ω[p]. This quantity sets the boundary in a frequency domain distinguishing between photons, which are
highly (photon frequency above plasma frequency) and rarely (photon frequency below plasma frequency) absorbed by plasma. The electron density corresponding to the plasma frequency is called critical
density ρ[crit]:
In the equation above m[e] corresponds to the electron rest mass, e is the electron charge and ε is the dielectric permitivity. Using the critical density it is possible to show that the plasma
contribution to the dielectric permitivity is negative. This means that the presence of free electrons lowers the air refractive index, namely:
Because the MPI rate is a power function of the laser intensity (the power index corresponds to the multi-photon absorption order) the most dense plasma will be created in the central part of the
self-focusing beam. In this way the electron density gradient is built, which as schematically depicted in Fig. 1, tends to defocus the beam forming akin of diverging lens counteracting Kerr-lens
focusing. Taking into account both Kerr and plasma contribution the effective refractive index of air can be expressed in a form:
Fig. 2 A digital camera image of
multiple filamentation within a
fs-laser beam
Propagation of the fs-laser pulse in the air with modified refractive index is much different compared to the linear propagation. At first, the beam diameter undergoes a gradual transverse
suppression initiated by Kerr-lens. Near the focus the E-field rapidly increases and efficient MPI takes place. Arising plasma retards Kerr-focusing and guides the energy in a stable filament, which
is clearly distinguishable from the rest of the beam. From this moment small-scale, consecutive convergence and divergence stabilizes the filament sustaining it on distances much longer then its
Rayleigh range. If the pulse power significantly exceeds P[crit] not only one, but many filaments are formed (Fig. 2). They are typically randomly arranged and their position is mainly determined by
fluctuations of a beam transverse profile. The fluctuations act as micro Kerr-lenses concentrating the energy, which is successively clamped by each of filaments.
It is worth noting that the air filaments are always similar to one another. Their diameter is variable between 70-100 μm. Light intensity inside a filament varies between 10^13-10^14 W/cm^2 and is
mainly set by the MPI. Finally, the plasma density inside the filament is approximately 10^16-10^17 cm^-3, which is only two orders of magnitude less than the air density.
Fig. 3 Conical emission resulting
from multiple filaments
Filament are always accompanied by many non-linear optical phenomena including odd harmonics generation, four wave mixing, self-phase modulation and spectrally broad, coherent white-light
supercontinuum. Supercontinuum is partially emitted in a form of concentric cones called conical emission (Fig. 3). These non-linear phenomena are very interesting for various applications. For
example, supercontinuum has been used for broadband atmospheric absorption spectroscopy, which as compared to Sun photometry is performed actively. | {"url":"https://www.physik.fu-berlin.de/einrichtungen/ag/ag-woeste/research/LiDaR/teramobile1/propagation.html","timestamp":"2024-11-06T05:17:45Z","content_type":"text/html","content_length":"33562","record_id":"<urn:uuid:000820a4-0c43-4657-8c20-5331a0e3eeaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00877.warc.gz"} |
Thinking Test: If you have Eagle Eyes Find the word Law in 15 Secs - EduViet Corporation
Thinking Test: If you have Eagle Eyes Find the word Law in 15 Secs
brain teasers
A brainteaser is a puzzle or problem that requires creative and unconventional thinking to solve. These puzzles are designed to challenge your cognitive abilities, including logic, reasoning, lateral
thinking, and sometimes even math or spatial skills. Brainteasers often have fun and interesting qualities that make them fun to play.
Brain teasers come in many forms, such as riddles, optical illusions, word puzzles, and more complex questions that require you to think outside the box
You are watching: Thinking Test: If you have Eagle Eyes Find the word Law in 15 Secs
Thought test: If you have eagle eyes, find the word “Law” in 15 seconds
It usually takes about 45 seconds for the average person to finally notice the word “law” in this brainteaser.
This image is designed to challenge those who like brain teasers and quizzes.
One word, “law,” is missing. If you look closely, you’ll see that it’s spelled strangely and violates the norm. So, can you spot this unique thing quickly? Or are you one of those people who needs a
sneak peek to crack the brainteasers? For helpful tips, focus on the lower right corner of the image.
Thought test: If you have eagle eyes, find the word “Law” in 15 seconds: Solution
See more : Brain Test: If you have Eagle Eyes Find the Word Kill among Hill in 15 Secs
If you’re racing against time and still struggling with brain teasers, turning your attention to row 13 from the top might be just the solution you need.
This brainteaser challenges you to figure out word rules, truly testing our brain’s incredible visual processing abilities. It highlights how easily our perceptions are influenced, and the challenge
of distinguishing reality from deception.
In a way, it’s a gentle reminder of the wonderful workings of human perception.
Indeed, you are right; the word “Law” (highlighted in the image below) matches the word you are looking for in this brainteaser.
Brain Teasers IQ Test: If 1=2, 2=4, 4=28, then 5=?
Embark on a brainteaser IQ test journey where numbers play with perception. According to this pattern, if 1 equals 2, 2 equals 4, and 4 equals 28, then the riddle arises: What does 5 equal?
Explore the answer: 1 equals 1+1! Equals 1+1 equals 2. Applying the same logic, 5 equals 5+5! Equal to 5+120, the result is 125. This puzzle utilizes addition and factorial concepts to design its
interesting sequences.
Brain teaser math speed test: 52÷2x(11+7)=?
See more : Brain Teaser: What Number should come Next? Math Puzzle
Delve deeper into the realm of brainteaser math speed tests using the following equation: 52 ÷ 2 x (11 + 7) = ? Your task is to solve the problem step by step, working meticulously to reveal the
final result.
To crack the code, first solve the addition inside the brackets: 11 + 7 equals 18. Next, divide 52 by 2 to get 26. Finally, multiply 26 by 18 to get the answer 52÷2x(11+7) = 26×18 = 468.
Brain teaser math puzzle: 1+2=10, 2+3=22, 4+5=?
Play a brain-teaser math puzzle: 1+2 equals 10 and 2+3 equals 22. Now the mystery deepens: What is the result of test 4+5?
Let’s break it down. In the first equation, 1+2² equals 5, which multiplied by 2 gives us 10. Applying the same logic to the second equation, 4+5² equals 29, then multiplying by 2 gives us the result
4+5 = 4 +5² = 29 → 29×2 = 58.
Brain Teasers IQ Test Math Test: 53-50÷10+6-55÷11=?
Enter the world of brainteasers IQ test math quiz using the following equation: 53 – 50 ÷ 10 + 6 – 55 ÷ 11 = ? Your task is to go through the operations and publish the final result.
We start by dividing: 50 ÷ 10 equals 5, 55 ÷ 11 equals 5. Substituting these results, we simplify the equation to 53 – 5 + 6 – 5. Continuing the math, we subtract 5 from 53 to get 48, and then add 6
to get 54. Finally, we complete the equation and find that 53 – 50 ÷ 10 + 6 – 55 ÷ 11 equals 49.
Brain Teasers Math IQ Test: Solve 35÷5×4+1-6
Take part in the brainteaser math IQ test by solving the equation: 35 ÷ 5 x 4 + 1 – 6. Your task is to carefully follow the order of operations and calculate the final result.
Following the order of operations, we first perform the division: 35 ÷ 5 equals 7. Then, multiplying, we get 7 x 4, which is equal to 28. Add 1 to 28 to get 29, then subtract 6 from the 29 result and
the final answer is 23. Therefore, equation 35 ÷ 5 x 4 + 1 – 6 =23.
Disclaimer: The above information is for general information purposes only. All information on this website is provided in good faith, but we make no representations or warranties, express or
implied, as to the accuracy, adequacy, validity, reliability, availability or completeness of any information on this website.
Source: https://truongnguyenbinhkhiem.edu.vn
Category: Brain Teaser
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Feasibility of regular polytopes
15 Jan 2022
There are five regular convex Euclidean polyhedra. Collectively known as the “Platonic solids,” these are:
• the tetrahedron {3, 3}
• the cube {4, 3}
• the octahedron {3, 4}
• the dodecahedron {5, 3}
• the icosahedron {3, 5}
The platonic solids can be identified by their Schläfli symbols, listed above in curly brackets. A Schläfli symbol {a, b} represents the polyhedron formed by attaching b a-gons at each vertex. For
example, the dodecahedron is written {5, 3} because each vertex has three pentagons, and the octahedron is written {3, 4} because each vertex has four triangles.
To be a Platonic solid, a polyhedron must be convex. If we remove this requirement, but maintain that the polyhedron cannot intersect itself, we obtain three more polyhedra, each with an infinite
number of faces. These are flat rather than convex, and are known as regular apeirohedra or regular tilings.
• the square tiling {4, 4}
• the triangular tiling {3, 6}
• the hexagonal tiling {6, 3}
What is it that makes these polyhedra flat and the Platonic solids convex? And why are there symbols such as {4, 5} that don’t work at all? Precisely, how can one determine from the Schläfli symbol
alone whether a polyhedron is 1) flat, 2) convex, or 3) impossible in Euclidean space?
I will define the feasibility of a polyhedron as D{a, b} = 1 - b/2 + b/a, where {a, b} is the Schläfli symbol of the polyhedron. For convex polyhedra, the output of D is positive. (For example, the
tetrahedron has a feasibility of 1/2.) For flat polyhedra, the value is 0. For impossible polyhedra such as {6, 4}, it is negative.
It isn’t difficult to show that these statements are true. Consider what happens when we try to fit some number of polygons around a vertex. If there is room left over (i.e. the angles of the
polygons add to less than 2π), the resulting polyhedron will be convex. If the angles add to exactly 2π, the polyhedron will be flat. If the angles add to more than 2π, then it is impossible to fit
that number of polygons around a single vertex, and the polyhedron cannot exist in Euclidean space. For a regular polygon with a edges, the angles are given by (a - 2) π/a. In a polyhedron {a, b},
there are b faces around each vertex, so the sum of the angles is obtained by multiplying the angle of a single polygon by b: (a - 2) πb/a = πb - 2πb/a. We then divide by 2π to get the total fraction
of the circle taken up by the polygons: b/2 - b/a. Finally, we subtract this from 1 to get the fraction not taken up by the polygons, i.e. the fraction of the circle left over after placing b a-gons
around a vertex: 1 - b/2 + b/a. This is the feasibility of the polygon. From the derivation of the formula, one can check that if the angle taken up by the polygons is 2π, the end result will be
zero, and if the angle is greater than 2π, the result will be negative.
If we plot the equation D{x, y} = 0 on the coordinate plane, the result is a hyperbolic curve. Every lattice point on the curve corresponds to a regular tiling, and every lattice point below the
curve (with both coordinates greater than 2) corresponds to one of the five Platonic solids. This is a neat graphical explanation of why there are exactly three regular tilings, and five Platonic
Points of the form (2, n) and (n, 2) are also located below the curve. When considered as solids in Euclidean space, these Schläfli symbols make no sense; it is impossible to have a 2-sided polygon.
But if you analyse convex polyhedra as tilings of the sphere, then symbols containing a “2” are perfectly valid; they represent the hosohedra and the dihedra respectively. This interpretation is
somewhat cleaner: lattice points on the curve represent tilings of the plane, and lattice points below the curve represent tilings of the sphere.
The concept of “feasibility” can be extended to higher dimensions. For a Schläfli symbol of dimension n, the feasibility is defined as:
\[D\{a_0, a_1, \ldots a_{n-1}, a_n\} = 1 - a_n \cdot \frac{\theta\{a_0, a_1, \ldots a_{n-1}\}}{2\pi}\]
where θ{a[0], a[1], … a[n-1]} represents the dihedral angle of each facet.
The exact formula for θ becomes increasingly complex in higher dimensions:
\[\begin{align*} & \theta\{a\} = (a - 2)\frac{\pi}{a} = 2\sin^{-1}\left( \cos\left(\frac{\pi}{a}\right) \right) ^\ast \\ & \theta\{a, b\} = 2\sin^{-1}\left( \frac{\cos(\frac{\pi}{b})}{\sin(\frac{\pi}
{a})} \right) \\ & \theta\{a, b, c\} = 2\sin^{-1}\left( \cos\left(\frac{\pi}{c}\right) \sin\left(\frac{\pi}{a}\right) \sqrt{\frac{1}{\sin(\frac{\pi}{b})^2 - \cos(\frac{\pi}{a})^2}} \right) ^{\ast\
ast} \end{align*}\]
* The equation (a - 2) π/a = 2 sin^-1(cos(π/a)) is true for a ≥ 1. This means we can write all three formulas as 2 sin^-1(Y), where Y is a function of π/a, π/b, and so on.
It might be possible to find a general recursive rule for θ using the techniques of linear algebra. This would allow you to calculate the feasibility of any regular polytope in any dimension from the
Schläfli symbol alone.
Finally, here is a table of feasibilities for regular polytopes in three, four, and five dimensions:
Schläfli symbol Name Feasibility
{3, 3} tetrahedron 1/2
{4, 3} cube 1/4
{3, 4} octahedron 1/3
{5, 3} dodecahedron 1/10
{3, 5} icosahedron 1/6
{3, 6} triangular tiling 0
{4, 4} square tiling 0
{6, 3} hexagonal tiling 0
{3, 3, 3} 5-cell 0.41226
{4, 3, 3} tesseract 1/4
{3, 3, 4} 16-cell 0.21635
{3, 4, 3} 24-cell 0.08774
{5, 3, 3} 120-cell 0.02862
{3, 3, 5} 600-cell 0.02043
{4, 3, 4} cubic honeycomb 0
{3, 3, 3, 3} 5-simplex 0.37064
{4, 3, 3, 3} 5-cube 1/4
{3, 3, 3, 4} 5-orthoplex 0.16086
{4, 3, 3, 4} tesseractic honeycomb 0
{3, 3, 4, 3} 16-cell honeycomb 0
{3, 4, 3, 3} 24-cell honeycomb 0 | {"url":"https://owenbechtel.com/blog/feasibility-of-regular-polytopes/","timestamp":"2024-11-01T22:55:59Z","content_type":"text/html","content_length":"14324","record_id":"<urn:uuid:d19e393e-1273-440d-ba47-2fc2567a0169>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00800.warc.gz"} |
Classes and functions for the construction of strain design MILPs
This module contains functions that help construct mixed-integer linear problems, mainly functions that facilitate the construction of LP and Farkas dual problems from linear problems of the type
A_ineq*x<=b_ineq, A_eq*x<=b_eq, lb<=x<=ub. The functions also help keeping track of the relationship of constraints and variables and their individual counterparts in dual problems, which is
essential when simulating knockouts in dual problems. Most of the time, the sparse datatype is used to store and edit matrices for improved speed and memory.
Module Contents
class straindesign.strainDesignProblem.ContMILP(A_ineq, b_ineq, A_eq, b_eq, lb, ub, c, z_map_constr_ineq, z_map_constr_eq, z_map_vars)[source]
Continuous representation of the strain design MILP.
This MILP can be used to verify computation results. Since this class also stores the relationship between intervention variables z and corresponding (in)equality constraints and variables in the
problem, it can be used to verify computed designs quickly and in a numerically stable manner.
class straindesign.strainDesignProblem.SDProblem(model: cobra.Model, sd_modules: List[straindesign.SDModule], *args, **kwargs)[source]
Strain design MILP
The construcor of this class translates a model and strain design modules into a mixed integer linear problem. This class, however, is the backbone of the strain design computation. Preprocessing
steps that enable gene, reaction and regulatory interventions, or network compression usually preceed the construction of an SDProblem-object and are integrated in the function
☆ model (cobra.Model) – A metabolic model that is an instance of the cobra.Model class.
☆ sd_modules ((list of) straindesign.SDModule) – Modules that specify the strain design problem, e.g., protected or suppressed flux states for MCS strain design or inner and outer objective
functions for OptKnock. See description of SDModule for more information on how to set up modules.
☆ ko_cost (optional (dict)) – (Default: None) A dictionary of reaction identifiers and their associated knockout costs. If not specified, all reactions are treated as knockout candidates,
equivalent to ko_cost = {‘r1’:1, ‘r2’:1, …}. If a subset of reactions is listed in the dict, all other are not considered as knockout candidates.
☆ ki_cost (optional (dict)) – (Default: None) A dictionary of reaction identifiers and their associated costs for addition. If not specified, all reactions are treated as knockout
candidates. Reaction addition candidates must be present in the original model with the intended flux boundaries after insertion. Additions are treated adversely to knockouts, meaning
that their exclusion from the network is not associated with any cost while their presence entails intervention costs.
☆ max_cost (optional (int)) – (Default: inf): The maximum cost threshold for interventions. Every possible intervention is associated with a cost value (1, by default). Strain designs
cannot exceed the max_cost threshold. Individual intervention cost factors may be defined through ki_cost and ko_cost.
☆ solver (optional (str)) – (Default: same as defined in model / COBRApy) The solver that should be used for preparing and carrying out the strain design computation. Allowed values are
‘cplex’, ‘gurobi’, ‘scip’ and ‘glpk’.
☆ M (optional (int)) – (Default: None) If this value is specified (and non-zero, not None), the computation uses the big-M method instead of indicator constraints. Since GLPK does not
support indicator constraints it uses the big-M method by default (with COBRA standard M=1000). M should be chosen ‘sufficiently large’ to avoid computational artifacts and ‘sufficiently
small’ to avoid numerical issues.
☆ essential_kis (optional (set)) – A set of reactions that are marked as addable and that are essential for at least one of the strain design modules. Providing such “essential knock-ins”
may speed up the strain design computation.
An instance of SDProblem containing the strain design MILP
Return type:
Generate module LP and z-linking-matrix for each module and add them to the strain design MILP
sd_module (straindesign.SDModule) – Modules to describe strain design problems like protected or suppressed flux states for MCS strain design or inner and outer objective functions for
OptKnock. See description of SDModule for more information on how to set up modules.
Connect binary intervention variables to variables and constraints of the strain design problem
Function that uses the maps between intervention indicators z and variables and constraints of the linear strain design (in)equality system (self.z_map_constr_ineq, self.z_map_constr_eq and
self.z_map_vars) to set up the strain design MILP.
MILP construction uses the following steps:
1. Translate equality-KOs/KIs to two inequality-KOs/KIs
2. Translate variable-KOs/KIs to inequality-KIs/KOs
3. Try to bound the problem with LPs
4. Use LP-determined bounds to link z-variables, where such bounds were found
(5) Translate remaining inequalities back to equalities when possible and link z via indicator constraints. If necessary, the solver interface will translate them to big-M constraints. (6)
Remove redundant equalities from static problem
straindesign.strainDesignProblem.LP_dualize(A_ineq_p, b_ineq_p, A_eq_p, b_eq_p, lb_p, ub_p, c_p, z_map_constr_ineq_p=None, z_map_constr_eq_p=None, z_map_vars_p=None) Tuple[scipy.sparse.csr_matrix,
Tuple, scipy.sparse.csr_matrix, Tuple, Tuple, Tuple, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix][source]
Translate a primal system to its LP dual system
The primal system must be given in the standard form: A_ineq x <= b_ineq, A_eq x = b_eq, lb <= x < ub, min{c’x}. The LP duality theorem defines a set of two problems. If one of the LPs is a
maximization and and optimum exists, the optimal value of this LP is identical to the minimal optimum of its LP dual problem. LP duality can be used for nested optimization, since solving the
primal and the LP dual problem, while enfocing equality of the objective value, guarantees optimality.
Construction of the LP dual:
Variables translate to constraints:
x={R} -> = x>=0 -> >= (new constraint is multiplied with -1 to translate to <= e.g. -A_i’ y <= -c_i) x<=0 -> <=
Constraints translate to variables:
= -> y={R} <= -> y>=0
☆ A_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ b_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ A_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ b_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ lb_p (list of float) – Upper and lower variable bounds in vector form.
☆ ub_p (list of float) – Upper and lower variable bounds in vector form.
☆ c_p (list of float) – The objective coefficient vector of the primal minimization-LP. z_map_constr_ineq_p, z_map_constr_eq_p, z_map_vars_p
☆ z_map_constr_ineq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or
other (in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without
returning information about these relationships.
☆ z_map_constr_eq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other
(in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without
returning information about these relationships.
☆ z_map_vars (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other (in)
equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or constraint
(j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality constraint j, there
is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without returning information
about these relationships.
(Tuple): The LP dual of the problem in the format: A_ineq, b_ineq, A_eq, b_eq, c, lb, ub and optionally also z_map_constr_ineq, z_map_constr_eq, z_map_vars
straindesign.strainDesignProblem.build_primal_from_cbm(model, V_ineq=None, v_ineq=None, V_eq=None, v_eq=None, c=None) Tuple[scipy.sparse.csr_matrix, Tuple, scipy.sparse.csr_matrix, Tuple, Tuple,
Tuple, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix][source]
Builds primal LP from constraint-based model and (optionally) additional constraints.
standard form: A_ineq x <= b_ineq, A_eq x = b_eq, lb <= x <= ub, min{c’x}. Additionally, this function also returns a set of matrices that associate each variable (and constraint) with reactions.
In the primal problems all variables correspond to reactions (z), therefore, the z_map_vars matrix is an identity matrix. The constraints correspond to metabolites, thus z_map_constr_ineq,
z_map_constr_eq are all-zero.
A_ineq, b_ineq, A_eq, b_eq, lb, ub, c, z_map_constr_ineq, z_map_constr_eq, z_map_vars. A constraint-based steady-state model in the form of a linear (in)equality system. The matrices
z_map_constr_ineq, z_map_constr_eq, z_map_vars contain the association between reactions and different parts of the LP, such as reactions, metabolites or other (in)equalities.
Return type:
straindesign.strainDesignProblem.farkas_dualize(A_ineq_p, b_ineq_p, A_eq_p, b_eq_p, lb_p, ub_p, z_map_constr_ineq_p=None, z_map_constr_eq_p=None, z_map_vars_p=None) Tuple[scipy.sparse.csr_matrix,
Tuple, scipy.sparse.csr_matrix, Tuple, Tuple, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix][source]
Translate a primal system of linear (in)equality to its Farkas dual
The primal system must be given in the standard form: A_ineq x <= b_ineq, A_eq x = b_eq, lb <= x < ub. Farkas’ lemma defines a set of two systems of linear (in)equalities of which exactly one is
feasible. Since the feasibility of one is a certificate for the infeasibility of the other one, this theorem can be used to set up problems that imply the infeasibility and thus exclusion of a
certain subspace. This priciple is used for MCS calculation (the SUPPRESS module).
Consider that the following is not implemented: In the case of (1) A x = b, (2) x={R}, (3) b~=0, Farkas’ lemma is special, because b’y ~= 0 is required to make the primal infeasible instead of
b’y < 0. 1. This does not occur very often. 2. Splitting the equality into two inequalities that translate to y>=0 would be posible, and yield b’y < 0 in the farkas’ lemma. Maybe splitting is
required, but I actually don’t think so. Using the special case of b’y < 0 for b’y ~= 0 should be enough.
☆ A_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ b_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ A_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ b_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ lb_p (list of float) – Upper and lower variable bounds in vector form.
☆ ub_p (list of float) – Upper and lower variable bounds in vector form.
☆ z_map_constr_ineq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or
other (in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without
returning information about these relationships.
☆ z_map_constr_eq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other
(in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without
returning information about these relationships.
☆ z_map_vars (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other (in)
equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or constraint
(j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality constraint j, there
is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated for the dualized LP, if not, the dual problem is constructed without returning information
about these relationships.
(Tuple): The Farkas dual of the linear (in)equality system in the format: A_ineq, b_ineq, A_eq, b_eq, lb, ub and optionally also z_map_constr_ineq, z_map_constr_eq, z_map_vars
straindesign.strainDesignProblem.prevent_boundary_knockouts(A_ineq, b_ineq, lb, ub, z_map_constr_ineq, z_map_vars) Tuple[scipy.sparse.csr_matrix, Tuple, Tuple, Tuple, scipy.sparse.csr_matrix][source]
Put negative lower bounds and positive upper bounds into (notknockable) inequalities
This is a helper function that puts negative lower bounds and positive upper bounds into (not-knockable) inequalities. Later on, one may simulate the knockouts by multiplying the upper and lower
bounds with a binary variable z. This functions prevents that
☆ A_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ b_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ lb_p (list of float) – Upper and lower variable bounds in vector form.
☆ ub_p (list of float) – Upper and lower variable bounds in vector form.
☆ z_map_constr_ineq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or
other (in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated. Otherwise, all reactions are assumed to be knockable and thus all
negative upper and positive lower bounds are translated into constraints.
☆ z_map_vars (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other (in)
equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or constraint
(j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality constraint j, there
is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated. Otherwise, all reactions are assumed to be knockable and thus all negative upper and
positive lower bounds are translated into constraints.
(Tuple): A linear (in)equality system in the format: A_ineq, b_ineq, A_eq, b_eq, lb, ub and optionally also updated z_map_constr_ineq, z_map_constr_eq
straindesign.strainDesignProblem.reassign_lb_ub_from_ineq(A_ineq, b_ineq, A_eq, b_eq, lb, ub, z_map_constr_ineq=None, z_map_constr_eq=None, z_map_vars=None) Tuple[scipy.sparse.csr_matrix, Tuple,
scipy.sparse.csr_matrix, Tuple, Tuple, Tuple, scipy.sparse.csr_matrix, scipy.sparse.csr_matrix][source]
Remove single constraints in A_ineq or A_eq in favor of lower and upper bounds on variables
Constraints on single variables instead translated into lower and upper bounds (lb, ub). This is useful to filter out redundant bounds on variables and keep the (in)equality system concise. To
avoid interference with the knock-out logic, negative upper bounds and positive lower bounds are not put into lb and ub, when reactions are flagged knockable with z_map_vars.
☆ A_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ b_ineq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear inequalities of the primal LP A_ineq_p*x <= b_ineq_p
☆ A_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ b_eq_p (sparse.csr_matrix and list of float) – A coefficient matrix and a vector that describe the linear equalities of the primal LP A_eq_p*x <= b_eq_p
☆ lb_p (list of float) – Upper and lower variable bounds in vector form.
☆ ub_p (list of float) – Upper and lower variable bounds in vector form.
☆ z_map_constr_ineq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or
other (in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated. Otherwise, all reactions are assumed to be notknockable and thus all
constraints on single variables put into lb and ub.
☆ z_map_constr_eq (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other
(in)equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or
constraint (j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality
constraint j, there is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated. Otherwise, all reactions are assumed to be notknockable and thus all
constraints on single variables put into lb and ub.
☆ z_map_vars (optional (sparse.csr_matrix)) – Matrices that contain the relationship between metabolic reactions and different parts of the LP, such as reactions, metabolites or other (in)
equalities. These matrices help keeping track of the parts of the LP that are affected by reaction knockouts and additions. When a reaction (i) knockout removes the variable or constraint
(j), the respective matrix contains a coefficient 1 at this position. -1 marks additions. E.g.: If the knockout of reaction i corresponds to the removal of inequality constraint j, there
is a matrix entry z_map_constr_ineq_(i,j) = 1. If these matrices are provided, they are updated. Otherwise, all reactions are assumed to be notknockable and thus all constraints on single
variables put into lb and ub.
(Tuple): A linear (in)equality system in the format: A_ineq, b_ineq, A_eq, b_eq, lb, ub and optionally also updated z_map_constr_ineq, z_map_constr_eq
straindesign.strainDesignProblem.worker_compute(i) Tuple[int, float][source]
Helper function for determining bounds on linear expressions
straindesign.strainDesignProblem.worker_init(A, A_ineq, b_ineq, A_eq, b_eq, lb, ub, solver, seed)[source]
Helper function for determining bounds on linear expressions | {"url":"https://straindesign.readthedocs.io/en/latest/autoapi/straindesign/strainDesignProblem/index.html","timestamp":"2024-11-10T11:25:41Z","content_type":"text/html","content_length":"66295","record_id":"<urn:uuid:55224521-eadc-4eaa-b73f-5224644faa6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00298.warc.gz"} |
Re: Tips for writing correct, non trivial Mathematica Libraries
• To: mathgroup at smc.vnet.net
• Subject: [mg124489] Re: Tips for writing correct, non trivial Mathematica Libraries
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Sat, 21 Jan 2012 05:18:21 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
On 1/20/12 at 1:46 AM, nehal.alum at gmail.com (l.i.b.) wrote:
>Thanks for your comments. My followups are inlined below:
>>>** Take for example the following: (Taken from the mathematica 8
>>>virtual book section "Applying Functions to Lists and Other
>>>Expressions") geom[list_] := Apply[Times, list]^(1/Length[list])
>>>So, this does a bad thing for geom[ x+ y] (returns (Sqrt[x y])
>>What were you expecting here? This looks correct to me
>the snippet for 'geom' is taken for the mathematica tutorial, and is
>meant to show the user how to define a function that implements the
>geometric mean. so for instance, geom[{x,y,z}] = (x y z
>)^(1/3). But geom[x+y] treats the expression as a list and
>returns Sqrt[x y],
>which is not what a new user would expect ( a better answer would be
If you want geom to return the geometric mean when either a list
or when given multiple arguments separated by commas try this:
In[8]:= geom[x_List] := (Times @@ x)^(1/Length[x])
In[9]:= geom[x__] := Times[x]^(1/Length[{x}])
In[10]:= geom[{x, y, z}]
Out[10]= (x*y*z)^(1/3)
In[11]:= geom[x, y, z]
Out[11]= (x*y*z)^(1/3)
>>>The built-in function GeometricMean properly complains when given
>>>this input,
>I'm saying GeometricMean does not get fooled by GeometricMean[x+y],
>which means the author thought of how to protect it from doing the
>wrong thing -- and it would be nice to see exactly what syntax they
Syntax for any built-in function is generally available by doing
That is doing ?GeometricMean returns something that immediately
tells you the intended argument is a list.
More information can be found in the Document Center. On the
Mac, selecting name and doing cmd-shift-f brings up the document
page for name. Or on Windows use F1.
>>If you want to look at examples of well written Mathematica
>>packages, a good place to start is to look at the packages that
>>ship with Mathematica.
>Actually that's exactly what I've started to do, once I nailed down
>the location of the bundled packages. Though I've yet to find the
>definition for GeometricMean in the install directory -- presumable
>it is a native command.
It is a built-in function. For anything defined in Mathematica
doing Context[name] will give the context the definition for
name appears in.
telling you it is a built-in symbol. Further, paying attention
to syntax coloring will tell you a lot. That is by default the
color will change from blue (an undefined symbol) to black (a
defined symbol) when the final letter of the symbol name is
typed. Anytime this happens you know what you typed must be
defined. So, if you haven't defined it yourself or loaded a
package, it must be a built-in symbol. | {"url":"http://forums.wolfram.com/mathgroup/archive/2012/Jan/msg00572.html","timestamp":"2024-11-10T05:18:35Z","content_type":"text/html","content_length":"32961","record_id":"<urn:uuid:d2863075-fe44-4a17-aed6-aa5d7ee40b18>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00366.warc.gz"} |
Inequalities From Sketches
Topic summary
When using sketches to find inequalities, the goal is to determine the ranges of values for which the function is greater than or less than zero (or another value), based on its graphical
Solve Inequalities From Sketches
Identify Key Points: Look at the graph to identify important points such as roots (where the graph crosses or touches the x-axis), maximum or minimum points, and the behaviour of the curve.
Analyze the Sign of the Function: Determine where the function is positive (above the x-axis) and where it is negative (below the x-axis). These regions help define the inequality.
Express the Inequality: Based on the graph, express the inequality using the appropriate symbols. | {"url":"https://www.onmaths.com/resource/inequalities-from-sketches/","timestamp":"2024-11-05T01:18:49Z","content_type":"text/html","content_length":"82978","record_id":"<urn:uuid:a3939091-eaa6-4009-9faf-7efa577fecd0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00744.warc.gz"} |
Transactions Online
Shangce GAO, Qiping CAO, Catherine VAIRAPPAN, Jianchen ZHANG, Zheng TANG, "An Improved Local Search Learning Method for Multiple-Valued Logic Network Minimization with Bi-objectives" in IEICE
TRANSACTIONS on Fundamentals, vol. E92-A, no. 2, pp. 594-603, February 2009, doi: 10.1587/transfun.E92.A.594.
Abstract: This paper describes an improved local search method for synthesizing arbitrary Multiple-Valued Logic (MVL) function. In our approach, the MVL function is mapped from its algebraic
presentation (sum-of-products form) on a multiple-layered network based on the functional completeness property. The output of the network is evaluated based on two metrics of correctness and
optimality. A local search embedded with chaotic dynamics is utilized to train the network in order to minimize the MVL functions. With the characteristics of pseudo-randomness, ergodicity and
irregularity, both the search sequence and solution neighbourhood generated by chaotic variables enables the system to avoid local minimum settling and improves the solution quality. Simulation
results based on 2-variable 4-valued MVL functions and some other large instances also show that the improved local search learning algorithm outperforms the traditional methods in terms of the
correctness and the average number of product terms required to realize a given MVL function.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E92.A.594/_p
author={Shangce GAO, Qiping CAO, Catherine VAIRAPPAN, Jianchen ZHANG, Zheng TANG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={An Improved Local Search Learning Method for Multiple-Valued Logic Network Minimization with Bi-objectives},
abstract={This paper describes an improved local search method for synthesizing arbitrary Multiple-Valued Logic (MVL) function. In our approach, the MVL function is mapped from its algebraic
presentation (sum-of-products form) on a multiple-layered network based on the functional completeness property. The output of the network is evaluated based on two metrics of correctness and
optimality. A local search embedded with chaotic dynamics is utilized to train the network in order to minimize the MVL functions. With the characteristics of pseudo-randomness, ergodicity and
irregularity, both the search sequence and solution neighbourhood generated by chaotic variables enables the system to avoid local minimum settling and improves the solution quality. Simulation
results based on 2-variable 4-valued MVL functions and some other large instances also show that the improved local search learning algorithm outperforms the traditional methods in terms of the
correctness and the average number of product terms required to realize a given MVL function.},
TY - JOUR
TI - An Improved Local Search Learning Method for Multiple-Valued Logic Network Minimization with Bi-objectives
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 594
EP - 603
AU - Shangce GAO
AU - Qiping CAO
AU - Catherine VAIRAPPAN
AU - Jianchen ZHANG
AU - Zheng TANG
PY - 2009
DO - 10.1587/transfun.E92.A.594
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E92-A
IS - 2
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - February 2009
AB - This paper describes an improved local search method for synthesizing arbitrary Multiple-Valued Logic (MVL) function. In our approach, the MVL function is mapped from its algebraic presentation
(sum-of-products form) on a multiple-layered network based on the functional completeness property. The output of the network is evaluated based on two metrics of correctness and optimality. A local
search embedded with chaotic dynamics is utilized to train the network in order to minimize the MVL functions. With the characteristics of pseudo-randomness, ergodicity and irregularity, both the
search sequence and solution neighbourhood generated by chaotic variables enables the system to avoid local minimum settling and improves the solution quality. Simulation results based on 2-variable
4-valued MVL functions and some other large instances also show that the improved local search learning algorithm outperforms the traditional methods in terms of the correctness and the average
number of product terms required to realize a given MVL function.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E92.A.594/_p","timestamp":"2024-11-06T23:48:47Z","content_type":"text/html","content_length":"63365","record_id":"<urn:uuid:0556fc34-95dc-4667-ab36-8f11753abdd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00887.warc.gz"} |
Course 2022-2023 a.y. - Universita' Bocconi
30591 - ELEMENTS OF REAL AND COMPLEX ANALYSIS
Department of Decision Sciences
Course taught in English
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05) -
(6 credits - II sem. - OP | MAT/05)
Suggested background knowledge
A sound knowledge of the main tools of calculus (limits, series, derivatives, integrals, also in the multivariable case) and of linear algebra, basic concepts of topology.
Mission & Content Summary
The course will provide the basic foundations of Complex and Fourier Analysis, introducing some fundamental tools for signal theory. Complex analysis deals with complex functions of a complex
variable and enlights remarkable links between complex differentiability, power series in the complex plane, line integral representations, conformal maps, and harmonic functions. Its main results
are particularly important to understand power series (which lie at the core of functional calculus for linear operators), to capture the geometric properties of conformal transformations in the
plane, to obtain integral representations of harmonic functions, and to understand the Laplace and the discrete Zeta and Fourier transforms. Fourier analysis is one of the most powerful tools to
analyze functions and it is the basic building block of signal theory. Starting from Fourier series dealing with periodic signals, which can be interpreted as an orthonormal decomposition in Hilbert
spaces, its range of applications is considerably expanded by the Fourier and the Laplace transforms, which cover the case of general signals. The course is meant to round up an adequate
undergraduate preparation in Mathematical Analysis and to give students a hint on more advanced issues, that find surprising and remarkable applications in several theoretical and applied fields.
The course is divided into two parts.
After a quick review of complex numbers, the first part of the course will concern with:
- holomorphic functions,
- power series expansions in the complex field,
- line integrals,
- index/winding number of a curve about a point,
- singularities and residuals, Laurent expansions
- the Theorems of Riemann and Cauchy, the residue Theorem,
- applications*: the argument and the maximum principles, the fundamental theorem of Algebra, the Zeta transform.
After a brief and informal recap of some properties of Hilbert spaces and the Lebesgue space L2, the second part of the course will be devoted to
- Fourier series in the framework of orthogonal systems of signals,
- properties and convergence of Fourier series,
- Fourier transform,
- applications*: Laplace transform, solutions to differential equations, Poisson summation formula and sampling theorem, the Heisenberg uncertainty principle.
(The choice of starred* topics will be adjusted according to the time available)
Intended Learning Outcomes (ILO)
At the end of the course student will be able to...
Understand the basic facts, tools, and techniques of complex analysis: recognize holomorphic functions and their singularities, identify power series and their convergence, describe their main
properties, recognize closed circuits and their winding numbers in the complex plane, explain the scope of the basic integral theorems and representation formulae.
Understand the basic structure of Hilbert spaces, the use of scalar products, and of orthonormal systems and expansions.
Know the basic properties of periodic signals and the trigonometric basis, understand the meaning of Fourier expansion, reproduce basic series expansions, and describe their convergence properties.
Understand Fourier transform, its inversion, and its link with Fourier series and Laplace transform.
At the end of the course student will be able to...
Manipulate simple power series in the complex plane.
Analyze singularities, compute residuals, and evaluate simple curvilinear integrals.
Use the power series expansions of fundamental functions. Use rational functions.
Solve simple exercises on holomorphic functions.
Compute Fourier series expansions of simple signals. Use Fourier series expansions for solving differential equations with periodic solutions. Estimate the behavior of the Fourier coefficients.
Compute the Fourier/Laplace transform of simple signals. Interpret the Fourier transform and its behavior. Use simple relations between a function and its Fourier/Laplace transform.
Teaching methods
• Face-to-face lectures
• Online lectures
• Exercises (exercises, database, software etc.)
Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face-to-face lectures and online lectures will mainly depend on external constraints.
Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.
Assessment methods
Continuous assessment Partial exams General exam
• Written individual exam (traditional/online) x x
Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways.
The exam can be split in two partial exams. Each partial may contain multiple-choice questions and/or open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice
questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer
questions mainly aim at evaluating:
• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
• The ability to apply mathematical notions to the solution of more complex problems and exercises.
The exam can also be taken as a single general exam, which may contain multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be
taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type
of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical
notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating:
• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
• The ability to apply mathematical notions to the solution of more complex problems and exercises.
We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.
Teaching materials
Steven Krantz: A guide to complex variables
The Mathematical Association of America, 2008
Henri Cartan: Elementary theory of analytic functions of one or several complex variables
Dover 1995
Elias Stein, Rami Shakarchi: Fourier Analysis, an introduction
Princeton University Press, 2002
Pierre Bremaud: Mathematical Principle of Signal Processing
Springer, 2002
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all in one ball mill pdf
WEBThis set of Mechanical Operations Multiple Choice Questions Answers (MCQs) focuses on "Ball Mill". 1. What is the average particle size of ultrafine grinders? a) 1 to 20 µm. b) 4 to 10 µm. c)
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WEBA ball mill is a crucial piece of machinery used in grinding and mixing materials in various industries. It works by rotating a cylinder with steel or ceramic balls, causing the balls to fall
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/radial runout. of the drive trains. power splitting. distances variable. load distribution. of the girth gear. gear is through hardened only, fatigue strength is limited. Dynamic behaviour. A
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WEBThis is about h per mill revolution. 5. Effect of rotation rate for the standard ball mill Charge behaÕiour Fig. 1 shows typical charge shapes predicted for our 'standard' 5 m ball mill and
charge Ždescribed above. filled to 40% Žby volume. for four rotation rates that span the typical range of operational speeds.
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Journées de topologie quantique
Based on the theory of quantum dilogarithms over locally compact Abelian groups, I will talk about a particular example of a quantum dilogarithm associated with a local field FFF which leads to a
generalized 3d TQFT based on the combinatorial input of ordered Δ\DeltaΔ-complexes. The associated invariants of 3-manifolds are expected to be specific counting invariants of representations of π1\
pi_1π1 into the group PSL2FPSL_2FPSL2F. This is an ongoing project in collaboration with Stavros Garoufalidis.
I will start by reviewing deformation quantisation of algebras, and explain how we in a similar spirit can define deformation quantisation of categories. The motivation is to understand how
deformation quantisation interacts with categorical factorization homology, or more explicitly: how deformation quantisation interacts with “gluing” local observables to obtain global observables.
One important and well-known example of factorization homology is given by skein categories, which I will briefly introduce. We generalise the theory of skein categories to fit into the deformation
quantisation-setting, and use it as a running example. This is based on joint work (in progress) with Corina Keller, Lukas Müller and Jan Pulmann.
Skein modules are invariants of 3-manifolds which were introduced by Józef H. Przytycki (and independently by Vladimir Tuarev) in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket
polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology, having strong
ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. One avenue in the study of skein modules
is determining whether they reflect the geometry or topology of the manifold, for example, whether the module detects the presence of incompressible or non-separating surfaces in the manifold.
Interestingly enough this presence manifests itself in the form of torsion in the skein module. In this talk we will discuss various skein modules which detect the presence of non-separating
surfaces. We will focus on the framing skein module and show that it detects the presence of non-separating 2-spheres in a 3-manifold by way of torsion. | {"url":"https://indico.math.cnrs.fr/event/9499/","timestamp":"2024-11-03T06:44:55Z","content_type":"text/html","content_length":"99452","record_id":"<urn:uuid:8be67835-fd18-40f6-b77d-6e38e8697e48>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00576.warc.gz"} |
What is a good field goal percentage in basketball?
What is a good field goal percentage in basketball?
In basketball, a FG% of . 500 (50%) or above is considered a good percentage, although this criterion does not apply equally to all positions. Guards usually have lower FG% than forwards and centers.
What is the average NBA player field goal percentage?
Overall, we estimate the league average field goal percentage to be 0.451, and the population standard deviation in true talent to be 0.0525.
Who has the highest FG %?
DeAndre Jordan
Rank Player FG%
1. DeAndre Jordan .6731
2. Rudy Gobert .6533
3. Clint Capela .6241
4. Montrezl Harrell .6199
Who has the lowest field goal percentage in the NBA?
John Mahnken has the worst career field-goal percentage, at 27.2 percent.
What is a good 3 point percentage in NBA?
KAT/Embiid/Cousins shooting between 33-36% is still good enough to space the floor even if it’s not EXACTLY what you want out of a high volume 3 point shooter. Great point. The average team in the
NBA averages about 1.1 points per true shot (i.e. including shooting fouls).
What is the average field goal percentage?
Heading into Tuesday night’s games, the field goal percentage league average sits at 44.6 — the lowest mark in the last four years through the first 102 games. The league average 3-point percentage
is also the lowest in the last four years through the first 102 games, sitting at 34.2 percent.
What is considered a good 3pt percentage?
KAT/Embiid/Cousins shooting between 33-36% is still good enough to space the floor even if it’s not EXACTLY what you want out of a high volume 3 point shooter. Great point. The average team in the
NBA averages about 1.1 points per true shot (i.e. including shooting fouls). A 35% 3pt shot only gets you 1.05 points.
What is a bad field goal percentage?
45% from the field, 33% from three, and 70% from the line is still acceptable. But a big man who only shoots at the rim is expected to make a better percentage than that, while someone who is a three
point specialist is expected to make 40%. If you are more versatile, you might be forgiven a lower percentage.
Has any NBA player shot 100%?
Wilt Chamberlain has scored the most points in a game with a field-goal percentage of 100.0, with 42 points against the Baltimore Bullets on February 24, 1967.
What percentage does Stephen Curry shooting from 3?
As a result, his season-long percentages are both at career-worst marks (aside from the 2019-20 season where he only played five games), shooting 41.8 percent from the field and 37.7 percent from 3.
To be clear, 37.3 percent from 3 is not bad by any means, especially on the volume he’s shooting.
What is the NBA average 3-point percentage?
The league average 3-point percentage is also the lowest in the last four years through the first 102 games, sitting at 34.2 percent.
What is field goal percentage in basketball?
Field goal percentage (FG%) is the percentage of successful shots (2 or 3 point) to attempted shots (2 or 3 point). It does not include free throw attempts or baskets. It is like “shooting
percentage” metrics found in Hockey or Soccer. FG% percentage is a performance statistic. It can help show a player’s ability independent of pure scoring.
What is a field goal attempt in basketball?
A “ Field Goal Attempt ” is an attempt at a basket that is not a free throw. A player has made 514 field goal attempts, and in that time has made a total of 261 field goals (both 2 and 3 points) FG%
= 0.5078 Therefore, this player has an FG% of .5078, or 50.78%.
What is effective field goal percentage (EFG%)?
Traditional field goal percentage counts all shots the same — it’s simply shots made divided by shots attempted. EFG% solves for that problem. The thesis behind effective field goal percentage has
really been the driver behind the NBA’s 3-point revolution. | {"url":"https://www.tonyajoy.com/2022/08/09/what-is-a-good-field-goal-percentage-in-basketball/","timestamp":"2024-11-08T07:55:18Z","content_type":"text/html","content_length":"51462","record_id":"<urn:uuid:59895f45-0e3d-42c4-80c7-ac4ab4af459e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00412.warc.gz"} |
Difference of Squares
How Do I Calculate the Difference between Two Squared numbers?
By Robert O
The difference of the two squares formula is useful when factoring quadratic expressions. In general form, a difference of two squares is (a² – b²). How does that come about?
If you are to expand (a – b)(a + b), the result is the difference of squares. Let us see how we get there.
This formula is very helpful in factoring expressions. Provided you can recognize that an expression involves the difference of two squares, you can apply the formula directly without even stating
it. Your examiner already knows it.
How to factor polynomials using the difference of two squares formula?
For the formula to work, there must be a subtraction (-) separating the two terms. Any other operation apart from subtraction is not applicable. For instance, (a² + b²) is not a difference between
the two squares. It may be a sum of two squares, but that does not exist.
We will use examples to help us in understanding this concept.
Example 1:
Factor x² – 4
This example is the simplest form of a difference of two squares. The first step is to confirm if it is indeed a difference between the two squares. That may not be obvious in some cases. So, you may
need to first pull out a common factor before proceeding with the identification. In this case, we have only 1 to factor out, which makes no difference.
After confirming that it is a difference between the two squares, we check the numbers that make up the squares. The first term is x² and the second term is 4. Find the square root of each term
independently. In this case, our square roots are x and 2. Note that if you cannot find a definite square root of individual terms, then that is not a difference between two squares.
Applying the general formula:
Example 2:
Factor 2x² – 98
A quick look at the problem tells you that it is not a difference of two squares. Do not fall into this trap. There is a common factor to pull out, and that is 2.
The terms in the bracket form a difference of two squares.
Example 3:
Factor 32 – 8x²
Does this look like a difference of two squares? Let us factor out 8 and see.
We now have a difference of two squares in between the braces.
The difference between the two squares formula makes it easier to factorize polynomials. This formula is also applicable in simplifying complex algebraic fractions plus many other applications. The
good news is that there is nothing complicated about it, and you can easily derive it when you need to use it.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and
assumed various leadership roles. He writes for Full Potential Learning Academy.
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Femtometers to Caliber Converter
β Switch toCaliber to Femtometers Converter
How to use this Femtometers to Caliber Converter π €
Follow these steps to convert given length from the units of Femtometers to the units of Caliber.
1. Enter the input Femtometers value in the text field.
2. The calculator converts the given Femtometers into Caliber in realtime β using the conversion formula, and displays under the Caliber label. You do not need to click any button. If the input
changes, Caliber value is re-calculated, just like that.
3. You may copy the resulting Caliber value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Femtometers to Caliber?
The formula to convert given length from Femtometers to Caliber is:
Length[(Caliber)] = Length[(Femtometers)] / 254000000001.016
Substitute the given value of length in femtometers, i.e., Length[(Femtometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in caliber, i.e.,
Calculation will be done after you enter a valid input.
Consider that the radius of a proton is about 0.84 femtometers.
Convert this radius from femtometers to Caliber.
The length in femtometers is:
Length[(Femtometers)] = 0.84
The formula to convert length from femtometers to caliber is:
Length[(Caliber)] = Length[(Femtometers)] / 254000000001.016
Substitute given weight Length[(Femtometers)] = 0.84 in the above formula.
Length[(Caliber)] = 0.84 / 254000000001.016
Length[(Caliber)] = 3.3071e-12
Final Answer:
Therefore, 0.84 fm is equal to 3.3071e-12 cl.
The length is 3.3071e-12 cl, in caliber.
Consider that the size of a neutron is approximately 1.1 femtometers.
Convert this size from femtometers to Caliber.
The length in femtometers is:
Length[(Femtometers)] = 1.1
The formula to convert length from femtometers to caliber is:
Length[(Caliber)] = Length[(Femtometers)] / 254000000001.016
Substitute given weight Length[(Femtometers)] = 1.1 in the above formula.
Length[(Caliber)] = 1.1 / 254000000001.016
Length[(Caliber)] = 4.3307e-12
Final Answer:
Therefore, 1.1 fm is equal to 4.3307e-12 cl.
The length is 4.3307e-12 cl, in caliber.
Femtometers to Caliber Conversion Table
The following table gives some of the most used conversions from Femtometers to Caliber.
Femtometers (fm) Caliber (cl)
0 fm 0 cl
1 fm 0 cl
2 fm 1e-11 cl
3 fm 1e-11 cl
4 fm 2e-11 cl
5 fm 2e-11 cl
6 fm 2e-11 cl
7 fm 3e-11 cl
8 fm 3e-11 cl
9 fm 4e-11 cl
10 fm 4e-11 cl
20 fm 8e-11 cl
50 fm 2e-10 cl
100 fm 3.9e-10 cl
1000 fm 3.94e-9 cl
10000 fm 3.937e-8 cl
100000 fm 3.937e-7 cl
A femtometer (fm) is a unit of length in the International System of Units (SI). One femtometer is equivalent to 0.000000000001 meters or 1 Γ 10^(-15) meters.
The femtometer is defined as one quadrillionth of a meter, making it a very small unit of measurement used for measuring atomic and subatomic distances.
Femtometers are commonly used in nuclear physics and particle physics to describe the sizes of atomic nuclei and the ranges of fundamental forces at the subatomic level.
Caliber is a unit of length used to describe the diameter of a firearm's barrel or the internal diameter of a projectile. One caliber is equivalent to 1/100 of an inch or approximately 0.254
The caliber is used to specify the size of bullets, guns, and artillery, providing a standard measure for weaponry and ammunition. For example, a firearm with a caliber of .45 means the barrel's
diameter is 0.45 inches.
Calibers are commonly used in firearms and ammunition industries to standardize measurements and ensure compatibility of projectiles with weapons. The unit is crucial for defining the specifications
and performance of firearms and ammunition.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Femtometers to Caliber in Length?
The formula to convert Femtometers to Caliber in Length is:
Femtometers / 254000000001.016
2. Is this tool free or paid?
This Length conversion tool, which converts Femtometers to Caliber, is completely free to use.
3. How do I convert Length from Femtometers to Caliber?
To convert Length from Femtometers to Caliber, you can use the following formula:
Femtometers / 254000000001.016
For example, if you have a value in Femtometers, you substitute that value in place of Femtometers in the above formula, and solve the mathematical expression to get the equivalent value in Caliber. | {"url":"https://convertonline.org/unit/?convert=femtometers-calibers","timestamp":"2024-11-09T17:43:11Z","content_type":"text/html","content_length":"90543","record_id":"<urn:uuid:a889c0f6-f215-405c-8645-5b7ab240df3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00046.warc.gz"} |
An image of the folks as mentioned above via the GAN de jour
First, as usual, i trust everyone is safe. Second, I’ve been “thoughting” a good deal about how the world is being eaten by software and, recently, machine learning. i personally have a tough time
with using the words artificial intelligence.
What Would Nash, Shannon, Turing, Wiener, and von Neumann Think of Today’s World?
The modern world is a product of the mathematical and scientific brilliance of a handful of intellectual pioneers who happen to be whom i call the Horsemen of The Digital Future. i consider these
humans to be my heroes and persons that i aspire to be whereas most have not accomplished one-quarter of the work product the humans have created for humanity. Among these giants are Dr. John Nash,
Dr. Claude Shannon, Dr. Alan Turing, Dr. Norbert Wiener, and Dr. John von Neumann. Each of them, in their own way, laid the groundwork for concepts that now define our digital and technological age:
game theory, information theory, artificial intelligence, cybernetics, and computing. But what would they think if they could see how their ideas, theories and creations have shaped the 21st century?
A little context.
John Nash: The Game Theorist
John Nash revolutionized economics, mathematics, and strategic decision-making through his groundbreaking work in game theory. His Nash Equilibrium describes how parties, whether they be countries,
companies, or individuals, can find optimal strategies in competitive situations. Today, his work influences fields as diverse as economics, politics, and evolutionary biology. NOTE: Computational
Consensus Not So Hard; Carbon (Human) Consensus Nigh Impossible.
The Nash equilibrium is the set of degradation strategies
such that, if both players adopt it, neither player can achieve a higher payoff by changing strategies. Therefore, two rational agents should be expected to pick the Nash equilibrium as their
If Nash were alive today, he would be amazed at how game theory has permeated decision-making in technology, particularly in algorithms used for machine learning, cryptocurrency trading, and even
optimizing social networks. His equilibrium models are at the heart of competitive strategies used by businesses and governments alike. With the rise of AI systems, Nash might ponder the implications
of intelligent agents learning to “outplay” human actors and question what ethical boundaries should be set when AI is used in geopolitical or financial arenas.
Claude Shannon: The Father of Information Theory
Claude Shannon’s work on information theory is perhaps the most essential building block of the digital age. His concept of representing and transmitting data efficiently set the stage for everything
from telecommunications to the Internet as we know it. Shannon predicted the rise of digital communication and laid the foundations for the compression and encryption algorithms protecting our data.
He also is the father of my favorite equation mapping the original entropy equation from thermodynamics to channel capacity:
The shear elegance and magnitude is unprecedented. If he were here, Shannon would witness the unprecedented explosion of data, quantities, and speeds far beyond what was conceivable in his era. The
Internet of Things (IoT), big data analytics, 5G/6G networks, and quantum computing are evolutions directly related to his early ideas. He might also be interested in cybersecurity challenges, where
information theory is critical in protecting global communications. Shannon would likely marvel at the sheer volume of information we produce yet be cautious of the potential misuse and the ethical
quandaries regarding privacy, surveillance, and data ownership.
Alan Turing: The Architect of Artificial Intelligence
Alan Turing’s vision of machines capable of performing any conceivable task laid the foundation for modern computing and artificial intelligence. His Turing Machine is still a core concept in the
theory of computation, and his famous Turing Test continues to be a benchmark in determining machine intelligence.
In today’s world, Turing would see his dream of intelligent machines realized—and then some. From self-driving cars to voice assistants like Siri and Alexa, AI systems are increasingly mimicking
human cognition human capabilities in specific tasks like data analysis, pattern recognition, and simple problem-solving. While Turing would likely be excited by this progress, he might also wrestle
with the ethical dilemmas arising from AI, such as autonomy, job displacement, and the dangers of creating highly autonomous AI systems as well as calling bluff on the fact that LLM systems do not
reason in the same manner as human cognition on basing the results on probabilistic convex optimizations. His work on breaking the Enigma code might inspire him to delve into modern cryptography and
cybersecurity challenges as well. His reaction-diffusion model called Turings Metapmorphsis equation, is foundational in explaining biological systems:
Turing’s reaction-diffusion system is typically written as a system of partial differential equations (PDEs):
In addition to this, his contributions to cryptography and game theory alone are infathomable.
In his famous paper, “Computing Machinery and Intelligence,” Turing posed the question, “Can machines think?” He proposed the Turing Test as a way to assess whether a machine can exhibit intelligent
behavior indistinguishable from a human. This test has been a benchmark in AI for evaluating a machine’s ability to imitate human intelligence.
Given the recent advances made with large language models, I believe he would find it amusing, not that they think or reason.
Norbert Wiener: The Father of Cybernetics
Norbert Wiener’s theory of cybernetics explored the interplay between humans, machines, and systems, particularly how systems could regulate themselves through feedback loops. His ideas greatly
influenced robotics, automation, and artificial intelligence. He wrote the books “Cybernetics” and “The Human Use of Humans”. During World War II, his work on the automatic aiming and firing of
anti-aircraft guns caused Wiener to investigate information theory independently of Claude Shannon and to invent the Wiener filter. (The now-standard practice of modeling an information source as a
random process—in other words, as a variety of noise—is due to Wiener.) Initially, his anti-aircraft work led him to write, with Arturo Rosenblueth and Julian Bigelow, the 1943 article ‘Behavior,
Purpose and Teleology. He was also a complete pacifist. What was said about those who can hold two opposing views?
If Wiener were alive today, he would be fascinated by the rise of autonomous systems, from drones to self-regulated automated software, and the increasing role of cybernetic organisms (cyborgs)
through advancements in bioengineering and robotic prosthetics. He, I would think, would also be amazed that we could do real-time frequency domain filtering based on his theories. However, Wiener’s
warnings about unchecked automation and the need for human control over machines would likely be louder today. He might be deeply concerned about the potential for AI-driven systems to exacerbate
inequalities or even spiral out of control without sufficient ethical oversight. The interaction between humans and machines in fields like healthcare, where cybernetics merges with biotechnology,
would also be a keen point of interest for him.
John von Neumann: The Architect of Modern Computing
John von Neumann’s contributions span so many disciplines that it’s difficult to pinpoint just one. He’s perhaps most famous for his von Neumann architecture, the foundation of most modern computer
systems, and his contributions to quantum mechanics and game theory. His visionary thinking on self-replicating machines even predated discussions of nanotechnology.
Von Neumann would likely be astounded by the ubiquity and power of modern computers. His architectural design is the backbone of nearly every device we use today, from smartphones to supercomputers.
He would also find significant developments in quantum computing, aligning with his quantum mechanics work. As someone who worked on the Manhattan Project (also Opphenhiemer), von Neumann might also
reflect on the dual-use nature of technology—the incredible potential of AI, nuclear power, and autonomous weapons to both benefit and harm humanity. His early concerns about the potential for mutual
destruction could be echoed in today’s discussions on AI governance and existential risks.
What Would They Think Overall?
Together, these visionaries would undoubtedly marvel at how their individual contributions have woven into the very fabric of today’s society. The rapid advancements in AI, data transmission,
computing power, and autonomous systems would be thrilling, but they might also feel a collective sense of responsibility to ask:
Where do we go from here?
Once again Oh Dear Reader You pre-empt me….
A colleague sent me this paper, which was the impetus for this blog:
My synopsis of said paper:
“The Tensor as an Informational Resource” discusses the mathematical and computational importance of tensors as resources, particularly in quantum mechanics, AI, and computational complexity. The
authors propose new preorders for comparing tensors and explore the notion of tensor rank and transformations, which generalize key problems in these fields. This paper is vital for understanding how
the foundational work of Nash, Shannon, Turing, Wiener, and von Neumann has evolved into modern AI and quantum computing. Tensors offer a new frontier in scientific discovery, building on their
theories and pushing the boundaries of computational efficiency, information processing, and artificial intelligence. It’s an extension of their legacy, providing a mathematical framework that could
revolutionize our interaction with quantum information and complex systems. Fundamental to systems that appear to learn where the information-theoretic transforms are the very rosetta stone of how we
perceive the world through perceptual filters of reality.
This shows the continuing relevance in ALL their ideas in today’s rapidly advancing AI and fluid computing technological landscape.
They might question whether today’s technology has outpaced ethical considerations and whether the systems they helped build are being used for the betterment of all humanity. Surveillance, privacy,
inequality, and autonomous warfare would likely weigh heavily on their minds. Yet, their boundless curiosity and intellectual rigor would inspire them to continue pushing the boundaries of what’s
possible, always seeking new answers to the timeless question of how to create the future we want and live better, more enlightened lives through science and technology.
Their legacy lives on, but so does their challenge to us: to use the tools they gave us wisely for the greater good of all.
Or would they be dismayed that we use all of this technology to make a powerpoint to save time so we can watch tik tok all day?
Until Then,
#iwishyouwater <- click and see folks who got the memo
Music To blog by: Bach: Mass in B Minor, BWV 232. By far my favorite composer. The John Eliot Gardiner and Monterverdi Choir version circa 1985 is astounding.
Dalle 3’s idea of an Abstract Syntax Tree in
If you would know strength and patience, welcome the company of trees.
~ Hal Borland
First, I hope everyone is safe. Second, I am changing my usual SnakeByte [] stance process. I am pulling this from a website I ran across. I saw the library mentioned, so I decided to pull from the
LazyWebTM instead of the usual snake-based tomes I have in my library.
As a Python developer, understanding and navigating your codebase efficiently is crucial, especially as it grows in size and complexity. Trust me, it will, as does Entropy. Traditional search tools
like grep or IDE-based search functionalities can be helpful, but they cannot often “‘understand” the structure of Python code – sans some of the Co-Pilot developments. (I’m using understand here
*very* loosely Oh Dear Reader).
This is where pyastgrep it comes into play, offering a powerful way to search and analyze your Python codebase using Abstract Syntax Trees (ASTs). While going into the theory of ASTs is tl;dr for a
SnakeByte[] , and there appears to be some ambiguity on the history and definition of Who actually invented ASTs, i have placed some references at the end of the blog for your reading pleasure, Oh
Dear Reader. In parlance, if you have ever worked on compilers or core embedded systems, Abstract Syntax Trees are data structures widely used in compilers and the like to represent the structure of
program code. An AST is usually the result of the syntax analysis phase of a compiler. It often serves as an intermediate representation of the program through several stages that the compiler
requires and has a strong impact on the final output of the compiler.
So what is the Python Library that you speak of? i’m Glad you asked.
What is pyastgrep?
pyastgrep is a command-line tool designed to search Python codebases with an understanding of Python’s syntax and structure. Unlike traditional text-based search tools, pyastgrep it leverages the
AST, allowing you to search for specific syntactic constructs rather than just raw text. This makes it an invaluable tool for code refactoring, auditing, and general code analysis.
Why Use pyastgrep?
Here are a few scenarios where pyastgrep excels:
1. Refactoring: Identify all instances of a particular pattern, such as function definitions, class instantiations, or specific argument names.
2. Code Auditing: Find usages of deprecated functions, unsafe code patterns, or adherence to coding standards.
3. Learning: Explore and understand unfamiliar codebases by searching for specific constructs.
I have a mantra: Reduce, Refactor, and Reuse. Please raise your hand of y’all need to refactor your code? (C’mon now no one is watching… tell the truth…). See if it is possible to reduce the code
footprint, refactor the code into more optimized transforms, and then let others reuse it across the enterprise.
Getting Started with pyastgrep
Let’s explore some practical examples of using pyastgrep to enhance your code analysis workflow.
Installing pyastgrep
Before we dive into how to use pyastgrep, let’s get it installed. You can install pyastgrep via pip:
(base)tcjr% pip install pyastgrep #dont actually type the tctjr part that is my virtualenv
Example 1: Finding Function Definitions
Suppose you want to find all function definitions in your codebase. With pyastgrep, this is straightforward:
pyastgrep 'FunctionDef'
This command searches for all function definitions (FunctionDef) in your codebase, providing a list of files and line numbers where these definitions occur. Ok pretty basic string search.
Example 2: Searching for Specific Argument Names
Imagine you need to find all functions that take an argument named config. This is how you can do it:
pyastgrep 'arg(arg=config)'
This query searches for function arguments named config, helping you quickly locate where configuration arguments are being used.
Example 3: Finding Class Instantiations
To find all instances where a particular class, say MyClass, is instantiated, you can use:
pyastgrep 'Call(func=Name(id=MyClass))'
This command searches for instantiations of MyClass, making it easier to track how and where specific classes are utilized in your project.
Advanced Usage of pyastgrep
For more complex queries, you can combine multiple AST nodes. For instance, to find all print statements in your code, you might use:
pyastgrep 'Call(func=Name(id=print))'
This command finds all calls to the print function. You can also use more detailed queries to find nested structures or specific code patterns.
Integrating pyastgrep into Your Workflow
Integrating pyastgrep into your development workflow can greatly enhance your ability to analyze and maintain your code. Here are a few tips:
1. Pre-commit Hooks: Use pyastgrep in pre-commit hooks to enforce coding standards or check for deprecated patterns.
2. Code Reviews: Employ pyastgrep during code reviews to quickly identify and discuss specific code constructs.
3. Documentation: Generate documentation or code summaries by extracting specific patterns or structures from your codebase.
Example Script
To get you started, here’s a simple Python script using pyastgrep to search for all function definitions in a directory:
import os
from subprocess import run
def search_function_definitions(directory):
result = run(['pyastgrep', 'FunctionDef', directory], capture_output=True, text=True)
if __name__ == "__main__":
directory = "path/to/your/codebase" #yes this is not optimal folks just an example.
Replace "path/to/your/codebase" with the actual path to your Python codebase, and run the script to see pyastgrep in action.
pyastgrep is a powerful tool that brings the capabilities of AST-based searching to your fingertips. Understanding and leveraging the syntax and structure of your Python code, pyastgrep allows for
more precise and meaningful code searches. Whether you’re refactoring, auditing, or simply exploring code, pyastgrep it can significantly enhance your productivity and code quality. This is a great
direct addition to your arsenal. Hope it helps and i hope you found this interesting.
Until Then,
#iwishyouwater <- The best of the best at Day1 Tahiti Pro presented by Outerknown 2024
𝕋𝕖𝕕 ℂ. 𝕋𝕒𝕟𝕟𝕖𝕣 𝕁𝕣. (@tctjr) / X
MUZAK to Blog By: SweetLeaf: A Stoner Rock Salute to Black Sabbath. While i do not really like bands that do covers, this is very well done. For other references to the Best Band In Existence ( Black
Sabbath) i also refer you to Nativity in Black Volumes 1&2.
[1] Basics Of AST
[2] The person who made pyastgrep
bulb 2 warez
Even if we crash and burn and loose everthing the experience is worth ten times the cost.
~ S. Jobs
As always, Oh Dear Readers, i trust this finds you safe. Second, to those affected by the SVB situation – Godspeed.
Third, i was inspired to write a blog on “Doing versus Thinking,” and then i decided on the title “Execution Is Everything”. This statement happens to be located at the top of my LinkedIn Profile.
The impetus for this blog came from a recent conversation where an executive who told me, “I made the fundamental mistake of falling in love with the idea and quickly realized that ideas are cheap,
it is the team that matters.”
i’ve written about the very issue on several occasions. In Three T’s of a Startup to Elite Computing, i have explicitly stated ideas are cheap, a dime a dozen. Tim Ferris, in the amazing book “Tools
Of Titans,” interviews James Altuchur, and he does this exercise every day:
This is taken directly from the book in his words, but condensed for space, here are some examples of the types of lists James makes:
• 10 olds ideas I can make new
• 10 ridiculous things I would invent (e.g., the smart toilet)
• 10 books I can write (The Choose Yourself Guide to an Alternative Education, etc).
• 10 business ideas for Google/Amazon/Twitter/etc.
• 10 people I can send ideas to
• 10 podcast ideas or videos I can shoot (e.g., Lunch with James, a video podcast where I just have lunch with people over Skype and we chat)
• 10 industries where I can remove the middleman
• 10 things I disagree with that everyone else assumes is religion (college, home ownership, voting, doctors, etc.)
• 10 ways to take old posts of mine and make books out of them
• 10 people I want to be friends with (then figure out the first step to contact them)
• 10 things I learned yesterday
• 10 things I can do differently today
• 10 ways I can save time
• 10 things I learned from X, where X is someone I’ve recently spoken with or read a book by or about. I’ve written posts on this about the Beatles, Mick Jagger, Steve Jobs, Charles Bukowski, the
Dalaï Lama, Superman, Freakonomics, etc.
• 10 things I’m interested in getting better at (and then 10 ways I can get better at each one)
• 10 things I was interested in as a kid that might be fun to explore now (like, maybe I can write that “Son of Dr. Strange” comic I’ve always been planning. And now I need 10 plot ideas.)
• 10 ways I might try to solve a problem I have. This has saved me with the IRS countless times. Unfortunately, the Department is Motor Vehicles is impervious to my superpowers
Is your brain tired of just “thinking” about doing those gymnastics?
i cannot tell you how many people have come to me and said “hey I have an idea!” Great, so do you and countless others. What is your plan of making it a reality? What is your maniacal passion every
day to get this thing off the ground and make money?
The statement “Oh I/We thought about that 3 years ago” is not a qualifier for anything except that fact you thought it and didn’t execute on said idea. You know why?
Creating software from an idea that runs 24/7 is still rather difficult. In fact VERY DIFFICULT.
“Oh We THOUGHT about that <insert number of days or years ago here>. i call the above commentary “THOUGHTING”. Somehow the THOUGHT is manifested from Ideas2Bank? If that is a process, i’d love to see
the burndown chart on that one. No Oh Dear Readers, THOUGHTING is about as useful as that overly complex PowerPoint that gets edited ad nauseam, and people confuse the “slideware” with “software”.
The only code that matters is this:
Code that is written with the smallest OPEX and Highest Margins thereby increasing Revenue Per Employee unless you choose to put it in open source for a wonderful plethora of reasons or you are
providing a philanthropic service.
When it comes to creating software, “Execution is everything.” gets tossed around just like the phrase “It Just Works” as a requirement. At its core, this phrase means that the ability to bring an
idea to life through effective implementation is what separates successful software from failed experiments.
The dynamic range between average and the best is 2:1. In software it is 50:1 maybe 100:1 very few things in life are like this. I’ve built a lot of my sucess on finding these truly gifted
~ S. Jobs
In order to understand why execution is so critical in software development, it’s helpful first to consider what we mean by “execution.” Simply put, execution refers to the process of taking an idea
or concept and turning it into a functional, usable product. This involves everything from coding to testing, debugging to deployment, and ongoing maintenance and improvement.
When we say that execution is everything in software development, what we’re really saying is that the idea behind a piece of software is only as good as the ability of its creators to make it work
in the real world. No matter how innovative or promising an idea may seem on paper, it’s ultimately worthless if it can’t be brought to life in a way that users find valuable and useful.
You can fail at something you dislike just as easily as something you like so why not choose what you like?
~ J. Carey
This is where execution comes in. In order to turn an idea into a successful software product, developers need to be able to navigate a complex web of technical challenges, creative problem-solving,
and user feedback. They need to be able to write code that is clean, efficient, and scalable. They need to be able to test that code thoroughly, both before and after deployment. And they need to be
able to iterate quickly and respond to user feedback in order to improve and refine the product continually.
The important thing is to dare to dream big, then take action to make it come true.
~ J. Girard
All of these factors require a high degree of skill, discipline, and attention to detail. They also require the ability to work well under pressure, collaborate effectively with other team members,
and stay focused on the ultimate goal of creating a successful product.
The importance of execution is perhaps most evident when we consider the many examples of software projects that failed despite having what seemed like strong ideas behind them. From buggy,
unreliable apps to complex software systems that never quite delivered on their promises, there are countless examples of software that fell short due to poor execution.
On the other hand, some of the most successful software products in history owe much of their success to strong execution. Whether we’re talking about the user-friendly interface of the iPhone or the
robust functionality of Paypal’s Protocols, these products succeeded not just because of their innovative ideas but because of the skill and dedication of the teams behind them.
The only sin is mediocrity^[1].
~ M. Graham
In the end, the lesson is clear: when it comes to software development, execution really is everything. No matter how brilliant your idea may be, it’s the ability to turn that idea into a functional,
usable product that ultimately determines whether your software will succeed or fail. By focusing on the fundamentals of coding, testing, and iterating, developers can ensure that their software is
executed to the highest possible standard, giving it the best chance of success in an ever-changing digital landscape.
So go take that idea and turn it into a Remarkable Viable Product, not a Minimum Viable Product! Who likes Minimum? (thanks R.D.)
Be Passionate! Go DO! Go Create!
Go Live Your Personal Legend!
A great video stitching of discussions from Steve Jobs on execution, and passion – click here-> The Major Thinkers Steve Jobs
Until then,
#iwishyouwater <- yours truly hitting around 31 meters (~100ft) on #onebreath
Muzak To Blog By: Todd Hannigan “Caldwell County.”
[1] The only sin is mediocrity is not true if there were a real Sin it should be Stupidity but the quote fits well in the narrative.
Got It?
We are organized like a startup. We are the biggest startup on the planet.
S. Jobs
First, i hope everyone is safe. Second, this blog is about something everyone seems to be asking me about and talking about, but no one seems to be able to execute the concept much like
interoperability in #HealthIT. Third, it is long-form content so in most cases tl;dr.
Let us look to Miriam-Websters OnLine Dictionary for a definition – shall we?
cul·ture <ˈkəl-chər>
a: the customary beliefs, social forms, and material traits of a racial, religious, or social group also : the characteristic features of everyday existence (such as diversions or a way of life)
shared by people in a place or time ; popular culture, Southern culture
b: the set of shared attitudes, values, goals, and practices that characterizes an institution or organization a corporate culture focused on the bottom line
c: the set of values, conventions, or social practices associated with a particular field, activity, or societal characteristic studying the effect of computers on print culture
d: the integrated pattern of human knowledge, belief, and behavior that depends upon the capacity for learning and transmitting knowledge to succeeding generations
a: enlightenment and excellence of taste acquired by intellectual and aesthetic training
b: acquaintance with and taste in fine arts, humanities, and broad aspects of science as distinguished from vocational and technical skills; a person of culture
3: the act or process of cultivating living material (such as bacteria or viruses) in prepared nutrient media also : a product of such cultivation
4: CULTIVATION, TILLAGE
5: the act of developing the intellectual and moral faculties especially by education
6: expert care and training; beauty culture
Wow, This sounds complicated. Which one to leave in and which one to leave out?
Add to this complexity the fact that creating and executing production software is almost an insurmountable task. i have said for years software creation is one of the most significant human
endeavors of all time. i also believe related to these concerns the interplay between comfort and solutions. Most if not all humans desire solutions however as far as i can tell solutions are never
comfortable. Solutions involve change most humans are homeostatic. Juxtapose this against the fact that humans love comfort. So what do you do?
So why does it seem like everyone is talking about kəl-chər? i consider this to be like Fight Club. 1st rule of kəl-chər is you don’t talk about culture. It should be an implicit aspect of your
organization. Build or Re-Build it at a first principles engineering practice. Perform root cause analysis of the behaviors within the company. If it does in fact need to be re-built start with you
and your leadership. Turn the mirror on you first. Understand that you must lead by example. Merit Not Inherit.
i’ve recently been asked how you change and align culture. Well here are my recommendations and it comes down to TRUST at ALL levels of the company.
Create an I^3 Lab: Innovation, Incubation, Intrapreneurship:
Innovation without code is just ideas and everyone has them. Ideas are cheap. Incubation without product market fit is a dead code base. Intrapreneurship is the spirit of a system that encourages
employees to think and act like individual entrepreneurs and empowers them to take action, embrace risk, and make decisions as if they had founded the company themselves. Innovate – create the idea –
Incubate – create the Maximum Viable Product (not minimum) – Intrapreneurship – spin out the Maximum Viable Product. As an aside Minimum Viable Product sounds like you bailed out making the best you
possibly could in the moment. Take that Maximum Viable product and roll it into a business vertical and go to market strategy – then spin the wheel again.
I think it’s very important to have a feedback loop, where you’re constantly thinking about what you’ve done and how you could be doing it better.
E. Musk
Value The Most Important Asset – Your People
Managing high-performance humans is a difficult task because most high-performance humans do not like to be managed they love to be led. Lead them by example. Value them and compensate them
accordingly. Knowledge workers love achievement and goals. Lead them into the impossible, gravitate toward dizzying heights, and be there for them. Be completely transparent and communicate. Software
is always broken. If anyone states differently they are not telling the truth. There is always refactoring, retargeting, more code coverage and nascent bugs. Let them realize you realize that however
let them know that if they do make a mistake escalate immediately. Under no circumstances can you tolerate surprises. Give them the framework with OKRs and KPIs that let them communicate openly,
efficiently and effectively and most important transparently. Great teams will turn pencils into Mount Blanc Fountain Pens. Let them do what they do best and reward them!
Process Doesn’t Make A Culture
Nor does it make great products. Many focus on some software process. Apple used and as far as i know still uses strict waterfall. As far as i am concerned, we are now trending towards a Holacracy
type of environment which is a self-organizing environment. However, this only can be achieved with the proper folks that appreciate the friction of creating great products from the best ideas. The
Process of evolving from an idea to a product is magic. You learn you evolve; you grow your passion for and into the product as it becomes itself the team that built the product. Your idea and
passion are inherent in that shipping software (or hardware).
What do you want me to do
To do for you to see you through?
The Grateful Dead
Empower Your People
Provide your people the ability to manage themselves and have autonomy. Set them free. Trust them to make the decisions that will drive the company and projects into world-class endeavors. Take a
chance with them, Let a new college graduate push some code to production. Let a new sales associate push a deal with a customer. Let your new marketing person design an area on the company site.
Allow them to evolve grow and be a part of the Great Endeavor. Put them in charge and provide the framework for autonomy to make decisions and when they deliver – award them not with something
ephemeral but volitional. Money and Stock work wonders. Empower. Align. Evolve.
Provide and Articulate a Common Vision
Provide a vision of the company or project. Two sentences that everyone understands. Most people who are empowered and given the frameworks to create within know what to do in these circumstances.
Articulate the vision and gain common alignment across the organization or project. That is leadership that high performance teams desire. Take this alignment then map it into the OKRs and KPIs then
in turn pick a process and let everyone know how this aligns to the vision. Create the environment that every line of code maps to that vision. Show commitment on this vision.
Give FeedBack
Communicate. Communicate. Communicate. Collaborate. Collaborate. Collaborate. Till you puke. i cannot emphasize this enough. You must be prepared everyday to manically interact with your teams and
have the hard friction filled uncomfortable discussions. You want to keep the top performers let them know where they stand, how they stand and why they stand in the rankings and how they are
contributing to the vision. Again attempt to create coder-metrics across your organization or project that exemplifies this performance. Interact with your most important asset your people. Over
communicate. We have the ability to reach everyone at anytime, email, zoom, slack, granite tablet where once used to message. Write the message and give feedback. Better yet go take a walk with them.
Have 1:1s. Listen to your people receptively and without bias and judgment about their concerns, passions, what scares them, what makes them happy, their joys, goals, and aspirations so they feel
validated and understood. Solicit feedback, shut up and listen.
What all of this comes down to what i call – Amplifying_Others[^TM]. This is easier said than done. Personally, i believe that you need to commit even to the point of possibly finding them a better
fit for a position at another company. This goes back to understanding what truly drives the only asset there is in technology the people. Always Be Listening, Always Be Networking, and Always Be
This brings up the next big question for your company – How do you attract the best right talent? Hmmmm… that might be another blog. Let me know your thought on these matters in the comments.
Until Then,
#IWishYouWater <- Psycho Session In Mentawis
Music To Blog By:
American Beauty by The Grateful Dead. Box of Rain and Ripple are amazing. Also if you haven’t heard Jane’s Addiction’s cover of Ripple check it out. i am not a Dead fan but the lyrics on some of
these songs are monumental.
References (click for purchase link):
The Psychology of Computer Programming
The Essence of Software: Why Concepts Matter for Great Design
The human body resonates at the same frequency as Mother Earth. So instead of only focusing on trying to save the earth, which operates in congruence to our vibrations, I think it is more
important to be one with each other. If you really want to remedy the earth, we have to mend mankind. And to unite mankind, we heal the Earth. That is the only way. Mother Earth will exist with
or without us. Yet if she is sick, it is because mankind is sick and separated. And if our vibrations are bad, she reacts to it, as do all living creatures.
Suzy Kassem, Rise Up and Salute the Sun: The Writings of Suzy Kassem
Image from: The Outer Limits (1963) S 2 E 15 “The Brain of Colonel Barham”
i have been considering writing a series of blogs on the coming age of Cybernetics. For those unaware Cybernetics was a term coined by one of my heroes Dr. Norbert Weiner. Dr. Weiner was a professor
of mathematics at MIT who early on became interested in stochastic and noise processes and has an entire area dedicated to him in the areas of Weiner Estimation Theory. However, he also is credited
as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could possibly be simulated by machines thereby coining the phrase “Cybernetics”. He
wrote two seminal books in this area: (1) “Cybernetics” (2) “The Human Use of Humans”. This brings us to the present issue at hand.
The catalyst for the blog came from a tweet:
More concerning Ted is how long before people start paying for upgrades. What effects will this have when you have achieved functional immortality?
This was in response to me RT’ing a tweet from @alisonBLowndes concerning the International Manifesto of the “2045” Strategic Social Initiative. An excerpt from said manifesto:
We believe that before 2045 an artificial body will be created that will not only surpass the existing body in terms of functionality but will achieve perfection of form and be no less attractive
than the human body.
2045 Manifesto
Now for more context. I am a proponent of using technology to allow for increased human performance as i am an early adopter if you will of the usage of titanium to repair the skeletal system. i have
staples, pins, plates and complete joints of titanium from pushing “Ye Ole MeatBag” into areas where it did not fair so well.
There are some objectives of the movement of specific interest is Number Two:
To create an international research center for cybernetic immortality to advance practical implementations of the main technical project – the creation of the artificial body and the preparation
for subsequent transfer of individual human consciousness to such a body.
2045 Manifesto
This is closely related to Transhumanism which is more of a philosophy than an execution. The way i frame it is Transhumanism sets the philosophical framework for cybernetics. The contemporary
meaning of the term “transhumanism” was foreshadowed by one of the first professors of futurology, a man who changed his name to FM-2030. In the 1960s, he taught “new concepts of the human” at The
New School when he began to identify people who adopt technologies, lifestyles and worldviews “transitional” to post-humanity as “transhuman“.
Coming from a software standpoint we could map the areas into pipelines and deploy as needed either material biological or conscious. We could map these areas into a CI/CD deployment pipeline. .
For a direct reference, i work with an amazing practicing nocturnist who is also a great coder as well as a medical 3D Printing expert! He prints body parts! It is pretty amazing to think that
something your holding that was printed that morning is going to enable someone to use their respective limbs or walk again. So humbling. By the way, the good doctor is also a really nice person.
Just truly a great human. Health practitioners are truly some of humanity’s rockstars.
He printed me a fully articulated purple octopus that doesn’t go in the body:
Building upon this edict and for those who have read William Gibson’s “Neuromancer,” or Rudy Ruckers The Ware Tetralogy: Software, Wetware, Realware, Freeware “it calls into question the very essence
of the use for the human body? Of the flesh is the only aspect we truly do know and associate with this thing called life. We continually talk about the ‘Big C Word” – Consciousness. However, we only
know the body. We have no idea of the mind. Carnality seems to rule the roost for the Humans.
In fact most of the acts that we perform on a daily basis trend toward physical pleasure. However what if we can upgrade the pleasure centers? What if the whole concept of dysphoria goes away and you
can order you a net-new body? What *if* this allows us as the above ponders to upgrade ad nauseam and live forever? Would you? Would it get tiresome and boring?
i can unequivocally tell you i would if given the chance. Why? Because if there *is* intelligent life somewhere then they have a much longer evolutionary scale that we mere humans on earth do not and
they have figured out some stuff let’s say that can possibly change the way we view life in longer time scales ( or time loops or even can we say Infinite_Human_Do_Loops? ).
i believe we are coming to an age where instead of the “50 is the new 30” we can choose our age – lets say at 100 i choose new core and legs and still run a 40-yard dash in under 5 seconds? i am all
for it.
What if you choose a body that is younger than your progeny?
What if you choose a body that isnt a representation of a human as we know it?
All with immortality?
i would love to hear folks thoughts on the matter in the comments below.
For reference again here is the link -> 2045 Manifesto.
Until then,
Be Safe.
Muzak To Blog By: Maddalena by Ennio Morricone
“What we’re doing here will send a giant ripple through the universe.”
Steve Jobs
I have an old mac laptop that was not doing anyone much use sitting around the house. i had formatted the rig and due to it only being an i7 Pentium series mac you could only roll up to the Lion OS.
Also, i wanted a “pure” Linux rig and do not like other form factors (although i do dig the System 76 rigs).
So i got to thinking why dont i roll Ubuntu on it and let one cat turn into another cat? See what i did there? Put a little shine on Ye Ole Rig? Here Kitty Kitty!
Anyways here are the steps that i found worked the most painless.
Caveat Emptor: these steps completely wipe the partition and Linux does run natively wiping out any and all OSes. You WILL lose your OS X Recovery Partition, so returning to OS X or macOS can be a
more long-winded process, but we have instructions here on how to cope with this: How to restore a Mac without a recovery partition. You are going All-In!
On that note i also don’t recommend trying to “dual-boot” OS X and Linux, because they use different filesystems and it will be a pain. Anyways this is about bringing new life to an old rig if you
have a new rig with Big Sur roll Virtual Box and run whatever Linux distro you desire.
What you need:
• A macintosh computer is the point of the whole exercise. i do recommend having NO EXTERNAL DRIVES connected as you will see below.
• A USB stick with at least 8 gig of storage. This to will be formatted and all data lost.
• Download your Linux distribution to the Mac. We recommend Ubuntu 16.04.4 LTS if this is your first Linux install. Save the file to your ~/Downloads folder.
• Download and install an app called Etcher from Etcher.io. This will be used to copy the Linux install .ISO file to your USB drive.
Steps to Linux Freedom:
• Insert your USB Thumb Drive. A reminder that the USB Flash drive will be erased during this installation process. Make sure you’ve got nothing you want on it.
• Open Etcher Click Select “Image”. Choose ubuntu-16.04.1-desktop-amd64.iso (the image you downloaded in Step 1). NOTE: i had some problems with 20.x latest release with wireless so i rolled back
to 16.0x just to get it running.
• Click “Change” under Select Drive.
• Pick the drive that matches your USB Thumb Drive in size. It should be /dev/disk1 if you only have a single hard drive in your Mac. Or /dev/disk2, /dev/disk3 and so on (if you have more drives
attached). NOTE: Do not pick /dev/disk0. That’s your hard drive! Pick /dev/disk0 and you’ll wipe your macOS hard drive HEED THY WARNING YOU HAVE BEEN WARNED! This is why i said its easier if you
have no external media.
• Click “Flash!” Wait for the iso file to be copied to the USB Flash Drive. Go browse your favorite socnet as this will take some time or hop on your favorite learning network and catch up on those
• Once it is finished remove the USB Flash Drive from your Mac. This is imperative.
• Now SHUTDOWN the mac and plug the Flashed USB drive into the mac.
• Power up and hold the OPTION key while you boot.
• Choose the EFI Boot option from the startup screen and press Return.
• IMMEDIATELY press the “e” key. i found you need to do this quickly otherwise the rig tries to boot.
• Pressing the “e” key will enter you into “edit mode” you will see a black and white screen with options to try Ubuntu and Install Ubuntu. Don’t choose either yet, press “e” to edit the boot
• This step is critical and the font maybe really small so take your time. Edit the line that begins with Linux and place the word "nomodeset" after "quiet splash". The whole line should read:
"linux /casper/vmlinuz.efifile=/cdrom/preseed/ubuntu.seed boot=casper quiet splash nomodeset --
• Now Press F10 on the mac.
• Now its getting cool! Your mac and Ubuntu boots into trial mode!
(Note: at this point also go browse your favorite socnet as this will take some time or hop on your favorite learning network and catch up on those certificates/badges.)
• Double-click the icon marked “Install Ubuntu”. (get ready! Here Kitty Kitty!)
• Select your language of choice.
• Select “Install this third-party software” option and click Continue. Once again important.
• Select “Erase disk and install Ubuntu” and click Continue.
• You will be prompted for geographic area and keyboard layout.
• You will be prompted to enter the name and password you want to use (make it count!).
• Click “Continue” and Linux will begin installing!
• When the installation has finished, you can log in using the name and password you chose during installation!
• At this point you are ready to go! i recommend registering for an ubuntu “Live Update” account once it prompts you.
• One side note: on 20.x update there was an issue with the Broadcom wireless adapter crashing which then reboots you without wireless. i am currently working through that and will get back you on
the fix!
Executing the command less /proc/cpuinfo will detail the individual cores. It looks like as i said the i7 Pentium series!
Happy Penguin and Kitten Time! Now you can customize your rig!
Screen shot of keybase running on my ubuntu mac rig!
And that is a wrap! As a matter of fact i ate my own dog food and wrote this blog on the “new” rig!
Until Then,
Muzak to blog by: Dreamways Of The Mystic by Bobby Beausoliel
“Joy, humor, and playfulness are indeed assets;”
~ Eric S. Raymond
As of late, i’ve been asked by an extreme set of divergent individuals what does “Open Source Software” mean?
That is a good question. While i understand the words and words do have meanings i am not sure its the words that matter here. Many people who ask me that question hear “open source” and hear or
think “free’ which is not the case.
Also if you have been on linkedin at all you will see #Linux, #LinuxFoundation and #OpenSource tagged constantly in your feeds.
Which brings me to the current blog and book review.
(CatB)as it is affectionately known in the industry started out and still is a manifesto as well accessible via the world wide web. It was originally published in 1997 on the world wide wait and then
in print form circa 1999. Then in 2001 was a revised edition with a foreword by Bob Young, the founding chairman and ceo of Redhat.
Being i prefer to use plain ole’ books we are reviewing the physical revised and extended paperback edition in this blog circa 2001. Of note for the picture, it has some wear and tear.
To start off as you will see from the cover there is a quote by Guy Kawasaki, Apple’s first Evangelist:
“The most important book about technology today, with implications that go far beyond programming.”
This is completely true. In the same train of thought, it even goes into the aspects of propriety and courtesy within conflict environments and how such environments are of a “merit not inherit”
world, and how to properly respond when you are in vehement disagreement.
To relate it to the book review: What is a cathedral development versus a bazaar environment?
Cathedral is a tip of the fedora if you will to the authoritarian view of the world where everything is very structured and there are only a few at most who will approve moving the codebase forward.
Bazaar refers to the many. The many coding and contributing in a swarm like fashion.
In this book, closed source is described as a cathedral development model and open source as a bazaar development model. A cathedral is vertically and centrally controlled and planned. Process and
governance rule the project – not coding. The cathedral is homeostatic. If you build or rebuild Basilica Sancti Petri within Roma you will not be picking it up by flatbed truck and moving it to
The forward in the 2001 edition is written by Bob Young co-founder and original CEO of RedHat. He writes:
“ There have always been two things that would be required if open-source software was to materially change the world; one was for open-source software to become widely used and the other was the
benefits this software development model supplied to its users had to be communicated and understood.”
Users here are an interesting target. Users could be developers and they could be end-users of warez. Nevertheless, i believe both conditions have been met accordingly.
i co-founded a machine learning and nlp service as a company in 2007 wherein i had the epiphany after my “second” read of Catb that the future is in fact open source. i put second in quotes as the
first time i read it back in 1998 it wasn’t really a read in depth nor having fully internalized it while i was working at Apple in the CPU software department on OS9/OSX and while at the same time
knowing full well that OSX was based on the Mach kernel. The Mach kernel is often mentioned as one of the earliest examples of a microkernel. However, not all versions of Mach are microkernels.
Mach’s derivatives are the basis of the operating system kernel in GNU Hurd and of Apple’s XNU kernel used in macOS, iOS, iPadOS, tvOS, and watchOS.
That being said after years of working with mainly closed source systems in 2007 i re-read Catb. i literally had a deep epiphany that the future of all development would be open source distributed
machine learning – everywhere.
Then i read it recently – deeply – a third time. This time nearly every line in the book resonates.
The third time with almost anything seems to be the charm. This third time through i realized not only is this a treatise for the open-source movement it is a call to arms if you will for the entire
developer community to behave appropriately with propriety and courtesy in a highly matrixed collaborative environment known as the bazaar.
The most obvious question is: Why should you care? i’m glad you asked.
The reason you care is that you are part of the information economy. The top market cap companies are all information-theoretic developer-first companies. This means that these companies build things
so others can build things. Software is truly eating the world. Think in terms of the recent pandemic. Work (code) is being created at an amazing rate due to the fact that the information work
economy is distributed and essentially schedule free. She who has distributed wins and she who can code anytime wins. This also means that you are interested in building world-class software and the
building of this software is now a decentralized peer reviewed transparent process.
The book is organized around Raymond’s various essays. It is important to note that just as software is an evolutionary process by definition so are the essays in this book. They can also be found
online. The original collection of essays date back to 1992 on the internet: “A Brief History Of Hackerdom.’
The book is not a “how-to” cookbook but rather what i call a “why to” map of the terrain. While you can learn how to hack and code i believe it must be in your psyche. The book also uses the term
“hacker” in a positive sense to mean one who creates software versus one who cracks software or steals information.
While the history and the methodology is amazing to me the cogent commentary on the types of the reasoning behind why hackers go into open source vary as widely as ice cream flavors.
Raymond goes into the theory of incentives with respect to the instinctive wiring of humans beings.
“The verdict of history seems to be free-market capitalism is the globally optimal way to cooperate for economic efficiency; perhaps in a similar way to cooperate for generating (and checking!)
high-quality creative work.”
He categorizes command hierarchy, exchange economy, and gift culture to address these incentives.
Command hierarchy:
Goods are allocated in a scarce economy model by one central authority.
Exchange Economy:
The allocation of scarce goods is accomplished in a decentralized manner allowing scale through trade and voluntary cooperation.
Gift Culture:
This is very different than the other two methods or cultures. Abundance makes command and control relationships difficult to sustain. In gift cultures, social status is determined not by what you
control but by what you give away.
It is clear that if we define the open source hackerdom it would be a gift culture. (It is beyond the current scope of this blog but it would be interesting to do a neuroscience project on the
analysis of open source versus closed source hackers brain chemistry as they work throughout the day)
Given these categories, the essays then go onto define the written and many times unwritten (read secrets) that operate within the open-source world via a reputation game. If you are getting the idea
it is tribal you are correct. Interestingly enough the open source world has in many cases very divergent views on all prickly things within the human condition such as religion and politics but one
thing is a constant – ship high-quality code.
Without a doubt the most glaring cogent commentary comes in a paragraph from the essay “The Magic Cauldron.” entitled “Open Source And Strategic Business Risk.”
Ultimately the reasons open source seems destined to become a widespread practice have more to do with customer demand and market pressures than with supply-efficiencies for vendors.”
And further:
“Put yourself for the moment in the position of a CTO at a Fortune 500 corporation contemplating a build or upgrade of your firm’s IT infrastructure. Perhaps you need to choose a network operating
system to be deployed enterprise-wide; perhaps your concerns involve 24/7 web service and e-commerce, perhaps your business depends on being able to field high-volume, high-reliability transaction
databases. Suppose you go the conventional closed-source route. If you do, then you put your firm at the mercy of a supplier monopoly – because by definition there is only one place you can go to
for support, bug fixes, and enhancements. If the supplier doesn’t perform, you will have no effective recourse because you are effectively locked by your initial investment.”
“The truth is this: when your key business processes are executed by opaque blocks of bits that you cant even see inside (let alone modify) you have lost control of your business.”
“Contrast this with the open-source choice. If you go this route, you have the source code, and no one can take that away from you. Instead of a supplier monopoly with a choke-hold on your business,
you now have multiple service companies bidding for your business – and you not only get to play them against each other, but you also have the option of building your own captive support
organization if that looks less expensive than contracting out. The market works for you.”
“The logic is compelling; depending on closed-source code is an unacceptable strategic risk So much so that I believe it will not be very long until closed-source single-vendor acquisitions when
there is an open source alternative available will be viewed as a fiduciary irresponsibility, and rightly grounds for a share-holder lawsuit.”
THIS WAS WRITTEN IN 1997. LOOK AROUND THE WORLD WIDE WAIT NOW… WHAT DO YOU SEE?
Open Source – full stop.
i will add that there was no technical explanation here only business incentive and responsibility to the company you are building, rebuilt, or are scaling. Further, this allows true software
malleability and reach which is the very reason for software.
i will also go on a limb here and say if you are a software corporation one that creates software you can play the monopoly and open-source models against each other within your corporation. Agility
and speed to ship code is the only thing that matters these days. Where is your github? Or why is this not shipping TODAY?
This brings me to yet another amazing prescient prediction in the book that Raymond says that applications are ultimately where we will land for monetary scale. Well yes, there is an app for that….
While i have never met Eric S. Raymond he is a legend in the field. We have much to thank him for in the areas of software. If you have not read CatB and work in the information sector do yourself
a favor: buy it today.
As a matter of fact here is the link: The Cathedral & the Bazaar: Musings on Linux and Open Source by an Accidental Revolutionary
Muzak To Blog To: “Morning Phase” by Beck
“I am putting myself to the fullest possible use, which is all I think any conscious entity can ever hope to do.” ~ HAL 9000
“If you want to make the world a better place take a look at yourself and then make a change.” ~ MJ.
First and foremost with this blog i trust everyone is safe. The world is in an interesting place, space, and time both physically and dare i say collectively – mentally.
A Laundry List
This past week we celebrated Earth Day. i believe i heard it was the 50th year of Earth Day. While I applaud the efforts and longevity for a day we should have Earth Day every day. Further just
“thoughting” about or tweeting about Earth Day – while it may wake up your posterior lobe of the pituitary gland and secret some oxytocin – creating the warm fuzzies for you it really doesn’t create
an action for furthering Earth Day. (much like typing /giphy YAY! In Slack).
As such, i decided to embark on a multipart blog that i have been “thinking” about what i call an Ecological Computing System. Then the more i thought about it why stop at Ecology? We are able to
model and connect essentially anything, we now have models for the brain that while are coarse-grained can account for gross behaviors, we have tons of data on buying habits and advertisement data
and everything is highly mobile and distributed. Machine learning which can optimize, classify and predict with extremely high dimensionality is no longer an academic exercise.
Thus, i suppose taking it one step further from ecology and what would differentiate it from other efforts is that <IT> would actually attempt to provide a compute framework that would compute The
Human Condition. I am going to call this effort Project Noumena. Kant the eminent thinker of 18th century Germany defined Noumena as a thing as it is in itself, as distinct from a thing as it is
knowable by the senses through phenomenal attributes and proposed that the experience was a product of the mind.
My impetus for this are manifold:
• i love the air, water, trees, and animals,
• i am an active water person,
• i want my children’s children’s children to know the wonder of staring at the azure skies, azure oceans and purple mountains,
• Maybe technology will assist us in saving us from The Human Condition.
i have waited probably 15+ years to write about this ideation of such a system mainly due to the technological considerations were nowhere near where they needed to be and to be extremely transparent
no one seemed to really think it was an issue until recently. The pandemic seems to have been a global wakeup call that in fact, Humanity is fragile. There are shortages of resources in the most
advanced societies. Further due to the recent awareness that the pollution levels appear (reported) to be subsiding as a function in the reduction of humans’ daily involvement within the environment.
To that point over the past two years, there appears to be an uptake of awareness in how plastics are destroying our oceans. This has a coupling effect that with the pandemic and other environmental
concerns there could potentially be a food shortage due to these highly nonlinear effects. This uptake in awareness has mainly been due to the usage of technology of mobile computing and social media
which in and of itself probably couldn’t have existed without plastics and massive natural resource consumption. So i trust the irony is not lost there.
From a technical perspective, Open source and Open Source Systems have become the way that software is developed. For those that have not read The Cathedral and The Bazaar and In The Beginning Was
The Command Line i urge you to do so it will change your perspective.
We are no longer hampered by the concept of scale in computing. We can also create a system that behaves at scale with only but a few human resources. You can do a lot with few humans now which has
been the promise of computing.
Distributed computing methods are now coming to fruition. We no longer think in terms of a monolithic operating system or in place machine learning. Edge computing and fiber networks are accelerating
this at an astonishing rate. Transactions now dictate trust. While we will revisit this during the design chapters of the blog I’ll go out on a limb here and say these three features are cogent to
distributed system processing (and possibly the future of computing at scale).
• Incentive models
• Consensus models
• Protocol models
We will definitely be going into the deeper psychological, mathematical, and technical aspects of these items.
Some additional points of interest and on timing. Microsoft recently released press about a Planetary Computer and announced the position of Chief Ecology Officer. While i do not consider Project
Nuomena to be of the same system type there could be similarities on the ecological aspects which just like in open source creates a more resilient base to work.
The top market cap companies are all information theoretic-based corporations. Humans that know the science, technology, mathematics and liberal arts are key to their success. All of these companies
are woven and interwoven into the very fabric of our physical and psychological lives.
Thus it is with the confluence of these items i believe the time is now to embark on this design journey. We must address the Environment, Societal factors and the model of governance.
A mentor once told me one time in a land far away: “Timing is everything as long as you can execute.” Ergo Timing and Execution Is Everything.
It is my goal that i can create a design and hopefully, an implementation that is utilizing computational means to truly assist in building models and sampling the world where we can adhere to goals
in making small but meaningful changes that can be used within what i am calling the 3R’s: recycle, redact, reuse. Further, i hope with the proper incentive models in place that are dynamic it has a
mentality positive feedback effect. Just as in complexity theory a small change – a butterfly wings – can create hurricanes – in this case positive effect.
Here is my overall plan. i’m not big on the process or gant charts. I’ll be putting all of this in a README.md as well. I may ensconce the feature sets etc into a trello or some other tracking
mechanism to keep me focused – WebSphere feel free to make recommendations in the comments section:
Action Items:
• Create Comparative Models
• Create Coarse-Grained Attributes
• Identify underlying technical attributes
• Attempt to coalesce into an architecture
• Start writing code for the above.
Humanity has come to expect growth as a material extension of human behavior. We equate growth with progress. In fact, we use the term exponential growth as it is indefinitely positive. In most cases
for a fixed time interval, this means a doubling of the relevant system variable or variables. We speak of growth as a function of gross national production. In most cases, exponential growth is
treacherous where there are no known or perceived limits. It appears that humanity has only recently become aware that we do not have infinite resources. Psychologically there is a clash between the
exponential growth and the psychological or physical limit. The only significance is the relevant (usually local) limit. How does it affect me, us, and them? This can be seen throughput most game
theory practices – dominant choice. The pattern of growth is not the surprise it is the collision of the awareness of the limit to the ever-increasing growth function is the surprise.
One must stop and ask:
Q: Are progress (and capacity) and the ever-increasing function a positive and how does it relate to 2nd law of thermodynamics aka Entropy? Must it always expand?
We are starting to see that our world can exert dormant forces that within our life can greatly affect our well being. When we approach the actual or perceived limit the forces which are usually
negative begin to gain strength.
So given these aspects of why i’ll turn now to start the discussion. If we do not understand history we cannot predict the future by inventing it or in most cases re-inventing it as it where.
I want to start off the history by referencing several books that i have been reading and re-reading on subjects of modeling the world, complexity, and models for collapse throughout this multipart
blog. We will be addressing issues concerning complex dynamics as are manifested with respect to attributes model types, economics, equality, and mental concerns.
These core references are located at the end of the blog under references. They are all hot-linked. Please go scroll and check them out. i’ll still be here. i’ll wait.
Checked them out? i know a long list.
As you can see the core is rather extensive due to the nature of the subject matter. The top three books are the main ones that have been the prime movers and guides of my thinking. These three books
i will refer to as The Core Trilogy:
The Collapse of Complex Societies
As i mentioned i have been deeply thinking about all aspects of this system for quite some time. I will be mentioning several other texts and references along the continuum of creation of this
We will start by referencing the first book: World Dynamics by J.W. Forrestor. World Dynamics came out of several meetings of the Rome Club a 75 person invite-only club founded by the President of
Fiat. The club set forth the following attributes for a dynamic model that would attempt to predict the future of the world:
• Population Growth
• Capital Investment
• Geographical Space
• Natural Resources
• Pollution
• Food Production
The output of this design was codified in a computer program called World3. It has been running since the 1970s what was then termed a golden age of society in many cases. All of these variables have
been growing at an exponential rate. Here we see the model with the various attributes in action. There have been several criticisms of the models and also analysis which i will go into in further
blogs. However, in some cases, the variants have been eerily accurate. The following plot is an output of the World3 model:
2060 does not look good
Issues Raised By World3 and World Dynamics
The issues raised by World3 and within the book World Dynamics are the following:
• There is a strong undercurrent that technology might not be the savior of humankind
• Industrialism (including medicine and public health) may be a more disturbing force than the population.
• We may face extreme psychological stress and pressures from a four-pronged dilemma via suppression of the modern industrial world.
• We may be living in a “golden age” despite a widely acknowledged feeling of malaise.
• Exhtortions and programs directed at population control may be self-defeating. Population control, if it works, would yield excesses thereby allowing further procreation.
• Pollution and Population seem to oscillate whereas the high standard of living increases the production of food and material goods which outrun the population. Agriculture as it hits a space
limit and as natural resources reach a pollution limit then the quality of life falls in equalizing population.
• There may be no realistic hope of underdeveloped countries reaching the same standard and quality of life as developed countries. However, with the decline in developed countries, the
underdeveloped countries may be equalized by that decline.
• A society with a high level of industrialization may be unsustainable.
• From a long term 100 years hence it may be unwise for underdeveloped countries to seek the same levels of industrialization. The present underdeveloped nations may be in better conditions for
surviving the forthcoming pressures. These underdeveloped countries would suffer far less in a world collapse.
Fuzzy Human – Fuzzy Model
The human mind is amazing at identifying structures of complex situations. However, our experiences train us poorly for estimating the dynamic consequences of said complexities. Our mind is also not
very accurate at estimating ad hoc parts of the complexities and the variational outcomes.
One of the problems with models is well it is just a model The subject-observer reference could shift and the context shifts thereof. This dynamic aspect needs to be built into the models.
Also while we would like to think that our mental model is accurate it is really quite fuzzy and even irrational in most cases. Also attempting to generalize everything into a singular model
parameter is exceedingly difficult. It is very difficult to transfer one industry model onto another.
In general parameterization of most of these systems is based on some perceptual model we have rationally or irrationally invented.
When these models were created there was the consideration of modeling social mechanics of good-evil, greed – altruism, fears, goals, habits, prejudice, homeostasis, and other so-called human
characteristics. We are now at a level of science where we can actually model the synaptic impulse and other aspects that come with these perceptions and emotions.
There is a common cross-cutting construct in most complex models within this text that consists of and mainly concerned with the concept of feedback and how the non-linear relationships of these
modeled systems feedback into one another. System-wide thinking permeates the text itself. On a related note from the 1940’s of which Dr Norbert Weiner and others such as Claude Shannon worked on
ballistic tracking systems and coupled feedback both in a cybernetic and information-theoretic fashion of which he attributed the concept of feedback as one of the most fundamental operations in
information theory. This led to the extremely famous Weiner Estimation Filters. Also, side note: Dr Weiner was a self-styled pacifist proving you can hold two very opposing views in the same instance
whilst being successful at executing both ideals.
Given that basic function of feedback, lets look at the principle structures. Essentially the model states there will be levels and rates. Rates are flows that cause levels to change. Levels can
accumulate the net level. Either addition or subtraction to that level. The various system levels can in aggregate describe the system state at any given time
The below picture is the model that grew out of interest from the initial meetings of the Club of Rome. The inaugural meeting which was the impetus for the model was held in Bern, Switzerland on June
29, 1970. Each of the levels presents a variable in the previously mentioned major structures. System levels appear as right triangles. Each level is increased or decreased by the respective flow. As
previously mentioned on feedback any closed path through the diagram is a feedback loop. Some of the closed loops given certain information-theoretic attributes be positive feedback loops that
generate growth and others that seek equilibrium will be negative feedback loops. If you notice something about the diagram it essentially is a birth and death loop. The population loop if you will.
For the benefit of modeling, there are really only two major variables that affect the population. Birth Rate (BR) and Death Rate (DR). They represent the total aggregate rate at which the population
is being increased or decreased. The system has coefficients that can initialize them to normal rates. For example, in 1970 BRN is taken as 0.0885 (88.5 per thousand) which is then multiplied by
population to determine BR. DRN by the same measure is the outflow or reduction. In 1970 it was 9.5% or 0.095. The difference is the net and called normal rates. The normale rates correspond to a
physical normal world. When there are normal levels of food, material standard of living, crowding, and pollution. The influencers are then multipliers that increase or decrease the normal rates.
Feedback and isomorphisms abound
As a caveat, there have been some detractors of this model. To be sure it is very coarse-grained however while i haven’t seen the latest runs or outputs it is my understanding as i said the current
outputs are close. The criticisms come in the shape of “Well its just modeling everything as a
So this is the first draft if you will as everything nowadays can be considered an evolutionary draft.
Then again isn’t really all of this just The_Inifinite_Human_Do_Loop?
until then,
(Note: They are all hotlinked)
The Collapse of Complex Societies
Thinking In Systems Donella Meadows
Designing Distributed Systems Brendan Burns
Introduction to Distributed Algorithms
A Pragmatic Introduction to Secure Multi-Party Computation
Reliable Secure Distributed Programming
Dynamic General Equilibrium Modeling
Advanced Information Systems Engineering
Introduction to Dynamic Systems Modeling
Technological Revolutions and Financial Capital
The Structure of Scientific Revolutions
Agent-Based Modelling In Economics
Blog Muzak: Brain and Roger Eno: Mixing Colors
“Wow must be nice, you’re lucky!”
For the successful entrepreneur how many times have you heard this? In the world of “ustas” I used to get really mad about that when someone would say this to me or someone that I knew. I suppose
jealousy and envy cloud the mind.
I have a good friend who is an accomplished martial artist and weapons expert with multiple level black belts in various forms. He is fond of telling his advanced students training up in the hills in
some remote local, “Luck has nothing to do with why you are here.”
There are some who say that having an exit for your startup is complete luck. Ok fine. So be it. We can argue that one later. What I want to discuss today is what happens post-acquisition with
the founders and principles of a startup that has been acquired with the proper frameworks in place. Lets take a recent example of the high profile IPO of Tesla. The PayPal Mafia is a great example
of how to take care of the people that make the company successful. It is also a great example of how the machine works. During my last startup here on the right coast I was continually asked why I
started the company? Well lets see: take an idea, create some software, sell it, make money. Money, Oh yea Money. Actually what I stated was to create a proper venture capital firm much like a
Y-Combinator but with more cash. A good rule of thumb is to not create a company specifically for the company but create it for the next scenario. the paypal founders rewarded their people. Musk
only had 11.5% at the time of the Push Me – Pull You.
PLEASE READ: REWARD THE ELITE PERFORMERS – HEAVILY – THEY WILL NOT LET YOU DOWN!
During the last go around I had a good friend of mine who has amazing experience come up from a semi-retirement from St. Marteen Island. He called me from the harbor club where he landed and said, “I
dont need a salary or contract I just want Equity. We can figure it out later, lets start coding.”
Take that to the bank all day. I gave him multi-digit percentages. So for those that are trying to get something off the ground or after years cannot scale because you cannot find the right people.
Give the blood. Give the equity. It is not luck.
I’ll leave you with a quote:
” The inferior man’s reasons for hating knowledge are not hard to discern. He hates it because it is complex — because it puts an unbearable burden upon his meager capacity for taking in ideas. Thus
his search is always for short cuts. ” ~ H.L. Mencken
Until Then,
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Chapter 9 – Combining Results Using Meta-Analysis
Jonathan J. Deeks, Richard D. Riley, Julian P.T. Higgins
Meta-analysis of controlled trials is usually a two-stage process involving the calculation of an appropriate summary statistic of the intervention effect for each trial, followed by the combination
of these statistics into a weighted average. Two models for meta-analysis are in widespread use. A fixed-effect meta-analysis (also known as common-effect meta-analysis) assumes that the intervention
effect is the same in every trial. A random-effects meta-analysis additionally incorporates an estimate of between-trial variation of the intervention effect (heterogeneity) into the calculation of
the weighted average, providing a summary effect that is the best estimate of the average intervention effect of a distribution of possible intervention effects. Selection of a meta-analysis method
for a particular analysis should reflect the data type, choice of summary statistic (considering the consistency of the effect and ease of interpretation of the statistic), and expected
heterogeneity, and the known limitations of the computational methods.
In this chapter we consider the general principles of meta-analysis, and introduce the most commonly used methods for performing meta-analysis. We shall focus on meta-analysis of randomized trials
evaluating the effects of an intervention, but much the same principles apply to other comparative studies, notably case–control and cohort studies evaluating risk factors. An important first step in
a systematic review of controlled trials is the thoughtful consideration of whether it is appropriate to combine all (or perhaps some) of the trials in a meta-analysis, to yield an overall statistic
(together with its confidence interval) that summarizes the effect of the intervention of interest. Decisions regarding the “combinability” of results should largely be driven by consideration of the
similarity of the trials (in terms of participants, experimental and comparator interventions, and outcomes), but statistical investigation of the degree of variation between individual trial
results, which is known as heterogeneity, can also contribute.
There are currently no corrections for this chapter.
See practicals below.
The exercises and script included in the zip folder take you through the analysis, including synthesis of data and presentation of results.
The file diuretic.xls include data from the meta-analysis published by Collins et al. British Medical Journal, 290, 17-23. They examine the prevention of pre-eclampsia with diuretics.
Datasets streptok.xls and magnes.xls give the effects (from all known trials) of streptokinase and magnesium respectively on the prevention of mortality after acute myocardial infarction.
Author affiliations
Jonathan G. Deeks
Test Evaluation Research Group, Institute of Applied Health Research, University of Birmingham, Birmingham, UK
Richard D. Riley
Centre for Prognosis Research, School of Medicine, Keele University, Newcastle-under-Lyme, UK
Julian P.T. Higgins
Population Health Sciences, Bristol Medical School, University of Bristol, Bristol, UK
National Institute of Health Research, Applied Research Collaboration West, University Hospital Bristol and Weston NHS Foundation Trust, Bristol, UK
How to cite this chapter?
For the printed version of the book
Deeks, J.J., Riley, R.D. and Higgins, J.P.T. (2022). Chapter 9. Combining results using meta-analysis. In: Systematic Reviews in Health Research: Meta-analysis in Context (eds M. Egger, J.P.T.
Higgins and G. Davey Smith), pp 159-184. Hoboken, NJ : Wiley.
For the electronic version of the book
Deeks, J.J., Riley, R.D. and Higgins, J.P.T. (2022). Chapter 9. Combining results using meta-analysis. In: Systematic Reviews in Health Research: Meta-analysis in Context (eds M. Egger, J.P.T.
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