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PyBites Bite 331. Convolution in Neural Networks
Finally a Bite about deep learning! At least, about one of the major building blocks of modern deep learning architectures: the Convolutional Neural Network (CNN or convnet for short).
The main idea of the convolutional layer in a convnet is to apply a filter, also called kernel, to a certain area of the input image. This will produce the activation or filter value for this
area. Next, the filter will be moved by a step size, called stride, on to the next area. Overall, the filter is passed over the entire input image to create a feature map (output matrix) for the
next layer.
This article will help you understand everything you need to complete this task. You do not have to read the part about backpropagation.
Interestingly, convolution is not only useful for deep learning but also the basis of Image Filtering. Previous to deep learning, it was the job of experts to define meaningful filters by hand.
Nowadays, a convnet starts with randomly initialized filters and learns the best filter values during the learning process. The function that you will implement in this bite can be used to apply
a classic filter to an image, so you can blur or sharpen the image or detect edges within it.
Here is what you need to know to implement the convolution operation that will create the feature map:
- Convolution operation: A convolution takes the sum of the element-wise product of the filter and a chunk of the input image. So, for example, if we have a 3x3 filter that is applied to a 3x3
chunk of the input image, the first element of the filter is multiplied by the first element of the image chunk, the second element of the filter is multiplied by the second element of the image
chunk, and so on for all nine elements. Finally, the sum of all element-wise multiplications is taken. This is the output value of the convolution operation and the value of the filter map.
For the example in the figure this means that you have to calculate:
- The filter is moved over the input image by a step length called stride. The default stride value is one, which means that the filter is moved exactly one pixel to the right or down, which
results in a bigger feature map size and a larger overlap of the receptive fields (the are of the filter around its center pixel). A larger stride reduces the feature map size and the overlap of
the receptive fields.
- By default, the filter map will be of smaller size than the input image because the filter is normally larger than 1x1 and even with a stride of one we will calculate 2 values less than the
original size in each dimension. For example, if we have a 9x9 image and a 3x3 filter and move it with a stride of one over the image, we start with the first 3x3 chunk, then go on to the 4th,
5th, 6th, and finally 7th pixel, but because we have a 3x3 filter, the area of the 7th pixel will already cover the 8th and 9th pixel, so we cannot move the filter further to the right.
Thus, we have a feature map size of only 7x7 instead of the original 9x9, our output has gotten smaller! To solve this (we do not want to get smaller outputs after each convolution, at least not
without explicitly saying so), we can use a technique called padding: we add a border around the image with pixels of value 0.
Thus, when moving the filter over our padded image, the feature map size will match the original image. To do so, we have to apply a padding of p=(f-1)/2. For our example, this means a p of 1,
because the filter size is 3. So we add a border of one pixel around our image, increasing its size to 11x11, which will result in a feature map size of 9x9, as the original image.
- Adding everything together, you can calculate the size of the feature map (output matrix) by the following formula:
n_in is the input dimension of the image (we only consider inputs with equal height and width), p is the padding, f is the filter size, and s is the stride.
- Implement the function convolution2D that calculates the convolution of an input image and some filter.
- Input validation: raise an appropriate exception for every violation of the expected input format.
- Assert that both the image and the filter are 2d numpy arrays.
- Assert that both the image and the filter are of quadratic sizes, so the height equals the width.
- Assert that both the image and the filter contain numeric values.
- Assert that the filter has an odd (3x3, 5x5) filter size (dimensionality) so there is a clear center for every filter.
- Assert that the filter size is less equal the image size
- Assert that padding, if given, is an integer greater equal zero
- Assert that stride, if given, is an integer greater equal 1
- If no padding is given, there should be a default padding that preserves the dimensions of the input image.
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Extra Dimensions – How to Think About Them
Matt Strassler 11/07/11
When I’m chatting with a non-physicist, and the topic turns to the possibility that space might have additional dimensions that we aren’t aware of, the most common question that I get is this one:
“How do you folks think about extra dimensions? I can only imagine three and have no idea how you would go beyond that; it doesn’t make any sense to me.”
Well, what we physics folks don’t do (at least, no one I know claims to do this) is visualize extra dimensions. My brain is just as limited as yours, and while that brain effortlessly creates a
three-space-dimensional image of a world that I can move around in, I can’t make it bring to mind a picture of a four- or five-dimensional world any more than yours can. My survival didn’t depend on
being able to imagine anything like that, so perhaps it isn’t surprising that my brain isn’t wired to do it.
Instead, what I do (and what I am pretty sure most of my colleagues do, based on how we all exchange ideas) is develop intuition based on a combination of analogies, visualization tricks, and
calculations. We’ll skip the calculations here, but a lot of the analogies and visualization tricks aren’t that hard to explain.
There are actually two parts to learning to think about extra dimensions.
• The first is easy: learning to represent or describe a world with extra dimensions. You already know how to do this in several different ways, even though you may not realize it — and you can
learn some more.
• The second is harder: learning how things work in a world with extra dimensions. How do you thread a needle in four dimensions rather than three, for instance; would planets orbit a sun with six
space dimensions; would protons form, and would atoms? Here you need to learn tricks that are unfamiliar, thinking about how different a world with only one or two dimensions would be from the
three-dimensional world we know, and working up by analogy.
So I’ll start with helping you learn to represent a world with extra dimensions. To answer that requires we think about how to represent any dimension of any sort. Let’s start at the beginning.
• A zero dimensional world is a point. There’s not much to say about it right now, but we’ll make use of it later.
• But a one-dimensional world is already pretty interesting. Click here to learn more.
• Here’s an article on two-dimensional worlds. A lot more going on here!
• It’s important to avoid confusion about spatial dimensions and the more general concepts of the word “dimension” as used in English and even in mathematics or statistics. A few comments about
that are here.
• And then here are some examples of extra dimensions, emphasizing what the “extra” really means conceptually, and how it could be that the world has spatial dimensions of which we are completely
After these articles, you can start to learn how we might figure out that extra dimensions actually exist, by clicking here.
10 Responses
1. One thing that you can do to bring a union to your company, particularly to the
corporate level, is by sharing unique corporate Christmas gifts.
If you are giving wine or a wine gift basket, it is better
to ask permission or get details regarding whether the person actually drinks (or might
drink too much and not want temptation). Most of the suppliers offer home delivery feature within a few days of placing order.
2. would any particle,or system existing in our known observable dimension,,,display the same properties in multiple dimensions???in other words could the same paricle appear in multiple
dimensions,,,with the same Me??all at once,,,or would any system in one dimension,be missing from another…???and if subatomic particles,are multidimentional,and so far all appear as +me in our
dimension,,,what would that imply about other unobservable dimensions?
1. You should read the following article and the articles that it links to
3. In reference to ” …from the three-dimensional world we know…”.
I thought we lived in a 4-D world (time) it’s linear and has length.
1. We’re both right: we live in a world with 3 spatial dimensions, 1 time dimension, and, a total of 4 space-time dimensions. Obviously one has to be clear about whether one is referring to
spatial dimensions only or to the full space-time. These types of confusions have to be carefully avoided even by professionals. Here, in these articles, I have rigorously stuck to talking
about the number of spatial dimensions only. If you want to talk about space-time that’s fine — just add one.
4. Okay, let me see if I get this…The way in which I picture three dimensional space is LWH or the world around us. When I add a fourth dimension, I have always “pictured” a seed (a point on a
line). As that seed grows, it gains width, height and lemgth. And as my imaginary tree is a Semper virens, it also gains age. Though I like the threading the needle. Can we look at the whole of
threading a needle as being a 3 dimensional space, while the actual process of feeding the thread through the eye as the fourth?
1. Hmm… I don’t think I understand enough what you’re saying here to reply usefully…
5. How does curvature show up? By geodesic convergence (attraction!), or geodesic divergence (repulsion!). Thus we can tell people who are not in the know this: each time one experiences a force,
one can suspect there is/are extra dimension(s) at work.
6. Can I hope for a post on Hilbert space?
7. Hi Prof. Matt Strassler,
Following along is hard……triangles on a sphere or saddle are ways off for sure, ….but yes following along systemically needs to be understood by myself. Would this sort of be like the evolution
of geometry in concert with building a dimensional world??
“A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean
geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry. ”
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N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number
« N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number»
“A concrete representation of a 4D Klein bottle has been desired by many but has never been presented. So, I decided to dive into the Klein bottle. Working with the Klein bottle was my first
opportunity to practice with this system. To my surprise, the ease with which it allowed me to create 4D Klein bottles was remarkable. The 4D Klein bottles were generated smoothly without the
slightest hitch. My video animations of the rotating 4D Klein bottle in 4D space, as well as the 3D slices ascending in the 4D space, were also computed effortlessly.”
Abstract: While a 3D complex number would be useful, it does not exist. Recently, I have constructed the N-complex number, which has demonstrated high efficiency in computations involving
high-dimensional geometry. The N-complex number provides arithmetic operations and polar coordinates for N-dimensional spaces, akin to the classic complex number. In this paper, we will explain how
these systems work and present studies on 4D Klein bottles and hyperspheres to illustrate the advantages of these systems.
The classic complex number system is a remarkable mathematical tool because it allows for the addition and rotation of vectors in two-dimensional space, following the same rules as real numbers for
addition and multiplication. However, in three-dimensional space, it is impossible to manipulate vectors with similarly intuitive arithmetic operations because such a system does not currently exist.
The development of a three-dimensional complex number system, analogous to the two-dimensional one, would represent a significant advancement in mathematics.
In 2022, I constructed a system of complex numbers for spaces with any number of dimensions, which I call the “N-complex number system.” Edgar Malinovsky used this system to create many beautiful 3D
objects (see «Rendering of 3D Mandelbrot, Lambda and other sets using 3D complex number system»[4]). Figure 1 shows the 3D Mandelbrot set he created. Computing 3D fractal objects is very
time-consuming; he would not have succeeded in this work without the 3-complex number system. His work demonstrates that the 3-complex number system significantly accelerates computations in 3D
I have worked on 4D Klein bottles by extending a 3D Klein bottle (see Figure 2) into 4D space. I rotated the 4D Klein bottles in 4D space and showcased the rotation in my video animation “Observing a
4D Klein Bottle in 4-Dimension” [5]. This work would have been impossible without the 4-complex number system. In addition to N-complex numbers, the new system provides a polar coordinate system for
N-dimensional spaces, which was previously missing in mathematics.
See « N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number» for more detail.
Kuan Peng
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June 2016 LSAT Question 23 Explanation
Which one of the following, if substituted for the condition that Waite's audition must take place earlier than the t...
Could we please have a game setup and also explanation for the last question in this game, thanks.
The game requires us to determine the order of the auditions and whether they are recorded or not. The game involves six singers - K L T W Y Z. Two of their auditions are recorded - K & L and four
not recorded - T W Y Z.
__ __ __ __ __ __
The following rules apply:
(1) The 4th audition cannot be recorded, the fifth must be.
This rule allows us to infer that the fifth audition must be either K or L, and 4th is neither K nor L.
R X
U X
__ __ __ __ K/L __
(2) W audition must take place earlier than two recorded auditions.
This rule tells us that W must audition before both K & L, thus W cannot be #6. and audition #1 cannot be recorded since W (unrecorded) must precede both recorded auditions.
W > K & L
(3) K audition must take place earlier than T audition.
This rule tells us that K cannot be #6, and allows us to infer that #5 and #6 auditions cannot both be recorded because it would leave no space for T.
Combined with the previous rule, another interesting inference is that W can only be #1 and #2 since we know that #4 and #6 are unrecorded auditions, the earliest the second recorded audition could
be is #2 -K or L, hence we can infer that W must be #1 or #2, and K/L (second recorded audition) must be #2 or #3.
We can also infer that if K is #5, then T must be #6 and that T cannot be #1 or #2 because T must precede K and the earliest K could be is #2.
R X
U X X X
/W /W /K/L __ K/L __
~T ~T ~K
(4) Z audition must take place earlier than Y audition.
Z> Y
This rule allows us to infer that Z cannot be #6 and Y cannot be #1. Combined with the previous rules we can also infer that #1 must be either W or Z, and #6 must be T or Y.
R X
U X X X
Z/W /W /K/L __ K/L T/Y
~T ~T
~Y ~Z
Question six asks us which of the following if substituted for the condition that W audition must take place earlier than the two recorded auditions would have the same effect in determining the
order of the auditions?
Looking back at the rules we see that this condition tells us that W can only be #1 or #2 because the other rules allow us to infer that auditions #4 and #6 are unrecorded, so the second recorded
audition aside from #5 could only be #2 or #3, thus W must be #1 and or #2 respectively, and it also allows us to infer that the first audition cannot be recorded (it cannot be L/K). We need a
substitute rule that captures both of these inferences. (A) tells us that Z is the only one that can take place earlier than W - what it means in practice that W is either #1 or Z is #1 and W is #2,
combining both of the above inferences.
Let me know if this helps and if you have any further questions.
Why does B not work in the rule substitution question? W has to go 1 or 2. If W is 1, then Z is 2 and if W is 2, Z must be 1. Thank you.
I get it now. Space 1 is restricted to w/z; but z has some flexibility after that; e.g., W K T/Z Z/T L Y does not violate any of the rules (W-KT; W-K; Z-Y; K OR L IN 5 AND K OR L NOT IN 4)
We would be removing the rule that forces w to be so constrained, and only saying it must be next to Z doesn't limit W as much as the initial rule.
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Gravity Wave Signals are being Analyzed to Detect Gravitational Memory Effect
Oct 16, 2023
An Ongoing Meta-analysis of Gravitational Wave Signals may soon Prove that Space Remembers: permanent memory imprints in spacetime may soon be detected, which will be a validation of Nassim Haramein
and our research team’s prediction that space has the property of memory, in which we described how the informational imprint of memory in space is what holographically generates time—that is to say
that 4D spacetime is a holographic projection of a 3D voxel information network—as well as ordering properties underlying dynamics of organized matter. The gravitational wave memory effect is a
prediction of general relativity, and physicists have devised a test of this interesting spacememory effect via a meta-analysis of gravitational wave detector data. The presence of memory effects in
gravitational wave signals not only provides the chance to test an important aspect of general relativity, but also represents a potentially non-negligible contribution to the waveform for certain
gravitational wave events. As well, memory properties of space will have far-reaching implications, from probing theories of quantum gravity and unified physics to potential applications in
telecommunications technologies.
By: William Brown, scientist at the International Space Federation
Gravitational waves, oscillations or more colloquially “ripples” in the substantive medium of space-time, were first predicted by Albert Einstein's theory of general relativity over a century ago.
These waves may be quite ubiquitous, what is known as the gravitational wave background (GWB), similar to the cosmic microwave background (CMB). The gravitational waves that have been detected
to-date are generated by some of the most violent and energetic processes in the cosmos, such as the merging of black holes and neutron stars. The ability to detect and analyze gravitational waves
opens an entire new observational field potentially enabling scientists to probe into some of the most enigmatic regions of our universe, like the surface of Neutron stars and the event horizons of
black holes. Gravitational waves could even offer insight into fundamental questions in physics; one possible probe of gravitational waves is to see if space itself has memory, referred to as the
gravitational memory effect. As a lesser-known phenomenon related to gravitational waves, the gravitational memory effect has recently gained attention within the scientific community, and proposals
have been formulated to analyze gravitational waves to detect potential spacememory effects. This article delves into the intriguing concept of gravitational memory and its implications for our
understanding of the fundamental forces of the universe.
Understanding Gravitational Waves
Gravitational waves are disturbances in the curvature of space-time, propagating as waves at the speed of light. They are generated when massive celestial objects accelerate asymmetrically, leading
to oscillations in the gravitational field that radiate outwards. These ripples carry information about their origins and travel across the universe, allowing astronomers to probe the cosmos in a
unique way.
The Gravitational Memory Effect
The gravitational memory effect is a manifestation of the persistent change in relative distances between test particles due to the passage of gravitational waves. Unlike the oscillatory nature of
gravitational waves, the memory effect produces a permanent change in the separation of objects in its path. This phenomenon is a consequence of the non-linear nature of gravity in Einstein's theory
of general relativity.
Types of Gravitational Memory
The gravitational memory effect can be classified into two types: positive and negative. A positive memory results in an increase in the separation of test particles, while a negative memory leads to
a decrease. These changes occur along the direction of propagation of the gravitational wave.
History of Empirical Analysis of Gravitational Waves and Prediction of Memory Effect
Joseph Weber, an American physicist, claimed to have discovered gravitational radiation in the 1960s [1]. He developed highly sensitive detectors known as Weber bars, which were designed to detect
tiny vibrations caused by passing gravitational waves. His claim was based on the observations made using these detectors.
Weber's work garnered significant attention and excitement within the scientific community and the media. The potential discovery of gravitational radiation was groundbreaking, as it would have
provided experimental confirmation of a fundamental prediction of Albert Einstein's theory of general relativity.
Weber's experiments involved aluminum cylinders, or "Weber bars," which were designed to resonate at specific frequencies when exposed to gravitational waves. Weber claimed to have detected
gravitational waves emanating from various cosmic events, including supernova remnants and binary star systems.
However, over time, skepticism regarding the validity of Weber's results began to grow. Other researchers attempted to replicate his findings but faced challenges in reproducing the results with the
same level of consistency and statistical significance. Physicists such as Yakov Zeldovich, a prominent Soviet theoretical physicist and cosmologist—who was instrumental in the formulation of how
wave resonance can convert electromagnetic waves to gravitational waves—ran calculations that explicitly demonstrated how Weber’s bars would need to be 100 million times more sensitive than reported
in order to detect even the largest theoretically possible gravitational wave sources, like from a super-dense highly-interacting cluster of stars.
However, the analysis utilized in proving Weber wrong led to a remarkable prediction. In the 1970’s in collaboration with his colleague Alexander Polnarev, Zeldovich predicted that passing
gravitational waves should result in a permanent change in the relative separation of test particles, like a record or memory of the passing gravitational radiation. Their work laid the foundation
for the theoretical understanding of the gravitational memory effect, emphasizing its potential significance in the study of gravitational waves and the implications for fundamental physics.
Zeldovich and Polnarev's theoretical analysis provided a framework for subsequent researchers to explore this intriguing phenomenon in more detail.
Zeldovich's insights into the behavior of gravitational waves and their impact on spacetime were instrumental in advancing our understanding of how these waves can induce lasting alterations in the
geometry of space. This pioneering work contributed to the development of experimental efforts to detect and study the gravitational memory effect.
For Weber, several factors contributed to the doubts surrounding his claims:
1. Replication Issues: Other research groups had difficulty replicating Weber's results, leading to concerns about the reliability and reproducibility of his experimental findings.
2. Statistical Significance: The statistical significance of Weber's results was a subject of debate. The detected signals were often near the threshold of detectability, raising questions about the
reliability of the data.
3. Noise and Interference: The sensitivity of Weber's instruments made them susceptible to various sources of noise and interference, including seismic activity and thermal fluctuations.
Distinguishing true gravitational wave signals from noise proved to be a significant challenge.
4. Lack of Correlation: Weber's observations did not consistently correlate with expected astrophysical events that should have produced gravitational waves, casting doubt on the legitimacy of his
As the years progressed, subsequent advancements in gravitational wave detection technologies, such as the Laser Interferometer Gravitational-Wave Observatory (LIGO), provided more accurate and
reliable evidence for the existence of gravitational waves. LIGO's success in directly detecting gravitational waves in 2015 through the observation of merging black holes effectively discredited
Weber's claims.
In retrospect, while Joseph Weber's work was influential in laying the groundwork for gravitational wave detection, subsequent advancements in technology and the success of more precise experiments
have reshaped our understanding of gravitational waves and reinforced the accuracy of Einstein's general theory of relativity.
Black hole Memory Effect
The investigation of Weber’s claim of gravitational wave detection and subsequent elucidation and prediction of gravitational wave memory effects by Yakov Zeldovich and Alexander Polnarev led them to
another related prediction of a similar effect occurring in the spacetime geometry of black hole’s event horizons—a “black hole memory effect”. This effect is a consequence of the non-linear nature
of general relativity and arises when gravitational waves passing through a region of space near a black hole causes a distortion in the geometry of spacetime. This distortion leads to a change in
the orbits and dynamics of particles in the vicinity of the black hole.
Even after the gravitational waves have passed, this change persists, creating a lasting memory in the structure of spacetime. In the black hole memory effect, it can be considered that an imprint is
left on the event horizon of a black hole even after the event, like the passage of gravitational waves, has occurred. The propensity for memory to be veritably encoded in the spacetime structure
around a black hole has important significance for some of the most outstanding problems currently in physics, such as the information loss paradox. Conventional theory regards the event horizon as
the demarcation surface in spacetime around a black hole on the outside of which no information can be retrieved from the internal volume. So, in conventional theory if an object falls into the black
hole, once pass the event horizon all information about the object is thought to be irretrievably lost. Although we have discussed numerous ways that this information loss paradox is resolved, the
black hole memory effect, related to gravitational memory, is another possible solution since there may be memory imprints remaining in the structure of spacetime around a black hole preserving the
information about any objects whose trajectories have taken them within the event horizon.
In more technical terms, the black hole memory effect is related to the so-called "asymptotic symmetries" of gravity. These are transformations that affect the geometry of spacetime at infinity and
can leave a permanent mark on the space surrounding a black hole. It is an important area of study in gravitational wave physics, helping researchers understand the lasting impact of gravitational
waves on the fabric of the universe and its implications for astrophysics and fundamental physics.
Detecting the Non-linear Gravitational Memory Effect
By combining data from the gravitational wave detectors—large highly sensitive laser interferometers—LIGO, the Virgo detector in Italy, and the Kamioka detector in Japan it may be possible to tease
out a tell-tale signal of gravitational wave memory effects from the meta-analysis of data [2]. Such analysis is underway, with new observations rolling in every week, pushing the current total to
over 100 and counting. At this rate, experimentalists hope they will detect gravitational memory within a few years. There is as well recent proposals to detect the gravitational memory effect in
LISA using triggers from ground-based detectors, which will obviate the signal-to-noise (SNR) problem that arises because the memory effect is one or two orders below the detector noise background
level [3], and to use the TianQin proposed space-based gravitational wave observatory designed to detect and study gravitational waves with high precision and sensitivity [4].
TianQin is conceived as a space-based gravitational wave observatory that aims to observe gravitational waves with lower frequencies (millihertz to hertz range). This complements ground-based
observatories like LIGO and Virgo, which detect higher-frequency gravitational waves. The TianQin observatory relies on a constellation of three spacecraft forming an equilateral triangle in space.
These spacecraft will be equipped with highly sensitive lasers and accelerometers to measure the tiny displacements caused by passing gravitational waves. TianQin's lower frequency range allows it to
detect gravitational waves from different sources, such as massive binary systems (e.g., merging supermassive black holes), extreme mass ratio inspirals (e.g., a small compact object orbiting a
massive black hole), and other astrophysical events.
[TianQin is a proposed space-based gravitational wave detector, like the Laser Interferometer Space Antenna (LISA) pictured above in Artist's impression. Credit: ESA]
Being in space, TianQin is free from the seismic noise and other disturbances that can affect ground-based detectors. This enables it to detect lower frequency gravitational waves with greater
sensitivity and accuracy. During its five years of operation, for the gravitational wave signals that could be detected by TianQin, about 0.5–2.0 signals may contain displacement memory effect with
SNR ratios greater than 3. This suggests that the chance for TianQin to detect the displacement memory effect directly is low but not fully negligible.
Implications and Application
Studying the gravitational memory effect allows physicists to probe the fundamental nature of gravity and its behavior in extreme conditions. This phenomenon holds promise in enhancing our
understanding of the intricate interplay between gravity and other fundamental forces in the universe.
Gravitational waves, including their memory effect, provide a powerful tool for studying astrophysical phenomena. They offer insights into the dynamics of compact binary systems, the properties of
merging black holes, neutron stars, and the early universe. Gravitational memory can be a valuable addition to our observational toolkit for understanding cosmic events.
1. Fundamental Physics and Gravity Understanding: The discovery and study of the gravitational memory effect could significantly contribute to our understanding of fundamental physics, particularly
gravity. It provides an avenue to test and verify the non-linear nature of gravitational interactions, shedding light on the complexities of the gravitational field.
2. Validation of General Relativity: Gravitational memory serves as an additional validation of Einstein's theory of general relativity, which has already been remarkably successful in explaining
the behavior of gravity and the curvature of space-time. Confirming the gravitational memory effect would further bolster the credibility of the theory.
3. New Gravitational Wave Detection Techniques: Successfully detecting and characterizing the gravitational memory effect necessitates the development of sensitive measurement techniques. The
pursuit of such techniques could lead to advancements in gravitational wave detection technologies, potentially enhancing our ability to study other aspects of these waves and the events that
generate them.
4. Insights into Extreme Astrophysical Events: The study of gravitational memory can provide valuable insights into the nature of extreme astrophysical events that generate gravitational waves, such
as black hole mergers and neutron star collisions. Understanding the gravitational memory associated with these events could deepen our understanding of their dynamics and the properties of the
celestial bodies involved.
5. Cosmology and Early Universe Dynamics: Gravitational waves, including their memory effect, offer a unique observational tool to study the early universe and its dynamics. Insights gained from
studying the gravitational memory could help researchers develop a more accurate and detailed understanding of the early cosmos, including its formation and evolution.
6. Technological Innovation and Applications: The pursuit of gravitational memory research may drive technological advancements in instrumentation and measurement devices, potentially leading to
applications beyond astrophysics. These innovations could find applications in precision sensing technologies and possibly influence fields such as telecommunications and navigation. As discussed
in our previous article Gravity Control via Wave Resonance [4], high-frequency gravitational waves could be generated and utilized for absolutely unimpeded high-fidelity wireless communications,
and understanding the subtle yet permanent perturbations induced in spacetime geometry by these energetic oscillations could have highly interesting corollary applications.
Unified Science- in perspective:
The near-undetectable perturbation effect of gravitational wave memory is a relatively subtle indication of the memory attribute of space arising from the vacuum’s substantive properties as a
physical medium, but it is not the only mechanism by which the memory properties of space can potentially be manifest. For example, gravitational interaction is proposed to mediate such
quintessentially quantum mechanical behavior as entanglement, such as in the ERb=EPR holographic correspondence conjecture by Susskind and Maldacena [5]. As such, the entanglement nexus of
spacememory may be integral in encoding naturally occurring qubit memory states of interacting systems all around us [6].
In terms of what investigating gravitational waves can help us to learn about astrophysics, cosmology, and quantum gravity, it is important to note that within our unified physics approach
gravitational waves are not just being generated by massively high-energy events like black hole mergers. We predict that gravitational waves will be a quite ubiquitous phenomenon occurring at many
scales. Gravitational waves probably emanate from fundamental particles like the proton and are even generated at the Planck scale. The role of such oscillatory radiation of spacetime itself will
form a significant contribution to the energy dynamics of material systems, and the complex interplay of many-bodied radiative sources generating constructive and destructive interference wave forms
and wave resonance may very well be a significant factor in the holographic memory properties of space.
The gravitational memory effect, a consequence of gravitational waves, remains a fascinating and relatively unexplored aspect of general relativity. As gravitational wave detection technology
advances, scientists are eager to unravel the mysteries surrounding this phenomenon. Unveiling the gravitational memory effect not only expands our understanding of the fundamental forces shaping the
cosmos but also holds promise for a deeper comprehension of the intricate dance of the universe.
[1] J. Weber, “Evidence for Discovery of Gravitational Radiation,” Phys. Rev. Lett., vol. 22, no. 24, pp. 1320–1324, Jun. 1969, doi: 10.1103/PhysRevLett.22.1320.
[2] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman, and Y. Chen, “Detecting Gravitational-Wave Memory with LIGO: Implications of GW150914,” Phys. Rev. Lett., vol. 117, no. 6, p. 061102, Aug. 2016,
doi: 10.1103/PhysRevLett.117.061102.
[3] S. Sun, C. Shi, J. Zhang, and J. Mei, “Detecting the gravitational wave memory effect with TianQin,” Phys. Rev. D, vol. 107, no. 4, p. 044023, Feb. 2023, doi: 10.1103/PhysRevD.107.044023.
[4] W. Brown, “Gravity Control via Wave Resonance.” Sep. 2023. Accessed: Oct. 16, 2023. [Online]. Available: https://www.resonancescience.org/blog/gravity-control-via-wave-resonance
[5] B. Kain, “Probing the Connection between Entangled Particles and Wormholes in General Relativity,” Phys. Rev. Lett., vol. 131, no. 10, p. 101001, Sep. 2023, doi: 10.1103/PhysRevLett.131.101001.
[6] W. Brown, “Unified Physics and the Entanglement Nexus of Awareness,” NeuroQuantology, vol. 17, no. 7, pp. 40–52, Jul. 2019, doi: 10.14704/nq.2019.17.7.2519.
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Graph Representation in Memory
Graphs can be represented in memory as objects and pointers, adjacency lists and adjacency matrices.
CLRS page 589, DDIA location 1392.
Graph Input Size
The input size of a graph is given by two parameters: the number of vertices (n) and the number of edges (m). For a connected, undirected graph with n vertices, the number of edges ranges from n - 1
and n⋅(n - 1)/2.
Adjacency Lists
An adjacency list graph is a data structure that keeps track of vertices and edges as independent entities. The structure can be used to store both directed and undirected graphs.
In its most generic form, it has an array or a list of vertices and an array or a list of edges. These two arrays cross-reference each other. Each edge has two pointers, one for each of its vertex
endpoints. For directed graphs we keep track of which one is the head and which one is the tail. Each vertex points to all of the edges of which it is a member. For directed graphs, a vertex keeps
track of each of the edges it is the tail.
A practical implementation (see CLRS page 590) consists of an array adj of n lists, where the n is the number of vertices. Each array element corresponds to one of the graph's vertices, numbered from
0 to n-1, and contains an adjacency list for that vertex: the list that contains all the vertices in the graph for which there's an edge from the current vertex. That is, adj[i] contains all the
vertices adjacent to i in the graph.
The space requirements for this data structure is Θ(n + m). An adjacency list representation provides a memory-efficient way of representing sparse graphs, so it is usually the method of choice.
Adjacency lists are very well suited for graph search.
Adjacency Lists and Undirected Graphs
For an undirected graph, the adjacency list for a given vertex v contains all the vertices u for which there is an edge (v, u).
⚠️ Because the edge is undirected, it counts as valid edge for both vertices v and u, so v will also show up in the adjacency list for vertex u. Each edge is represented twice. For a directed graph,
that is not the case, as u → v ≠ v → u.
Adjacency Lists and Directed Graphs
For a directed graph, an adjacency list for a vertex v contain all vertices u for which there is a directed edge v → u, incident from v. v is the tail of the edges, and the vertices u are the heads.
Such a structure allows returning head vertices for all directed edges for which a vertex v is the tail in an O(1).
However, if we need all the tails for which a given vertex is the head, all adjacency lists must be scanned in an O(m) operation.
Adjacency Lists in Java
Adjacency Matrices
The edges of a graph are represented using a matrix. This representation works both with directed and undirected graphs. The n x n square matrix is denoted by A. n is the number of vertices in the
graph. The semantics is A[ij]=1 if and only if there's an edge between vertices i and j in the graph. It can be extended for parallel arcs and weighted graphs. For parallel arcs, A[ij] denotes the
number of edges. For weighted graphs, A[ij] denotes the weight of the edge. For directed graphs, if the arc is directed from i to j, A[ij]=+1 and if it is directed from j to i, A[ij]=-1.
An adjacency matrix requires space quadratic in the number of vertices Θ(n^2). For dense graphs this is fine but for sparse graphs, it is wasteful. Adjacency matrix representation is preferred for
dense graphs or when we need to tell quickly if there is an edge connecting two given vertices.
The same graph algorithms that are used on adjacency lists can be performed on adjacency matrices, but they may be somewhat less efficient. The adjacency list representation allow iteration over the
neighbors (adjacent nodes). In the adjacency matrix representation, you will need to iterate through all nodes to identify a node's neighbors.
More details in CLRS page 590.
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[EM] How to find the voters' honest preferences
Forest Simmons fsimmons at pcc.edu
Sat Sep 7 17:50:24 PDT 2013
The following method makes use of two ballots for each voter. The first
ballot is a three slot ballot with allowed ratings of 0, 1, and 2. The
second ballot is an ordinal preference ballot that allows equal rankings
and truncations as options.
The three slot ballot is used to select two finalists: one of them is the
candidate rated at two on the greatest number of ballots. The other one is
the candidate rated zero on the fewest ballots.
The runoff between them is decided by the voters' pairwise preferences as
expressed on the three slot ballots (when these finalists are not rated
equally thereon), or (otherwise) on the ordinal ballots when the three slot
ballot makes no distinction between them.
[Giving priority to the three slot pairwise preference over the ordinal
ballot preferences is necessary to remove the burial incentive.]
Note that there is no strategic advantage for insincere rankings on the
ordinal ballots.
(1) What are some near optimal strategies for voters to convert their
complete cardinal ratings into three slot ratings in this context?
(2) We have a "sincere approval" method of converting cardinal ratings into
two slot ballots. What is the analogous "sincere three slot" method?
[Sincere approval works by topping off the upper ratings with the lower
ratings; think of the ratings as full or partially full cups of rating
fluid next to each candidate's name. If you rate a candidate at 35%, then
that candidate's cup is 35% full of rating fluid. Empty all of the rating
fluid into one big pitcher and use it to completely fill as many cups as
possible from highest rated candidate down. Approve the candidates that
end up with full cups. This is called "sincere approval" because
generically (and statistically) the total approval (over all voters) for
each candidate turns out to be the same as the total rating would have
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14.7 Error-correcting conditions | Introduction to Quantum Information Science
14.7 Error-correcting conditions
We can summarise the notion of a stabiliser code that we have defined rather succinctly: everything is determined by picking a stabiliser group, i.e. an abelian subgroup \mathcal{S} of the Pauli
group \mathcal{P}_n that does not contain -\mathbf{1}. From this, we define the codespace to be the stabiliser subspace V_\mathcal{S}, the codewords to be a choice of basis vectors, the logical
operators to be the cosets of \mathcal{S}\triangleleft N(\mathcal{S}), and the error families to be the cosets of N(\mathcal{S})\triangleleft\mathcal{P}_n.
By setting up some ancilla qubits and constructing appropriate quantum circuits^305, we can enact any logical operator in such a way that we also measure an error syndrome, which points at a specific
error family. But unlike in our study of the Steane code in Section 14.3, we can no longer simply apply the corresponding operator to fix the error, because the error is a whole coset — it contains
many individual Pauli operators.
To fix an example to keep in mind, we return yet again to the three-qubit code. In Figure 14.8 we draw a diagram grouping together all the elements of \mathcal{P}_3 into the coset structure induced
by \mathcal{S}=\langle ZZ\mathbf{1},\mathbf{1}ZZ\rangle. This is analogous to the diagrams that we saw back in Exercise 7.8.2, but with the simplification of ignoring phase.^306
As we can see by looking at Figure 14.8, if we somehow measure an error syndrome pointing to the error family [X\mathbf{1}\mathbf{1}], for example, then there are 16 possible errors that could have
occurred! We said that the stabiliser formalism would be better than our previous approach, so why do things seem so much worse now? Well, we are forgetting one key assumption that we made before
that we have yet to impose in the stabiliser formalism: up until now, we have only studied single-qubit errors. Thinking back to our introduction of the three-qubit code in Section 13.1, we were
specifically trying to deal with single bit-flip errors, i.e. only X\mathbf{1}\mathbf{1}, \mathbf{1}X\mathbf{1}, and \mathbf{1}\mathbf{1}X (as well as the trivial error \mathbf{1}\mathbf{1}\mathbf
{1}, which we must not forget about, as we shall see). If we look back at Figure 14.8 with this in mind, we notice something particularly nice: each of these single X-type errors lives in a different
error family, and each error family contains exactly one of these errors.
In other words, if we assume that only single bit-flip errors can occur, then the stabiliser formalism describes errors in exactly the same way as before, since the error families are in bijection
with the physical errors. But here is where the power of the stabiliser formalism can really shine through, since it allows us to understand what type of error scenarios our code can actually deal
with in full generality. That is, rather than thinking about a code as something being built to correct for a specific set of errors, the stabiliser formalism lets us say “here is a code”, and then
ask “for which sets of errors is this code actually useful?”. The answer to this question lies in understanding how any set of physical errors is distributed across the error families, and we can
draw even simpler versions of the diagram in Figure 14.8 to figure this out.
Returning to the scenario where we assume that only single bit-flip errors can occur, we can mark the corresponding physical errors in Figure 14.8 — namely \mathbf{1}, X\mathbf{1}\mathbf{1}, \mathbf
{1}X\mathbf{1}, and \mathbf{1}\mathbf{1}X — with a dot. We do this in Figure 14.9, which is the first of many more diagrams of this form, which we call error-dot diagrams. Although we are working
with the specific example of the three-qubit code in mind, these diagrams are meant to be understood more generally as applying to any stabiliser code. As we shall soon see, we don’t really need to
worry about making sure that we have the right number of rows in each small rectangle (i.e. the right number of cosets of \mathcal{S} inside N(\mathcal{S})), and in some sense we don’t even really
need to worry about what the physical errors are.
As we said above, if each error family (i.e. coset) contains exactly one physical error (i.e. Pauli operator), then we already know how to apply corrections based on the error-syndrome measurements.
In terms of the diagram in Figure 14.9, this rule becomes rather simple: if each error family contains exactly one dot, then we can error correct.
But can we say something more interesting than this? Well, let’s consider what happens if we have a diagram that looks like this:
That is, we’re considering a scenario where there are two possible physical errors that can occur for a physical error syndrome. In the example of the three-qubit code, we’re looking at the scenario
where any single bit-flip error can occur, but also the operator YZ\mathbf{1} might affect our computation, enacting a bit-phase-flip on the first qubit and a phase-flip on the second. What would
then happen if we measured the error syndrome |01\rangle? We know (from Section 13.7) that this corresponds to the error family [X\mathbf{1}\mathbf{1}], but both X\mathbf{1}\mathbf{1} and YZ\mathbf
{1} live in this coset, so we’re back to the question posed at the end of Section 14.5: how do we pick which operator to use to correct the error?
Here’s the fantastic fact: in this case, it doesn’t matter! Say we pick X\mathbf{1}\mathbf{1}, but the physical error that had actually affected our qubits, originally in some encoded state |\psi\
rangle, was YZ\mathbf{1}. Then by applying the “correction” X\mathbf{1}\mathbf{1} our qubits would be in the state (X\mathbf{1}\mathbf{1})(YZ\mathbf{1})|\psi\rangle = (ZZ\mathbf{1})|\psi\rangle
(where, once again, we ignore global phases). But |\psi\rangle is, by construction, some codeword, which exactly means that it is stabilised by ZZ\mathbf{1}, and so (X\mathbf{1}\mathbf{1})(YZ\mathbf
{1})|\psi\rangle = |\psi\rangle. We can fully generalise this to improve upon the previous rule: if all the dots in any given error family are all in the same row, then we can perfectly error correct
To prove this, we just return to the definition of cosets and the properties of the Pauli group.^307 If two physical errors P_1 and P_2 are in the same row inside some family E\cdot N(\mathcal{S}),
then by definition they both come from the same coset P\cdot\mathcal{S}, i.e. \begin{aligned} P_1 &= EP'_1 \\P_2 &= EP'_2 \end{aligned} where P'_1,P'_2\in P\cdot\mathcal{S}. Then EP corrects both P_1
and P_2, since (again, we ignore global phase, which means that Pauli operators commute) \begin{aligned} (EP)P_i &= (EP)(EP'_i) \\&= E^2 PP'_i \\&= PP'_i \in\mathcal{S} \end{aligned} because Pauli
operators square to \mathbf{1}, and P'_i\in P\cdot\mathcal{S}.
We also get the converse statement from this argument: if any family contains dots in different rows, then we cannot error correct. This is because we need EP to correct for some errors, and some
different EP' to correct for others, and we have no way of choosing which one to correct with when we measure the error syndrome for E without already knowing which physical error took place.^308
So is this the whole story? Almost, but one detail is worth making explicit, concerning maybe the most innocuous looking error of all: the identity error family. Consider a scenario like the
In the case of the three-qubit code, this corresponds to the possible physical errors being single phase-flips Z\mathbf{1}\mathbf{1}, \mathbf{1}Z\mathbf{1}, and \mathbf{1}\mathbf{1}Z. But here we see
how misleading it is to omit mention of the identity error \mathbf{1}\mathbf{1}\mathbf{1}, because the single phase-flips all live in the same N(\mathcal{S}) coset as \mathbf{1}\mathbf{1}\mathbf{1},
but different \mathcal{S} cosets. That is, they are in the same error family, but a different row. By our above discussion, this means that we cannot correct for these errors — indeed, if we measure
the error syndrome corresponding to “no error”, then we don’t know whether there truly was no error or if one of these single phase-flips happened instead. To put it succinctly, we nearly always make
the assumption that no errors at all might occur, which is exactly the same as saying that the trivial error \mathbf{1} might occur. This means that we cannot correct for any errors that are found in
the normaliser of \mathcal{S} but not in \mathcal{S} itself. Although this is technically a sub-rule of the previous rule, it’s worth pointing out explicitly.
An error-dot diagram describes a perfectly correctable set of errors if and only if the following two rules are satisfied:
1. In any given error family, all the dots are in the same row.
2. Any dots in the bottom error family are in the bottom row.
(The second rule follows from the first as long as the scenario in question allows the possibility for no errors to occur.)
Of course, we can state these conditions without making reference to the dot-error diagrams, instead using the same mathematical objects that we’ve been using all along. Proving the following version
of the statement is the content of Exercise 14.11.12.
Let \mathcal{E}\subseteq\mathcal{P}_n be a set of physical errors such that \mathbf{1}\in\mathcal{E}. Then the stabiliser code defined by \mathcal{S} can perfectly correct for all errors in \mathcal
{E} if and only if E_1^\dagger E_2 \not\in N(\mathcal{S})\setminus\mathcal{S} for all E_1,E_2\in\mathcal{E}.
Sometimes we might not specify that \mathbf{1}\in\mathcal{E}, but this is always meant to be assumed. In other words, the error correction scenario specified by \mathcal{E} is the following: any one
single operator in \mathcal{E} could affect our state, or no error at all could happen. In particular, we are not considering that multiple errors could happen; if we want to allow for this, then we
should do something like replace \mathcal{E} with the group that it generates.
You might notice that we’ve been sometimes been saying “perfectly correctable” instead of just “correctable”. This is because there might be scenarios where we are happy with being able to correct
errors not perfectly, but instead merely with some high probability.
These dot-error diagrams are also able to describe more probabilistic scenarios. We have been saying “single-qubit errors”, but we could just have well have been saying “lowest-weight errors”, and
then the assumption that errors are independent of one another means that higher-weight errors happen with lower probability. But the stabiliser formalism (and thus the error-dot diagrams) don’t care
about this “independent errors” assumption! What this means is that we could refine our diagrams: instead of merely drawing dots to denote which errors can occur, we could also label them with
specific probabilities. So we could describe a scenario where, for example, one specific high-weight error happens annoyingly often.
One last point that is important for those who care about mathematical correctness concerns our treatment of global phases.^309 We do need to care about global phases in order to perform
error-syndrome measurements, but once we have the error syndrome we can forget about them. In other words, we need the global phase in order to pick the error family, but not to pick a representative
within it.
305. We will see these circuits soon, starting in Section 14.9.↩︎
306. Formally, we can think of ignoring phase as looking at the quotient of \mathcal{P}_3 by the subgroup \langle\pm\mathbf{1},\pm i\rangle, which results in an abelian group.↩︎
307. This is one of those arguments where it’s easy to get lost in the notation. Try picking two physical errors P_1 and P_2 in the same row somewhere in Figure 14.8 and following through the
argument, figuring out what E, P, P'_1, and P'_2 are as you go.↩︎
308. Just to be clear, if we knew which physical errors took place, then we wouldn’t have to worry about error correction at all, because we’d always know how to perfectly recover the desired state.
And remember that we can’t measure to find out which physical error took place, since this would destroy the state that we’re trying so hard to preserve!↩︎
309. We are being slightly informal with the way we draw these dot-error diagrams: cosets of \mathcal{S} itself inside \mathcal{P}_n don’t make sense, as we’ve said, because \mathcal{S} is generally
not normal inside \mathcal{P}_n. Also, when we quotient by \{\pm1,\pm i\} (by drawing just a single sheet, instead of four as in the diagrams in Exercise 7.8.2), we make \mathcal{P} abelian, and
this makes the normaliser no longer the actual normaliser.↩︎
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Three common hyperparametric tuning methods and codes
Super parameter optimization methods: grid search, random search, Bayesian Optimization (BO) and other algorithms.
reference material: Three super parameter optimization methods are explained in detail, as well as code implementation
Experimental basic code
import numpy as np
import pandas as pd
from lightgbm.sklearn import LGBMRegressor
from sklearn.metrics import mean_squared_error
import warnings
from sklearn.datasets import load_diabetes
from sklearn.model_selection import KFold, cross_val_score
from sklearn.model_selection import train_test_split
import timeit
import os
import psutil
# In sklearn Datasets's diabetes dataset demonstrates and compares different algorithms to load it.
diabetes = load_diabetes()
data = diabetes.data
targets = diabetes.target
n = data.shape[0]
random_state = 42
# Time occupation s
start = timeit.default_timer()
# Memory usage mb
info_start = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
train_data, test_data, train_targets, test_targets = train_test_split(data, targets,
test_size=0.20, shuffle=True,
num_folds = 2
kf = KFold(n_splits=num_folds, random_state=random_state, shuffle=True)
model = LGBMRegressor(random_state=random_state)
score = -cross_val_score(model, train_data, train_targets, cv=kf, scoring="neg_mean_squared_error", n_jobs=-1).mean()
print('score: ', score)
end = timeit.default_timer()
info_end = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
print('This program takes up memory' + str(info_end - info_start) + 'mB')
print('Running time:%.5fs' % (end - start))
The experimental results are as follows:
Least square error: 3897.5550693355276
This program takes up 0.5mB of memory
Running time:1.48060s
For comparison purposes, we will optimize the model that only adjusts the following three parameters:
n_estimators: from 100 to 2000
max_depth: 2 to 20
learning_rate: from 10e-5 to 1
1, Grid search
Grid search is probably the simplest and most widely used hyperparametric search algorithm. It determines the optimal value by finding all points in the search range. If a larger search range and a
smaller step size are used, the grid search has a high probability of finding the global optimal value. However, this search scheme consumes a lot of computing resources and time, especially when
there are many super parameters to be tuned.
Therefore, in the process of practical application, the grid search method generally uses a wider search range and larger step size to find the possible location of the global optimal value; Then
narrow the search range and step size to find a more accurate optimal value. This operation scheme can reduce the time and computation required, but because the objective function is generally non
convex, it is likely to miss the global optimal value.
The grid search element corresponds to the element in sklearn GridSearchCV Module: sklearn model_ selection. GridSearchCV
(estimator, param_grid, , scoring=None, n_jobs=None, iid='deprecated', refit=True, cv=None, verbose=0, pre_dispatch='2n_jobs', error_score=nan, return_train_score=False). See blog for details.
1.1 GridSearch algorithm code
from sklearn.model_selection import GridSearchCV
# Time occupation s
start = timeit.default_timer()
# Memory usage mb
info_start = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
param_grid = {'learning_rate': np.logspace(-3, -1, 3),
'max_depth': np.linspace(5, 12, 8, dtype=int),
'n_estimators': np.linspace(800, 1200, 5, dtype=int),
'random_state': [random_state]}
gs = GridSearchCV(model, param_grid, scoring='neg_mean_squared_error',
n_jobs=-1, cv=kf, verbose=False)
gs.fit(train_data, train_targets)
gs_test_score = mean_squared_error(test_targets, gs.predict(test_data))
end = timeit.default_timer()
info_end = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
print('This program takes up memory' + str(info_end - info_start) + 'mB')
print('Running time:%.5fs' % (end - start))
print("Best MSE {:.3f} params {}".format(-gs.best_score_, gs.best_params_))
Experimental results:
This program takes up 12.79296875mB of memory
Running time:6.35033s
Best MSE 3696.133 params {'learning_rate': 0.01, 'max_depth': 5, 'n_estimators': 800, 'random_state': 42}
1.2 visual interpretation
import matplotlib.pyplot as plt
gs_results_df = pd.DataFrame(np.transpose([-gs.cv_results_['mean_test_score'],
columns=['score', 'learning_rate', 'max_depth',
gs_results_df.plot(subplots=True, figsize=(10, 10))
We can see, for example, max_depth is the least important parameter and it will not significantly affect the score. However, we are searching for max_ Eight different values of depth, and any fixed
value is searched on other parameters. Obviously a waste of time and resources.
2, Random search
The idea of random search is similar to network search, except that all values between the upper and lower bounds are no longer tested, but sample points are randomly selected within the search
range. His theoretical basis is that if the sample point set is large enough, the global optimal value or approximate value can be roughly found through random sampling. Random search is generally
faster than web search. When searching for super parameters, if the number of super parameters is small (three or four or less), we can use grid search, an exhaustive search method. However, when the
number of super parameters is large, we still use grid search, and the search time will increase exponentially.
Therefore, someone proposed a random search method, which randomly searches hundreds of points in the hyperparametric space, among which there may be relatively small values. This method is faster
than the above sparse grid method, and experiments show that the random search method is slightly better than the sparse grid method. RandomizedSearchCV uses a method similar to the GridSearchCV
class, but instead of trying all possible combinations, it selects a specific number of random combinations of a random value of each super parameter. This method has two advantages:
If you run a random search, say 1000 times, it will explore 1000 different values of each super parameter (instead of just a few values of each super parameter, as in grid search)
You can easily control the calculation amount of super parameter search by setting the search times.
The use method of randomized searchcv is actually the same as that of GridSearchCV, but it replaces GridSearchCV's grid search for parameters by sampling randomly in the parameter space. For
parameters with continuous variables, randomized searchcv will sample them as a distribution, which is impossible for grid search, and its search ability depends on the set n_iter parameter, the same
code is given.
Random search for sklearn in sklearn model_ selection. RandomizedSearchCV
(estimator, param_distributions, *, n_iter = 10, score = None, n_jobs = None, iid = 'deprecated', refine = true, cv = None, verbose = 0, pre_dispatch = '2 * n_jobs', random_state = None, error_score
= nan, _return_train_score = False) The parameters are roughly the same as GridSearchCV, but there is an additional n_ ITER is the number of iteration rounds.
from sklearn.model_selection import RandomizedSearchCV
from scipy.stats import randint
param_grid_rand = {'learning_rate': np.logspace(-5, 0, 100),
'max_depth': randint(2, 20),
'n_estimators': randint(100, 2000),
'random_state': [random_state]}
# Time occupation s
start = timeit.default_timer()
# Memory usage mb
info_start = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
rs = RandomizedSearchCV(model, param_grid_rand, n_iter=50, scoring='neg_mean_squared_error',
n_jobs=-1, cv=kf, verbose=False, random_state=random_state)
rs.fit(train_data, train_targets)
rs_test_score = mean_squared_error(test_targets, rs.predict(test_data))
end = timeit.default_timer()
info_end = psutil.Process(os.getpid()).memory_info().rss / 1024 / 1024
print('This program takes up memory' + str(info_end - info_start) + 'mB')
print('Running time:%.5fs' % (end - start))
print("Best MSE {:.3f} params {}".format(-rs.best_score_, rs.best_params_))
This program takes up 17.90625mB of memory
Running time:3.85010s
Best MSE 3516.383 params {'learning_rate': 0.0047508101621027985, 'max_depth': 19, 'n_estimators': 829, 'random_state': 42}
It can be seen that the effect is better than that of GridSearchCV when running 50 rounds, and the time is shorter.
3, Bayesian Optimization (BO)
Grid search is slow, but it works well in searching the whole search space, while random search is fast, but it may miss important points in the search space. Fortunately, there is a third option: Bayesian optimization. In this paper, we will focus on an implementation of Bayesian Optimization called hyperopt of Python modular. Bayesian optimization algorithm is used to find the optimal and maximum parameters. It adopts a completely different method from grid search and random search. When testing a new point, grid search and random search will ignore the information of the previous point, while Bayesian optimization algorithm makes full use of the previous information. Bayesian optimization algorithm learns the shape of the objective function and finds the parameters that improve the objective function to the global optimal value.
Specifically, the method of learning objective function is to assume a search function according to a priori distribution; Then, every time a new sampling point is used to test the objective function, this information is used to update the prior distribution of the objective function; Finally, the algorithm tests the points where the global maximum value given by the a posteriori distribution may appear. For Bayesian optimization algorithm, one thing to note is that once a local optimal value is found, it will continuously sample in the region, so it is easy to fall into the local optimal value. In order to make up for this shortcoming, Bayesian algorithm will find a balance between exploration and utilization. Exploration is to obtain sampling points in the area that has not been sampled, while utilization will sample in the most likely global maximum area according to the posterior distribution.
We will use[hyperopt library](http://hyperopt.github.io/hyperopt/#documentation) to handle this algorithm. It is one of the most popular libraries for hyperparametric optimization. See the blog for details.
(1)TPE algorithm:
TPE is the default algorithm for Hyperopt. It uses Bayesian method for optimization. At each step, it tries to establish the probability model of the function and select the most promising parameters
for the next step. This kind of algorithm works as follows:
1. Generate random initial point x
2. Calculate F(x)
3. Try to establish conditional probability model P(F|x) by using test history
4. Selecting xi according to P(F|x) is most likely to lead to better F(xi)
5. Calculate the actual value of F(xi)
6. Repeat steps 3-5 until one of the stop conditions is met, such as I > max_ evals
from hyperopt import fmin, tpe, hp, anneal, Trials
def gb_mse_cv(params, random_state=random_state, cv=kf, X=train_data, y=train_targets):
# the function gets a set of variable parameters in "param"
params = {'n_estimators': int(params['n_estimators']),
'max_depth': int(params['max_depth']),
'learning_rate': params['learning_rate']}
# we use this params to create a new LGBM Regressor
model = LGBMRegressor(random_state=random_state, **params)
# and then conduct the cross validation with the same folds as before
score = -cross_val_score(model, X, y, cv=cv, scoring="neg_mean_squared_error", n_jobs=-1).mean()
return score
# State space, minimize the value range of params of the function
space={'n_estimators': hp.quniform('n_estimators', 100, 2000, 1),
'max_depth' : hp.quniform('max_depth', 2, 20, 1),
'learning_rate': hp.loguniform('learning_rate', -5, 0)
# trials will record some information
trials = Trials()
#Time occupation s
#Memory usage mb
info_start = psutil.Process(os.getpid()).memory_info().rss/1024/1024
best=fmin(fn=gb_mse_cv, # function to optimize
algo=tpe.suggest, # optimization algorithm, hyperotp will select its parameters automatically
max_evals=50, # maximum number of iterations
trials=trials, # logging
rstate=np.random.RandomState(random_state) # fixing random state for the reproducibility
# computing the score on the test set
model = LGBMRegressor(random_state=random_state, n_estimators=int(best['n_estimators']),
tpe_test_score=mean_squared_error(test_targets, model.predict(test_data))
info_end = psutil.Process(os.getpid()).memory_info().rss/1024/1024
print('This program takes up memory'+str(info_end-info_start)+'mB')
print('Running time:%.5fs'%(end-start))
print("Best MSE {:.3f} params {}".format( gb_mse_cv(best), best))
Experimental results:
This program takes up 2.5859375mB of memory
Running time:52.73683s
Best MSE 3186.791 params {'learning_rate': 0.026975706032324936, 'max_depth': 20.0, 'n_estimators': 168.0}
4, Conclusion
We can see that even in the next steps, TPE and annealing algorithm will actually improve the search results over time, while random search randomly found a good solution at the beginning, and then
only slightly improved the results. The current difference between TPE and RandomizedSearch results is small, but hyperopt can significantly improve time / score in some real-world applications with
a more diversified range of super parameters
|
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Dynamics of flow in a branching channel
Issue Mechanics & Industry
Volume 22, 2021
Application of Experimental and Numerical Methods in Fluid Mechanics and Energy
Article Number 25
Number of page(s) 14
DOI https://doi.org/10.1051/meca/2021014
Published online 09 April 2021
© V. Uruba et al., published by EDP Sciences 2021
1 Introduction
The branched channels are applied in many practical situations. In general, the main goal of branching is distribution of the flowing fluid across the various locations. However, the flow-rate
distribution across the branches is irregular even if theirs geometry is similar − see e.g. [1].
There are extensive knowledge resources in available literature, many authors address this problem both from theoretical and practical point of view. In the classical engineering book [2] there are a
lot of variants and configurations. Many papers deal with the flow-pattern in channels of rectangular cross-section. The so called “secondary flow of second kind” in a channel of square or
rectangular cross-section was first observed by Nikuradse [3] and has been studied both experimentally [4] and theoretically [5]. Recently there are many numerical studies on the problem with
detailed analysis of the flow-field available, see e.g. [6,7]. However, the flow-field dynamics is not studied in this case, the authors of the presented paper recently published the paper [8], where
some preliminary results are shown. The novelty of the present study is detailed analysis of dynamical structures in the channel branches. The typical topologies and frequencies are to be presented.
The presented study was motivated by geometry of cooling channels in rotor of a power generator.
2 Experimental setup
The experimental model was designed and fabricated from the Plexiglas to allow optical access to the flow. Experiments were performed using the time-resolved PIV technique.
2.1 Channel geometry
The main channel was of the cross-section 25×30 mm^2, 1450mm long with the dead-end. The 13 branches (A, B, …, M) of reduced cross-sections 38×4 mm^2 and 127mm in length, are distributed
regularly along the main channel perpendicularly to the main channel axis, as shown in schematic picture in Figure 1.
The 3 channels were selected for detailed analysis, the first branch A, G in the main channel middle and the last but one L. The selected branches are denoted by red letters in Figure 1.
In the inlet, fully turbulent and developed channel flow was present. The air flow in the main channel inlet is characterized by Reynolds number of 11 300, velocity of the air in the input is about
6.4m/s. The volumetric flow is distributed into individual branches, the flow-rates V are shown in Figure 2.
Detailed description of the channel modelused for the experiments is given in [1,9].
Schematics of a singlebranch under study is in Figure 3, Cartesian coordinate system is introduced for each branch. The measuring plane was located in the mid-span of each channel. The area of a
branch was divided into 2 parts: zones I and II for technical reasons, the zones are interconnected in the x=65mm.
The coordinates in all graphs presented in this paper are given in millimetres.
2.2 Instrumentation
The time-resolved PIV method was used for the experiments in the channel model. The measuring system DANTEC consists of the double-pulse laser with cylindrical optics and CMOS camera. The software
Dynamic Studio 3.4 was used for velocity-fields evaluation. Laser New Wave Pegasus Nd:YLF, double head, wavelength 527nm, maximal frequency 10kHz, a shot energy is 10mJ for 1kHz (corresponding
power 10W per head). Camera Phantom V711 with maximal resolution 1 280×800 pixels and corresponding maximal frequency 3000 double-snaps per second. For the presented measurements, the frequency
1kHz and 4000 double-snaps in sequence corresponding to 4s of record for mean evaluation was acquired. As particles the oil droplets generated by SAFEX fog generator were used.
Detailed description of the measuring system could be find e.g. in [1].
2.3 Analysis methods
Instantaneous velocity fields in the measuring plane have been evaluated using “Adaptive PIV” method. The averaged velocity field has been evaluated first to obtain the flow statistics and remove the
mean flow. Mean flow structure is evaluated as well as velocity components variances and correlation coefficient distributions.
For study of dynamical properties of the flow-field the Oscillation Pattern Decomposition method (OPD) was adopted resulting in series of OPD mode. Each OPD mode is characterized by topology in
complex form (consisting of real and imaginary parts), frequency and attenuation of the pseudo-periodic (oscillating) behaviour. Attenuation or amplitude decay is described by e-folding time
representing mean time period of the mode amplitude decay by “e”. The other decay characteristic is “periodicity” which express the e-folding time in multiples of periods.
The details on OPD method could be find in [9].
3 Results
The results are divided into two parts. The first part is oriented on time-mean results (chapter 3.1), while the second part characterizes the flow dynamics in the 3 selected branches (chapter 3.2).
The zones I and II are presented separately as the dynamical analysis using the OPD method was applied separately to both zones. However the time-mean characteristics presented in the next chapter
3.1 follow-up between the zones satisfactorily.
3.1 Time-mean characteristics
The averaged characteristics will be presented in each figure both for the zone I on the left-hand the zone II on the right-hand sides, the division value of the x coordinate is 65mm. The
distributions of mean characteristics follow from one zone to the other satisfactorily. Please note that the dimensions of a branch are 127×38 mm^2.
The presentation of results starts with the mean velocities distributions, for the sake of clarity the vector-lines in green are added arbitrarily.
The mean velocity vector fields are shown in Figure 4a–f.
In the mean-flow fields the back-flow regions associated with a vortex located in left-hand part I could be detected. The back-flow results from separation of the flow on the sharp corner in [0,38]
position on the branch inlet from the main channel. This phenomenon was studied in details for all branches. The positions of the vortices centres [x[c]; y[c]] have been evaluated for all branches.
The results are shown in Figure 5, the branches are marked with numbers 1–13. The branch A is marked with 1, G with 7 and L with 12 respectively. In the last branch M (no. 13) no back-flow and
associated vortex were detected.
The back-flow regions were identified within the branches. In Figure 6 the lines dividing the flow-field into back- and forward-flow regions are shown. The back-flow is locatedin the area above the
limit line, below it there is forward flow. The limit line is defined as zero x-component mean velocity location.
The positions of the reattachment points on the upper branch wall are detected for branches E–L. In the branches A–D the flow does not reattach the upper wall, while in the last branch M no
separation and thus no reattachment occur. In Figure 7 the x[ra] positions are shown.
To indicate regions with high dynamical behaviour the sum of variances x and y velocity components is to be presented next in Figure 8a–f. In Figures the dark-blue colour represents low dynamical
activity regions, the red colour corresponds to excessive dynamics due to frequent errors in velocity evaluation using the standard PIV evaluation procedure in near-wall regions. Finally, the
light-blue and green regions are those with high dynamical activity in the flow itself.
To estimate turbulence generation, the correlation coefficient of the velocity components is evaluated in Figure 9a–f.
The correlation coefficient value close to 0 indicates low or even no turbulence generation, while negative (or positive) values approaching±1 indicate turbulence production regions. The sign of the
correlation depends on the coordinate system definition.
3.2 Dynamics
The OPD method has been applied on the zones I and II of the selected branches respectively. The analysis of both zones is completely independent to each other and OPD modes from one zone could not
be linked directly to those of the other zone.
In Figure 10 there are OPD modes frequency and periodicity combinations evaluated by the standard procedure for the branch A. The results for the zone I are in blue, while the red points indicate the
zone II. Numbers of the modes ordered according to the e-folding time (descending order) are indicated within the points.
The periodicity values for the zone I are very low, this indicates that the modes decay very quickly in time. The value of periodicity 0.3 is considered to be the marginal value, below it the process
could be considered as aperiodic, as the amplitude decay in one period is more than 10 – see [9]. On the other hand in the zone II the selected modes show much higher periodicity, higher than 2, the
modes are well periodical and relatively very stable.
A few modes have been selected with the highest value of periodicity for the detailed analysis. Those modes could be considered as oscillating or pseudo-periodic, the others, with low periodicity are
decaying as they decay too quickly. All parameters of the selected modes includingfrequency, e-folding time and periodicity are given in Tables 1–3 respectively. The Strouhal number is determined
using the mode frequency, mean velocity in the given branch calculated from a branch flow-rate (see Fig. 2) and channel width (38mm). The parameters related to the flow dynamics in the branch A are
shown in Table 1.
Topology of the selected modes is to be shown next. Each mode topology consists of real and imaginary parts respectively. The real part corresponds to the phase angle 0, while the imaginary part
corresponds to the phase angle $π / 2$, than phase π is characterized by negative real part and the phase $3 π / 2$ by the negative imaginary part. The process is pseudo-periodical with decaying
Topology is shown as vector fields. For clearness the vector lines are added arbitrarily, for the real part in red and for the imaginary part in blue colours.
The topology of the mode 1, branch A, zone I is shown in Figure 11.
The OPD mode 1 for AI represents the train of vortices moving in x direction, spacing s=38.4mm and the period corresponding to the frequency 18.4Hz is 54.34ms. Thus, the velocity of the vortex
train is about 0.71m/s.
Topology of the mode 2 in Figure 12 is similar to the mode 1, however the vortices in train have smaller spacing s=23.4mm and the frequency is higher in the same time f=42.4Hz. Resulting
velocity is similar but higher, about 0.99m/s. Obviously, the imaginary part of the mode topology is shifted in space about a quarter of the period.
The OPD mode 4, AI in Figure 13 is again a train of vortices propagating in the green arrow direction by the velocity 0.95m/s.
So, for the dynamics of the first zone of the branch A we could conclude, that it is populated by trains of vortices configured in a single row moving approximately in the streamwise direction x with
the velocity 0.7–1m/s. The orientations of the vortices in the row are alternating.
The dynamical flow-field of the zone AII is populated by smaller vortical structures forming more complicated configuration and filling lower 2 thirds of the area. Again, the structures are
propagating in waves in the streamwise direction. As an example the OPD mode AII 8 is shown in Figure 14.
The other highly periodical modes within the AII are 14, 16 and 17, in Figure 15 only real parts of the modes are shown. The modes consist of one or more rows of vortices oriented and moving in x
direction. The imaginary partsare very similar, only the structures are shifted in space about 1/4 of the period in streamwise direction, which is the direction of movement. The typical velocities of
the vortex trains within the AII are about 1.4m/s.
Overall the flow dynamics in the first branch A is characterized by vortex trains propagating along the channel. In the branch inlet, zone I, the vortices are relatively big filling the whole
channel, the frequency is low, Strouhal number typically smaller than 1. Further downstream, in the zone II, the instability is developed close to the back-flow region located in the lower half of
the channel producing strong vortex train. Those vortices are much smaller, well developed and the frequency is much higher, Strouhal number is 1.2–2.9. Stability of pseudo-periodical vortex trains,
characterized by periodicity, is much higher in the second part of the branch.
Next, the flow dynamics in the branch G located in the middle of the main channel is to be shown.
In Figure 16 the frequency − periodicity graph is given. Here, in the branch G, both zones I and II exhibit highly periodical OPD modes. The relevant values for the selected modes with the
periodicity higher than 1 are given in Table 2.
The topology of the selected modes is introduced bellow. In the zone I of the branch G the typical topology is represent consisting of vortex train with alternating orientation located in the lower
domain third. Only the real parts of the modes are to be shown here, as the imaginary parts are only shifted in space, as mentioned above.
In Figure 17 there are modes 6, 11, 12 and 14, the typical convection velocity of vortices is about 1.5m/s. The frequencies relatively high, the Strouhal number between 2.6 and 5.8.
The zone II of the branch G is to be presented now. The mode 1 is characterized by a train of big vortices filling the whole domain. Train of the smaller vortices in x direction is detected in the
mode no. 11, see Figure 18. The velocity of the vortices is 0.75m/s for the mode no. 1 and 1.7m/s for the mode 11.
The mode 4, branchG, zone II is characterized by two rows of vortices, the first in the upper half consisting of 3 vortices and the second in the lower half consisting of 2 vortices only. Both
systems are moving in the x direction, however the propagation velocity is different: 0.9m/s for the upper system, while 1.7m/s for the lower system. Both real and imaginary part of the mode 4,
branchG, zone II topology are shown in Figure 19.
The dynamical situation in the branch G is in some aspect opposite to this in the branch A. In the branch G, in the inlet the frequencies of vortex trains are very high, the Strouhal number reaches
5.8. The structures are located in the lower third of the channel close to the wall. Further downstream, in the zone II, the vortex trains form two-row structure filling the whole channel (the mode
4). The frequencies are lower.
Last branch to be analysed for the flow dynamics is the branch L, the last but one in the order. In the branch L the flowdynamics is weak, the flow is rather steady. The frequency-periodicity graph
is in Figure 20 and the selected modes are specified in Table 3.
The modes in the branch L, zone I, nos. 3 and 10 are formed by system of saddle lines of oblique orientation − see the real parts of the modes topology in Figure 21a and b. In the upper part there is
an indication of vortices. The saddle lines are moving in the vertical direction, perpendicularly to the flow. The mode 3 is very stable and well pronounced, the frequency is moderate, the Strouhal
number 0.72.
The vortex train is well detectable in the branch L, zone I mode no. 14, it is located in the upper half of the domain, velocity of propagation in streamwise direction about 1.36m/s.
The mode no. 6 in the branch L, zone II represents pairs of contra-rotating vortices travelling in the middle of the domain in the x direction, the velocity is close to 1m/s, see Figure 22a and b.
The mode no. 10 in the branch L, zone II is shown in Figure 23a and b, both real and imaginary parts. The mode consists of elongated vortices stretched in oblique direction. The vortices are
convected perpendicularly to the elongation direction − see the green arrow in Figure 23b. The convection velocity is very small, about 0.28m/s.
In the L branch, the last but one, typical dynamic of moving structures in transversal direction has been detected. The typical dynamical structures are in the form of elongated vortices and the
predominant saddle lines. The corresponding frequencies are moderate, typical Strouhal number below 1.
The heat transfer process between the wall and fluid is strongly affected by the flow structure close to the wall. Both mean velocity and the flow dynamics play important role in this process, as
shown e.g. in [10].
In general, the stable, well profound vortical trains occurring close to the walls and high mean velocity will promote heat transfer. In our complex case such regions could be detected close to the
lower walls of the AII region and all the G branch. In the case of the L branch, both lower and upper walls are surrounded by dynamical structures and thus in good thermal communication with the
On the other hand, the low-velocity flow-regions with weak fluctuations prevent the heat transfer. The typical such regions are the separation regions affecting the upper walls of the A and G branch
4 Conclusions
In the paper the flow structure of selected branches of a sample branched channel is shown as a result of PIV experiments. The flow is separated creating a back-flow region. In the shear layer
created, the instability effects take place.
Both statistical methods based on averaging and special methods for vector field frequency analysis are applied. Typical pseudo-periodic dynamical structures are detected. The typical dynamical
structures are trains of vortices of alternative orientation aligned in the x direction and propagating in the streamwise direction and moving saddle lines. However, the dynamical structures differ
substantially in individual branches.
The flow dynamics in the branches depends strongly on the position on the main channel. Three distinct regions were identified in this context.
Close to the main channel inlet, the branches flow-dynamics is characterized by vortex trains propagating along the channel. In the branch inlet the vortices are relatively big filling the whole
channel, the frequency is low, Strouhal number typically smaller than 1. Further downstream the instability is developed close to the back-flow region located in the lower half of the channel
producing strong vortex train. Those vortices are much smaller, well developed and the frequency is much higher than in the first partof the branch. Stability of pseudo-periodical vortex trains,
characterized by periodicity value, is much higher in the second part of the branch.
In the region of the main channel middle part, the dynamical situation in branches is nearly opposite to this in the inlet branches. Here, close to the branch inlet, the frequencies of vortex trains
are very high, Strouhal numbers up to 5.8. The structures are located in the lower third of the channel close to the lower wall. Further downstream the vortex trains form two-row structure filling
the whole channel and the frequencies are again lower.
Approaching the main channel end, the typical flow dynamics in the branches is characterized by the structures moving in transversal direction. The typical dynamical structures are in the form of the
predominant saddle lines and elongated vortices. The corresponding frequencies are moderate, Strouhal numbers bellow 1.
The flow-field close to the channel walls affects heat transfer process between the wall and fluid, high velocities and intensive fluctuations generated by dynamical structures promote the heat
x, y [m]: Cartesian coordinates
x[ra] [m]: Reattachment position
x[c], y[c] [m]: Position of vortex centre
A, B, C, D, E, F, G, H, I, J, K, L, M: Branches indicators (11)
OPD: Oscillation Pattern Decomposition
PIV: Particle Image Velocimetry
This work was supported by the Grant Agency of the Czech Republic, projects Nos. 17-01088S and 19-02288J and by the Technology Agency of the Czech Republic, project No. TK03020057.
All Tables
All Figures
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Quick RatioQuick Ratio (Acid-Test): Calculation, Importance & Financial Insights
Quick Ratio: A Key Liquidity Metric for Businesses
The Quick Ratio, often referred to as the Acid-Test Ratio, is a financial metric that evaluates a company’s short-term liquidity. It measures the ability of a business to meet its short-term
obligations using its most liquid assets, without relying on the sale of inventory. This metric is crucial for investors and stakeholders as it provides insight into the financial health of a
• Current Assets: These are assets that are expected to be converted into cash or used up within one year. They include cash, cash equivalents, accounts receivable and other short-term assets.
• Inventory: Unlike the current ratio, the Quick Ratio excludes inventory from current assets. This is because inventory may not always be easily converted into cash in the short term.
• Current Liabilities: These are obligations that the company needs to pay off within one year, including accounts payable, short-term loans and other similar liabilities.
The Quick Ratio is calculated using the formula:
\(\text{Quick Ratio} = \frac{\text{Current Assets} - \text{Inventory}}{\text{Current Liabilities}}\)
This formula provides a more stringent assessment of a company’s liquidity compared to the current ratio, which includes inventory.
Recently, there has been a growing focus on liquidity metrics like the Quick Ratio, especially amid economic uncertainties. Companies are increasingly prioritizing their liquidity positions to ensure
they can weather financial storms.
Investors are also paying more attention to the Quick Ratio as it provides a quick snapshot of a company’s financial health, especially in industries where inventory turnover is slow or
While the standard Quick Ratio remains widely used, some variations include:
• Acid-Test Ratio: This is effectively the same as the Quick Ratio but emphasizes the exclusion of inventory even more.
• Modified Quick Ratio: This version might adjust the components further, depending on the industry’s specific practices or the company’s unique operational model.
Imagine a company with the following financials:
• Current Assets: $500,000
• Inventory: $200,000
• Current Liabilities: $300,000
Using the Quick Ratio formula, we can calculate:
\(\text{Quick Ratio} = \frac{500,000 - 200,000}{300,000} = \frac{300,000}{300,000} = 1.0\)
This indicates that the company has a Quick Ratio of 1.0, meaning it can cover its current liabilities with its liquid assets.
• Increase Liquid Assets: Companies can improve their Quick Ratio by increasing cash reserves or receivables.
• Reduce Current Liabilities: This can be achieved by paying off short-term debts or renegotiating payment terms with creditors.
• Efficient Inventory Management: Although inventory is excluded from the Quick Ratio, managing it efficiently ensures that it does not become a financial burden.
The Quick Ratio is a vital financial metric that offers valuable insights into a company’s liquidity position. By understanding its components and implications, stakeholders can make informed
decisions regarding their investments and financial strategies. Regularly monitoring the Quick Ratio can help businesses maintain healthy liquidity and navigate economic uncertainties effectively.
What is the Quick Ratio and why is it important?
The Quick Ratio is a financial metric that measures a company’s short-term liquidity position, indicating its ability to meet short-term obligations without relying on inventory sales.
How do you calculate the Quick Ratio?
To calculate the Quick Ratio, use the formula: (Current Assets - Inventory) / Current Liabilities. This gives a clearer picture of liquidity than the current ratio.
More Terms Starting with Q
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Python in Mathematics
22nd April 2004
Python in the Mathematics Curriculum by Kirby Urner is something of a sprawling masterpiece. It really comes in four parts: the first is a history of computer science in education, the second an
appraisal of the impact of open source on education and the world at last, the third a dive in to the things that make Python so suitable for enhancing the mathematics curriculum and the fourth a
discussion of how computer science and traditional mathematics are likely to play off against each other in the field of high school education.
It’s a long read, but well worth it. Kirby drops in numerous short Python code samples, such as this neat little implementation of Euclid’s algorithm for finding the greatest common denominator of
two numbers:
def gcd(a,b):
while b:
a,b = b, a % b
return a
His thoughs on open source and general geek culture are worth digging out even if the main topic of the paper has no interest for you. Here’s a sample:
Additionally, I think a key cultural phenomenon is the evolving perception of geek culture as a whole. What many students discover is a global network of loosely organized, yet talented
individuals, including many free spirits. The network is cosmopolitan and guided by some newly articulated principles regarding how some forms of intellectual assets should remain freely
accessible and reusable. While these values might seem another ideological pipe dream, were they expressed in merely political terms, in this case the lingua franca of the movement is source
code, and licensing agreements designed to protect it against leaking off into the proprietary sector. Even though Python may be used in proprietary ways, Python itself remains free.
Kirby presented the talk at Python DC ’04 back in March. I wish I’d been there, but the conference was too close to SxSW for me to make it to both.
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Binary Cross Entropy Explained - Sparrow Computing
The most common loss function for training a binary classifier is binary cross entropy (sometimes called log loss). You can implement it in NumPy as a one-liner:
def binary_cross_entropy(yhat: np.ndarray, y: np.ndarray) -> float:
"""Compute binary cross-entropy loss for a vector of predictions
An array with len(yhat) predictions between [0, 1]
An array with len(y) labels where each is one of {0, 1}
return -(y * np.log(yhat) + (1 - y) * np.log(1 - yhat)).mean()
Why does this work?
Good question! The motivation for this loss function comes from information theory. We’re trying to minimize the difference between the y and yhat distributions. That is, we want to minimize the
difference between ground truth labels and model predictions. This is an elegant solution for training machine learning models, but the intuition is even simpler than that.
Binary classifiers, such as logistic regression, predict yes/no target variables that are typically encoded as 1 (for yes) or 0 (for no). When the model produces a floating point number between 0 and
1 (yhat in the function above), you can often interpret that as p(y == 1) or the probability that the true answer for that record is “yes”. The data you use to train the algorithm will have labels
that are either 0 or 1 (y in the function above), since the answer for each record in your training data is known.
To train a good model, you want to penalize predictions that are far away from their ground truth values. That means you want to penalize values close to 0 when the label is 1 and you want to
penalize values close to 1 when the label is 0.
The y and (1 - y) terms act like switches so that np.log(yhat) is added when the true answer is “yes” and np.log(1 - yhat) is added when the true answer is “no”. That would move the loss in the
opposite direction that we want (since, for example, np.log(yhat) is larger when yhat is closer to 1 than 0) so we take the negative of the sum instead of the sum itself.
Here’s a plot with the first and second log terms (respectively) when they’re switched on:
Notice the log function increasingly penalizes values as they approach the wrong end of the range.
A couple other things to watch out for:
• Since we’re taking np.log(yhat) and np.log(1 - yhat), we can’t use a model that predicts 0 or 1 for yhat. This is because np.log(0) is -inf. For this reason, we typically apply the sigmoid
activation function to raw model outputs. This allows values to get close to 0 or 1, but never actually reach the extremes of the range.
• We typically divide by the number of records so the value is normalized and comparable across datasets with different sizes. This is the purpose of the .mean() method call in the implementation
In practice
Of course, you probably don’t need to implement binary cross entropy yourself. The loss function comes out of the box in PyTorch and TensorFlow. When you use the loss function in these deep learning
frameworks, you get automatic differentiation so you can easily learn weights that minimize the loss. You can also use the same loss function in scikit-learn.
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CSN Assumption Names
• This document provides standardized assumption names or descriptors for use in Model Coupling Metadata (MCM) files. They are organized into groups which are (for the most part) mutually exclusive
and which intend to span the types that are needed to describe a model's underlying physics. They cannot yet be said to be exhaustive but they are illustrative. The ones collected here already
illustrate various language patterns that are commonly used to describe assumptions.
• Note that assumption is meant to be taken as a broad term that can include things like conditions, simplifications, approximations, limitations, conventions, provisos and other forms of
• CSDMS encourages model developers to include as many <assume> tags in their Model Coupling Metadata (MCM) file as they feel apply to their model or to a particular input or output variable name.
XML tag nesting determines the scope of an <assume> tag. For someone familiar with a particular modeling domain, the terms that have been collected here should be easily recognized and understood
as part of that domain's standard terminology.
• Given a collection of models that have Model Coupling Metadata (MCM) files which include a standardized listing of assumptions it will be straightforward to write software that allows the CSDMS
modeling framework to automatically check whether two components to be coupled are compatible and alert users to potential problems or mismatches. Reports can also be generated automatically that
quantify the degree of compatibility. Similarly, opportunities for valid model coupling can then also be automatically identified.
Boundary Condition Assumptions
periodic_boundary_condition (same as "wrap-around")
stefan_boundary_condition (See: Stefan problem.)
toroidal_boundary_condition (same as "doubly periodic" ??)
Conserved Quantity Assumptions
• These names all end in "_conserved". See Conservation Law and links therein.
• These names would usually be used within an <object> tag block which would make it clear what is being conserved (e.g. water or sediment).
Coordinate Systems
• These end in "coordinate_system".
cartesian_coordinate_system (same as "rectilinear")
cylindrical_coordinate_system (same as "polar" if 2D)
projected_coordinate_system (i.e. map projections)
terrain_following_vertical_coordinate_system (Same as "sigma coordinates"?? See: S-coordinate models, Sigma coordinates.)
Georeferencing Assumptions
• Standard names for projections, ellipsoids and datums are available in the EPSG Registry.
• Standard names for projections, ellipsoids and datums are also used in the GeoTIFF Spec.
Sign and Angle Conventions
clockwise_from_north_azimuth_convention (all "bearing" angles, e.g. wind "from" angles)
counter-clockwise_from_x_axis_azimuth_convention (standard math)
z-axis_directed_downward (positive_downward)
z-axis_directed_upward (positive_upward)
Dimensionality Assumptions
• If the "richards_equation" assumption is used for infiltration, keep 1D and 3D out of the assumption name and give one of these with a separate <assume> tag.
• What about things like "1.5-dimensional" models?
Equations, Laws and Principles
• An <assume> tag should be provided for each equation that a model (or model component) uses. Most equations have standard names, as shown in the examples below.
• These all end in "_equation", "_law" or "_principle", except for the "law of the wall".
adams_williamson_equation (See: Adams-Williamson equation.)
conservation_of_energy_law ###
conservation_of_mass_law ### (same as continuity_equation)
conservation_of_momentum_law ###
darcy_law (See: Darcy's law).
darcy_weisbach_equation (See: Darcy-Weisbach).
ehrenfest_equations (See: Ehrenfest equations).
eikonal_equation (See: Eikonal equation.)
euler_equation (inviscid flow)
gibrat_law (See: Gibrat's law.)
glen_stress_strain_law (Glen's Law for glacier flows, Glen (1955); nonnewtonian fluid)
### heat_equation (use "diffusion_equation" instead)
huygens_fresnel_principle (See: Huygens-Fresnel principle.)
ideal_gas_law (See: Ideal gas law. Also an "ideal_gas_model" ??)
kirchoff_circuit_laws Kirchoff circuit laws
law_of_the_wall (flow resistance)
manning_equation (flow resistance)
nernst_equation (See: Nernst equation.)
nonlinear_diffusion_equation ###
richards_equation (infiltration theory)
stiff_equation #### (a type, vs. a named equation)
zipf_law (See: Zipf's law.)
• These all end in "_inequality".
• These all end in "_identity".
• These all end in "_approximation" and some can be found in other sections.
boussinesq_approximation (ocean modeling)
coopmans_approximation (See: Coopmans approximation.)
diophantine_approximation (of real numbers by rationals)
infiltrated-depth_approximation (infiltration modeling)
pade_approximation (of functions by rational functions)
perturbation_series_approximation (See: Perturbation theory.)
rigid_lid_approximation (## maybe not here?)
small-angle_approximation (also, paraxial approximation)
wavelet_series_approximation ####
Flow-Type Assumptions
• These names all end in "_flow".
axisymmetric_flow (in cylindrical coordinates, all theta derivatives are zero)
couette_flow (See: Couette flow. Really a "flow model"?)
critical_flow (Froude number = 1. See subcritical & supercritical.)
depth_integrated_flow (for 3D flow to 2D flow; vs. vertically_integrated_flow)
drag_induced_flow (e.g. Couette flow)
fully_developed_flow (i.e. derivatives of velocity with distance in the flow direction vanish)
geostrophic_flow ###### CHECK
hele_shaw_flow (See: Hele-Shaw flow. Really a "flow model"?)
inviscid_flow (of an ideal fluid with no viscosity)
isentropic_flow (both adiabatic and reversible; see isentropic_process)
nonaccelerating_flow (i.e. the nonlinear inertial term is negligible compared to others)
no_radial_flow (i.e. in cylindrical coordinates, the r component of velocity is zero)
no_swirl_flow (i.e. in cylindrical coordinates, the theta component of velocity is zero; also non-swirling)
plug_flow (See: Plug flow. Sometimes called "piston flow".)
poiseuille_flow (See: [1]. Really a "flow model"?)
potential_flow (irrotational and inviscid, as around airfoils; See Potential flow).
pressure_induced_flow (e.g. Poiseuille flow)
steady_flow (all time derivatives equal zero)
stokes_flow (same as "creeping flow"; See: Stokes flow).
subcritical_flow (Froude number < 1; see: Froude number.)
supercritical_flow (Froude_number > 1)
taylor_couette_flow (Really a flow model?)
taylor_dean_flow (Really a flow model?)
variable_area_flow (include this one? see converging and diverging flow; nozzles)
Note: "reynolds_averaged" is used in "reynolds_averaged_navier_stokes_equation".
Fluid-Type Assumptions
• The word "material" is often used instead of "fluid" or "solid", especially in the case of material types or models that may occur in either fluid or solid form.
• Most of these assumptions correspond to a particular functional relationship that describes how a fluid or material responds to an applied shear stress. (See: stress-strain curve.) These
typically involve some combination of (1) shear stress (often denoted by tau or sigma), (2) time derivative of shear stress, and (3) shear strain rate (time derivative of the strain), sometimes
abbreviated to "shear rate" or "strain rate" Note that strain is dimensionless and often denoted as epsilon. Shear stress (like pressure) has SI units of Pa. Shear rate (same as strain rate) has
SI units of (1/s).
• Glen's Law is a power-law relationship that expresses the shear strain rate as the shear stress to a power, where the power is often n=3. It may be a special case of one of the nonnewtonian fluid
types listed here.
bingham_plastic_fluid (See: Bingham plastic).
boger_fluid (See: [2]).
carreau_fluid (See: [3]).
casson_fluid (industry standard model for molten milk chocolate)
cross_fluid (See: [4]).
dilatant_fluid (shear thickening fluid or STF)
herschel_bulkley_fluid (See: [5]).
kelvin_voigt_fluid (a linear viscoelastic model; same as "kelvin_material" ?? ######)
maxwell_fluid (a linear viscoelastic model. See: Maxwell material.)
newtonian_fluid (linear relation between shear stress and strain rate that goes through origin)
oldroyd_fluid (a linear viscoeleastic model; see Oldroyd-B model.
power_law_fluid (generalized Newtonian, Ostward-de Waele)
pseudoplastic_fluid (shear thinning)
quemada_fluid (Used to model blood. See: Hemorheology.)
super_fluid (See: Superfluid).
thixotropic_fluid (See: Thixotropy.)
viscoelastic_fluid (See: Viscoelastic.)
viscoplastic_fluid (See: Viscoplastic.)
Note: Use "inviscid_flow" vs. "inviscid_fluid" and "viscous_flow" vs. "viscous_fluid".
Material-Type Assumptions
• These names all end in "_material".
• There is sometimes a blurred semantic distinction between a "material model" (e.g. Arruda-Boyce model) and just a "material'. There are many named models (see separate section) for mathematical
models of materials.
• Some types of materials can exist as either a solid or a fluid, and an extra assumption tag should be used to specify if one or the other is assumed.
amorphous_material (e.g. gel, glass; also noncrystalline_material. See: [6].)
auxetic_material (See: Auxetics.)
bio_material (See: Biomaterial.)
cauchy_elastic_material (same as simple elastic material)
ceramic_material (See: Ceramic materials.)
composite_material (See: Composite materials.)
crystalline_material (or solid?)
elastic_material (See: Elasticity.)
elastoviscoplastic_material (or solid?)
glass_material (amorphous solid that exhibits a glass transition. See: Glass.)
hyperelastic_material (See: Hyperelastic material. Also called green elastic material and special case of cauchy elastic material.)
hypoelastic_material (See: Hypoelastic material).
kelvin_voigt_material (See: Kelvin-Voigt Material.)
maxwell_material (See: [7].)
mohr_coulomb_material (See: Mohr-Coulomb theory. Model or material type?)
plastic_material (See: Plastic. Compare to polymeric material.)
polymeric_material (See: Polymer.)
solid_material (for cases where material may be fluid or solid)
viscoelastic_material (See: Viscoelastisticity.)
viscoplastic_material (See: Viscoplasticity.)
Function-Type Assumptions
• These names all end in "_function".
• See Geometry and Shape Assumptions.
daubechies_d2_wavelet_function (actually a whole family, with d2, d4, ..., d20)
holomorphic_function (very similar to "analytic_function")
log_spiral_function (See: Log spiral.)
logit_function (See: Logit function.)
nondecreasing_function (distinct from "increasing_function")
parabola_function (same as "quadratic_function")
probit_function (See: Probit function.)
ricker_wavelet_function ("mexican hat wavelet")
square_wave_function (See: square wave.)
step_function (See: Heaviside step function.)
See: List of mathematical functions (Wikipedia).
Note that "multivalued_function" is a misnomer. See: multivalued function.)
• These are already included with Probability Distributions
• We could have a similar section for surfaces.
Geometric Assumptions
• Most of these names end in "_shaped".
• The polygons here are assumed to be regular polygons. If they aren't, insert the adjective "irregular".
• See: Geometry.
ellipsoid_shaped (e.g. for earth)
semicircle_shaped (e.g. for a channel_cross_section)
trapezoid_shaped (e.g. for a channel_cross_section)
concave_upward (long profiles)
• These are objects or effects that are neglected or excluded from consideration in a model.
• Only relevant/important exclusions should be reported.
• Most of these names start with "no_".
no_aerosols (1 in CF)
no_antrhopogenic_land_use_change (###### 1 in CF; excluding_anthropogenic_land_use_change)
no_baseflow (hydrology) (1 in CF; excluding_baseflow)
no_clouds (1 in CF)
no distributaries
no_interception (hydrology)
no_litter (on forest floor) (1 in CF; excluding_litter)
no_radial_flow (explained and duplicated in "Flow type assumptions")
no_swirl_flow (explained and duplicated in "Flow type assumptions")
no_snow (1 in CF)
no_tides (2 in CF)
## no_viscosity (use inviscid_flow)
Named Model-Type Assumptions (by Domain)
• These names all end in "_model".
Aerodynamics Models
Agent-Based Models
agent_based_model (See: Agent-based model.)
schelling_segregation_model Schelling segregation model
Atmosphere and Radiation Models
boussinesq_approximation (not in CF, but see for_*)
clear_sky (23 in CF)
deep_snow (1 in CF)
horizontal_plane_topography (for clear-sky radiation calculation, not in CF)
(OR zero_slope_terrain, OR no_sloped_terrain OR nonsloped_terrain ???
rigid_lid (in CF; always related to boussinesq approximation ??)
sea_level_for_geoid (4 in CF)
standard_pressure (not in CF)
standard_temperature (not in CF)
Chemistry Models
nuclear_shell_model (See: nuclear shell model.)
vespr_model (See: VESPR theory.)
Cosmological Models
baum_frampton_model (a cyclic model)
big_bang_model (See: Big bang).
dark_energy_model (and dark_mass_model ?)
lambda_cdm_model (standard model of Big Bang cosmology)
steinhardt_turok_model (a cyclic model)
Earthquake Models
travelling_wave_model (include the word "earthquake"? ####)
Ecological Models
food_web_model Food web
lotka_volterra_model Lotka-Volterra
natural_selection_model Natural selection
trophic_cascade_model Trophic cascade
Fluid Dynamics Models
free_vortex_model (irrotational, velocity proportional to 1/r)
rigid_body_vortex_model (velocity proportional to r)
trailing_vortex_model (or wing_tip_vortex_model)
General Physics Models
double_pendulum_model (See: Double pendulum.)
foucault_pendulum_model (See: Foucault Pendulum.)
simple_pendulum_model (harmonic oscillator ? gravity pendulum?)
Geodynamics Models
Hydrology: Channelized Flow Models
diffusive_wave_model ####
d_infinity_surface_flow_model ####
hydraulic_geometry_downstream_model #### (Leopold et al.)
law_of_the_wall_flow_resistance_model ##### ???
manning_flow_resistance_model ##### ???
muskingum_flow_routing_model (routing flow through a channel network)
Hydrology: Evaporation (and sometimes Transpiration) Process Models
(See: Methods for estimating ET.)
asce_standardized_evaporation_model (in CUAHSI HIS HydroModeler)
debruin_evaporation_model (lakes and ponds)
hargreaves_evaporation_model (remove the "s" in hargreaves ??)
kohler_nordenson_fox_evaporation_model (lakes and reservoirs)
shuttleworth_evaporation_model (a modified penman model)
stewart_rouse_evaporation_model (lakes and ponds)
thornthwaite_water_balance_model ########
Hydrology: Ground Water and Infiltration Modeling Assumptions
(also see dupuit_forschheimer in Modeling Methods.)
brooks_corey_soil_model ? ####
confined_aquifer ####
homogeneous_medium (separate from isotropic ??)
horizontal_flowlines (and vertical equipotential lines)
impermeable_horizontal_base (or impermeable_boundary_at_base)
impermeable_lower_boundary (or impermeable_base)
steady_state_recharge ???
transitional_brooks_corey_soil_model ? ####
unconfined_aquifer ####
van_genuchten_soil_model ? ####
Hydrology: Infiltration Process Models
beven_infiltration_model (assumes Ks decays exponentially)
infiltrated_depth_approximation (not in CF) (Used by Green-Ampt and Smith-Parlange)
(or infiltrability_depth_approximation)
scs_curve_number_infiltration_model (remove "curve number"?)
Hydrology: Snowmelt Models
Hydrology: Soil Models
darcy_soil_model ?###### See Equations, Laws, Etc.
Hydrology: Surface Water Modeling Assumptions
bankfull_flow (or maximum inbank flow)
convergent_or_divergent_topography ###### ?
hydrologically_sound (applied to a DEM)
impermeable_surface ???
inbank_flow (an accepted term; contrast with overbank and bankfull flow)
kinematic_wave (hydraulic_slope_equals_channel_slope)
law_of_the_wall (also listed with equations)
liquid_water_equivalent (used to clarify a quantity like precipitation_rate)
manning_equation (also listed with equations)
#### instantaneous_unit_hydrograph idea ???
Illumination and Shading Models
See: List of common shading algorithms.
lambert_illumination_model (lambert vs. lambertian)
Infiltration Models (Ventilation Models)
Nonlinear Science Models
aperiodic_tiling_model (See: Aperiodic tiling.)
bond_percolation_model (what type of lattice ??)
dimer_model (and "double_dimer_model" ?)
lattice_gas_model (includes: lattice_gas_automata_model and lattice_boltzmann_model. See: Lattice gas automaton.)
potts_model (See: Potts model.)
sandpile_model (Per Bak, self-organized criticality)
site_percolation_model (what type of lattice ??)
Ocean Models
kelvin_wave (coastal or equatorial)
passive_scalar (e.g. temperature and salinity, perhaps suspended sediment)
### per_unit_length_of_wave_crest
shore_parallel_contours (not in CF)
Sediment Transport Models
bagnold_sediment_transport_model (distinguish total load and bedload ####)
komar_longshore_sediment_transport_model ####
Thermodynamics Models
black_body_model (See: Black body.)
carnot_heat_engine_model (See: Carnot heat engine).
Turbulence and Turbulence Closure Models
See: Turbulence modeling.
detached_eddy_simulation_model (DES)
direct_numerical_simulation_model (DNS) (Navier-Stokes solved without a turbulence model)
eddy_viscosity_model (due to Boussinesq, 1887)
k_epsilon_model (due to Jones and Launder)
k_omega_model (due to Kolmogorov ??)
large_eddy_simulation_model (LES) ?? #####
prandtl_mixing_length_model (due to Prandtl)
reynolds_averaged_navier_stokes_model (or reynolds_shear_stress_model)
smagorinsky_model (due to Smagorinsky, 1964; for sub-grid scale eddy viscosity)
Water Wave Models
airy_wave_model Airy waves
capillary_wave_model (type of wave vs. model for waves?)
cnoidal_wave_model Cnoidal waves
kelvin_wave_model [8]
Models Not Yet Grouped
hagen_poiseuille (pressure drop in a pipe; laminar, viscous, incompressible)
harmonic_function (solution to Laplace equation)
power_law ####
unnamed_empirical_law ####
VSEPR (to compute molecular geometry)
dispersion_relation (could be linear)
Thermodynamic Process Assumptions
adiabatic_process (See: Adiabatic).
endothermic_process (better as adjective? absorbs energy)
exothermic_process (better as adjective? releases energy)
isentropic_process (also called "reversible" ?; See: Isentropic).
isenthalpic_process (also called "isoenthalpic"; See: Isenthalpic).
quasistatic_process (reversible implies quasistatic, but not conversely)
thermal_equilibrium #### (See "black_body_model".) (See: Thermal equilibrium).
Stochastic Model Assumptions
• Many of these end with the word "_process", which is part of the standard terminology. Many others end with "_distribution".
#### independent_and_identically_distributed (use both)
negative (process?)
nonnegative (process?)
positive (process?)
random_multiplicative_cascade_process (is there "additive", too?)
random_walk_process (symmetric or unsymmetric)
renewal_process (generalization of Poisson point process)
schramm_loewner_evolution_process (See: SLE process).
shot_noise_process (e.g. raindrops on a roof)
Probability Distributions
• Many of these end with the word "_distribution",
• Also see: List of probability distributions (Wikipedia).
• Most of these are named distributions, but some are a type of distribution (e.g. discrete distribution).
arcsine_distribution (See: Arcsine distribution.)
bates_distribution (See: Bates distribution.)
benford_distribution (See: Benford's law.)
bernoulli_distribution (See: Bernoulli distribution.)
beta_distribution (See: Beta distribution.)
beta-binomial_distribution (See: Beta-binomial distribution.)
beta-prime_distribution (See: Beta prime distribution.)
bimodal_distribution (See: Multimodal distribution.)
binomial_distribution (See: Binomial distribution.)
boltzmann_distribution (See: Boltzmann distribution.)
borel_distribution (See: Borel distribution.)
burr_distribution (See: Burr distribution.)
cauchy_distribution (See: Cauchy distribution.)
champernowne_distribution (See: Champernowne distribution.)
continuous_uniform_distribution (See: Uniform distribution (continuous).)
dagum_distribution (See: Dagum distribution.)
dirichlet_distribution (See: Dirichlet distribution.)
dirichlet-multinomial_distribution (See: Dirichlet-multinomial distribution.)
discrete_uniform_distribution (See: Uniform distribution (discrete).)
elliptical_distribution (See: Elliptical distribution.)
erlang_distribution (See: Erlang distribution.)
exponential_distribution (See: Exponential distribution.)
first-contact_distribution (See: Spherical contact distribution.)
frechet_distribution (See: Frechet distribution.)
gamma_distribution (See: Gamma distribution.)
gaussian_distribution (See: Normal distribution.)
generalized_extreme_value_distribution (See: GEV distribution.)
geometric_distribution (See: Geometric distribution.)
geometric-stable_distribution (See: Geometric stable distribution.)
gompertz_distribution (See: Gompertz function.)
gumbel_distribution (See: Gumbel distribution.)
half-normal_distribution (See: Half-normal distribution.)
hitting-time_distribution (See: Hitting time.)
holtsmark_distribution (See: Holtsmark distribution.)
hyperbolic_distribution (See: Hyperbolic distribution.)
hyperbolic-secant_distribution (See: Hyperbolic secant distribution.)
hypergeometric_distribution (See: Hypergeometric distribution.)
identically_distributed ####
independently_distributed #####
inverse-gamma_distribution (See: Inverse-gamma distribution.)
inverse-gaussian_distribution (See: Inverse Gaussian distribution.)
irwin-hall_distribution (See: Irwin-Hall distribution.)
joint-probability_distribution (See: Joint probability distribution.)
kent_distribution (See: Kent distribution.)
landau_distribution (See: Landau distribution.)
laplace_distribution (See: Laplace distribution.)
levy_distribution (See: Levy distribution.)
log-cauchy_distribution (See: Log-Cauchy distribution.)
log-logistic_distribution (See: Log-logistic distribution.)
log-normal_distribution (See: Log-normal distribution.)
logarithmic_distribution (See: Logarithmic distribution.)
logistic_distribution (See: Logistic distribution.)
logit-normal_distribution (See: Logit-normal distribution.)
lomax_distribution (See: Lomax distribution.)
maximum_entropy_probability_distribution (See: Max entropy pdf.)
maxwell-boltzmann_distribution (See: Maxwell-Boltzmann distribution.)
mixture_distribution (See: Mixture distribution.)
multimodal_distribution (See: Multimodal distribution.)
multinomial_distribution (See: Multinomial distribution.)
nakagami_distribution (See: Nakagami distribution.)
negative-binomial_distribution (See: Negative binomial distribution.)
parabolic-fractal_distribution (See: Parabolic fractal distribution.)
pareto_distribution (See: Pareto distribution.)
pascal_distribution (special case of negative binomial.)
pearson_distribution (See: Pearson distribution.)
poisson_distribution (See: Poisson distribution.)
poisson-binomial_distribution (See: Poisson binomial distribution.)
polya_distribution (special case of negative binomial)
rademacher_distribution (See: Rademacher distribution.)
rayleigh_distribution (See: Rayleigh distribution.)
rayleigh-mixture_distribution (See: Rayleigh mixture distribution.)
reciprocal_distribution (See: Reciprocal distribution.)
rice_distribution (See: Rice distribution.)
skellam_distribution (See: Skellam distribution.)
skew-normal_distribution (See: Skew normal distribution.)
slash_distribution (See: Slash distribution.)
stable_distribution (See: Stable distribution.)
student-t_distribution (See: Student's t-distribution.)
symmetric_distribution (See: Symmetric distribution.)
tracy-widom_distribution (See: Tracy-Widom distribution.)
triangular_distribution (See: Triangular distribution.)
truncated_distribution (See: Truncated distribution.)
tukey-lambda_distribution (See: Tukey lambda distribution.)
u-quadratic_distribution (See: U-quadratic distribution.)
voigt_distribution (See: Voigt profile.)
von-mises_distribution (See: von Mises distribution.)
von-mises-fisher_distribution (See: von Mises-Fisher distribution.)
weibull_distribution (See: Weibull distribution.)
yule-simon_distribution (See: Yule-Simon distribution.)
zeta_distribution (See: Zeta distribution.)
zipf_distribution (See: Zipf's law.)
Statistical Operation Assumptions
• Perhaps this should be generalized to something like "Data Transformation Assumptions"?
• These names currently all end with "averaged".
• For ones that start with a unit of time, one of those units is assumed. A number can be inserted in front, when necessary, as in "two_day_averaged".
Mathematical Assumptions
algebraic (equation)
bounded (set)
closed (set, curve)
compact (set)
constant_coefficients (equation or polynomial)
continuum (continuum_hypothesis ?)
differential (equation)
multiple_valued_function ### (misnomer)
Numerical Grid Assumptions
• Most of these end with the word grid.
• The word "grid" is used to include the word "mesh".
arakawa_a_grid (unstaggered)
arakawa_b_grid (staggered)
arakawa_c_grid (staggered)
arakawa_d_grid (staggered, rotated 90 degrees)
arakawa_e_grid (staggered, rotated 45 degrees)
arakawa_u_component (attached to an input var)
arakawa_v_component (attached to an input var)
arakawa_w_component (attached to an input var)
boundary-fitted_grid (also called "body-fitted")
staggered_grid (###### already in arakawa system ??)
Numerical Method Assumptions
• These are used to describe the numerical method that a model uses to solve the equations it uses to compute variables of interest. The equations could be ODEs, PDEs, algebraic equations (e.g.
root finding), etc. We probably don't need separate assumption names like "ode" and "pde" because that is implied by the equation name. See Equations, Laws and Principles for a standardized list
of equation names.
• Most of these names end with "_method", "_scheme" or "_grid".
adaptive_mesh_refinement_method (See: Adaptive mesh refinement.)
adaptive_stepsize_method (See: Adaptive stepsize.)
adjoint_state_method (See: Adjoint state method.)
analytic_element_method (See: Analytic element method).
backward_euler_method (See: Backward Euler method.)
boundary_element_method (See: Boundary element method.)
central_difference_scheme (See: Central differencing scheme.)
characteristics_method (known as "method of characteristics")
collocation_method (See: Collocation method.)
conjugate_gradient_method (See: Conjugate gradient method.)
crank-nicolson_method (See: Crank-Nicolson method.)
discrete_element_method (See: Discrete element method.)
discrete_event_simulation (See: Discrete event simulation.)
dynamic_relaxation_method (See: Dynamic relaxation.)
euler_method (See: Euler method.) (distinguish between "forward" and "backward" with a prefix?)
explicit_method (See: Explicit and implicit methods.)
fast_marching_method (See: Fast marching method, a type of level_set_method.)
finite_difference_method (See: Finite difference method.)
finite_element_method (See: Finite element method.)
finite_volume_method (See: Finite volume method.)
five-point_stencil_method (See: Five-point stencil.)
forward_time_centered_space_scheme (FTCS scheme)
galerkin_method (See: Galerkin method.)
gauss-legendre_method (See: Gauss-Legendre method.)
gauss-seidel_method (See: Gauss-Seidel method.)
halley_method (See: Halley's method.)
heun_method (See: Heun's method.)
implicit_method (See: Explicit and implicit methods.)
interior_point_method (See: Interior point method.)
iterative_method (See: Iterative method.)
l-stable_method (See: L-stability.)
landweber_iteration_method (See: Landweber iteration.)
lattice_boltzmann_method (See: Lattice Boltzmann methods.)
lax-friedrichs_method (See: Lax-Friedrichs method.)
lax-wendroff_method (See: Lax-Wendroff method.)
level_set_method (See: Level set method.)
linear_multistep_method (See: Linear multistep method.)
maccormack_method (See: MacCormack method.)
meshfree_method (See: Meshfree method.)
midpoint_method (See: Midpoint method.)
multigrid_method (See: Multigrid method.)
newton_raphson_method (See: Newton's method; also see "halley_method".)
numerov_method (See: Numerov's method.)
particle-in-cell_method (See: Particle in cell.)
predictor-corrector_method (See: Predictor-corrector method.)
rayleigh-ritz_method (See: Rayleigh-Ritz method.)
relaxation_method (See: Relaxation (iterative method).)
runge_kutta_method (See: Runge-Kutta methods. There are several distinct types.)
shooting_method (See: Shooting method.)
spectral_method (See: Spectral method.)
split-step_method (See: Split-step method.)
successive_over_relaxation_method (See: Successive over-relaxation.)
trapezoidal_rule_method (See: Trapezoidal rule.)
upwind_difference_scheme (See: Upwind difference scheme.)
upwind_first-order_scheme (See: Upwind scheme.)
upwind_second-order_scheme (See: Upwind scheme.)
upwind_third-order_scheme (See: Upwind scheme.)
verlet_integration_method (See: Verlet integration.)
State of Matter Assumptions
• These can be provided when the model involves a substance (object) like water that could be in any of several possible states. See: States of matter.
• Note that "liquid_equivalent" can also be inserted in quantity names such as "liquid_equivalent_precipitation_rate" to create a single quantity that can accommodate multiple states of matter.
System State Assumptions
metastable (See: Metastability).
CF Convention Standard Name Assumptions
• CF Convention Standard Names often include additional information and assumptions in the name itself. The ones in this section were found in the list of CF Standard Names and the number of
occurrences found is listed in parentheses. It is not yet clear how some of these should be captured with standard assumption names.
• Many of these are Location Assumptions.
above_geoid (3 in CF)
above_land_surface (not in CF)
above_reference_datum (1 in CF)
above_reference_ellipsoid (5 in CF)
above_sea_floor (1 in CF)
above_sea_floor_surface (not in CF)
above_sea_level (1 in CF)
above_threshold (5 in CF)
at_*** (51 in CF)
at_cloud_base (1 in CFFl
at_cloud_top (3 in CF)
at_equilibrium (not in CF)
at_freezing_level (1 in CF)
at_land_surface (not in CF; e.g. air pressure)
at_maximum_upward_derivative (1 in CF)
at_saturation (4 in CF)
at_sea_floor (3 in CF)
at_sea_floor_surface (not in CF; e.g. water pressure)
at_sea_ice_base (8 in CF)
at_sea_level (1 in CF)
at_top_of_*** (3 in CF)
at_bottom_*** (not in CF)
assuming_*** (33 in CF)
assuming_clear_sky (24 in CF)
assuming_deep_snow (1 in CF, for surface_albedo)
assuming_no_aerosol_or_cloud (1 in CF)
assuming_no_snow (1 in CF, for surface_albedo)
assuming_no_tide (2 in CF)
assuming_sea_level_for_geoid (4 in CF)
below_geoid (1 in CF)
below_sea_level (1 in CF)
below_sea_surface (1 in CF)
below_surface (1 in CF)
below_threshold (3 in CF)
between_air_and_sea_water (1 in CF)
between_sea_water_and_air (2 in CF)
due_to_*** (399 in CF)
due_to_all_land_processes (2 in CF)
due_to_convective_cloud (1 in CF)
due_to_diffusion (18 in CF)
due_to_dry_convection (1 in CF)
due_to_dry_deposition (35 in CF)
due_to_dry_troposphere (1 in CF)
due_to_dust_ambient_aerosol (2 in CF)
due_to_emission_from_grazing (in CF)
excluding_anthropogenic_land_use_change (in CF)
excluding_baseflow (in CF)
excluding_litter (in CF)
expressed_as_*** (140 in CF)
expressed_as_carbon (67 in CF)
expressed_as_chlorine (7 in CF)
expressed_as_nitrogen (24 in CF)
for_*** (13 in CF)
for_biomass_growth (1 in CF)
for_biomass_maintenance (1 in CF)
for_boussinesq_approximation (1 in CF)
for_momentum (2 in CF; both "for_momentum_in_air")
per_unit_area (already used in CF)
per_unit_mass (already used in CF, and synonym for "specific")
per_unit_time ??
per_unit_width (e.g. discharge_per_unit_width)
(see CF: sea_water_transport_across_line,
and "transport_across_unit_distance")
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, in the most general meaning,
is a subdivision of a geometric object into
. In particular, in the plane it is a subdivision into triangles, hence the name.
Different branches of geometry use slightly differing definitions of the term.
A triangulation T of R^n+1 is a subdivision of R^n+1 into (n+1)-dimensional simplices such that:
1. any two simplices in T intersect in a common face or not at all;
2. any bounded set in R^n+1 intersects only finitely many simplices in T.
of a
set of points
is a triangulation of
such that the set of points that are vertices of the subdividing simplices coincides with
In geometry, in the most general meaning, triangulation is a subdivision of a geometric object into simplices. In particular, in the plane it is a subdivision into triangles, hence the name.
Different branches of geometry use slightly differing definitions of the term.
A triangulation T of R^n+1 is a subdivision of R^n+1 into (n+1)-dimensional simplices such that:
1. any two simplices in T intersect in a common face or not at all;
2. any bounded set in R^n+1 intersects only finitely many simplices in T.
of a
set of points
is a triangulation of
such that the set of points that are vertices of the subdividing simplices coincides with
The following definitions are used in Computational geometry.
A triangulation of a polygon P is its partition into triangles. In the strict sence, these triangles may have vertices only in the vertices of P. In non-strict sense, it is allowed to add more points
to serve as vertices of triangles.
Also, a triangulation of a set of points P is sometimes taken to be the triangulation of the convex hull of P.
See also: Delaunay triangulation
Topology generalizes this notion in a natural way as follows. A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K->X.
Triangulation is useful in determining the properties of a topological space.
is the process of finding a distance by calculating the length of one side of a triangle, given a deterministic combination of angles and sides of the triangle. It uses mathematical identities from
Some identities often used:
Triangulation is used for many purposes, including
binocular vision
and gun direction of
See: Parallax.
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HP Forums
The inverse F distribution is throwing a Domain Error before achieving convergence with a few sample problems I've tried. For example, df1 = 5, df2 = 10, p = 0.75, as well as the complementary
problem of df1 = 10, df2 = 5, p = 0.25. I don't think there is anything too taxing about either of these samples, and I am pretty sure that I got correct results the last I checked them quite a few
revisions ago.
I know there have been challenges with the Newton refiner, and I wonder if things have cropped up again. I haven't run into this with the inverse normal, chi-squared, or t, though admitted the
inverse problem can seems slow at times if the underlying CDF, like the t, is pretty complicated to calculate each time.
This isn't a quest for obsessive accuracy. Just concern that something that was once okay seems now broken.
Edited: 17 May 2012, 3:58 p.m.
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Hey! Pass this quiz: Translations Reflections and Rotations - Geometric Transformations!
Created from Youtube video: https://www.youtube.com/watch?v=GqHWdTLL8Qwvideo
Concepts covered:translations, reflections, rotations, geometric transformations, quadrants
The video discusses geometric transformations including translations, reflections, and rotations. It explains how to perform these transformations on figures in different quadrants using specific
examples and provides step-by-step instructions for each type of transformation.
Reflections over X and Y Axes
Concepts covered:Reflections, X-axis, Y-axis, Quadrants, Coordinates
The chapter discusses reflections over the x-axis and y-axis, explaining how figures move and flip when reflected. It covers the process of reflecting shapes in different quadrants and provides
examples with clear instructions on multiplying coordinates for accurate reflections.
Question 1
Where does a point move reflecting over x-axis from Quadrant II?
Question 2
Which axis reflection changes the x-coordinate?
Question 3
How does x-coordinate change in y-axis reflection?
Rotating Triangles in Quadrants
Concepts covered:Triangle, Quadrants, Rotation, Clockwise, Counterclockwise
Rotating a triangle in quadrant one by 90 degrees clockwise moves it to quadrant four, changing the positions of points A, B, and C. The hypotenuse becomes the side of the right triangle. Rotating
the triangle 90 degrees counterclockwise shifts it from quadrant one to quadrant two, altering the positions of points A, B, and C.
Question 4
Where does a triangle end after 90 degrees counterclockwise rotation?
Question 5
What happens when a triangle rotates 90 degrees clockwise?
Question 6
Which quadrant does clockwise rotation of 90 degrees reach?
Rotating Triangles in Quadrants
Concepts covered:Triangle, Quadrants, Rotation, Clockwise, Counterclockwise
Rotating a triangle in quadrant one by 90 degrees clockwise moves it to quadrant four, changing the positions of points A, B, and C. The hypotenuse becomes the side of the right triangle. Rotating
the triangle 90 degrees counterclockwise shifts it from quadrant one to quadrant two, altering the positions of points A, B, and C.
Question 7
Where does a triangle move with a 90-degree counterclockwise rotation?
Question 8
Which rotation direction moves a shape from quadrant one to four?
Question 9
What quadrant does a 90-degree clockwise rotation reach?
Rotating Figures by 180 Degrees
Concepts covered:rotation, 180 degrees, coordinates, reflection, origin
Rotating a figure by 180 degrees results in the same shape as reflecting it across the origin. Changing the coordinates by multiplying each by -1 achieves a 180-degree rotation.
Question 10
What happens to a point after 180-degree rotation?
Question 11
How to transform point (x, y) by 180 degrees?
Question 12
Effect on (x, y) after 90-degree counterclockwise rotation?
Rotating Figures by 180 Degrees
Concepts covered:rotation, 180 degrees, coordinates, reflection, origin
Rotating a figure by 180 degrees results in the same shape as reflecting it across the origin. Changing the coordinates by multiplying each by -1 achieves a 180-degree rotation.
Question 13
What happens to a point after 180-degree rotation?
Question 14
How to transform point (x, y) by 180 degrees?
Question 15
Transform (x, y) for 90-degree clockwise rotation?
Would you like to create and run this quiz?
Created with Kwizie
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Re: [tlaplus] Inductive Invariants and Counterexamples in TLAPS
Have there been any previous attempts to incorporate counterexample generation into TLAPS? For example, say we are trying to prove an inductive invariant, Ind. If it is not inductive, it is helpful
to get a counterexample to induction i.e. some state that satisfies Ind but which can reach a state violating Ind via some protocol transition. Recently, I have been using the probabilistic method of
checking inductive invariance described in [1], and it has seemed to work surprisingly well. Of course, it only works on finite protocol instances, but this is still very helpful when debugging an
inductive invariant.
I am curious about whether this could be done with TLAPS, though, since TLAPS can produce an SMT encoding of a TLA+ invariant and transition relation. In theory, it seems that the probabilistic
method is solving something at least as hard as satisfiability, since it first needs to generate states that satisfy some arbitrary predicate, Ind. It takes a completely "dumb" approach, though, and
just randomly samples states and checks to see if they satisfy Ind. I would think, though, that state of the art SMT solvers would be able to outperform the efficiency of this technique. So, it would
seem TLAPS would be well suited to handle this problem, since it is able to generate an SMT encoding that can be handed to a solver.
Presumably, Apalache [2] is another candidate for this task, since it also produces an SMT encoding that can be given to an SMT solver. Their OOPSLA '19 paper [3] claims that it can prove certain
inductive invariants automatically, and that it can detect invariant violations quickly as well. I suppose it would be interesting to compare the performance of the probabilistic method for finding
inductiveness violations with Apalache, and/or compare Apalache with TLAPS. I suppose the details of the SMT encoding can make a significant difference in performance here? Tools like Ivy [4] are
supposed to be fast at inductive invariant checking because they restrict the input language in a rather spartan way. It would seem that TLA+ seems cannot utilize similar ideas due to its
In general, I would be interested to hear others' thoughts on this, if they worked on or considered these problems in the past.
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A relative nodal coordinate formulation for finite element nonlinear analysis
Nodal coordinates are referred to a fixed configuration in the conventional equations of equilibrium. Nodal coordinates are referred to the initial configuration in the total Lagrangian formulation
and to the last calculated configuration in the updated Lagrangian formulation. This research proposes to use the relative nodal coordinates in representing the position and orientation for a node.
Since the nodal coordinates are measured relative to its adjacent nodal reference frame, they are still small for a structure undergoing large deformations if the element sizes are small. As a
consequence, many element formulations developed under small deformation assumptions are still valid for structures undergoing large deformations, which significantly simplifies the equations of
equilibrium. A structural system is represented by a graph to systematically develop the governing equations of equilibrium for general systems. A node and an element are represented by a node and an
edge in graph form, respectively. Closed loops are opened to form a tree topology by cutting edges. Two computational sequences are defined in a graph. One is the forward path sequence that is used
to recover the Cartesian nodal deformations from relative nodal displacements and traverses a graph from the base node towards the terminal nodes. The other is the backward path sequence that is used
to recover the nodal forces in the relative coordinate system from the known nodal forces in the absolute coordinate system and traverses from the terminal nodes towards the base node. One open loop
and one closed loop structure undergoing large displacements are analyzed to demonstrate the efficiency and validity of the proposed method.
Conference 18th Biennial Conference on Mechanical Vibration and Noise
Country/Territory United States
City Pittsburgh, PA
Period 9/09/01 → 12/09/01
Bibliographical note
Funding Information:
This research was supported by Advanced Highway Research Center, Hanyang University, sponsored by KOSEF.
Dive into the research topics of 'A relative nodal coordinate formulation for finite element nonlinear analysis'. Together they form a unique fingerprint.
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Module Contents¶
pymor.algorithms.gram_schmidt.gram_schmidt(A, product=None, return_R=False, atol=1e-13, rtol=1e-13, offset=0, reiterate=True, reiteration_threshold=0.9, check=True, check_tol=0.001, copy=True)
Orthonormalize a VectorArray using the modified Gram-Schmidt algorithm.
The VectorArray which is to be orthonormalized.
The inner product Operator w.r.t. which to orthonormalize. If None, the Euclidean product is used.
If True, the R matrix from QR decomposition is returned.
Vectors of norm smaller than atol are removed from the array.
Relative tolerance used to detect linear dependent vectors (which are then removed from the array).
Assume that the first offset vectors are already orthonormal and start the algorithm at the offset + 1-th vector.
If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.
If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.
If True, check if the resulting VectorArray is really orthonormal.
Tolerance for the check.
If True, create a copy of A instead of modifying A in-place.
The orthonormalized VectorArray.
The upper-triangular/trapezoidal matrix (if compute_R is True).
pymor.algorithms.gram_schmidt.gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshold=0.1, check=True, check_tol=0.001, copy=True)[source]¶
Biorthonormalize a pair of VectorArrays using the biorthonormal Gram-Schmidt process.
See Algorithm 1 in [BKohlerS11].
Note that this algorithm can be significantly less accurate compared to orthogonalization, in particular, when V and W are almost orthogonal.
V, W
The VectorArrays which are to be biorthonormalized.
The inner product Operator w.r.t. which to biorthonormalize. If None, the Euclidean product is used.
If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.
If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.
If True, check if the resulting VectorArray is really orthonormal.
Tolerance for the check.
If True, create a copy of V and W instead of modifying V and W in-place.
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Homogenization of eigenvalue problems in perforated domains
Vanninathan, M. (1981) Homogenization of eigenvalue problems in perforated domains Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 90 (3). pp. 239-271. ISSN 0253-4142
PDF - Publisher Version
Official URL: http://www.ias.ac.in/j_archive/mathsci/90/3/239-27...
Related URL: http://dx.doi.org/10.1007/BF02838079
In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain
increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of
each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ[ε]=ε^−2λ+λ[0]+O (ε), Stekloff: λ[ε]=[ε]λ[1]+O (ε^2), Neumann: λ[ε]=λ[0]+[ε]λ[1]+O (ε^2).
Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.
Item Type: Article
Source: Copyright of this article belongs to Indian Academy of Sciences.
Keywords: Homogenization; Correctors; Eigenvalues; Eigenvectors
ID Code: 55288
Deposited On: 18 Aug 2011 07:03
Last Modified: 18 May 2016 07:37
Repository Staff Only: item control page
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Will Quantum Computers Break Bitcoin? - 2024
Bitcoin has become a household name over the past decade. As the first and most prominent cryptocurrency, Bitcoin introduced the revolutionary concept of decentralized digital money powered by
blockchain technology. However, concerns have emerged regarding Bitcoin’s vulnerability to attack from quantum computers. Let’s examine the potential threat quantum computing poses to Bitcoin’s
Key Takeaways:
• Quantum computers possess the raw computational capacity that could theoretically breach current cryptography, including Bitcoin’s SHA-256 and ECDSA.
• However, significant engineering barriers around scaling qubit counts and error correction must first be overcome before quantum computers reach cryptographically relevant scales, likely at least
a decade away.
• Even with technical capacity, immense economic costs around successfully attacking Bitcoin’s global network may deter exploitation of cryptographic breaks.
• Bitcoin demonstrates antifragility – its growing adoption, hash rate, longevity suggest flexibility to adapt to quantum threats through technical countermeasures like quantum resistant
• While speculative quantum attack vectors like private key exposure or blockchain reorganization may emerge, Bitcoin’s consensus structure mitigates many risks.
• Quantum computing simultaneously opens doors to advanced cryptography like post quantum algorithms and quantum key distribution that could ultimately harden Bitcoin’s defenses.
• Rather than an inevitably catastrophic risk, the quantum computing age shapes up as an opportunity for Bitcoin to evolve, display robustness, and lead the push into next generation quantum
resistant security infrastructures.
What Is Quantum Computing?
To understand if quantum computers can break Bitcoin, we must first understand what quantum computing is. Quantum computers utilize quantum mechanics phenomena like superposition and entanglement to
perform calculations fundamentally differently from classical computers. They leverage quantum bits (qubits) that can represent 1, 0, or any quantum superposition of those two states simultaneously.
This enables quantum computers to solve certain problems exponentially faster than classical computers by essentially trying all possible solutions at once.
A quantum computer utilizes qubits and quantum phenomena like superposition to perform specialized computations
How Could Quantum Computers Pose a Threat?
Most cryptography today relies on certain mathematical functions being extremely difficult for classical computers to invert or crack within a reasonable timeframe. This includes Bitcoin’s underlying
SHA-256 hashing and public key ECDSA encryption. However, Shor’s algorithm enables quantum computers to efficiently crack these functions. So if a scalable, fault tolerant quantum computer is ever
built, it could theoretically break Bitcoin’s cryptography and undermine its security.
Assessing the Risk Quantum Computers Pose
Let’s examine factors like the state of quantum computing today versus Bitcoin’s staying power to assess whether quantum computers are likely to break Bitcoin’s security:
Development of Quantum Computers
Year Progress & Projections on Quantum Computers
2011 D-Wave releases first commercial quantum annealer (controversy over whether it demonstrates “quantum supremacy”)
2017 IBM makes first 5 qubit universal quantum computer prototype available via cloud access
2019 Google achieves “Quantum Supremacy” on 53 qubit computer (controversy over benchmark’s meaningfulness)
2024 IBM projects a 1,121+ qubit system, pushing closer to fault tolerance
2030s Potential achievement of fault tolerance in quantum computers with cryptographic breaking capacity
While quantum computing has advanced rapidly, most experts believe we are still more than a decade away from quantum computers that threaten cryptography. However, some researchers warn we could
reach an inflection point sooner.
Bitcoin’s Staying Power
Metric Bitcoin’s Current State
Longevity Created in 2009, has persisted over 13 years
Hash Rate Hash rate exceeds 250 exahashes per second world’s largest computing network
Value Market cap recently exceeded $1 trillion
Community Over 100 million estimated users worldwide
Security Has proven essentially impenetrable to cyberattacks thus far
Bitcoin has demonstrated impressive resilience and security over more than a decade of operation. This staying power indicates Bitcoin may have flexibility to adapt to potential quantum computing
Could Quantum Computers Really Break Bitcoin’s Encryption?
While quantum computers theoretically possess the raw power to break Bitcoin’s cryptography, critical technical and economic challenges remain.
Technical Hurdles
• Achieving fault tolerance with millions of qubits is extremely difficult from an engineering perspective.
• Bitcoin’s SHA 256 algorithm requires an impractically high qubit count exceeding 4000.
• Technical countermeasures may enhance Bitcoin’s quantum resistance.
Economic Challenges
• Successfully attacking Bitcoin’s network would require enormous financial resources.
• An attack causing loss of confidence could ruin Bitcoin’s value, undermining incentives.
• If progress toward cryptographically relevant quantum computers became evident, Bitcoin could implement protocol changes to use quantum resistant cryptography.
So while quantum computers could potentially break Bitcoin’s encryption someday, there are still major hurdles to realize this threat practically.
Grover’s Algorithm and Brute Forcing Addresses
The cryptographic schemes used throughout the Bitcoin protocol could theoretically succumb to Grover’s quantum algorithm for quickly searching unsorted databases.
Rather than try every possible key sequentially, Grover’s algorithm allows a quantum computer to isolate the sought key after just O(N^1/2) operations where N is the size of the search space. For
spaces containing 2^256 possible elements such as Bitcoin private keys, Grover’s algorithm offers over a 2^128 speedup!
With such efficiency, an adversary with a sufficiently powerful quantum computer could use Grover’s algorithm to brute force randomly guess the private key that unlocks any Bitcoin address after
testing just 2^128 possibilities on average. Then they could swiftly empty that wallet.
Shor’s Algorithm and Breaking ECDSA
In addition to brute forcing private keys, quantum computers pose another critical threat to cracking Bitcoin’s Elliptic Curve Digital Signature Algorithm (ECDSA) which protects transaction
The security of ECDSA rests on the conjectured computational difficulty of solving the discrete logarithm problem on elliptic curves. Quantum computers could employ Shor’s quantum algorithm for
efficiently solving such discrete logarithms and related problems that underpin public key crypto schemes including ECDSA.
By leveraging Shor’s algorithm, scalable quantum computers might render ECDSA obsolete, thereby allowing bad actors to fake signed messages and “hack” the blockchain.
Current Quantum Capabilities
In light of these threats, how far away is the quantum computing apocalypse for Bitcoin? The truth is, no quantum computer today comes even remotely close to breaking any cryptocurrency scheme, but
rapid advancements suggest we must take the threat seriously.
Most Qubits Operational ~127 (IBM Eagle Processor)
Required Qubits to Break ECDSA 1500-3000+
Required Qubits to Break Symmetric Crypto >500
Logical Qubits Required for Scalability Millions
While the qubit counts of leading machines currently number in the low hundreds, manufacturers plan to begin building fault tolerant quantum computers composed of millions of logical qubits later
this decade. Such capacity could threaten symmetric cryptography within the next 10 years and hash based signatures within 20 years.
Solving elliptic curves and breaking ECDSA requires even more scale. Realistically, Bitcoin is not likely to face an existential quantum threat before 2030. However, the 2040s look dicey if quantum
progress maintains momentum and proactive defenses haven’t been enacted.
Mitigating Quantum Risk
While quantum computers may one day upend cryptography as currently practiced, new quantum resistant schemes can mitigate these risks if deployed judiciously. Post-quantum cryptography offers a rich
area of research for blockchain engineers to combat tomorrow’s quantum foe.
Leading proposals for post quantum signatures include hash-based schemes (e.g. SPHINCS), code based schemes, and lattice based schemes. Each offers tradeoffs across complexity, security, performance,
and features. Forking to implement post-quantum signatures remains Bitcoin’s strongest long-term quantum defense with complexity and some technical debt the primary short term barrier.
On the horizon, novel methods like quantum money may even employ quantum principles themselves to create cryptography intrinsically secure against quantum attack. More speculatively, migrating
Bitcoin to a quantum secure hashgraph structure could offer robustness.
Such solutions must be implemented well in advance ofconcrete quantum threat to allow sufficient testing, deployment, and transition. Therefore prioritizing crypto agility should rank among Bitcoin
stakeholders’ top priorities this decade if not sooner.
Evaluating the Risk of Quantum Attack Vectors
Besides direct cryptographic breaking, quantum computers might also introduce new attack vectors against Bitcoin. However, Bitcoin’s consensus mechanism helps defend against many attack scenarios.
Let’s examine two of the possible quantum attack risks to Bitcoin:
Private Key Exposure
• A quantum computer could retroactively break encryption on Bitcoin keys stored online or generated insecurely in the past, enabling theft.
• However, Bitcoin’s public key cryptography means exposed private keys pose less of an aggregate threat compared to symmetric cryptography.
Blockchain Reorganization
• A quantum computer could theoretically override blockchain consensus to double spend coins or reverse transactions.
• However, successfully executing attacks would likely require prohibitive resources while providing little financial incentive.
• Bitcoin’s ever growing hash rate makes reorganizing the blockchain increasingly difficult over time.
So while quantum computers introduce new potential attack vectors, Bitcoin’s blockchain consensus protocol helps mitigate risks. And ongoing research efforts continue focusing on quantum threat
The Critical Role of Quantum Cryptography
Quantum computing is also spurring important innovations in cryptography and security research:
• Post-quantum cryptography (PQC) algorithms like hash based, code based, and lattice based cryptography offer different approaches to quantum resistance using specialized mathematical problems or
asymmetric encryption techniques.
• Quantum key distribution (QKD) enables theoretically unbreakable encryption through leveraging quantum entanglement principles to establish shared keys.
Ongoing research and standardization efforts for these quantum resistant cryptographic schemes hold promise for keeping information secure in a world with scalable quantum computers. These
innovations could provide vital tools for protecting Bitcoin long-term.
Conclusion: Quantum Computing Brings Risks and Opportunities
In summary, while quantum computers do pose a long-term threat to Bitcoin’s cryptography, we still have years to prepare through research and open-source developer contributions. Bitcoin has already
demonstrated impressive resilience and antifragility over its first decade. So the advent of cryptographically relevant quantum computers may ultimately introduce healthy opportunities to harden
Bitcoin’s security and usher the broader world into developing next-generation quantum resistant encryption standards. Rather than just a risk, quantum computing can become a rising tide that lifts
all boats if society responds thoughtfully.
How soon could quantum computers break Bitcoin?
Experts estimate we are likely a decade or more away from quantum computers large enough and reliable enough to break Bitcoin’s encryption. However, it depends on the pace of advances in quantum
If someone broke Bitcoin’s encryption with a quantum computer, could they steal all the Bitcoin?
In theory, someone could access individual private keys to drain associated Bitcoin wallets. But breaking enough keys to significantly damage confidence in Bitcoin globally would be extremely
challenging in practice.
Could Bitcoin implement a soft fork to enable quantum resistance?
Yes, the Bitcoin open source developer community could potentially implement soft forks to introduce quantum resistant cryptography like hash based signatures when the threat becomes more imminent.
Doesn’t the long term quantum threat make Bitcoin risky to invest in?
Bitcoin clearly still carries risks, but it has so far demonstrated impressive security and antifragility while growing enormous global adoption and hash rate. And potential protocol changes provide
paths to navigate the quantum threat.
Can’t governments also use quantum computing to crack cryptocurrencies?
Governments likely have quantum computing development programs for cryptography breaking purposes. However, Bitcoin’s public blockchain still provides more transparency and access than potential
central bank digital currencies. Protocol upgrades may also counter state level threats.
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If the difference between the mean and variance of a binomial d... | Filo
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Question Text If the difference between the mean and variance of a binomial distribution for 5 trails is then the distribution is
Updated On Jun 16, 2023
Topic Permutation-Combination and Probability
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 129
Avg. Video Duration 3 min
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Trimmed Mean: Meaning, Examples & Step-By-Step Calculation | Finschool By 5paisa
Trimmed Mean is a statistical measure that aims to provide a more accurate dataset representation by removing extreme values or outliers. It is commonly used in various fields, including economics,
finance, and data analysis. In this article, we will delve into the concept of Trimmed Mean, its definition, understanding, and practical applications. We will also explore step-by-step examples of
calculating a Trimmed Mean and its relationship with inflation rates. So, let’s dive in and explore the fascinating world of Trimmed Mean!
What is Trimmed Mean?
Trimmed Mean, also known as truncated Mean or truncated average, is a statistical measure that calculates the average of a dataset by excluding a certain percentage of the highest and lowest values.
By removing outliers, the Trimmed Mean provides a more robust measure of central tendency less influenced by extreme values.
Simply put, Trimmed Mean trims off the tails of a dataset, eliminating extreme values that might skew the overall average. This trimming process helps to reduce the impact of outliers and provides a
more accurate representation of the central tendency of the data.
Understanding a Trimmed Mean
Let’s consider an example to gain a deeper understanding of Trimmed Mean. Imagine you have a dataset of 100 data points representing the prices of houses in a particular neighborhood. Some houses
might be exceptionally expensive or extremely cheap due to location, condition, or size. These extreme values could significantly affect the overall average if included in the calculation.
Using a Trimmed Mean, you can exclude, for instance, the top 10% and bottom 10% of the house prices. This means you disregard the most expensive and the least expensive houses. Doing so lets you
focus on most prices within a reasonable range, providing a more accurate estimate of the average price for houses in that neighborhood.
Trimmed Mean is particularly useful when the dataset contains outliers or extreme values that do not reflect the overall pattern or characteristics of the data. By eliminating these outliers, the
Trimmed Mean allows for a more reliable analysis and interpretation of the dataset.
Trimmed Means and Inflation Rates
One exciting application of Trimmed Mean is in the calculation of inflation rates. The inflation rate is the rate at which the standard level of prices for goods and services rises and, subsequently,
purchasing power is falling. It is a crucial economic indicator that affects individuals, businesses, and governments.
When calculating inflation rates, statisticians often use a Trimmed Mean to remove the effects of extreme price changes. By focusing on the core or underlying inflation, which excludes the most
volatile price movements, policymakers can obtain a more accurate measure of inflation that reflects the long-term trend.
The Trimmed Mean helps identify the persistent price changes that are likely to have a lasting impact on the economy. Policymakers can make more informed decisions regarding monetary policy, interest
rates, and other economic measures by excluding temporary fluctuations.
Example of a Trimmed Mean
Let’s consider an example to illustrate the concept of a Trimmed Mean. Suppose you have a dataset of 50 monthly returns for a particular stock. The returns range from -20% to 30%. To calculate a 10%
Trimmed Mean, you would exclude the highest 10% and lowest 10% of the returns.
After trimming the dataset, you would calculate the average of the remaining returns. This trimmed average provides a more accurate representation of the typical monthly return for the stock, as it
eliminates the influence of extreme positive or negative returns.
Trimmed Mean Example with Step-by-Step Calculation
To further clarify the process of calculating a Trimmed Mean, let’s walk through a step-by-step example:
1. Sort the dataset in ascending order.
2. Determine the percentage to be trimmed. Let’s use 10% for this example.
3. Count the total number of data points.
4. Calculate the number of data points to be trimmed from both ends. For a 10% Trimmed Mean, you would trim 10% of the total data points from each end.
5. Exclude the specified number of data points from the highest and lowest ends.
6. Calculate the average of the remaining data points.
Consider a dataset of 20 values: [2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40]. To calculate a 10% trimmed mean, we need to trim 10% of the data points from both ends.
As 10% of 20 is 2, we will exclude the highest and lowest values: [6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30].
The average of the remaining values is (6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30) / 13 = 20.92. Therefore, the 10% Trimmed Mean of this dataset is approximately 20.92.
In conclusion, Trimmed Mean is a statistical measure that allows for a more accurate estimation of the central tendency of a dataset by excluding extreme values or outliers. It is particularly useful
in situations where outliers can significantly skew the average and affect the interpretation of the data. Trimmed Mean finds applications in various fields, including economics, finance, and data
analysis. Trimmed Mean helps researchers, analysts, and policymakers make more informed decisions based on reliable statistical insights by providing a more robust measure of central tendency.
Frequently Asked Questions(FAQs)
The 10% trimmed mean is a statistical measure that calculates the average of a dataset by excluding the highest 10% and lowest 10% of the values. It provides a more robust estimate of central
tendency by removing outliers or extreme values.
Truncated Mean is another term for Trimmed Mean. It refers to a statistical measure that calculates the average of a dataset by excluding a certain percentage of the highest and lowest values.
o find the truncated Mean, you need to follow these steps:
1. Sort the dataset in ascending order.
2. Determine the percentage to be trimmed.
3. Exclude the specified number of data points from both ends.
4. Calculate the average of the remaining data points.
Following these steps, you can find the truncated Mean of a given dataset.
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Note (a) for Implications for Mathematics and Its Foundations: A New Kind of Science | Online by Stephen Wolfram [Page 1163]
[Examples of] unprovable statements
After the appearance of Gödel's Theorem a variety of statements more or less directly related to provability were shown to be unprovable in Peano arithmetic and certain other axiom systems. Starting
in the 1960s the so-called method of forcing allowed certain kinds of statements in strong axiom systems—like the Continuum Hypothesis in set theory (see page 1155)—to be shown to be unprovable. Then
in 1977 Jeffrey Paris and Leo Harrington showed that a variant of Ramsey's Theorem (see page 1068)—a statement that is much more directly mathematical—is also unprovable in Peano arithmetic. The
approach they used was in essence based on thinking about growth rates—and since the 1970s almost all new examples of unprovability have been based on similar ideas. Probably the simplest is a
statement shown to be unprovable in Peano arithmetic by Laurence Kirby and Jeff Paris in 1982: that certain sequences g[n] defined by Reuben Goodstein in 1944 are of limited length for all n, where
g[n_] := Map[First, NestWhileList[{f[#] - 1, Last[#] + 1} &, {n, 3}, First[#] > 0 &]]
f[{0, _}] = 0; f[{n_, k_}] := Apply[Plus, MapIndexed[#1 k^f[{#2〚1〛 - 1, k}] &, Reverse[IntegerDigits[n, k - 1]]]]
As in the pictures below, g[1] is {1, 0}, g[2] is {2, 2, 1, 0} and g[3] is {3, 3, 3, 2, 1, 0}. g[4] increases quadratically for a long time, with only element 3×2^402653211 - 2 finally being 0. And
the point is that in a sense Length[g[n]] grows too quickly for its finiteness to be provable in general in Peano arithmetic.
The argument for this as usually presented involves rather technical results from several fields. But the basic idea is roughly just to set up a correspondence between elements of g[n] and possible
proofs in Peano arithmetic—then to use the fact that if one knew that g[n] always terminated this would establish the validity of all these proofs, which would in turn prove the consistency of
arithmetic—a result which is known to be unprovable from within arithmetic.
Every possible proof in Peano arithmetic can in principle be encoded as an ordinary integer. But in the late 1930s Gerhard Gentzen showed that if proofs are instead encoded as ordinal numbers (see
note above) then any proof can validly be reduced to a preceding one just by operations in logic. To cover all possible proofs, however, requires going up to the ordinal ε[0]. And from the
unprovability of consistency one can conclude that this must be impossible using the ordinary operation of induction in Peano arithmetic. (Set theory, however, allows transfinite
induction—essentially induction on arbitrary sets—letting one reach such ordinals and thus prove the consistency of arithmetic.) In constructing g[n] the integer n is in effect treated like an
ordinal number in Cantor normal form, and a sequence of numbers that should precede it are found. That this sequence terminates for all n is then provable in set theory, but not Peano arithmetic—and
in effect Length[g[n]] must grow like [ε[0]][n].)
In general one can imagine characterizing the power of any axiom system by giving a transfinite number κ which specifies the first function [κ] (see note above) whose termination cannot be proved in
that axiom system (or similarly how rapidly the first example of y must grow with x to prevent ∃[y] p[x, y] from being provable). But while it is known that in Peano arithmetic κ = ε[0], quite how to
describe the value of κ for, say, set theory remains unknown. And in general I suspect that there are a vast number of functions with simple definitions whose termination cannot be proved not just
because they grow too quickly but instead for the more fundamental reason that their behavior is in a sense too complicated.
Whenever a general statement about a system like a Turing machine or a cellular automaton is undecidable, at least some instances of that statement encoded in an axiom system must be unprovable. But
normally these tend to be complicated and not at all typical of what arise in ordinary mathematics. (See page 1167.)
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10 Math Concepts You Can't Ignore - dummies
Math itself is one big concept, and it's chock full of so many smaller mathematical concepts that no one person can possibly understand them all — even with a good dose of studying. Yet certain
concepts are so important that they make the Math Hall of Fame:
Sets and set theory
A set is a collection of objects. The objects, called elements of the set, can be tangible (shoes, bobcats, people, jellybeans, and so forth) or intangible (fictional characters, ideas, numbers, and
the like). Sets are such a simple and flexible way of organizing the world that you can define all of math in terms of them.
Mathematicians first define sets very carefully to avoid weird problems — for example, a set can include another set, but it can't include itself. After the whole concept of a set is well-defined,
sets are used to define numbers and operations, such as addition and subtraction, which is the starting point for the math you already know and love.
Prime numbers go forever
A prime number is any counting number that has exactly two divisors (numbers that divide into it evenly) — 1 and the number itself. Prime numbers go on forever — that is, the list is infinite — but
here are the first ten: 2 3 5 7 11 13 17 19 23 29 . . .
It may seem like nothing, but . . .
Zero may look like a big nothing, but it's actually one of the greatest inventions of all time. Like all inventions, it didn't exist until someone thought of it. (The Greeks and Romans, who knew so
much about math and logic, knew nothing about zero.)
The concept of zero as a number arose independently in several different places. In South America, the number system that the Mayans used included a symbol for zero. And the Hindu-Arabic system used
throughout most of the world today developed from an earlier Arabic system that used zero as a placeholder. In fact, zero isn't really nothing — it's simply a way to express nothing mathematically.
And that's really something.
Have a big piece of pi
Pi (π): The symbol π (pronounced pie) is a Greek letter that stands for the ratio of the circumference of a circle to its diameter. Here's the approximate value of π:
π ≈ 3.1415926535…
Although π is just a number — or, in algebraic terms, a constant — it's important for several reasons:
Geometry just wouldn't be the same without it. Circles are one of the most basic shapes in geometry, and you need π to measure the area and the circumference of a circle.
Pi is an irrational number, which means that no fraction that equals it exactly exists. Beyond this, π is a transcendental number, which means that it's never the value of x in a polynomial equation
(the most basic type of algebraic equation).
Pi is everywhere in math. It shows up constantly (no pun intended) where you least expect it. One example is trigonometry, the study of triangles. Triangles obviously aren't circles, but trig uses
circles to measure the size of angles, and you can't swing a compass without hitting π.
Equality in mathematics
The humble equals sign (=) is so common in math that it goes virtually unnoticed. But it represents the concept of equality — when one thing is mathematically the same as another — which is one of
the most important math concepts ever created. A mathematical statement with an equals sign is an equation. The equals sign links two mathematical expressions that have the same value and provides a
powerful way to connect expressions.
Bringing algebra and geometry together
Before the xy-graph (also called the Cartesian coordinate system) was invented, algebra and geometry were studied for centuries as two separate and unrelated areas of math. Algebra was exclusively
the study of equations, and geometry was solely the study of figures on the plane or in space.
The graph, invented by French philosopher and mathematician René Descartes, brought algebra and geometry together, enabling you to draw solutions to equations that include the variables x and y as
points, lines, circles, and other geometric shapes on a graph.
The function: a mathematical machine
A function is a mathematical machine that takes in one number (called the input) and gives back exactly one other number (called the output). It's kind of like a blender because what you get out of
it depends on what you put into it. Suppose you invent a function called PlusOne that adds 1 to any number. So when you input the number 2, the number that gets outputted is 3:
PlusOne(2) = 3
Similarly, when you input the number 100, the number that gets outputted is 101:
PlusOne(100) = 101
It goes on, and on, and on . . .
The very word infinity commands great power. So does the symbol for infinity (∞). Infinity is the very quality of endlessness. And yet mathematicians have tamed infinity to a great extent. In his
invention of calculus, Sir Isaac Newton introduced the concept of a limit, which allows you to calculate what happens to numbers as they get very large and approach infinity.
Putting it all on the line
Every point on the number line stands for a number. That sounds pretty obvious, but strange to say, this concept wasn't fully understood for thousands of years. The Greek philosopher Zeno of Elea
posed this problem, called Zeno's Paradox: To walk across the room, you have to first walk half the distance across the room. Then you have to go half the remaining distance. After that, you have to
go half the distance that still remains).
This pattern continues forever, with each value being halved, which means you can never get to the other side of the room. Obviously, in the real world, you can and do walk across rooms all the time.
But from the standpoint of math, Zeno's Paradox and other similar paradoxes remained unanswered for about 2,000 years.
The basic problem was this one: All the fractions listed in the preceding sequence are between 0 and 1 on the number line. And there are an infinite number of them. But how can you have an infinite
number of numbers in a finite space? Mathematicians of the 19th century — Augustin Cauchy, Richard Dedekind, Karl Weierstrass, and Georg Cantor foremost among them — solved this paradox. The result
was real analysis, the advanced mathematics of the real number line.
Numbers for your imagination
The imaginary numbers (numbers that include the value i = √ - 1) are a set of numbers not found on the real number line. If that idea sounds unbelievable — where else would they be? — don't worry:
For thousands of years, mathematicians didn't believe in them, either. But real-world applications in electronics, particle physics, and many other areas of science have turned skeptics into
So, if your summer plans include wiring your secret underground lab or building a flux capacitor for your time machine — or maybe just studying to get a degree in electrical engineering — you'll find
that imaginary numbers are too useful to be ignored.
About This Article
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Is an obtuse angle smaller or larger than a right angle? - Alea Quiz
Is an obtuse angle smaller or larger than a right angle?
Is an obtuse angle smaller or larger than a right angle?
An obtuse angle is a salient angle greater than the right angle, in other words an angle whose measurement in degrees is between 90° and 180°.
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Minimal Surfaces Close to a Plane and Two-Valued Harmonic Functions
Spencer Becker-Kahn, University of Washington
Suppose $M$ is a minimal surface (a surface which is a critical point for the area functional) in the unit ball and that it is very close to the flat unit disc in the $x$-$y$ plane. If the area of
$M$ is close to the area of the disc, then $M$ must be a smooth graph over the disc and many useful estimates hold. This is a kind of regularity result for minimal surfaces and results like this are
prototypical in the wider world of geometric variational problems.
An open and fundamental problem in regularity theory relates to the situation in which $M$ is close to the disc but where the area of $M$ is close to $2\ \times$ the area of the disc. In this
situation, it is not known exactly how complicated $M$ can be. In a forthcoming series of papers, written jointly with Neshan Wickramasekera, a special case is analyzed using techniques from
geometric measure theory: the case in which $M$ corresponds to a two-sheeted Lipschitz graph (over some other plane). Central to the method is a graphical linearization procedure (in the spirit of
that performed by \emph{e.g.} W. Allard, F. Almgren, or L. Simon) which produces a class of `two-valued harmonic functions' that must be studied in detail. Significant new results are obtained in
this case and in the course of so doing, more is revealed - at a technical level at least - about the general case (when $M$ is not a two-sheeted graph). I will give an overview of the problem and
this work.
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Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco? - Computing Learner
Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San
Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?
Relevant definitions for this exercise:
THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2
ways to do the second task, then there are n1n2 ways to do the procedure.”
THE SUM RULE: “If a task can be done either in one of n1 ways or in one of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 +n2 ways to do the
The definitions were taken from the textbook Discrete Mathematics and its Applications by Rosen.
Now we can solve the exercise.
We want to go from New York to San Francisco via Denver picking an airline for the flight to Denver, and an airline for the flight from Denver to San Francisco.
We can choose an airline to travel from New York to Denver in 6 different ways. The problem description states that 6 different airlines flights on that route.
For each of the 6 different ways that we choose to fly from New York to Denver, there are 7 ways we can fly from Denver to San Francisco (as per the exercise description).
Therefore, by the product rule, there are 42 different pairs of airlines that we can choose to book a trip from New York to San Francisco via Denver when we pick 1 airline to go to Denver, and
another one to go to San Francisco.
Notice that we should always give a final answer to the exercise. Sometimes is not enough just to write a number as an answer, we need to elaborate an answer according to the question we were asked.
Notice the final answer to this exercise and the question we were asked. The rest is what we did to give the final answer.
Related exercises:
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Christian Wüthrich :: Description of Dissertation
My dissertation studies the foundations of loop quantum gravity (LQG), a candidate for a quantum theory of gravity based on classical general relativity. At the outset, I discuss two—and I claim
separate—questions: first, do we need a quantum theory of gravity at all; and second, if we do, does it follow that gravity should or even must be quantized? My evaluation of different arguments
either way suggests that while no argument can be considered conclusive, there are strong indications that gravity should be quantized.
LQG attempts a canonical quantization of general relativity and thereby provokes a foundational interest as it must take a stance on many technical issues tightly linked to the interpretation of
general relativity. Most importantly, it codifies general relativity's main innovation, the so-called background independence, in a formalism suitable for quantization. This codification pulls
asunder what has been joined together in general relativity: space and time. It is thus a central issue whether or not general relativity's four-dimensional structure can be retrieved in the
alternative formalism and how it fares through the quantization process. I argue that the rightful four-dimensional spacetime structure can only be partially retrieved at the classical level. What
happens at the quantum level is an entirely open issue in the literature and I sketch how background independence could be understood there.
Known examples of classically singular behaviour which gets regularized by quantization evoke an admittedly pious hope that the singularities which notoriously plague the classical theory may be
washed away by quantization. This works scrutinizes pronouncements claiming that the initial singularity of classical cosmological models vanishes in quantum cosmology based on LQG and concludes that
these claims must be severely qualified. In particular, I explicate why casting the quantum cosmological models in terms of a deterministic temporal evolution fails to capture the concepts at work
adequately. Finally, a scheme is developed of how the re-emergence of the smooth spacetime from the underlying discrete quantum structure could be understood.
Last modified on 6 March 2008.
Created and maintained by
Christian Wüthrich
URL: http://philosophy.ucsd.edu/faculty/wuthrich/pub/dissertationdescription.html
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I've written up a
CEKS machine
semantics for the essential core of Javascript, which just for fun I'm calling
. It demonstrates the strange scoping and store semantics of the language, the confusing meaning of
, and the prototype-based inheritance model. It doesn't correctly treat the primitive types like objects in quite the way it's supposed to, but that wouldn't be too hard to add. Oh, I also don't have
a rule for starting up the machine, which I ought to, because it would explain the "global" object and create the initial
objects. I'll do that at some point.
I imagine you could use this to reason about a compiler that generated Javascript, if you were pretty ambitious, but mostly I just did it to help myself understand the language. It would also make it
really easy to write an interpreter. A
easier than trying to write to the
, which is written in a combination of English prose and BASIC-like pseudocode (ouch).
If I do say so myself, this turned out to be a whole lot more concise than the 75 or so pages of the spec specifying the core semantics of the language (that's the part
any description of the standard library).
5 comments:
Maybe your semantics capture this, but consider this stupid accumulator:
var j = 1;
function accum() {
var n = 1;
function inc() {return j*n++;}
return inc;
j isn't put in inc's closure, n is, but wouldn't your MakeClosure rule put j in the closure for inc? I'm probably missing something. It's here, too.
I'm retarded Dave, sorry, disregard my other comment...
If you aren't already aware of them, there are some really neat object calculi where extension-via-protoyping and method reference is all you have. And when I say "all", its significant, because
all the methods take only one argument, which is (wait for it) this. That's it, no extra arguments.
So how do I code up addition, or anything where you want to take arguments other than this? Yeah, I puzzled over that one too. Its pretty cool stuff.
(I don't know that much about it; Carl is the expert on this stuff at the moment.)
I'm responding to the question, "So how do I code up addition, or
anything where you want to take arguments other than this?", from
http://calculist.blogspot.com/2005/06/classicjavascript.html --- with
regard to Abadi and Cardelli's object calculi.
This is explained in a slightly different notation than Abadi and
Cardelli use for their object calculus. I'm leaving out the syntactic
sugar I usually use in order to keep this explanation simple for those
who haven't read Abadi and Cardelli.
{} is an object with no methods.
self.foo = self.bar + 1 is a method whose definition is "call bar
on the same object, then add 1." Abadi and Cardelli write this as
"foo = sigma(self) self.bar + 1".
x.foo = x.bar + 1 is the same method (or, if you prefer, another
method that behaves identically.) This expression lexically binds
the name 'x' within the method definition to the "self" parameter
--- the object the method is called on.
x{ self.foo = self.bar + 1 } is an override expression; it means
"an object exactly like x, except that it has this new foo
method," which is defined as above.
I'll separate methods inside objects or override expressions with
newlines or commas.
I've also added new methods to objects in override expressions
without restraint; you can rewrite this code so that it only ever
overrides existing methods, as in Abadi and Cardelli's work,
without much effort, simply by adding useless method definitions
in objects whose descendants do this.
So here are booleans:
booleans.true = {
self.negated = booleans.false
self.ifTrue = { x.then = 1, x.else = 0, x.val = x.then }
self.ifFalse = self.negated.ifTrue
booleans.false = booleans.true {
self.negated = booleans.true
self.ifTrue = booleans.ifTrue { x.val = x.else }
If "hungry" is a variable that might be one of booleans.true or
booleans.false, we can say
hungry.ifTrue{ x.then = "eat", x.else = "don't eat" }.val
Suppose hungry is the above 'true'. This resolves to
self.negated = booleans.false
self.ifTrue = { x.then = 1, x.else = 0, x.val = x.then }
self.ifFalse = self.negated.ifTrue
}.ifTrue{ x.then = "eat", x.else = "don't eat" }.val
and then
x.then = 1, x.else = 0, x.val = x.then
x.then = "eat", x.else = "don't eat"
which reduces to
{ x.then = "eat", x.else = "don't eat", x.val = x.then }.val
Which resolves to 'x.then', with 'x' being the above-constructed
object whose 'then' is defined as "eat". Supposing instead that
hungry were 'false'; instead we get
booleans.true {
self.negated = booleans.true
self.ifTrue = booleans.ifTrue { x.val = x.else }
}.ifTrue{ x.then = "eat", x.else = "don't eat" }.val
If we expand out the booleans.true here, we get
self.negated = booleans.false
self.ifTrue = { x.then = 1, x.else = 0, x.val = x.then }
self.ifFalse = self.negated.ifTrue
self.negated = booleans.true
self.ifTrue = booleans.ifTrue { x.val = x.else }
}.ifTrue{ x.then = "eat", x.else = "don't eat" }.val
And then if we evaluate the first override expression, we get
self.negated = booleans.true
self.ifTrue = booleans.ifTrue { x.val = x.else }
self.ifFalse = self.negated.ifTrue
}.ifTrue{ x.then = "eat", x.else = "don't eat" }.val
since the only method from 'true' not overridden in 'false' is
ifFalse. Now we can evaluate the .ifTrue method call and get:
booleans.ifTrue { x.val = x.else }{ x.then = "eat", x.else = "don't eat" }.val
which expands out to
x.then = 1, x.else = 0, x.val = x.then
x.val = x.else
x.then = "eat", x.else = "don't eat"
We can collapse the first override and get
x.then = 1, x.else = 0, x.val = x.else
x.then = "eat", x.else = "don't eat"
and then the second, and get
x.then = "eat", x.else = "don't eat", x.val = x.else
And that reduces to just 'x.else', which is defined as "don't eat".
This is exactly the same as the last reduction step for when hungry
was 'true', except that we inherited a different definition for
Now, for numbers. A real implementation of this object-calculus on a
computer would use the CPU's rapid number-handling machinery rather
than implementing its own math primitives, and the method I am about
to explain is a terribly inefficient way of implementing them anyway;
its purpose is to demonstrate that the object-calculus can
theoretically perform any computable computation on its own. The
following technique is just a transliteration of the lambda-calculus's
Church numerals.
Here's an object containing numeric primitives; I assume the earlier
booleans object is available under the name 'booleans'.
numbers.zero = {
self.isZero = booleans.true
self.plus = numbers.sum { x.firstArgument = self }
self.succ = self {
succ.isZero = booleans.false
succ.pred = self
numbers.lessThan = {
self.firstArgument = numbers.zero
self.secondArgument = numbers.zero.succ
self.val = self.secondArgument.isZero.ifTrue {
x.then = booleans.false
x.else = self.firstArgument.isZero.ifTrue {
x.then = booleans.true
x.else = numbers.lessThan {
child.firstArgument = self.firstArgument.pred
child.secondArgument = self.secondArgument.pred
numbers.sum = {
self.firstArgument = numbers.zero.succ
self.secondArgument = numbers.zero.succ
self.val = self.firstArgument.isZero.ifTrue {
x.then = self.secondArgument
x.else = self {
child.firstArgument = self.firstArgument.pred
The above definitions should evaluate
numbers.zero.succ.succ.succ.plus {
x.secondArgument = numbers.zero.succ.succ.succ.succ
to the same thing as numbers.zero.succ.succ.succ.succ.succ.succ.succ.
But I may have made a mistake.
So the short answer is that methods often return an object not derived
from self, but some of whose method definitions are closed over self.
As I've implied previously (in "functional programming for amateurs in
an outliner", before I read Abadi and Cardelli's book --- thank you
David Gibson!) I think this is the theoretical underpinning for
dramatically more usable and productive programming environments.
> Abadi and Cardelli's object calculus is described in the first three
pages of "A Theory of Primitive Objects" at
> "functional programming for amateurs in an outliner" is at
> "towards a modern programming environment" is at
> Also related is Jonathan Edwards' Subtext:
> And Wouter van Oortmerssen wants "abstractionless programming", the
same thing:
> And Dynamic Aspects' "domain/object" platform seems to be a larger
idea including the same kernel:
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Master the Digits of Pi at the MaThCliX Annual Digits of Pi Contest
Three Strategies to Conquer MaThCliX Digits of Pi Contest
On March 14th, MaThCliX will be hosting our third annual Pi Day, which is filled with a variety of activities for students of all ages. The most anticipated event of the day is the “Digits of Pi”
Contest. The rules are simple: whoever wants to participate merely has to recite as many digits of Pi, the famous irrational number, as they can (in order, of course), in other words, master the
digits of Pi. The person that says the most digits of Pi accurately wins a Pi Day t-shirt and a pie/cake! Good luck to everyone competing; I hope you find these tips useful! (P.S. make sure you have
the correct digits of Pi pulled up on your phone or computer while attempting to memorize it.)
Strategy #1:
One way to memorize the digits of Pi effectively is through auditory learning. Look at the first 5 digits of Pi and say each one of them out loud. Repeat the process four more times while still
looking at the correct form of Pi to guide you. Then look away and try to say the five digits by memory. If you get it correct the first time, then repeat it four more times while looking away.
However, if you get it wrong the first time, look back at the correct form and repeat the five digits five times while looking again. Next, attempt to say it five times without looking (successfully
this time, hopefully). Repeat these steps until you feel like you have those five digits glued to your brain. If you can use this strategy every day for 10 days before the contest, you will have
memorized the first 50 digits of Pi!
Strategy #2:
Carry around a piece of paper with Pi written on it. Whenever you have a minute to spare either in the classroom or at home, take out the piece of paper and begin writing the digits of Pi by memory,
as many as you can do. Then look at the correct form of Pi and assess how you did. Next, write it again, maybe this time adding one or two digits on to the end. If you make this a habit for a week or
two before the contest, you are bound for success.
Strategy #3:
This final strategy is based off the idea that it is easier to remember numbers that have a purpose rather than a random sea of numbers. What you do is assign phone numbers to each set of ten digits
in Pi and then attempt to memorize each phone number. It helps to set patterns within the phone numbers to better remember them: make the first letter of the name for the first phone number an “A”,
the first letter of the name for the second phone number a “B”, etc. Also, try making the numbers of letter in each name correspond with the first number in that phone number. Try memorizing one
phone number every 2 days, and in 10 days you will know 50 digits.
Everyone is different, so a technique that works for one person might not work for another. Experiment with different memorization techniques and find which one works best for YOU. Also, just a
reminder: last year’s winner recited 108 digits of Pi. Good luck, and we’ll see you on March 14th!
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Finding a Mall Parking Spot Using Mathematics – Part II
If you read the previous article on this topic, then I imagine you were quite piqued by the nature of its contents. How we use mathematics to find a mall parking spot is not a typical thing you would
hear people discussing at their Christmas parties. Yet I think anyone with a modicum of human interest would find this a most curious topic of conversation. The reaction I usually get is one of “Wow.
How do you do that?”, or “You can really use mathematics to find a parking spot?”
As I mentioned in the first article, I was never content to get my degrees in mathematics and then not do anything with them other than to leverage job opportunities. I wanted to know that this newly
found power that I studied feverishly to obtain could actually inure to my personal benefit: that I would be able to be an effective problem solver, and not just for those highly technical problems
but also for more mundane ones such as the case at hand. Consequently, I am constantly probing, thinking, and searching for ways of solving everyday problems, or using mathematics to help optimize or
streamline an otherwise mundane task. This is exactly how I stumbled upon the solution to the Mall Parking Spot Problem.
Essentially the solution to this question arises from two complementary mathematical disciplines: Probability and Statistics. Generally, one refers to these branches of mathematics as complementary
because they are closely related and one needs to study and understand probability theory before one can endeavor to tackle statistical theory. These two disciplines aid in the solution to this
Now I am going to give you the method (with some reasoning–fear not, as I will not go into laborious mathematical theory) on how to go about finding a parking spot. Try this out and I am sure you
will be amazed (Just remember to drop me a line about how cool this is). Okay, to the method. Understand that we are talking about finding a spot during peak hours when parking is hard to come
by–obviously there would be no need for a method under different circumstances. This is especially true during the Christmas season (which actually is the time of the writing of this article–how
Ready to try this? Let’s go. Next time you go to the mall, pick an area to wait that permits you to see a total of at least twenty cars in front of you on either side. The reason for the number
twenty will be explained later. Now take three hours (180 minutes) and divide it by the number of cars, which in this example is 180/20 or 9 minutes. Take a look at the clock and observe the time.
Within a nine minute interval from the time you look at the clock–often quite sooner–one of those twenty or so spots will open up. Mathematics pretty much guarantees this. Whenever I test this out
and especially when I demonstrate this to someone, I am always amused at the success of the method. While others are feverishly circling the lot, you sit there patiently watching. You pick your
territory and just wait, knowing that within a few minutes the prize is won. How smug!
So what guarantees that you will get one of those spots in the allotted time. Here is where we start to use a little statistical theory. There is a well-known theory in Statistics called the Central
Limit Theory. What this theory essentially says is that in the long run, many things in life can be predicted by a normal curve. This, you might remember, is the bell-shaped curve, with the two tails
extending out in either direction. This is the most famous statistical curve. For those of you who are wondering, a statistical curve is a chart off of which we can read information. Such a chart
allows us to make educated guesses or predictions about populations, in this case the population of parked cars at the local mall.
Charts like normal curve tell us where we stand in height, let us say, with respect to the rest of the country. If we are in the 90th percentile in regard to height, then we know that we are taller
than 90% of the population. The Central Limit Theorem tells us that eventually all heights, all weights, all intelligence quotients of a population eventually smooth out to follow a normal curve
pattern. Now what does “eventually” mean. This means that we need a certain size population of things for this theorem to be applicable. The number that works very well is twenty-five, but for our
case at hand, twenty will generally be sufficient. If you can get twenty-five cars or more in front of you, the better the method works.
Once we have made some basic assumptions about the parked cars, statistics can be applied and we can start to make predictions about when parking spots might become available. We cannot predict which
one of the twenty cars will leave first but we can predict that one of them will leave within a certain time period. This process is similar to the one used by a life insurance company when it is
able to predict how many people of a certain age will die in the following year, but not which ones will die. To make such predictions, the company relies on so-called mortality tables, and these are
based on probability and statistical theory. In our particular problem, we assume that within three hours all twenty of the cars will have turned over and be replaced by another twenty cars. To
arrive at this conclusion, we have used some basic assumptions about two parameters of the Normal Distribution, the mean and standard deviation. For the purposes of this article I will not go into
the details regarding these parameters; the main goal is to show that this method will work very nicely and can be tested next time out.
To sum up, pick your spot in front of at least twenty cars. Divide 180 minutes by the number of cars–in this case 20–to get 9 minutes (Note: for twenty-five cars, the time interval will be 7.2
minutes or 7 minutes and 12 seconds, if you really want to get precise). Once you have established your time interval, you can check your watch and be sure that a spot will become available in at
most 9 minutes, or whatever interval you calculated depending on the number of cars you are working with; and that because of the nature of the Normal curve, a spot will often become available sooner
than the maximum allotted time. Try this out and you will be amazed. At the very least you will score points with friends and family for your intuitive nature.
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Mismatch Between MPS and MPO sizes
I've made a SiteSet named DP that is similar to the one provided by the CustomSpin class, and I want to use DMRG to find the ground state of the Hamiltonian
p \sum S{-}S{+} + q \sum S{+}S{-}
The code in C++ is:
#include "itensor/all.h"
#include "DP.h"
using namespace itensor;
int main() {
// lattice parameters
double p = 0.2;
double q = 0.7;
int capacity = 1;
int num_sites = 3;
auto args = Args("capacity=", capacity);
auto lattice = DP(num_sites, args)
// Hamiltonian
auto ampo = AutoMPO(lattice);
for (int i = 1; i <= lattice.length()-1; ++i) {
ampo += p, "S-", i, "S+", i+1;
ampo += q, "S+", i, "S-", i+1;
auto H = toMPO(ampo);
// MPS
auto state = InitState(lattice);
auto psi0 = randomMPS(state);
print("MPS \n");
print("MPO \n");
// DMRG
auto sweeps = Sweeps(5);
sweeps.maxdim() = 10, 20, 100, 100, 200;
sweeps.cutoff() = 0.0001;
auto [energy, psi] = dmrg(H, psi0, sweeps);
return 0;
I receive an error "davidson: size of initial vector should match linear matrix size". However, the printout of the MPS and MPO (if I'm not misunderstanding them) say that they match:
ITensor ord=2: (dim=2|id=672) (dim=1|id=834|"l=1,Link")
{norm=0.89 (Dense Real)}
ITensor ord=3: (dim=1|id=834|"l=1,Link") (dim=2|id=67) (dim=1|id=475|"l=2,Link")
{norm=1.01 (Dense Real)}
ITensor ord=2: (dim=1|id=475|"l=2,Link") (dim=2|id=393)
{norm=0.91 (Dense Real)}
ITensor ord=3: (dim=4|id=436|"l=1,Link") (dim=2|id=672) (dim=2|id=672)'
{norm=1.59 (Dense Real)}
ITensor ord=4: (dim=4|id=436|"l=1,Link") (dim=4|id=372|"l=2,Link") (dim=2|id=67) (dim=2|id=67)'
{norm=2.56 (Dense Real)}
ITensor ord=3: (dim=4|id=372|"l=2,Link") (dim=2|id=393) (dim=2|id=393)'
{norm=2.00 (Dense Real)}
How should I properly construct the MPO/MPS?
Hi, thanks for the question. So looking over your code and the error message, the error is not referring to the length of your MPO or MPS. It is referring instead to a numerical problem that occurred
inside of the innermost loop of the DMRG algorithm. Based on the information provided, it's not clear what really led to this error.
Here are two things that would help:
- showing how the DP sites are defined, because there may be a mistake in some of the operator definitions that is e.g. leading to the Hamiltonian not being Hermitian
- trying a system of length 4 or greater, because the 3-site case may have some slight issue as it is an unusually small system & we don't often test the code on such small sizes
If those two approaches don't pinpoint the issue, then the thing to do is to run it in a debugger and/or print out some intermediate quantities to see what's going on inside the code. It could be a
bug in ITensor even.
Let's keep discussing –
Hi Miles,
Thanks for the clear advice. The operators on the DP sites are defined so that the Hamiltonian is non-Hermitian, like you pointed out. I will try your suggestions for adapting DMRG to use the Arnoldi
algorithm (http://itensor.org/support/2367/non-hermitian-dmrg-in-julia).
Thanks very much for your help!
I see - so you were intentionally studying a non-Hermitian Hamiltonian? If so, yes you would need to adapt the DMRG algorithm which by default requires a Hermitian Hamiltonian.
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The multiplier process; Inflationary and deflationary gaps
The multiplier process refers to the economic phenomenon in which an initial change in spending or investment has a greater impact on the overall economy than the initial change itself.
• It explores how changes in one component of aggregate demand, such as consumption or investment, can lead to multiple rounds of subsequent spending and income generation.
• The multiplier represents the ratio of the total change in national income or output to the initial change in spending.
• It measures the magnification effect that occurs when an injection of spending enters the economy and circulates through various sectors.
Multiplier = change in equilibrium income / change in injection
= 1 / marginal propensity to withdraw
Multiplier = change in equilibrium income / change in injection
= 1 / marginal propensity to withdraw
• Calculation of the Multiplier in a Closed Economy (Without Government)
□ In a closed economy without government involvement, the multiplier is determined by the marginal propensity to consume (MPC).
☆ The MPC represents the portion of each additional unit of income that is spent on consumption.
□ The formula to calculate the multiplier in a closed economy is:
Multiplier = 1 / (1 - MPC) = 1 / MPS
□ For example, if the MPC is 0.8, the multiplier would be:
Multiplier = 1 / (1 - 0.8) = 5
☆ This implies that a $1 increase in investment or spending would result in a $5 increase in the national income or output.
• Calculation of the Multiplier in a Closed Economy (With Government):
□ In a closed economy with government involvement, the multiplier formula incorporates both the marginal propensity to save (MPS) and the marginal rate of tax (MRT), which represents the
portion of each additional unit of income that is saved rather than spent. The formula to calculate the multiplier in this case is:
Multiplier = 1 / (MPS + MRT)
□ For example, if the MPS is 0.2 and the MRTis 0.1, the multiplier would be:
Multiplier = 1 / (0.2 + 0.1) = 3.33
☆ This indicates that a $1 increase in investment or spending would lead to a $3.33 increase in the national income or output.
• Calculation of the Multiplier in an Open Economy with government
□ In an open economy, which involves international trade, the multiplier calculation takes into account the marginal propensity to import (MPI), representing the portion of each additional unit
of income that is spent on imports.
□ The formula to calculate the multiplier in an open economy is
Multiplier = 1 / (MPS + MRT + MPI)
□ For instance, if the MPS is 0.1, the MRT is 0.1 and the MPI is 0.1, the multiplier would be:
Multiplier = 1 / (0.2 + 0.1 + 0.1) = 2.5
☆ This means that a $1 increase in investment or spending would lead to a $2.5 increase in the national income or output, considering the impact of imports and taxation.
The AD approach emphasizes the relationship between aggregate demand and national income. It states that the total spending in an economy determines the level of output and income. Aggregate demand
consists of four main components: consumption (C), investment (I), government spending (G), and net exports (X - M).
• The AD equation is given by: AD = C + I + G + (X - M)
• The equilibrium is achieved at
AD = Y
Aggregate Expenditure = Income
To understand the multiplier process, we need to consider the concept of induced expenditure.
• Induced expenditure refers to the additional spending that occurs when income increases. It includes consumption, which is the largest component of aggregate demand.
• The multiplier effect occurs because an initial change in spending leads to a chain reaction of increased income and further spending.
• This process is driven by the marginal propensity to consume (MPC), which is the fraction of additional income that people spend on consumption.
• The multiplier (K) represents the ratio of the change in national income to the initial change in spending.
• It is calculated as the reciprocal of the marginal propensity to save (MPS), which is the fraction of additional income that people save rather than spend.
The formula for the multiplier is: K = 1 / MPS
• The multiplier effect works as follows:
1. An initial increase in autonomous spending, such as an increase in investment or government spending, leads to an increase in aggregate demand.
2. This increase in aggregate demand stimulates production and leads to an increase in national income.
3. As income rises, people's consumption increases, creating additional rounds of spending.
4. The increased consumption leads to further increases in production and income.
5. This process continues in a cumulative fashion, with each round of increased income leading to additional spending, amplifying the initial change in autonomous expenditure.
• The multiplier process has a magnifying effect on changes in aggregate demand. For example, if the initial increase in spending is $100 and the multiplier is 5, the total increase in national
income will be $500 (5 times the initial change in spending).
Assume we have an economy where the marginal propensity to consume (MPC) is 0.8. This means that for every additional dollar of income, people tend to spend 80 cents and save 20 cents.
Now, suppose there is an increase in private capital investment by $100 million. We want to calculate the effect of this increase on the national income using the multiplier.
1. Calculate the multiplier:
□ The multiplier (K) is the reciprocal of the marginal propensity to save (MPS), which is equal to 1-MPC.
□ In this case, MPS would be 1−0.8=0.2. Therefore, the multiplier is 1/0.2 = 5.
2. Determine the total change in national income: Multiply the initial change in autonomous expenditure (government spending) by the multiplier.
□ In this example, the initial change is $100 million, and the multiplier is 5.
□ So, the total change in national income is $100 million × 5 = $500 million.
3. Determine the final level of national income:
□ Add the total change in national income to the initial level of national income. Let's say the initial level of national income was $1 billion.
□ The final level of national income would be $1 billion + $500 million = $1.5 billion.
Therefore, the increase in government spending by $100 million leads to an increase in national income of $500 million, resulting in a final level of $1.5 billion.
• It's important to note that this example assumes simplified conditions and a closed economy without leakages like taxes or imports. In reality, the multiplier effect can be influenced by various
factors, and the actual impact on national income can be more complex.
The calculation of the effect of changing AD on national income using the multiplier helps us understand how changes in spending can have a multiplied effect on the overall level of income in an
economy. It demonstrates the interplay between spending, consumption, and the resulting impact on national income.
• The full employment level of national income refers to the level of real GDP (gross domestic product) that an economy can produce when all available resources, such as labor and capital, are
fully utilized.
□ At this level, the economy is operating at its maximum productive capacity, and there is no cyclical unemployment. It represents the point where the economy is producing goods and services to
meet the demands of all willing and able workers.
• An inflationary gap occurs when the equilibrium level of national income exceeds the full employment level of national income.
□ In other words, aggregate demand is higher than the economy's productive capacity.
□ This situation can lead to upward pressure on prices as demand outstrips supply. It may result in inflationary pressures in the economy, as firms may increase prices to match the increased
• A deflationary gap occurs when the equilibrium level of national income falls below the full employment level of national income.
□ In this case, aggregate demand is insufficient to support the economy's productive capacity.
□ This situation can lead to downward pressure on prices as supply exceeds demand. It may result in deflationary pressures, as firms may lower prices to stimulate demand and reduce excess
The concepts of full employment level of national income and equilibrium level of national income help economists analyze the state of an economy and identify any gaps between actual output and
potential output. Understanding inflationary and deflationary gaps is crucial for policymakers to implement appropriate measures to stabilize the economy, such as fiscal or monetary policies, to
bring the economy back to equilibrium and promote stable economic growth.
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critical root value pada data nonstasioner
Aziz, Abdul (2011) Analisis critical root value pada data nonstasioner. Cauchy: Jurnal Matematika Murni dan Aplikasi, 2 (1). pp. 1-6. ISSN 2086-0382
Text (full text)
3730.pdf - Published Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.
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A stationery process can be done t-test, on the contrary at non stationery process t-test cannot be done again because critical value of this process isn’t t-distribution. At this research, we will
do simulation of time series AR(1) data in four non stationery models and doing unit root test to know critical value at ttest of non stationery process. From the research is yielded that
distribution of critical point for t-test of non stationery process comes near to normal with restating simulation of random walk process which ever greater. Result of acquirement of this critical
point has come near to result of Dickey-Fuller Test. From this research has been obtained critical point for third case which has not available at tables result of Dickey-Fuller Test.
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A t-test is a statistical test that helps compare whether the average values of two groups of data are significantly different from each other. It is used to obtain a measure of the difference
between the means (averages) of the groups, relative to the spread of data within each group.
The t-test helps decide whether a difference in mean values between two groups is due to random chance in a sample selection.
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Рабочее место участника
Ограничения: время – 2s/4s, память – 256MiB Ввод: input.txt или стандартный ввод Вывод: output.txt или стандартный вывод
Послать решение Blockly Посылки Темы Где Обсудить (0)
Mirko is a party animal, so he has decided to organise an endless amount of parties for his friends. To satisfy the party's needs, he has decided to set up N tables with candy on them. We know the
number of candies `b_i` on each table. On the first day of the rest of eternity, Mirko is going to invite one friend per table, on the second day he will invite two friends per table, on the third
day three friends… In general, obviously, on the `k`th day he is going to invite `k`friends per each table.
When his friends enter the room, `k` people will sit down at each table and they will divide the candies on their table in `k` as large as possible equal pieces, and get rid of the possible remains.
After the candy division, because of jealousy and various other reasons, only tables with the same amount of candy per capita will socialise together. Mirko has all eternity to study the social
dynamics of his parties. Firstly, he wants to know the answer to the following question: given an s between 1 and `N`, what is the earliest day when there is a group of exactly `s` tables socialising
As usual, Mirko is incapable of solving his own problems, so every few days he comes to you and asks you what the required number is, given an s. Alas, he has all eternity to ask questions, but you
don't. Therefore, you are going to write a programme which outputs Mirko's required answers for each `s` from 1 to `N`.
Please note: Before each party, Mirko renews the candy supply on each table, meaning the supplies are equal to those before the first party. Additionally, all people leave from the current party
before the next one starts.
The first line of input contains the integer `N` (`1\ ≤\ N\ ≤\ 100`).
The second line of input contains `N` integers, the `i`th number marking the number of candy on the `i`th table. The numbers are from the interval `[1,\ 10^8]`.
Output `N` lines, each line containing a single integer.
The `s`th line should contain the required number for a group sized `s` or –1 if there will never be a group of that size.
Sample Input #1
Sample Output #1
Sample Input #3
Sample Output #3
Clarification of the first example: On the first day, each table will socialise only with itself so the answer for groups sized 1 is 1. Already on the second day, people sitting at tables 1 and 2 are
going to get 5 candies per capita and socialise together, so the answer for a group sized 2 is 2. On the third day, tables 1, 2 and 3 will socialise (because they all have 3 candies per capita). On
the sixth day, tables 1, 2, 3 and 4 will socialise (because they now have 1 candy per capita). Finally, on the twelfth day, all tables will socialise together because they will all get zero candy per
Clarification of the second example: All tables have the same amount of candy per capita, so a group sized less than 3 will never exist.
Source: COCI 2013/2014, contest #4
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Multivariable Calculus Problems And Solutions | Hire Someone To Do Calculus Exam For Me
Multivariable Calculus Problems And Solutions It’s fun, guys. Today I’m going to be talking about Calculus problems, and I’ll tell you a little bit about the general idea of the Calculus problem.
First of all, this is a problem of calculus. It’s actually a sort of anachronism, a sort of theory of mathematical logic: that is, it’s a kind of logic that can be written down in one language, but
sometimes it can’t. And this is one way of thinking about it. In other words, it”s a kind-of theory of math. And so by means of this Calculus problem in mathematics, you can think about the problem
in a different way than what I’ve done so far. And so you can think of it as an analog of the calculus problem of your school, or to be more precise, what’s called the calculus problem. You can’ve a
lot of problems in a database of mathematical formulas. And the general principle is this: if you’re trying to find a formula that exactly matches a given formula, you’ll have to find it yourself.
And by using this principle, you can have many, many problems in a relatively short time. And so then you can write down a formula. So for example when you have a formula, you can’re in a database
and it’ll say that you have a given formula. And so you can have a formula that matches all the formulas in it. And so it’d be, in a database, you can write a formula in a way that matches all of the
formulas in the database. And so that’s the problem, so this is the Calculus Problem, as I said before. The Calculus Problem of Mathematics So when you say, “what’s going to be the problem of
calculus?” this is a kind of problem of calculus, but it’’s also a kind of mathematical problem. And it can be written in one language. It”s this kind of math that forms a kind of a kind of theory of
mathematics, so we can think of calculus as a kind of kind of theory. So it”’s not really a kind of the mathematical problem.
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It� George Hill has made it clear to me that when you come to calculus, you start in school, and you’ve got a lot of the things that you have to do, that you just have to work your way up through the
mathematics. And so he says, “I”m going to do these problems in many different ways. And so I”m trying to be a little bit more precise about visit the site problem. And so we”re going to be working
from the beginning. I”ll talk about this in a bit more detail. In his book, A Theory of Mathematics, he uses the same basic concept to treat mathematics. So he”s going to look at one of the most
fundamental problems in mathematics. In this book, he”ll use the word ”strictly”. And he””ll say that he”d like to do a special kind of theory that is called ”the deductive theoryMultivariable
Calculus Problems And Solutions Calculus problems and solutions are often encountered in mathematics as examples of the see here now of how to solve them. These problems are known as the Calculus
Problem(s) and Calculus Problems Solution(s). For a mathematician, it is often possible to solve a problem by solving it in a given form, and there are a lot of tools available to this task. Examples
of Calculus Problems and Solutions are illustrated in Figure 1. Figure 1 Calculus problem and solutions. Calculating the answer to the problem The problem of how can one solve a given problem can be
calculated using the following formulas. The solution The formula We can solve the following problem Which equation is the equation that tells the equation to be solved? The equation is that it is
the solution of the equation. How can I solve the equation? Well, let’s try to solve the equation. For this problem, we have to find the solution of 1. The solution $$\label{eq:1} \frac{x^x}{x-x^0} =
\frac{1}{x-1} \;\;\; \text{or}\;\; \;\frac{1-x^x }{1+x-x}$$ 2. The solution can be computed by $$\label {eq:2} \text{for} \; \; x = 1+i \,\;\text{and}\;\, \;\text{\rm such}\;\ \;\, \frac{x^{x-1}}
{x-1}\;\text {is}\;\ \;\ \text{a solution of} \;x^x \text{. }$$ Note that for any $x$ and $x-1$, $\text{such}\;\:$ is a solution of equation $\;x^1 \;\times\;\,x^1\;\times$ 3.
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The solution is a solution. Now we can solve the equation $$\label{equ:3} y = \frac{\sqrt{2}}{x^2} + \frac{5}{x} \;$$ 4. The solution of equation $y = \sqrt{x^3} + \sqrt{\sqrt{\frac{x-2}{x}}+\frac
{5-x^3}}$ 5. The solution in equation $y=\sqrt{1+2x}$ is look at this site second solution, the third is a solution for the equation $y+\sqrt{\left(x-2\right)^2} = 0$ 6. We can solve the problem $$\
frac{y^2}{x^3}\; \; \text{and} \; y = \sq \frac{3}{x}$$ $$\label else$$ $$\text{where} \; y = 1+2\sqrt x + 2\sqrt \frac{2x-x^{3}}{x}$$ and $$\text{\boldsymbol{R}} = \text{\boldstyle\begin{matrix} \
text{sgn} & \text{latin} \\ \text{null} & \end{matrix}\;\quad}$$ We now have the equation in the equation $$\text{y =}\sqrt{(1-x)^2-3x^2}\;\Rightarrow\;\quad \sqrt \text{x+x^{2}} = 1+x\sqrt\text{x-x}
\;\Leftrightarrow\; \text{\rm company website =} \sqrt x\;\Rightrightarrow\quad\text{\text{such} that}\; \sqrt\left(1-\frac{3x}{x} + \frac{\left[\frac{2\sqr}{x}+\frac{\left(1+2\right)\sqrt\frac{6}{x}
\right]^3Multivariable Calculus Problems And Solutions We all know, and I know you know, that calculus is a very complex method that is difficult to master. This chapter is an attempt to explain how
to solve the problems of calculus problems. First, we will need to explain how we solve these problems. If you are new to calculus, you might have some experience in the field. But if you are new and
you are not familiar with the field then you might think “no, you are not new yet”. In the end, it is in your best interests to take that knowledge and do the best you can. Let’s start by defining
the simplicial polynomials. Suppose we have the simplicial ring $R=SL(n,K)$ with $n \in \mathbb{Z}$, and $f : R \rightarrow R$. If we denote by $f^*$ the natural homomorphism from $R$ to $R$, then we
can write the $(n+1)$-dimensional simplicial poine as $f^*:R\rightarrow R$ with the homomorphism given by $\begin{pmatrix} a & b\\ c & d\end{pmat.}=\begin{pm} f^*(a,b,c) + f(c,a,d) + f^* (b,c,d) \end
{pm}$. Now we will need the following definitions. \[def:opf\] We say that $f$ is a map from $R \rightarrow \mathbb R$ if $\begin{smallmatrix} a && b\\ c && d\end{\smallmatrix}\in R$ and $\begin
{equation} f^*: \mathbb Z \rightarrow \mathbb N$ is a homomorphism if $\mathbb Z$ is a finite dimensional space, or if $\mathcal{F}$ is a field. To define the simplicial map, we have to define the
map $f^{\text{op}}$ as follows. We say that $g\in R\cap \mathbb O$ is a $*$-homomorphism if $gf\in \mathcal{O}_\mathbb Q$ and $f^\text{op}(g)$ is a $(n+2)$-simplex in $R\cap \left(\mathbb Z\right)$.
We can define the simplicially homogeneous polynomially homogeneous convex function $f^k$ by the formula $$\label{eqn:fk} f^k(x)=\begin{cases} 0 & \text{if $x\in \left(\left[0,1\right]\right)\
setminus \left(\{0\} \right)\cup \left[0\right]$,}\\ \frac{1}{n} & \text {if $x$ is in $\left[0-1\right]=\{1\}$,}\\\frac{n}{2} & \ \text {else.} \end{cases}$$ \[[@bib:mapping]\] It is known that the
map $k^*$ is a polynomial homomorphism of $\mathbb R^{n+1}$ into $\mathbb N$, where $n=\text{dim}(\mathbb R)$.
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However, in order to see that the map is a poxial homomorphism, we have the following corollary. The map $k$ is a monomorphism of $R \times \mathbb T$ into $\text{Spf}(R)$, where $\text{spf}(C)$ is
the subspace of $C$ consisting of polynomial linear functionals on $C$. The authors of [@bib(1)] have shown that there are five classes of $*$ homomorphisms from $\mathbb T^2$ into $\widetilde{R}$
with the following properties: 1. The homogeneous poxial map $k(x,y)=f^*
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Journal of the European Optical Society-Rapid Publications
Issue J. Eur. Opt. Society-Rapid Publ.
Volume 19, Number 2, 2023
Article Number 44
Number of page(s) 10
DOI https://doi.org/10.1051/jeos/2023042
Published online 07 December 2023
J. Eur. Opt. Society-Rapid Publ. 2023,
, 44
Research Article
Three-dimensional temperature field measurement method based on light field colorimetric thermometry
School of Optoelectronic Engineering, Xidian University, Xi’an, Shaanxi 710071, China
^* Corresponding author: yuanying@xidian.edu.cn
Received: 4 October 2023
Accepted: 10 November 2023
This paper proposes a novel method for three-dimensional (3D) temperature measurement using light field colorimetric thermometry, aiming to overcome the challenges associated with the intricate
system structure and the limited availability of 3D information in traditional radiation temperature measurement methods. Firstly, the correlation between corresponding image points and the positions
of 3D object in the light field image system is analyzed using the ray tracing method. The 3D position acquisition model and the light field colorimetric thermometry model are established, enabling
simultaneous acquisition of the spatial coordinates and radiometric information of the 3D object. Then, the light field camera radiation calibration experiment was conducted, and the 3D temperature
field will be obtained by employing colorimetric thermometry for each corresponding image point of the same object point. Finally, the experiment employed a light field camera for temperature
measurement and reconstruction of candle flames. The accuracy of the temperature measurement is 3.31%, thus confirming the feasibility and effectiveness of the proposed method.
Key words: Temperature measurement / Optical field imaging / Colorimetric thermometry / 3D reconstruction / Radiometric calibration
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The radiometric thermometry technology, characterized by its fast response speed, high upper limit of temperature measurement, and wide dynamic range, holds great potential for various applications
in the field of temperature measurement [1–3]. The traditional single camera system, while simple in structure and easy to implement, faces challenges in acquiring and reconstructing 3D spatial
information of the radiation field. The camera array thermometry system [4] involves the arrangement of multiple cameras at various locations and angles around the temperature field, enabling
simultaneous imaging to capture radiation images from multiple perspectives. This system offers the benefit of superior spatial resolution; however, it is characterized by high system complexity and
challenging calibration.
Light field imaging technology [5–7] encompasses the acquisition and data processing of optical fields. By incorporating a microlens array (MLA) onto the optical path of a conventional camera, it
becomes feasible to capture the complete optical radiation information in a single image. Through data processing techniques such as transformation and integration, the position, direction, radiation
spectrum, and other pertinent information of the 3D object can be calculated. This innovative approach presents a novel concept for 3D temperature field measurement, with the potential to surpass the
limitations of point and surface temperature measurements and achieve breakthroughs in 3D temperature measurement.
In recent years, numerous scholars have conducted research on methods for measuring and reconstructing 3D radiation temperature. Daniel et al. [8] proposed a spectral measurement method utilizing
curve fitting techniques to accurately measure the temperature of a target within the range of 800–1200 K. Hertz and Faris [9] employed the principles of spectroscopy to simultaneously capture
radiation projections of the transient fireball temperature field from multiple directions. They further integrated the MART algorithm to achieve a two-dimensional reconstruction of the radiation
intensity. Bheemul et al. [10] employed three CCD cameras to simultaneously capture target emission intensity projections in three directions, enabling quantitative reconstruction of gas radiation
properties. Upton et al. [11] developed a 12-direction high-temperature fireball radiation intensity acquisition system based on six CCD cameras, facilitating 3D reconstruction of the radiation
field. Floyd et al. [12] utilized an experimental circular table with 5 CCDs positioned at 36° intervals, allowing for simultaneous projection of light field information in 2 directions onto one CCD.
Consequently, projection information in 10 directions could be obtained simultaneously, achieving high spatial and temporal resolution 3D reconstruction of the turbulent impulsive temperature field.
Ishino et al. [13] constructed an emission spectral tomography device with 40 projection acquisition directions, equipped with 40 imaging lenses, enabling 3D reconstruction of the propane temperature
field. Furthermore, Ishino and Ohiwa [14] stacked the imaging lenses into 4 layers, resulting in a device with 158 imaging lenses, a number that set a world Guinness record, and further enhancing the
reconstruction accuracy of the transient combustion field. Sun et al. [15] propose a novel geometric calibration method for focused light field cameras to accurately track the paths of flame radiance
and reconstruct the 3D temperature distribution of a flame. Huang et al. [16] introduce a reconstruction technique for determining the 3D temperature distribution and radiative properties of
participating media using the multi-spectral light-field imaging technique. Yuan et al. [17] and Li et al. [18] simplify the reconstruction process by simulating the light field imaging of nonuniform
temperature distributions using a previously developed multi-focus plenoptic camera model. The aforementioned research achievements have laid a solid foundation for the study of light field
colorimetric temperature measurement method in this paper.
This paper proposes a 3D temperature field measurement method based on light field colorimetric thermometry. The mathematical relationship between the spatial positions of 3D object and corresponding
image points of elemental images array (EIA) in the integral imaging system was theoretically analyzed. Corresponding image points refers to multiple points on the focal plane that are generated by a
point in the object space being captured by the MLA in an integral imaging system. These different corresponding image points capture the light field information of the same 3D objects from different
angles, thus earning the name corresponding image points. Additionally, the 3D temperature distribution of the object was calculated by combining colorimetric temperature measurement with optical
field imaging system parameters. The rest of the paper is organized as follows. The theoretical analysis of colorimetric temperature measurement with light field camera is derived in Section 2.
Section 3 introduces the blackbody radiation calibration principle of the light field camera and gives the experimental calibration results. Experimental results are given in Sections 4 and 5
concludes this paper.
2 Principle of light field imaging colorimetric temperature measurement
2.1 Principles of 3D information acquisition in light field imaging
Light field imaging incorporates an MLA between the main lens and the image sensor, enabling the projection of light from various directions of the object onto the image sensor, thereby generating an
EIA. This technique allows for the capture of single-view images of objects from multiple perspectives, thereby providing a ray recording and acquisition technology that offers enhanced dimensions
and a more comprehensive information representation. This paper uses a focused light field camera to obtain passive 3D information and radiation information. As shown in Figure 1 is the schematic
diagram of the 3D information acquisition of the focused light field camera. In contrast to conventional light field cameras, the MLA of a focused light field camera is positioned away from the image
plane of the main lens. The image plane of the main lens, which is not captured by the image sensor, is referred to as the virtual image plane. This virtual image plane is the conjugate plane of the
object plane with respect to the main lens, while the CCD plane of the image sensor is situated at the position of the virtual image plane in relation to the conjugate plane of the MLA. Therefore,
the imaging process of a focused light field camera can be divided into two distinct and separate imaging processes. The first process involves the imaging of the object plane in relation to the main
lens, while the second process involves the imaging of the virtual image plane in relation to the MLA. By employing this approach, focused light field cameras are able to effectively reduce
directional sampling and enhance the resolution of spatial sampling.
Figure 1
Schematic diagram of 3D information acquisition technology of focused light field camera.
In Figure 1, the x–y–z 3D coordinate system is established with the coordinate origin O as the center. The point A [1] in 3D space is projected through the main lens and MLA to obtain an EIA. Among
them, L [1] represents the distance from the object point to the main lens, L [2] represents the distance from the main lens to the virtual image plane, L [3] represents the distance from the virtual
object point A [2] to the lens array, and L [4] represents the distance from the element image array to the lens array. L [1] and L [2] are conjugate to the main lens, while L [3] and L [4] are
conjugate to the MLA. The focal length of the main lens array is denoted as f [main], and the focal length of the MLA is denoted as f [MLA]. D(m, n) represents the sub lens located at (m, n)th in the
MLA. Image (m, n) represents the elemental image corresponding to D(m, n). For point A [1], it is projected from D(m, n) to Image (m, n) at point A [3](m, n), and the set of all A [3](m, n) in the
element image array is defined as the image point of A [1] with the same name.
This section employs the ray tracing method to thoroughly analyze the correlation between pixels sharing the same name. The coordinates of point A [1] are denoted as A [1](X [ A1], Y [ A1], Z [ A1]),
while the coordinates of the virtual point A [2] are represented as A [2](X [ A2], Y [ A2], Z [ A2]). The optical center coordinates of the (m, n)th microlens, denoted as D(m, n), within the MLA are
given by (X [ D(m, n)], Y [ D(m, n)], Z [ D (m, n)]). Furthermore, the coordinates of the corresponding homonymous image points A [3](m, n) of object point A [1] and virtual object point A [2] in the
element image corresponding to D(m, n) are expressed as (X [ A3 (m, n)], Y [ A3 (m, n)], Z [ A3 (m, n)]). The microlens is considered to be a square lens, with adjacent microlenses spaced p apart in
the MLA. Based on the imaging equation of the main lens, the relationship between A [1], O, and A [2] can be described as follows:${ X O - X A 1 X A 2 - X O = Z O - Z A 1 Z A 2 - Z O = L 1 L 2 = β 1
, Y O - Y A 1 Y A 2 - Y O = Z o - Z A 1 Z A 2 - Z O = L 1 L 2 = β 1 , 1 Z O - Z A 1 + 1 Z A 2 - Z O = 1 f main ,$(1)where β [1] = L [1]/L [2] represents the magnification of the main lens.
According to the lens imaging equation, the coordinate relationship between A [2], D(m, n), and A [3](m, n) is:${ X D ( m , n ) - X A 2 X A 3 ( m , n ) - X D ( m , n ) = Z D ( m , n ) - Z A 2 Z A 3 (
m , n ) - Z D ( m , n ) = L 3 L 4 = β 2 , Y D ( m , n ) - Y A 2 Y A 3 ( m , n ) - Y D ( m , n ) = Z D ( m , n ) - Z A 2 Z A 3 ( m , n ) - Z D ( m , n ) = L 3 L 4 = β 2 , 1 Z A 3 ( m , n ) - Z D ( m ,
n ) + 1 Z D ( m , n ) - Z A 2 = 1 f MLA ,$(2)where β [2] = L [3]/L [4] represents the magnification of the MLA.
For periodically arranged MLA, the relationship between the optical center coordinates of D(m, n) and D(m + i, n + j) within the MLA can be expressed as follows:${ X D ( m + i , n + j ) = X D ( m , n
) + i × p , Y D ( m + i , n + j ) = Y D ( m , n ) + j × p , Z D ( m + i , n + j ) = Z D ( m , n ) .$(3)
The aforementioned equation represents the correlation between the optical center coordinates of any two microlenses within an MLA. By using equations (1)–(3), the coordinates of the corresponding
pixel points A [3](m, n) and A [3](m + i, n + j) formed by two sub lenses D(m, n) and D(m + i, n + j) can be derived as follows:${ X A 3 ( m , n ) = X D ( m , n ) - X A 2 β 2 + X D ( m , n ) , X A 3
( m + i , n + j ) = X D ( m + i , n + j ) - X A 2 β 2 + X D ( m + i , n + j ) .$(4)
By simultaneously solving equations (3) and (4), the relationship between any two corresponding image points can be obtained:${ X A 3 ( m , n ) = X A 3 ( m + i , n + j ) - ip × ( 1 + 1 / β 2 ) , Y A
3 ( m , n ) = Y A 3 ( m + i , n + j ) - jp × ( 1 + 1 / β 2 ) .$(5)
When i = 1 and j = 1, the relationship between two corresponding image points obtained by adjacent lenses passing through the origin O can be derived as follows:${ X A 3 ( m + 1 , n + 1 ) = X A 3 ( m
, n ) + p × ( 1 + 1 / β 2 ) , Y A 3 ( m + 1 , n + 1 ) = Y A 3 ( m , n ) + p × ( 1 + 1 / β 2 ) ,$(6)where X [ A3(m, n)] and X [ A3(m+1, n+1)] represent the coordinates of the corresponding image
points A [3(m,n)] and A [3(m+1,n+1)], respectively. By simultaneously solving equations (2)–(6), the coordinates of the virtual object point A [2] can be obtained.${ X A 2 = X D ( m , n ) - p ( X A 3
( m , n ) - X D ( m , n ) ) X A 3 ( m + 1 , n + 1 ) - X A 3 ( m , n ) - p , Y A 2 = Y D ( m , n ) - p ( Y A 3 ( m , n ) - Y D ( m , n ) ) Y A 3 ( m + 1 , n + 1 ) - Y A 3 ( m , n ) - p , Z A 2 = Z D (
m , n ) + f MLA ( Z A 3 ( m , n ) - Z D ( m , n ) ) f MLA + Z D ( m , n ) - Z A 3 ( m , n ) ,$(7)where the coordinates (X [ A3(m, n)], Y [ A3(m, n)], Z [ A3(m, n)]) of A [3](m, n) and the coordinates
(X [ A3(m+1, n+1)], Y [ A3(m+1, n+1)], Z [ A3(m+1, n+1)]) of A [3](m + 1, n + 1) can be obtained by extracting the corresponding elements from the image array. The parameters p and L [4] can be
directly measured. Given the specified focal point coordinates X [ D(m, n)] and Y [ D(m, n)] of D(m, n), the spatial coordinates (X [ A1], Y [ A1], Z [ A1]) of object point A [1] can be obtained by
simultaneously solving equation (1).${ X A 1 = X O + f main ( X A 2 - X O ) f main + Z O - Z A 2 , Y A 1 = Y O + f main ( Y A 2 - Y O ) f main + Z O - Z A 2 , Z A 1 = Z O + f main ( Z A 2 - Z O ) f
main + Z O - Z A 2 .$(8)
Based on the aforementioned analysis, it can be inferred that the acquisition of the spatial coordinates of object point A [1] can be achieved by obtaining the focal length of the light field camera
system, the spacing between microlenses in the MLA (p), and the distance from the image array to the MLA. This facilitates the implementation of 3D shape acquisition through light field imaging.
2.2 Principle of 3D temperature field colorimetric temperature measurement
In order to measure the 3D temperature distribution of the object’s surface, a colorimetric temperature measurement method is employed for each corresponding pixel on the light field image sensor.
According to Planck’s law, the equation for the spectral radiance of non-blackbody radiation is as follows:$L λ = C 1 π λ 5 ε ( λ , T ) ( e C 2 / λ T - 1 ) - 1 ,$(9)where ε(λ,T) < 1 represents the
emissivity of non-blackbody spectral radiation. L [ λ ] is the spectral radiance at wavelength λ. T is the blackbody temperature value. C [1] = 3.7418 × 10^−16 W ∙ m^2 and C [2] = 1.4388 × 10^−2 m ∙
K are the first and second radiation constants, respectively. When the temperature of an object changes, the peak wavelength of the object’s spectral radiance also changes accordingly. The higher the
temperature, the smaller the peak wavelength. When T is less than 3000 K, that is, C [2]/λT significantly greater than 1, equation (9) can be simplified as:$L λ = C 1 π λ 5 ε ( λ , T ) e - C 2 / λ T
The peak spectral radiance increases with the temperature, and the peak spectral radiance of a certain temperature can only be detected within specific wavelength bands. The wavelength range of the
CCD sensor is 380–780 nm, with red, green, and blue as the three primary colors. In computer graphics, the primary colors R, G, and B of a single pixel are represented by values ranging from 0 to
255. By combining different brightness values of the R, G, and B primary colors in specific proportions, any desired brightness value L can be achieved.$L = l ( R ) + l ( G ) + l ( B ) ,$(11)where l(
R), l(G), l(B) represent the brightness values of the three primary colors. The range of binary grayscale values is [0,1]. Assuming the spectral response function of the visible light CCD sensor in
the light field is denoted as Y(λ), the grayscale value of the light field image can be represented as H, which can be defined as follows:$H = 1 4 ⋅ η μ t ⋅ [ 2 a f ′ ] 2 ⋅ ∫ 380 780 K T ( λ ) L ( λ
, T ) Y ( λ ) d λ ,$(12)where η represents the conversion coefficient between the input current of the CCD and the grayscale value of the image; μ denotes photoelectric conversion coefficient of the
CCD sensor; a is the conversion coefficient between the output voltage of the photosensitive unit and the image grayscale value; t represents the integration time of the camera; a is the aperture of
the light field camera; f denotes focal length; K [ T ](λ) represents the optical transmittance of the main lens of the light field camera; L(λ,T) represents the radiance value of the CCD image
sensor. At a specific wavelength λ, K [ T ](λ) undergoes minimal changes and is generally considered as a constant. Therefore, equation (12) can be rewritten as:$H = A ∫ 380 780 L ( λ , T ) Y ( λ ) d
λ , A = 1 4 ⋅ η μ t ⋅ [ 2 a f ′ ] 2 .$(13)According to the spectral response functions associated with the R, G, and B channels of the visible light CCD image sensor, the correlation between the
grayscale values of the three primary color signals and the temperature of the measured object can be derived as follows:${ R = A ∫ 380 780 L ( λ , T ) r ( λ ) d λ , G = A ∫ 380 780 L ( λ , T ) g ( λ
) d λ B = A ∫ 380 780 L ( λ , T ) b ( λ ) d λ , ,$(14)where r(λ), g(λ), b(λ) represent the spectral response functions of R, G, and B channels. The optical field colorimetric temperature measurement
method involves comparing the spectral radiance values at two wavelengths. The expression for this method is as follows:$L 1 ( λ 1 , T 1 ) L 1 ( λ 2 , T 1 ) = L ( λ 1 , T C ) L ( λ 2 , T C ) ,$(15)
where T [1] represents the actual temperature of the non-blackbody thermal radiation material, and T [ c ] represents the temperature of the blackbody. By combining equation (10) with the expression
for color temperature (15), the temperature of the object point corresponding to any pixel of light field image can be determined as follows:$T = C 2 ( 1 λ 2 - 1 λ 1 ) ln L ( λ 1 , T ) L ( λ 2 , T )
- ln ε ( λ 1 , T ) ε ( λ 2 , T ) - 5 ln λ 2 λ 1 ,$(16)where L(λ [1], T) and L(λ [2], T) represent the monochromatic radiance at different wavelengths. For substances that can be considered equivalent
to gray body, their spectral emissivity remains relatively constant within a narrow band range of the same spectrum. Therefore, it is reasonable to assume that the spectral emissivity expression for
gray body is 0. The CCD colorimetric temperature measurement model does not require precise measurement of emissivity. Instead, it utilizes the emissivity ratio at two wavelengths to eliminate
uncertainties arising from unknown variables, such as the measurement environment. Based on this assumption, the CCD colorimetric temperature measurement equation for a gray body is as follows:$T = C
2 ( 1 λ 2 - 1 λ 1 ) ln L ( λ 1 , T ) L ( λ 2 , T ) - 5 ln λ 2 λ 1 .$(17)
This method acquires 3D information of object points through light field imaging and utilizes the colorimetric temperature measurement method to obtain radiation field temperature information. By
capturing both the 3D position information of the target object in a single shot and the radiation temperature information, it overcomes the limitations of existing technologies that necessitate
multiple measurements from different perspectives to obtain both position and radiation information of the same target. This approach offers the advantages of 3D imaging and multispectral detection,
while maintaining a simple system structure and convenient data sampling.
3 Spectral radiation calibration of light field camera
A blackbody can serve as a standard radiation source for calibrating the flame radiation temperature. By employing a light field camera to capture a light field image of a blackbody with a known
temperature, the radiance values received by each pixel in the image and the temperature values of the blackbody can be related through equation (10) based on Planck’s law. Subsequently, the
grayscale value of the image can be fitted with the corresponding radiance value at that temperature, thereby enabling the calibration of the image sensor of the light field camera for radiation
This paper utilizes a Raytrix R8 focused field camera for data acquisition. The Raytrix R8 image sensor’s R, G, and B channels have response wavelengths of 610 nm, 530 nm, and 460 nm, respectively.
The experiment used the Lumasense M360 standard blackbody, which features a cavity made of graphite tube targets. The blackbody has an effective emissivity of 1.0 and can measure temperatures ranging
from 5 °C to 1200 °C. Radiation calibration experiments are conducted under dark conditions without any illumination to eliminate the potential influence of stray light. As depicted in Figure 2, the
light field camera is positioned directly in front of the central cavity of the blackbody at the distance of 370 mm.
Figure 2
Photos of blackbody calibration device.
Due to the use of a pentagon diaphragm in the camera, the smallest image unit captured is also a pentagon. This leads to some gaps between the original light field images. To avoid collecting invalid
grayscale data, only the grayscale values within the pentagon area are read, and the average grayscale value is taken based on the recorded data. When the exposure time set by the camera is deemed
unreasonable, it may lead to overexposure of the captured image, thereby causing image distortion. To prevent overexposure of the captured raw light field image, an exposure time of 0.3 ms was
employed. In the dark calibration laboratory at a room temperature of 22.5 °C, the blackbody calibration temperature ranged from 800 °C to 1000 °C, with images of the central cavity of the blackbody
furnace being captured at intervals of 50 °C, as shown in Figure 3.
Figure 3
Original light field image of the center of the blackbody furnace.
Figure 4 shows three channel images of R, G, and B in the center cavity of the blackbody. According to the colorimetric temperature measurement model, it is essential to extract multiple grayscale
values of the R and G channels from each color image within the central calibration area of the blackbody. Subsequently, the average grayscale values of different temperature images under the two
channels are calculated, as illustrated in Table 1.
Figure 4
Three channel images of R, G, and B in the center cavity of the blackbody.
Table 1
Grayscale values of the central cavity image of blackbody at different temperatures.
By utilizing the grayscale values presented in Table 1, it is possible to establish a correlation between image grayscale values and radiation intensity through the following fitting process:$I R = -
134660 + 1.07 × 1 0 6 × R + 9.46 × 1 0 7 × R 2 - 4.61 × 1 0 7 × R 3 , I G = - 18173 + 1.59 × 1 0 6 × G - 6.05 × 1 0 6 × G 2 + 8.27 × 1 0 6 × G 3 , I B = - 17595 + 6.603 × 1 0 5 × B - 1.69 × 1 0 6 × B
2 + 1.91 × 1 0 6 × B 3 ,$(18)where R, G, and B represent the binary grayscale values corresponding to the three channel images, while I [ R ], I [ G ] and I [ B ] represent the radiation intensity
values corresponding to the three channels. The colorimetric temperature measurement equation for the optical field temperature measurement system can be expressed as follows:$T = C 2 ( 1 λ g - 1 λ r
) ln I R I g - 5 ln λ g λ r ,$(19)where λ [ r ] = 610 nm, λ [ g ] = 530 nm. The colorimetric temperature value of the light field temperature measurement system can be directly obtained based on the
grayscale values of the R and G channel images.
4 Results and discussion
The light field imaging colorimetric temperature measurement system comprises three main components: the main lens, MLA and CCD sensors, and a computer image processing system. As depicted in Figure
5, the light field temperature measurement system captures the spectral radiation brightness information of candle flames, converts it into digital signals of flame images through the photoelectric
and analog-to-digital conversion processes of the light field CCD sensor, and subsequently transmits these signals to the computer image processing system using USB3.0 data technology, ultimately
presenting the original flame light field image. This system employs a main lens with a short focal length of 35 mm and an aperture coefficient of 1.4. The F-number of the MLA is 2.8, and the single
pixel size of the CCD is 2.24 μm.
Figure 5
Schematic diagram of candle flame temperature measurement experiment.
The candle was positioned at the distance of 300 mm from the light field camera. To mitigate the impact of stray visible light interference, the experiment was conducted in a controlled laboratory
environment with complete darkness conditions. The impact of environmental stray light and absorption effects on the results was found to be minimal. The light field colorimetric thermometry method
can also be applied for long-distance measurements. However, it requires the selection of a telephoto lens. When conducting experimental measurements at long distances in outdoor settings, it is
important to consider the influence of environmental stray light and atmospheric absorption effects in order to improve measurement accuracy.
Similar to the blackbody calibration experiment, a standardized exposure time of 0.3 ms was employed for the light field camera. Figure 6a displays the original flame light field image captured by
the light field camera. The pentagonal aperture of the main lens results in a corresponding pentagonal micro unit image projected onto the CCD sensor of the camera through the MLA.
Figure 6
Light field imaging acquisition and reconstruction image: (a) original flame light field image, (b) R channel grayscale image, (c) G channel grayscale image, (d) B channel grayscale image and (e)
flame refocused color image.
The measurement and reconstruction of flame temperature field in this paper consists of the following three steps:
• Step 1: Capture the original light field image, as depicted in Figure 6a. The original light field image encompasses the R, G, and B channels of the CCD sensor.
• Step 2: Utilizing the 3D information creation method outlined in Section 2.1, generate monochromatic 3D reconstructed images for the R, G, and B channels of the light field images. These images
are depicted in Figures 6b–6d. Subsequently, employ the RGB color space model to fuse the individual channel images of R, G, and B, resulting in the flame refocused color image displayed in
Figure 6e.
• Step 3: By utilizing the grayscale values of the R and G channels, in conjunction with the radiation calibration equation (18) and the colorimetric temperature measurement model, the temperature
values of each pixel within the flame image can be derived. Figure 7a shows the plane distribution of the flame temperature field, and Figure 7b shows the 3D distribution of the flame temperature
field. The flame temperature field underwent isothermal layering treatment, with the layering temperature aligning with the actual temperature distribution of the flame. This alignment indicates
the rationality of reconstructing the temperature field distribution.
Figure 7
Reconstruction results of flame temperature field: (a) plane distribution diagram of flame temperature field and (b) 3D distribution diagram of flame temperature field.
To validate the accuracy of temperature measurements obtained from the system, a thermocouple thermometer with a temperature measurement accuracy of 0.2% was employed to measure the contact
temperature of the identical flame. Figure 8a illustrates the configuration of the thermocouple flame temperature measurement device, which comprises two components: a thermocouple instrument and a
thermocouple detection line. The detection line consists of a total of 8 sub-lines, enabling simultaneous temperature measurements at 8 different positions within the flame. The measurement point’s
position is depicted in Figure 8b. Multiple measurements were conducted at each temperature measurement point, and the average value was considered as the actual temperature value for that specific
point. Table 2 presents a comparison between the reconstructed flame temperature field and the measured data. The maximum value observed in the flame colorimetric temperature measurement method is 35
K. Furthermore, the maximum rate of the colorimetric temperature values measured by the system is 3.31%, thereby confirming the accuracy of the colorimetric temperature measurement model in
comparison to the monochromatic temperature measurement model.
Figure 8
(a) Thermocouple contact temperature measurement and (b) schematic diagram of 8 temperature point locations.
Table 2
Comparison results between reconstructed flame temperature field and the thermocouple contact temperature measurement.
5 Conclusion
In this work, we present a novel method for measuring the 3D temperature field using light field colorimetric thermometry. This method enables the simultaneous acquisition of both 3D position
information and radiation temperature data of the objects being measured. The mathematical relationship between the spatial positions of 3D object and corresponding image points of EIA in the
integral imaging system is theoretically analyzed. The 3D temperature distribution of the object is calculated by combining colorimetric temperature measurement with optical field imaging system
parameters. The effectiveness and feasibility of the proposed method have been validated through experimental verification. In response to the limitations identified in this study, future research
efforts will primarily concentrate on addressing the following two areas: (1) the light field temperature measurement method proposed in this study has been investigated primarily for objects that
exhibit characteristics similar to gray bodies. The subsequent phase of research will involve measuring and studying non-gray objects, with the objective of establishing a temperature measurement
model suitable for such bodies. (2) While the maximum error rate of the proposed light field temperature measurement method in this paper is 3.31%, representing an improvement over traditional
monochromatic light field temperature measurement methods, there remains significant scope for further advancements to achieve high-precision 3D temperature measurement systems.
Conflict of interest
The authors declare no conflict of interest.
This research is supported by National Natural Science Foundation of China (62005204, 62075176 and 62005206) and the Fundamental Research Funds for the Central Universities (ZYTS23124, SY22033I,
ZYTS23127, QTZX22004, JPJC2112).
All Tables
Table 1
Grayscale values of the central cavity image of blackbody at different temperatures.
Table 2
Comparison results between reconstructed flame temperature field and the thermocouple contact temperature measurement.
All Figures
Figure 1
Schematic diagram of 3D information acquisition technology of focused light field camera.
In the text
Figure 2
Photos of blackbody calibration device.
In the text
Figure 3
Original light field image of the center of the blackbody furnace.
In the text
Figure 4
Three channel images of R, G, and B in the center cavity of the blackbody.
In the text
Figure 5
Schematic diagram of candle flame temperature measurement experiment.
In the text
Figure 6
Light field imaging acquisition and reconstruction image: (a) original flame light field image, (b) R channel grayscale image, (c) G channel grayscale image, (d) B channel grayscale image and (e)
flame refocused color image.
In the text
Figure 7
Reconstruction results of flame temperature field: (a) plane distribution diagram of flame temperature field and (b) 3D distribution diagram of flame temperature field.
In the text
Figure 8
(a) Thermocouple contact temperature measurement and (b) schematic diagram of 8 temperature point locations.
In the text
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negative zero
Hi Guys, i found a bug? not sure if this -0 (negative zero) is a feature or somewhat useful on other scenarios, kindly try out this code
x = 1
t = -100
function _update()
im using t to animate colors or sprites but i noticed that when t is approaching to 0 from a negative value, it stays negative even tho its 0, and i thought, "of course it does not matter because
it's still zero" but it does matter because somehow it's 1 off to 0, at least that's what i think happens base on the colors
im doing a workaround where i check if it's less than 0 and set it to 0. but it costs tokens :<<< i hope this gets resolved. thanks
Yeah, it looks like T is not actually 0 or -0 but something between -0.00003 and -0.000016:
Each of those returns 0xffff.ffff when you get to the point in your code that your print shows t = -0.
The smarter people here probably have a better idea, but this is probably you bumping up against the precision limitations in PICO-8. I think once your halving operation gets below 0x0.000016 it just
stops doing anything maybe and you get stuck at -0xffff.ffff instead of ever getting to 0? The manual says:
"Numbers in PICO-8 are all 16:16 fixed point. They range from -32768.0 to 32767.99999"
No bug, just fixed floating point weirdness :
when t is at 0xffff.ffff with is the biggest (nearest to zero) negative value a 16.16bits variable can hold, printing it shows -0 witch is a very accurate if weird looking approximation in base ten.
when you do t*=0.5, the result is exactly the mid point between
0xffff.ffff and 0x0000.0000, so a rounding to any of those two values is valid.
If you want to reach 0 without using extra tokens, you can use division instead of multiplication to get the rounding towards zero.
t*=0.5 becomes t/=2
0xffff.ffff/2 gives 0x0000.0000
0xffff.ffff*0.5 gives 0xffff.ffff
oohhhhh. Thanks 2bitchuck and RealShadowCaster for these infos, really appreciate it.
I tried t/=2 but something new is happening.
there is a bit of delay on x-t but for x+t there is no delay,
x = 1
t = 100 -- i changed t to 100 instead of -100
function _update()
what i did was to use t\=2 to just floor it hahahaha. either way thank you so much guys, I learned something new today especially with those weird things in floating point
The delay is just because of adding vs subtracting.
For instance, consider what happens when t=1. Then
x+t = 2, and
x-t = 0
As t gets smaller and closer to 0 both of those values approach 1 but from different directions. When t=0.9:
x+t = 1.9, and
As colours those will just be considered as 1 and 0 respectively. So when t becomes less than one, x+t goes immediately to colour 1. But x-t won't go to colour 1 until the very end when t=0. Hence
the delay.
If you want symmetrical signed behavior, you can do this to circumvent the 2's complement hazard with negative rounding behavior:
-- long form
if t<0 then
t = -(-t*.5)
t = t*.5
-- shorthand
t = t<0 and -(-t*.5) or t*.5
This will always trend towards true 0.
This is why true IEEE-style floating point numbers, unlike PICO-8's signed 2's-complement 16.16 fixed-point numbers, don't actually use 2's complement to represent the mantissa. The mantissa is
always positive and there is a separate bit flag to say if the mantissa should be treated as negative.
As an aside, you could also do >>1 instead of *.5. Not a very useful optimization for something you only do once per frame, but hey, worth knowing at least, maybe for other use cases. 🙂
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Trading Stocks using Bollinger Bands, Keltner Channel, and RSI in Python
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time. Now, Bollinger Bands are trend lines plotted above and below the SMA of the given stock at a specific standard deviation level. To understand Bollinger Bands better, have a look at the
following chart that represents the Bollinger Bands of the Apple stock calculated with SMA 20.
Bollinger Bands are great to observe the volatility of a given stock over a period of time. The volatility of a stock is observed to be lower when the space or distance between the upper and lower
band is less. Similarly, when the space or distance between the upper and lower band is more, the stock has a higher level of volatility. While observing the chart, you can observe a trend line named
‘MIDDLE BB 20’ which is the SMA 20 of the Apple stock. The formula to calculate both upper and lowers bands of stock is as follows:
UPPER_BB = STOCK SMA + SMA STANDARD DEVIATION * 2
LOWER_BB = STOCK SMA - SMA STANDARD DEVIATION * 2
Keltner Channel (KC)
First introduced by Chester Keltner, the Keltner Channel is a technical indicator that is often used by traders to identify volatility and the direction of the market. The Keltner Channel is composed
of three components: The upper band, the lower band, and the middle line. Now, let’s discuss how each of the components is calculated.
Before diving into the calculation of the Keltner Channel it is essential to know about the three important inputs involved in the calculation. First is the ATR (Average True Range) lookback period,
which is the number of periods that are taken into account for the calculation of ATR. Secondly, the Keltner Channel lookback period. This input is more or less similar to the first one but here, we
are determining the number of periods that are taken into account for the calculation of the Keltner Channel itself. The final input is the multiplier which is a value determined to multiply with the
ATR. The typical values that are taken as inputs are 10 as the ATR lookback period, 20 as the Keltner Channel lookback period, and 2 as the multiplier. Keeping these inputs in mind, let’s calculate
the readings of the Keltner Channel’s components.
The first step in calculating the components of the Keltner Channel is determining the ATR values with 10 as the lookback period.
The next step is calculating the middle line of the Keltner Channel. This component is the 20-day Exponential Moving Average of the closing price of the stock. The calculation can be represented as
MIDDLE LINE 20 = EMA 20 [ C.STOCK ]
EMA 20 = 20-day Exponential Moving Average
C.STOCK = Closing price of the stock
The final step is calculating the upper and lower bands. Let’s start with the upper band. It is calculated by first adding the 20-day Exponential Moving Average of the closing price of the stock by
the multiplier (two) and then, multiplied by the 10-day ATR. The lower band calculation is almost similar to that of the upper band but instead of adding, we will be subtracting the 20-day EMA by the
multiplier. The calculation of both upper and lower bands can be represented as follows:
UPPER BAND 20 = EMA 20 [ C.STOCK ] + MULTIPLIER * ATR 10
LOWER BAND 20 = EMA 20 [ C.STOCK ] - MULTIPLIER * ATR 10
EMA 20 = 20-day Exponential Moving Average
C.STOCK = Closing price of the stock
MULTIPLIER = 2
ATR 10 = 10-day Average True Range
That’s the whole process of calculating the components of the Keltner Channel. Now, let’s analyze a chart of the Keltner Channel to build more understanding of the indicator.
The above chart is a graphical representation of Apple’s 20-day Keltner Chanel. We could notice that two bands are plotted on either side of the closing price line and those are the upper and lower
band, and the grey-colored line running in between the two bands is the middle line, or the 20-day EMA. The Keltner Channel can be used in an extensive number of ways but the most popular usages are
identifying the market volatility and direction.
The volatility of the market can be determined by the space that exists between the upper and lower band. If the space between the bands is wider, then the market is said to be volatile or showing
greater price movements. On the other hand, the market is considered to be in a state of non-volatile or consolidating if the space between the bands is narrow. The other popular usage is identifying
the market direction. The market direction can be determined by following the direction of the middle line as well as the upper and lower band.
While looking at the chart of the Keltner Channel, it might resemble the Bollinger Bands. The only difference between these two indicators is the way each of them is calculated. The Bollinger Bands
use standard deviation for its calculation, whereas, the Keltner Channel utilizes ATR to calculate its readings.
Relative Strength Index
Before moving on, let’s first gain an understanding of what an oscillator means in the stock trading space. An oscillator is a technical tool that constructs a trend-based indicator whose values are
bound between a high and low band. Traders use these bands along with the constructed trend-based indicator to identify the market state and make potential buy and sell trades. Also, oscillators are
widely used for short-term trading purposes, but there are no restrictions in using them for long-term investments.
Founded and developed by J. Welles Wilder in 1978, the Relative Strength Index is a momentum oscillator that is used by traders to identify whether the market is in a state of overbought or oversold.
Before moving on, let’s explore what overbought and oversold is. A market is considered to be in the state of overbought when an asset is constantly bought by traders, moving it to an extremely
bullish trend and bound to consolidate. Similarly, a market is considered to be in the state of oversold when an asset is constantly sold by traders, moving it to a bearish trend and tends to bounce
Being an oscillator, the values of RSI are bound between 0 to 100. The traditional way to evaluate a market state using the Relative Strength Index is that an RSI reading of 70 or above reveals a
state of overbought, and similarly, an RSI reading of 30 or below represents the market is in the state of oversold. These overbought and oversold can also be tuned concerning which stock or asset
you choose. For example, some assets might have constant RSI readings of 80 and 20. So in that case, you can set the overbought and oversold levels to be 80 and 20 respectively. The standard setting
of RSI is 14 as the lookback period.
RSI might sound more similar to Stochastic Oscillator in terms of value interpretation, but the way it’s being calculated is quite different. There are three steps involved in the calculation of RSI:
• Calculating the Exponential Moving Average (EMA) of the gain and loss of an asset: A word on Exponential Moving Average. EMA is a type of Moving Average (MA) that automatically allocates greater
weighting (nothing but importance) to the most recent data point and lesser weighting to data points in the distant past. In this step, we will first calculate the returns of the asset and
separate the gains from losses. Using these separated values, the two EMAs for a specified number of periods are calculated.
• Calculating the Relative Strength of an asset: The Relative Strength of an asset is determined by dividing the Exponential Moving Average of the gain of an asset from the Exponential Moving
Average of the loss of an asset for a specified number of periods. It can be mathematically represented as follows:
RS = GAIN EMA / LOSS EMA
RS = Relative Strength
GAIN EMA = Exponential Moving Average of the gains
LOSS EMA = Exponential Moving Average of the losses
• Calculating the RSI values: In this step, we will calculate the RSI itself by making use of the Relative Strength values we calculated in the previous step. To calculate the values of RSI of a
given asset for a specified number of periods, there is a formula that we need to follow:
RSI = 100.0 - (100.0 / (1.0 + RS))
RSI = Relative Strength Index
RS = Relative Strength
Now let’s analyze a chart of RSI plotted along with Apple’s historical stock data to gain a strong understanding of the indicator.
The above chart is separated into two panels: The above panel with the closing price of Apple and the lower panel with the calculated RSI 14 values of Apple. While analyzing the panel plotted with
the RSI values, it can be seen that the trend and movement of the calculated values follow the same as the closing price of Apple. So, we can consider that RSI is a directional indicator. Some
indicators are non-directional meaning that their movement will be inversely proportional to the actual stock movement and this can sometimes confuse traders and be hard to understand too.
While observing the RSI chart, we are able to see that the plot of RSI reveals trend reversals even before the market does. Simply speaking, the RSI shows a downtrend or an uptrend right before the
actual market does. This shows that RSI is a leading indicator. A leading indicator is an indicator that takes into account the current value of a data series to predict future movements. RSI being a
leading indicator helps in warning the traders about potential trend reversions before in time. The opposite of leading indicators is called lagging indicators. Lagging indicators are indicators that
represent the current value by taking into account the historical values of a data series.
Trading Strategy
Now that we have built some basic understanding of the three indicators, let’s discuss the trading strategy that we are going to implement in this article. Basically, our strategy is a squeeze
trading strategy which is confirmed by RSI. We will go long (buy the stock) whenever the Lower Keltner Channel is below the Lower Band, the Upper Keltner Channel is above the Upper Band, and the RSI
shows a reading of below 30. Similarly, we go short (sell the stock) whenever the Lower Keltner Channel is below the Lower Band, the Upper Keltner Channel is above the Upper Band, and the RSI shows a
reading of above 70. Our strategy can be represented as follows:
IF LOWER_KC < LOWER_BB AND UPPER_KC > UPPER_BB AND RSI < 30 ==> BUY
IF LOWER_KC < LOWER_BB AND UPPER_KC > UPPER_BB AND RSI > 70 ==> SELL
That’s it! This concludes our theory part and let’s move on to the programming part where we will use Python to first build the indicators from scratch, construct the discussed trading strategy,
backtest the strategy on Apple stock data, and finally compare the results with that of SPY ETF. Let’s do some coding! Before moving on, a disclaimer: This article’s sole purpose is to educate people
and must be considered as an information piece, but not as investment advice.
Implementation in Python
The coding part is classified into various steps as follows:
1. Importing Packages
2. Extracting Stock Data using EODHD
3. Bollinger Bands Calculation
4. Keltner Channel Calculation
5. RSI Calculation
6. Creating the Trading Strategy
7. Creating our Position
8. Backtesting
9. SPY ETF Comparison
We will be following the order mentioned in the above list and buckle up your seat belts to follow every upcoming coding part.
Step-1: Importing Packages
Importing the required packages into the Python environment is a non-skippable step. The primary packages are going to be Pandas to work with data, NumPy to work with arrays and for complex
functions, Matplotlib for plotting purposes, and Requests to make API calls. The secondary packages are going to be Math for mathematical functions and Termcolor for font customization (optional).
Python Implementation:
import numpy as np
import requests
import pandas as pd
import matplotlib.pyplot as plt
from math import floor
from termcolor import colored as cl
plt.rcParams['figure.figsize'] = (20,10)
Now that we have imported all the required packages into our Python. Let’s pull the historical data of Apple with EODHD’s OHLC split-adjusted data API endpoint.
Step-2: Extracting Stock Data using EODHD
In this step, we are going to pull the historical stock data of Apple using the OHLC split-adjusted API endpoint provided by EODHD. Before that, a note on EOD Historical Data: EOD Historical Data
(EODHD) is a reliable provider of financial APIs covering a huge variety of market data ranging from historical data to economic and financial news data. Also, ensure that you have an EODHD account,
only then you will be able to access your personal API key (a vital element to extract data with an API).
Python Implementation:
def get_historical_data(symbol, start_date):
api_key = 'YOUR API KEY'
api_url = f'https://eodhistoricaldata.com/api/technical/{symbol}?order=a&fmt=json&from={start_date}&function=splitadjusted&api_token={api_key}'
raw_df = requests.get(api_url).json()
df = pd.DataFrame(raw_df)
df.date = pd.to_datetime(df.date)
df = df.set_index('date')
return df
aapl = get_historical_data('AAPL', '2010-01-01')
Code Explanation: The first thing we did is to define a function named ‘get_historical_data’ that takes the stock’s symbol (‘symbol’) and the starting date of the historical data (‘start_date’) as
parameters. Inside the function, we are defining the API key and the URL and stored them in their respective variable. Next, we are extracting the historical data in JSON format using the ‘get’
function and stored it in the ‘raw_df’ variable. After doing some processes to clean and format the raw JSON data, we are returning it in the form of a clean Pandas dataframe. Finally, we are calling
the created function to pull the historic data of Apple from the start of 2010 and stored it into the ‘aapl’ variable.
Step-3: Bollinger Bands calculation
In this step, we are going to calculate the components of the Bollinger Bands by following the methods and formulas we discussed before.
Python Implementation:
def sma(data, lookback):
sma = data.rolling(lookback).mean()
return sma
def get_bb(data, lookback):
std = data.rolling(lookback).std()
upper_bb = sma(data, lookback) + std * 2
lower_bb = sma(data, lookback) - std * 2
middle_bb = sma(data, lookback)
return upper_bb, middle_bb, lower_bb
aapl['upper_bb'], aapl['middle_bb'], aapl['lower_bb'] = get_bb(aapl['close'], 20)
Code Explanation: The above can be classified into two parts: SMA calculation, and the Bollinger Bands calculation.
SMA calculation: Firstly, we are defining a function named ‘sma’ that takes the stock prices (‘data’), and the number of periods (‘lookback’) as the parameters. Inside the function, we are using the
‘rolling’ function provided by the Pandas package to calculate the SMA for the given number of periods. Finally, we are storing the calculated values in the ‘sma’ variable and returned them.
Bollinger Bands calculation: We are first defining a function named ‘get_bb’ that takes the stock prices (‘data’), and the number of periods as parameters (‘lookback’). Inside the function, we are
using the ‘rolling’ and the ‘std’ function to calculate the standard deviation of the given stock data and stored the calculated standard deviation values in the ‘std’ variable. Next, we are
calculating Bollinger Bands values using their respective formulas, and finally, we are returning the calculated values. We are storing the Bollinger Bands values into our ‘aapl’ dataframe using the
created ‘bb’ function.
Step-4: Keltner Channel Calculation
In this step, we are going to calculate the components of the Keltner Channel indicator by following the methods we discussed before.
Python Implementation:
def get_kc(high, low, close, kc_lookback, multiplier, atr_lookback):
tr1 = pd.DataFrame(high - low)
tr2 = pd.DataFrame(abs(high - close.shift()))
tr3 = pd.DataFrame(abs(low - close.shift()))
frames = [tr1, tr2, tr3]
tr = pd.concat(frames, axis = 1, join = 'inner').max(axis = 1)
atr = tr.ewm(alpha = 1/atr_lookback).mean()
kc_middle = close.ewm(kc_lookback).mean()
kc_upper = close.ewm(kc_lookback).mean() + multiplier * atr
kc_lower = close.ewm(kc_lookback).mean() - multiplier * atr
return kc_middle, kc_upper, kc_lower
aapl['kc_middle'], aapl['kc_upper'], aapl['kc_lower'] = get_kc(aapl['high'], aapl['low'], aapl['close'], 20, 2, 10)
Code Explanation: We are first defining a function named ‘get_kc’ that takes a stock’s high (‘high’), low (‘low’), and closing price data (‘close’), the lookback period for the Keltner Channel
(‘kc_lookback’), the multiplier value (‘multiplier), and the lookback period for the ATR (‘atr_lookback’) as parameters. The code inside the function can be separated into two parts: ATR calculation,
and the Keltner Channel calculation.
ATR calculation: To determine the readings of the Average True Range, we are first calculating the three differences and storing them in their respective variables. Then we are combining all three
differences into one dataframe using the ‘concat’ function and took the maximum values out of the three collective differences to determine the True Range. Then, using the ‘ewm’ and ‘mean’ functions,
we are taking the customized Moving Average of True Range for a specified number of periods to get the ATR values.
Keltner Channel calculation: Utilizing the previously calculated ATR values, we are first calculating the middle line of the Keltner Channel by taking the EMA of ATR for a specified number of
periods. Then comes the calculation of both the upper and lower bands. We are substituting the ATR values into the upper and lower bands formula we discussed before to get the readings of each of
them. Finally, we are returning and calling the created function to get the Keltner Channel values of Apple.
Step-5: RSI Calculation
In this step, we are going to calculate the values of RSI with 14 as the lookback period using the RSI formula we discussed before.
Python Implementation:
def get_rsi(close, lookback):
ret = close.diff()
up = []
down = []
for i in range(len(ret)):
if ret[i] < 0:
up_series = pd.Series(up)
down_series = pd.Series(down).abs()
up_ewm = up_series.ewm(com = lookback - 1, adjust = False).mean()
down_ewm = down_series.ewm(com = lookback - 1, adjust = False).mean()
rs = up_ewm/down_ewm
rsi = 100 - (100 / (1 + rs))
rsi_df = pd.DataFrame(rsi).rename(columns = {0:'rsi'}).set_index(close.index)
rsi_df = rsi_df.dropna()
return rsi_df[3:]
aapl['rsi_14'] = get_rsi(aapl['close'], 14)
aapl = aapl.dropna()
Code Explanation: Firstly, we are defining a function named ‘get_rsi’ that takes the closing price of a stock (‘close’) and the lookback period (‘lookback’) as parameters. Inside the function, we are
first calculating the returns of the stock using the ‘diff’ function provided by the Pandas package and stored it in the ‘ret’ variable. This function basically subtracts the current value from the
previous value. Next, we are passing a for-loop on the ‘ret’ variable to distinguish gains from losses and append those values to the concerning variable (‘up’ or ‘down’).
Then, we are calculating the Exponential Moving Averages for both the ‘up’ and ‘down’ using the ‘ewm’ function provided by the Pandas package and storing them in the ‘up_ewm’ and ‘down_ewm’ variables
respectively. Using these calculated EMAs, we are determining the Relative Strength by following the formula we discussed before and storing it into the ‘rs’ variable.
By making use of the calculated Relative Strength values, we are calculating the RSI values by following its formula. After doing some data processing and manipulations, we are returning the
calculated Relative Strength Index values in the form of a Pandas dataframe. Finally, we are calling the created function to store the RSI values of Apple with 14 as the lookback period.
Step-6: Creating the Trading Strategy:
In this step, we are going to implement the discussed Bollinger Bands, Keltner Channel, and Relative Strength Index trading strategy in Python.
Python Implementation:
def bb_kc_rsi_strategy(prices, upper_bb, lower_bb, kc_upper, kc_lower, rsi):
buy_price = []
sell_price = []
bb_kc_rsi_signal = []
signal = 0
for i in range(len(prices)):
if lower_bb[i] < kc_lower[i] and upper_bb[i] > kc_upper[i] and rsi[i] < 30:
if signal != 1:
signal = 1
elif lower_bb[i] < kc_lower[i] and upper_bb[i] > kc_upper[i] and rsi[i] > 70:
if signal != -1:
signal = -1
return buy_price, sell_price, bb_kc_rsi_signal
buy_price, sell_price, bb_kc_rsi_signal = bb_kc_rsi_strategy(aapl['close'], aapl['upper_bb'], aapl['lower_bb'], aapl['kc_upper'], aapl['kc_lower'], aapl['rsi_14'])
Code Explanation: First, we are defining a function named ‘bb_kc_rsi_strategy’ which takes the stock prices (‘prices’), Upper Keltner Channel (‘kc_upper’), Lower Keltner Channel (‘kc_lower’), Upper
Bollinger band (‘upper_bb’), Lower Bollinger band (‘lower_bb’), and the Relative Strength Index readings (‘rsi’) as parameters.
Inside the function, we are creating three empty lists (buy_price, sell_price, and bb_kc_rsi_signal) in which the values will be appended while creating the trading strategy.
After that, we are implementing the trading strategy through a for-loop. Inside the for-loop, we are passing certain conditions, and if the conditions are satisfied, the respective values will be
appended to the empty lists. If the condition to buy the stock gets satisfied, the buying price will be appended to the ‘buy_price’ list, and the signal value will be appended as 1 representing
buying the stock. Similarly, if the condition to sell the stock gets satisfied, the selling price will be appended to the ‘sell_price’ list, and the signal value will be appended as -1 representing
to sell the stock. Finally, we are returning the lists appended with values. Then, we are calling the created function and stored the values in their respective variables.
Step-7: Creating our Position
In this step, we are going to create a list that indicates 1 if we hold the stock or 0 if we don’t own or hold the stock.
Python Implementation:
position = []
for i in range(len(bb_kc_rsi_signal)):
if bb_kc_rsi_signal[i] > 1:
for i in range(len(aapl['close'])):
if bb_kc_rsi_signal[i] == 1:
position[i] = 1
elif bb_kc_rsi_signal[i] == -1:
position[i] = 0
position[i] = position[i-1]
kc_upper = aapl['kc_upper']
kc_lower = aapl['kc_lower']
upper_bb = aapl['upper_bb']
lower_bb = aapl['lower_bb']
rsi = aapl['rsi_14']
close_price = aapl['close']
bb_kc_rsi_signal = pd.DataFrame(bb_kc_rsi_signal).rename(columns = {0:'bb_kc_rsi_signal'}).set_index(aapl.index)
position = pd.DataFrame(position).rename(columns = {0:'bb_kc_rsi_position'}).set_index(aapl.index)
frames = [close_price, kc_upper, kc_lower, upper_bb, lower_bb, rsi, bb_kc_rsi_signal, position]
strategy = pd.concat(frames, join = 'inner', axis = 1)
Code Explanation: First, we are creating an empty list named ‘position’. We are passing two for-loops, one is to generate values for the ‘position’ list to just match the length of the ‘signal’ list.
The other for-loop is the one we are using to generate actual position values.
Inside the second for-loop, we are iterating over the values of the ‘signal’ list, and the values of the ‘position’ list get appended concerning which condition gets satisfied. The value of the
position remains 1 if we hold the stock or remains 0 if we sold or don’t own the stock. Finally, we are doing some data manipulations to combine all the created lists into one dataframe.
From the output being shown, we can see that in the first four rows, our position in the stock has remained at 1 (since there isn’t any change in the trading signal) but our position suddenly turned
to 0 as we sold the stock when the trading signal represents a buy signal (-1). Our position will remain -1 until some changes in the trading signal occur. Now it’s time to implement some backtesting
Step-8: Backtesting
Before moving on, it is essential to know what backtesting is. Backtesting is the process of seeing how well our trading strategy has performed on given stock data. In our case, we are going to
implement a backtesting process for our trading strategy over the Apple stock data.
Python Implementation:
aapl_ret = pd.DataFrame(np.diff(aapl['close'])).rename(columns = {0:'returns'})
bb_kc_rsi_strategy_ret = []
for i in range(len(aapl_ret)):
returns = aapl_ret['returns'][i]*strategy['bb_kc_rsi_position'][i]
bb_kc_rsi_strategy_ret_df = pd.DataFrame(bb_kc_rsi_strategy_ret).rename(columns = {0:'bb_kc_rsi_returns'})
investment_value = 100000
bb_kc_rsi_investment_ret = []
for i in range(len(bb_kc_rsi_strategy_ret_df['bb_kc_rsi_returns'])):
number_of_stocks = floor(investment_value/aapl['close'][i])
returns = number_of_stocks*bb_kc_rsi_strategy_ret_df['bb_kc_rsi_returns'][i]
bb_kc_rsi_investment_ret_df = pd.DataFrame(bb_kc_rsi_investment_ret).rename(columns = {0:'investment_returns'})
total_investment_ret = round(sum(bb_kc_rsi_investment_ret_df['investment_returns']), 2)
profit_percentage = floor((total_investment_ret/investment_value)*100)
print(cl('Profit gained from the BB KC RSI strategy by investing $100k in AAPL : {}'.format(total_investment_ret), attrs = ['bold']))
print(cl('Profit percentage of the BB KC RSI strategy : {}%'.format(profit_percentage), attrs = ['bold']))
Profit gained from the BB KC RSI strategy by investing $100k in AAPL : 165737.51
Profit percentage of the BB KC RSI strategy : 165%
Code Explanation: First, we are calculating the returns of the Apple stock using the ‘diff’ function provided by the NumPy package and we have stored it as a dataframe in the ‘aapl_ret’ variable.
Next, we are passing a for-loop to iterate over the values of the ‘aapl_ret’ variable to calculate the returns we gained from our RVI trading strategy, and these returns values are appended to the
‘bb_kc_rsi_strategy_ret’ list. Next, we are converting the ‘bb_kc_rsi_strategy_ret’ list into a dataframe and storing it into the ‘bb_kc_rsi_strategy_ret_df’ variable.
Next comes the backtesting process. We are going to backtest our strategy by investing a hundred thousand USD into our trading strategy. So first, we are storing the amount of investment into the
‘investment_value’ variable. After that, we are calculating the number of Apple stocks we can buy using the investment amount. You can notice that I’ve used the ‘floor’ function provided by the Math
package because, while dividing the investment amount by the closing price of Apple stock, it spits out an output with decimal numbers. The number of stocks should be an integer but not a decimal
number. Using the ‘floor’ function, we can cut out the decimals. Remember that the ‘floor’ function is way more complex than the ‘round’ function. Then, we are passing a for-loop to find the
investment returns followed by some data manipulation tasks.
Finally, we are printing the total return we got by investing a hundred thousand into our trading strategy and it is revealed that we have made an approximate profit of 165K USD in around
thirteen-and-a-half years with a profit percentage of 165%. That’s great! Now, let’s compare our returns with SPY ETF (an ETF designed to track the S&P 500 stock market index) returns.
Step-8: SPY ETF Comparison
This step is optional but it is highly recommended as we can get an idea of how well our trading strategy performs against a benchmark (SPY ETF). In this step, we will extract the SPY ETF data using
the ‘get_historical_data’ function we created and compare the returns we get from the SPY ETF with our trading strategy returns on Apple.
Python Implementation:
def get_benchmark(start_date, investment_value):
spy = get_historical_data('SPY', start_date)['close']
benchmark = pd.DataFrame(np.diff(spy)).rename(columns = {0:'benchmark_returns'})
investment_value = investment_value
benchmark_investment_ret = []
for i in range(len(benchmark['benchmark_returns'])):
number_of_stocks = floor(investment_value/spy[i])
returns = number_of_stocks*benchmark['benchmark_returns'][i]
benchmark_investment_ret_df = pd.DataFrame(benchmark_investment_ret).rename(columns = {0:'investment_returns'})
return benchmark_investment_ret_df
benchmark = get_benchmark('2010-01-01', 100000)
investment_value = 100000
total_benchmark_investment_ret = round(sum(benchmark['investment_returns']), 2)
benchmark_profit_percentage = floor((total_benchmark_investment_ret/investment_value)*100)
print(cl('Benchmark profit by investing $100k : {}'.format(total_benchmark_investment_ret), attrs = ['bold']))
print(cl('Benchmark Profit percentage : {}%'.format(benchmark_profit_percentage), attrs = ['bold']))
print(cl('BB KC RSI Strategy profit is {}% higher than the Benchmark Profit'.format(profit_percentage - benchmark_profit_percentage), attrs = ['bold']))
Benchmark profit by investing $100k : 159541.51
Benchmark Profit percentage : 159%
BB KC RSI Strategy profit is 6% higher than the Benchmark Profit
Code Explanation: The code used in this step is almost similar to the one used in the previous backtesting step but, instead of investing in Apple, we are investing in SPY ETF by not implementing any
trading strategies. From the output, we can see that our trading strategy has outperformed the SPY ETF by 6%. That’s awesome!
Final Thoughts!
After the immense process of crushing both theory and coding parts, we have successfully learned what the three indicators are all about and built a killer trading strategy out of them that manages
to surpass the returns of the SPY ETF in a 13-year timeframe. From the results, it is notable that by consolidating two or more indicators, we can gain better returns out of our investment than we
could with a single indicator trading strategy. Also, we can erase non-authentic signals from our strategy as much as possible.
With that being said, you’ve reached the end of the article. Hope you learned something new and useful from this article!
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What is a Decimal? (Review Video & Practice Questions)
Hi, and welcome to this review of decimals! Before we dive in, we will need to have a solid understanding of place value.
Place Value
Decimals and place value go hand in hand, so it can be tough to make sense of one without having some background experience with the other. Take a look at this place value chart.
With this example of 3,528.74, we can see that each movement to the right of the decimal point drops us down by a factor of 10. We move from tenths, to hundredths, to thousandths. On the other hand,
as we move to the left of the decimal point, we increase by a factor of 10 each time. We move from tens to hundreds to thousands. This base 10 number system is not the only number system in use
today, but it is very common and widely used around the world.
Let’s look at another example. This time, let’s use the number 136.289. This number as a value is read as “one hundred thirty-six, and two hundred eighty-nine thousandths.” Let’s break this value up
into its different parts in the terms of place value. In this example, we look at each digit and then take note of where it is in relation to the decimal point to determine the total value of the
The digit 2 is located in the tenths position, so it does not represent the value “2” but “2 tenths.” The digit 1 is located in the hundreds position, so it does not represent a value of “1” but a
value of “1 hundred.” When you break apart a decimal value into its different parts, you are essentially thinking about the number in its expanded form. You are looking at each digit, and identifying
how much this digit represents based on its location. The location of each digit reveals its value, or “place value.” This is similar to the process of writing a number in expanded form.
Our number 136.289 would be \(100 + 30 + 6 + 0.2 + 0.08 + 0.009\), in expanded form. This can sometimes be helpful in order to see each digit as an independent value.
The place value as a system allows us to write and express numbers with extreme accuracy. For example, instead of simply rounding the amount of fuel needed for an important mission to the moon as
approximately 950 gallons, we could instead say with complete accuracy and confidence that the amount needed for a safe trip would be exactly 950.458 gallons.
Another helpful way to visualize place value is by using money. We know that one dollar can be represented by 10 dimes. We also know that one dollar can be represented by 100 pennies.
This is similar to saying that one whole can be represented by 10 tenths or by 100 hundredths.
$1 = 10 dimes
One dime is one tenth of a dollar (\(\frac{1}{10}\))
$1 = 100 pennies
One penny is one hundredth of a dollar (\(\frac{1}{100}\))
With this understanding of place value, we are ready to dive into the topic of decimals.
Operations with decimals is a topic that most of us use in our daily lives, especially when dealing with money.
For example, 5 dollars, 2 dimes, and 5 pennies is represented by the decimal number $5.25. If we remember our place values, the 5 in the ones place represents 5 dollars, the 2 in the tenths position
represents 20 cents (or 2 tenths), and the 5 in the hundredths position represents 5 cents (or 5 hundredths).
In the real world, it is often convenient to work with fractions and mixed numbers. Other times, it makes more sense to work with decimals. If we want to talk about measuring amounts in a recipe, it
is generally helpful to use fractions. On the other hand, when we are dealing with something like temperatures, it is often more efficient to use decimals.
Using fractions, mixed numbers, or decimals does not change the amount, it simply changes the form. For example, if a recipe calls for 4/5 cups of flour, this is the same amount as the decimal 0.8.
These values are equivalent.
All decimal values can be written as fractions, mixed numbers included. For example, when writing the decimal value 5.25 as a mixed number, we simply need to look at each digit and then note its
With the value 5.25, we would represent the first 5 in the ones position as simply 5. Then, we look at the values to the right of the decimal. We see .25. This reaches out to the hundredths place, so
it is representing twenty-five hundredths, or as a fraction \(\frac{25}{100}\). So, 5.25 as a mixed number would be \(5\frac{25}{100}\). We can simplify the fraction, making it \(5\frac{1}{4}\).
As you can see, using decimals is something we do on a regular basis, and being able to convert between decimals and fractions can ensure that our number is as accurate as possible.
I hope this review was helpful! Thanks for watching, and happy studying!
Frequently Asked Questions
What is a decimal?
A decimal number is any number that uses a decimal point to show the part of the number that is less than one.
Ex. 16.275
How do you round decimals?
Round decimals the same way you round non-decimal numbers. Check the number in the place value one below the one you are rounding to. If it is greater than or equal to 5, round up. If it is less than
5, round down. Then, only write the numbers until you reach the place you are rounding to.
Ex. Round 8.715 to the nearest whole number.
Since 7 is greater than 5, round up. 8.715 rounds to 9.
Ex. Round 63.271 to the nearest hundredth.
Since 1 is less than 5, round down. 63.271 rounds to 63.27.
Can an integer be a decimal?
By the definition of an integer, no, an integer cannot have a fractional part. However, an integer can be turned into a decimal by adding a decimal point and zeroes after the number.
What is a repeating decimal?
A repeating decimal is a decimal that has a number that repeats forever. It is usually represented by placing a bar over the repeated number.
Ex. \(0.\overline{3}=0.33333…\)or \(0.\overline{16}=0.1616… \)
Are decimals rational numbers?
Yes, most decimal numbers are rational numbers. A rational number is any number that can be turned into a fraction, so any decimal number that ends or repeats. One notable decimal number that is not
rational is pi (3.14…) because it is a decimal that never ends or repeats.
Decimal Practice Questions
Question #1:
Which shows the number “three hundred forty three, and twenty five hundredths”?
The correct answer is C: 343.25.
The first part of the statement “three hundred forty three” is describing the part of the number before the decimal. The word “and” indicates the decimal and the word “hundredths” informs us the
number has two digits after the decimal.
Question #2:
Which shows the number 429.317 in expanded form?
The correct answer is D: \(400+20+9+0.3+0.01+0.007\).
With the place value chart in mind, for the number 429.317, the 4 is in the hundreds place, the 2 in the tens place, the 9 in the ones place, the 3 in the tenths place, 1 in the hundredths place, and
7 in the thousandths place.
Question #3:
What is 836.1792 rounded to the nearest hundredth?
The correct answer is C: 836.18.
The hundredths place is two digits after the decimal. The same rule for rounding whole numbers is used when rounding a decimal; therefore, since the number in the thousandths place is 9, the 7 is
rounded up to 8.
Question #4:
What is \(3\frac{6}{100}\) in decimal form?
The correct answer is C: 3.06.
When converting a mixed number to a decimal, the 3 in the ones place is converted to 3 and the fraction \(\frac{6}{100}\), or six hundredths, is converted to 0.06 when you divide the numerator by the
Question #5:
What is 2.75 in fraction form?
The correct answer is A: \(2\frac{3}{4}\).
When converting a decimal to a fraction, the 2 in the ones place remains as a 2. Since the .75 is to the hundredths place, we can convert it to \(\frac{75}{100}\), so the mixed number is \(2\frac{75}
{100}\), or \(2\frac{3}{4}\) when the fraction is simplified.
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Python Post Init: Initialize Your Data Classes Like a ProPython Post Init: Initialize Your Data Classes Like a Pro
Python Post Init: Initialize Your Data Classes Like a Pro Read it later
In OOP, classes are the fundamental building blocks that encapsulate data and behavior. When creating a class in Python, one important aspect is how to initialize it. In this blog, we will explore
how to use Python Post Init to initialize your Python data classes like a pro. We will also provide examples of how to implement post-init processing in Python dataclass.
Before directly jumping on what is post init in Python, let’s make ourselves comfortable with init in Python.
What is __init__ in Python?
In Python, __init__ is a special method that is automatically called when an instance of a class is created. It is used to initialize the attributes of the class.
__init__ stands for “initialize” and it is one of the most important methods in a Python class. The basic syntax for defining init is:
class MyClass:
def __init__(self, arg1, arg2):
self.arg1 = arg1
self.arg2 = arg2
The __init__ method initializes the attributes of the class, arg1 and arg2, with the values of the arguments.
Limitations of __init__
While __init__ is a useful method for initializing the attributes of a class, it has some limitations.
• Cannot set default values for class attributes that are not defined in the constructor signature.
• Cannot perform additional initialization logic that depends on the values of other attributes.
• Cannot modify the values of attributes after they have been initialized.
• Cannot return a value or raise an exception, as it is not a regular method.
What is Post Init Processing in Python?
Python Post Init is a special method in Python’s dataclasses module that automatically calls this function immediately after initializing the object.
When creating an object and setting its initial values, post init processing involves performing additional operations on the object.
Post init processing is beneficial when there is a need to perform further actions on the object after its creation.
Benefits of using Python Post Init
Using Python Post Init can bring several benefits, such as:
1. Improved readability and maintainability of code.
2. Simplified initialization logic, especially for classes with many attributes.
3. A cleaner separation of initialization logic from other class methods.
4. Improved flexibility and extensibility of the class.
How to use Python Post Init?
Python data classes provide a way to define simple classes that are used primarily for storing data. Data classes can be defined using the @dataclass decorator.
The Python decorator automatically generates several methods for the class, including an __init__() method. The __init__() method is called when an object of the class is created.
To implement post init in a Python data class, you can define a method that performs the additional processing and call it from the __post_init__() method.
Python dataclass post init arguments
The __post_init__ function can also be used with parameters.
For example, consider the following Python data class:
from dataclasses import dataclass
class Rectangle:
width: int
height: int
area: int = 0
def __post_init__(self, area_function):
if self.area == 0:
self.area = area_function(self.width, self.height)
In this dataclass, we have defined three attributes: width, height, and area. We have also defined a __post_init__ function that takes an additional parameter area_function.
This function checks if the area attribute is zero, and if it is, it calculates the area using the area_function and sets the area attribute accordingly.
def calculate_area(width, height):
return width * height
rectangle = Rectangle(width=10, height=20, area_function=calculate_area)
print(rectangle.area) # Output: 200
As we can see, the __post_init__ function has calculated the area of the rectangle using the calculate_area function and set the area attribute to 200.
Data Validation Using Python Post Init
One common use case for post-init processing involves performing data validation on the object after initializing it.
In this example, we will create a data class that represents a rectangle. We will define a __post_init__() method that checks whether the rectangle’s width and height are positive numbers.
from dataclasses import dataclass
class Rectangle:
width: float
height: float
def __post_init__(self):
if self.width <= 0 or self.height <= 0:
raise ValueError("Width and height must be positive numbers")
How to effectively use Python Post Init?
Here are some tips for using Python Post Init effectively:
1. Use it to set default values for class attributes that cannot be set in the __init__ method.
2. Use it to perform additional initialization logic that could not be implemented in the __init__ method.
3. Keep the __post_init__ method simple and focused on initialization logic only.
4. Document the purpose of the __post_init__ method to make it clear to other developers.
Common mistakes to avoid
Here are some common mistakes to avoid when using Python Post Init:
1. Overusing it and cluttering the code with too much initialization logic.
2. Not understanding the difference between __init__ and __post_init__ and using them interchangeably.
3. Not testing the class thoroughly after implementing the __post_init__ method.
4. Not documenting the class and its methods properly.
Python __init__ vs __post_init__ Method
Features __init__ Method __post_init__ Method
Purpose Initializes the class attributes with values passed as arguments to the Performs additional initialization logic and sets default values for attributes that cannot be set in
constructor the constructor
Executes Before the class object is fully constructed After the class object is fully constructed
Parameters self and any other parameters specified in the constructor signature self only
Return Value None None
Exception Handling Can raise exceptions Cannot raise exceptions
Attribute Can modify the values of attributes in the constructor signature Cannot modify the values of attributes initialized in the constructor signature
Access to Can access all attributes in the constructor signature Can access all attributes in the class object, even those not initialized in the constructor signature
Difference between Python __init__ and __post_init__ methods
Wrap Up
In conclusion, the __post_init__ function is a powerful feature of Python data classes that allows us to perform post-init processing on objects.
By using it effectively, we can create classes that are flexible, extensible, and easy to use. However, it is important to use it wisely and avoid common mistakes to ensure that our code is reliable
and robust.
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Describing The Exterior Angles Of A Triangle Worksheet Answer Key - Angleworksheets.com
Exterior Angles Of A Triangle Worksheet Answers – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral
triangles. You can use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle … Read more
Triangle Exterior Angles Worksheet Answer Key
Triangle Exterior Angles Worksheet Answer Key – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral
triangles. You can use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle Worksheet … Read more
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Median Revisited
To understand a quartile, let us revisit median. To compute the median, we cut off the data into two groups with equal number of points. Thus the middle value that separates these groups is the
median. In a similar fashion, if we divide the data into 4 equal groups now instead of 2, the first differentiating point is the first quartile, the second differentiating point is the second
quartile which is the same as the median and the third such differentiating point is the third quartile.
To further see what quartiles do, the first quartile is at the 25th percentile. This means that 25% of the data is smaller than the first quartile and 75% of the data is larger than this. Similarly,
in case of the third quartile, 25% of the data is larger than it while 75% of it is smaller. For the second quartile, which is nothing but the median, 50% or half of the data is smaller while half of
the data is larger than this value.
Interpreting Quartiles
As you know, the median is a measure of the central tendency of the data but says nothing about how the data is distributed in the two arms on either side of the median. Quartiles help us measure
Thus if the first quartile is far away from the median while the third quartile is closer to it, it means that the data points that are smaller than the median are spread far apart while the data
points that are greater than the median are closely packed together.
An Alternative View
Another way of understanding quartiles is by thinking of those as medians of either of the two sets of data points differentiated by the median. In this case, the first quartile is the median of the
data that is smaller than the full median while the third quartile is the median of the data that is larger than the full median. Here full median is used in the context of the median of the entire
set of data.
It should be noted that a quartile is not limited to discrete variables but also applies equally well to continuous variables. In this case, you will need to know the data distribution to figure out
the quartiles. If the distribution is symmetric, like normal distribution, then the first and third quartiles are equidistant from the median in either direction.
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Determination of Transformation Ratio of a Single Phase Transformer -
Determination of Transformation Ratio of a Single Phase Transformer
Experiment No. : 5
Experiment Name :
Determination of Transformation Ratio of a Single Phase Transformer.
To determine transformation ratio of a single phase tranformer.
Transformer is a static electrical a.c. powered device, which transfer the electrical power from one circuit to another circuit without changing its frequency.
The level of voltage may be stepped up or down depending upon the number of turns on low voltage side winding and high voltage side winding. If we connect the ac power supply with low voltage side
winding, then the winding is considered as Primary Side. And the remaining winding, i.e. high voltage winding, from which we take power output, is considered as Secondary Side.
And if we connect the ac power supply with high voltage side winding, then the winding is considered as Primary Side. And the remaining winding, i.e. low voltage winding, from which we take power
output, is considered as Secondary Side.
So, there is a ratio, depending on which, the level of voltage is stepped up or down, that is called, Transformation Ratio.
The transformation ratio is defined as the ratio of secondary voltage to primary voltage. The transformation ratio is indicated as K.
If the value of K is-
• greater than 1. Then the transformer is step up Transformer
• less than 1. Then the transformer is step down Transformer
• equal to 1. Then the transformer is isolator Transformer
As, K is the ratio of two same quantity, so it is unitless.
Circuit Diagram:
Observation Table:
Sl.No. Primary Voltage (V[1]) volt Secondary Voltage (V[2]) volt Transformation Ratio Remarks
(K= V[2] / V[1] )
1. 230 124.7 0.54 Step down transformer
2. 115 212.9 1.85 Step up transformer
Apparatus Used:
Sl. No. Name of the Apparatus Quantity Specification Maker’s Name
1. Single phase Transformer 1 230/115 V, Shell type, Air cooled
2. Digital Multimeter 1 0-10 MΩ, 0-750 V, 0-10 A Akademika
The transformation ratio is calculated by measuring the primary voltage and secondary voltage.
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Summer Undergraduate Research in Number Theory at Amherst 2018
Description: Amherst College undergraduate students are invited to apply to spend 7 weeks of summer 2018 working together as a small group under the supervision of Prof. Folsom on an original
research project in pure mathematics, in the area of number theory. See topic section below for more details. Approximately three students will participate.
Dates: June 11 - July 27, 2018. Participants are required to be in residence for the entire duration of the program. Minor exceptions may (or may not) be permitted; these should be discussed in
Funding: Participants will receive a stipend of $440/week for 7 weeks. Participants will also receive (at not cost to them) on-campus housing, subject to availability after formally applying to
Amherst College Housing. Included in the summer 2018 housing contract is a partial meal plan (number of meals TBD, at no cost to the student for the included meals). This funding is provided by NSF
CAREER Grant DMS-1449679.
Prerequisites: By the start of summer, applicants should have taken, and demonstrated strong ability in, at least two of Math 350 (Groups, Rings and Fields), Math 355 (Analysis), Math 345 (Complex
Variables), or Math 310 (Intro. to the Theory of Partitions). Math 460 (Analytic Number Theory), Math 281 (Combinatorics), or Math 250 (Number Theory) may also be useful, but are not required.
Participants will spend a portion of the summer reading and learning background material together.
Eligibility: The program is open to any full-time Amherst College undergraduate student. Current seniors who will graduate in Spring 2018 are not eligible, however. The program is a full-time
commitment; participants may not be involved in any other summer program, classes, jobs, research opportunities, etc., even if part time.
Application Form: A completed application form is due by email by February 21, 2018. The application form, with instructions, is available at this link . Applicants will have until early March to
accept or decline an offer.
Topic: Modular forms are central objects of study in number theory. Loosely speaking, they are complex-valued functions, which additionally obey certain symmetry properties with respect to a group
action. Here’s one example of a modular form:
m(q) := q^(-1/24)(1 + q + 2q^2 + 3q^3 + 5q^4 + 7q^5 + 11q^6 + 15q^7 + 22q^8 + . . . . . )
While interesting in their own right, modular forms are also often studied due to intrinsic combinatorial or algebraic information that they may possess. For example, consider the integer partitions
of a positive integer n, the different ways to write n as a non-increasing sum of positive integers (i.e. the partitions of the number n=4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1). It is well known that
integer partitions, which a priori are combinatorial in nature, are intimately connected to modular forms (in particular, to the modular form m(q) shown above). Moreover, the special values of
modular forms are known to play important roles (i.e. the values of m(q) and other modular forms can be of great algebraic interest when q is appropriately chosen). So called "q-series" (infinite
power series in the variable q) and their analytic properties are also studied independent of whether or not they are modular forms. Participants will spend a portion of the beginning of the summer
reading and learning background material on these topics, with the goal of later exploring these types of topics in an original research project.
Here are related papers you may wish to take a look at:
(1) An expository paper by Prof. Folsom: A. Folsom, What is...a mock modular form? , Notices of the Amer. Math. Soc. 57 issue 11 (2010), 1441-1443.
(2) Results of Summer 2015 research group: A. Folsom, C. Ki `17, Y.N. Truong Vu `17, B. Yang `18, Strange combinatorial quantum modular forms, Journal of Number Theory, 170 (2017), 315-346 .
(3) Results of the Summer 2017 research group: M. Barnett `18, A. Folsom, O. Ukogu `18, W. Wesley `18, H. Xu `18, Quantum Jacobi forms and balanced unimodal sequences, Journal of Number Theory 186
(2018), 16-34.
(4) Results of the SUMRY 2014 research group: A. Folsom, Y. Homma, J. Ryu, and B. Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 iss. 5 (2016), 1482-1506.
Questions? Feel free to email or see Prof. Folsom with any questions about the program or application.
Student participants: Greg Carroll ’20, James Corbett ’19, Ellie Thieu ’19.
Results/Paper: G. Carroll ’20, J. Corbett ’19, A. Folsom, and E. Thieu ’19. Universal mock theta functions as quantum Jacobi forms, Research in the Mathematical Sciences, 6:6 (2019), 15pp.
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Lecture 5: Methods for self-referential lists
Lecture 5: Methods for self-referential lists
Designing classes to represent lists. Methods on lists, including basic recursive methods, and sorting
Lecture outline
• Representing lists
• Basic methods
• Sorting
5.1 Representing lists
The following class diagram defines a class hierarchy that represents a list of books in a bookstore:
| ILoBook |<----------------------+
+--------------------------------+ |
+--------------------------------+ |
| int count() | |
| double salePrice(int discount) | |
| ILoBook allBefore(int y) | |
| ILoBook sortByPrice() | |
+--------------------------------+ |
| |
/ \ |
--- |
| |
----------------------------- |
| | |
+--------------------------------+ +--------------------------------+ |
| MtLoBook | | ConsLoBook | |
+--------------------------------+ +--------------------------------+ |
+--------------------------------+ +-| Book first | |
| int count() | | | ILoBook rest |-+
| double salePrice(int discount) | | +--------------------------------+
| ILoBook allBefore(int y) | | | int count() |
| ILoBook sortByPrice() | | | double salePrice(int discount) |
+--------------------------------+ | | ILoBook allBefore(int y) |
| | ILoBook sortByPrice() |
| +--------------------------------+
| Book |
| String title |
| String author |
| int year |
| double price |
| double salePrice(int discount) |
Let’s make some examples
//Books Book htdp = new Book("HtDP", "MF", 2001, 60);
Book lpp = new Book("LPP", "STX", 1942, 25);
Book ll = new Book("LL", "FF", 1986, 10);
// lists of Books ILoBook mtlist = new MtLoBook();
ILoBook lista = new ConsLoBook(this.lpp, this.mtlist);
ILoBook listb = new ConsLoBook(this.htdp, this.mtlist);
ILoBook listc = new ConsLoBook(this.lpp,
new ConsLoBook(this.ll, this.listb));
ILoBook listd = new ConsLoBook(this.ll,
new ConsLoBook(this.lpp,
new ConsLoBook(this.htdp, this.mtlist)));
5.2 Basic list computations
Given this preceding class diagram, we would like to design methods to answer the following questions
• Count how many books we have in this list of books
• Compute the total sale price of all books in this list of books, at a given discount rate
• Given a date (year) and this list of books, produce a list of all books in this list that were published before the given year
• Produce a list of the same books as this list, but sorted by their price
Each of the four questions concerns a list of books, and so we start by designing the appropriate method headers and purpose statements in the interface ILoBook:
// In ILoBook // ------- // count the books in this list int count();
// produce a list of all books published before the given date // from this list of books ILoBook allBefore(int year);
// calculate the total sale price of all books in this list for a given discount double salePrice(int discount);
// produce a list of all books in this list, sorted by their price ILoBook sortByPrice();
We now have to define these methods in both the class MtLoBook and in the class ConsLoBook. (You may find it helpful to recall similar functions in DrRacket. Remember from last lecture the pattern
given to us by virtue of Dynamic dispatch: we take each clause of the cond that checked for a particular variant and “move” the right-hand sides of those clauses into the methods defined in the
corresponding class, then eliminate the cond altogether.)
The design recipe asks us to make examples. For clarity, we write them in an abbreviated manner, just showing the actual computation and the expected outcome:
// Examples for the class MtLoBook // ---------------------------- mtlist.count() => 0
mtlist.salePrice(0) => 0
mtlist.allBefore() => mtlist
and our methods become:
// In MtLoBook: // ---------
// count the books in this list public int count() { return 0; }
// produce a list of all books published before the given date // from this empty list of books public ILoBook allBefore(int year) { return this; }
// calculate the total sale price of all books in this list for a given discount public double salePrice(int discount) { return 0; }
Notice that the values produced by these methods are the base case values we have been in DrRacket for the empty lists. The count for an empty list is zero; the salePrice of no Books is zero as well;
and starting with an empty list there are no Books at all, let alone any before a given year.
Note: We will return to the sort method later.
Of course, there will be more work to do in the ConsLoBook class. First, examples!
// Examples // -------- lista.count() => 1
listc.count() => 3
lista.salePrice(0) => 25
listd.salePrice(0) => 95
lista.allBefore(2000) => lista
listb.allBefore(2000) => mtlist
listc.allBefore(2000) => new ConsLoBook(lpp, new ConsLoBook(ll,mtlist))
The design recipe asks us now to derive the template. A template serves as a starting point for any method inside ConsLoBook:
/* TEMPLATE: --------- Fields: ... this.first ... -- Book ... this.rest ... -- ILoBook Methods: ... this.count() ... -- int ... this.salePrice(int discount) ... -- double ... this.allBefore(int
year) ... -- ILoBook Methods for Fields: ... this.rest.count() ... -- int ... this.rest.salePrice(int discount) ... -- double ... this.rest.allBefore(int year) ... -- ILoBook */
count and salePrice
In the template, this.rest.count() produces the count of all books in the rest of this list — and so the method body in the class ConsLoBook becomes:
// count the books in this list public int count() { return 1 + this.rest.count(); }
In the template, this.rest.salePrice(discount) produces the total sale price of all books in the rest of this list for the given discount — and so the method body in the class ConsLoBook just adds to
this value the price of the first book in the list:
// calculate the total sale price of all books in this list for the given discout public double salePrice(int discount) {
return this.first.salePrice(discount) + this.rest.salePrice(discount);
Did you notice how similar this method body is to the one above for
? In Fundies 1, we had a terser way of expressing this sort of computation. What kind of operation on lists are we computing here? We will see in
Lecture 13: Abstracting over behavior
how to improve this code.
In the template, this.rest.allBefore(year) produces a list of all books in the rest of this list published before the given date. The only work that remains is to decide whether the first book of
this list belongs in the output list, and either add it to the result or not. If only we could determine whether that first Book was published before the given year! (Look carefully at the template:
we cannot access this.first.year, because we do not have access to fields of fields.) So we add a method to our wish list, and we will delegate the job of deciding this question to the Book class
itself. The method body in the class ConsLoBook becomes:
// produce a list of all books published before the given date // from this empty list of books ILoBook allBefore(int year) {
if (this.first.publishedBefore(year)) {
return new ConsLoBook(this.first, this.rest.allBefore(year));
else {
return this.rest.allBefore(year);
(This method introduces a new piece of syntax: if statements. An if-statement always follows the same basic template:
if (some condition) {
//...statements to execute if condition was true... }
else {
//...statements to execute if condition was false... }
where only one of the branches of the if executes its statements. Note that unlike DrRacket, an if in Java is not an expression, and does not produce a value. In the code for allBefore above, the if
statement itself does not return a value; the return statements inside it do.)
We’re not quite done; we have a method remaining on our wish list, so we must add to the class Book the method
// was this book published before the given year? boolean publishedBefore(int year) {
return this.year < year;
Flesh out the rest of the design of this method, adding examples and tests.
Of course, for all of these methods, we end the design process by making sure all tests run. The actual test methods will be:
// tests for the method count boolean testCount(Tester t) {
t.checkExpect(this.mtlist.count(), 0) &&
t.checkExpect(this.lista.count(), 1) &&
t.checkExpect(this.listd.count(), 3);
// tests for the method salePrice boolean testSalePrice(Tester t) {
// no discount -- full price t.checkInexact(this.mtlist.salePrice(0), 0.0, 0.001) &&
t.checkInexact(this.lista.salePrice(0), 10.0, 0.001) &&
t.checkInexact(this.listc.salePrice(0), 95.0, 0.001) &&
t.checkInexact(this.listd.salePrice(0), 95.0, 0.001) &&
// 50% off sale -- half price t.checkInexact(this.mtlist.salePrice(50), 0.0, 0.001) &&
t.checkInexact(this.lista.salePrice(50), 5.0, 0.001) &&
t.checkInexact(this.listc.salePrice(50), 47.5, 0.001) &&
t.checkInexact(this.listd.salePrice(50), 47.5, 0.001);
// tests for the method allBefore boolean testAllBefore(Tester t) {
t.checkExpect(this.mtlist.allBefore(2001), this.mtlist) &&
t.checkExpect(this.lista.allBefore(2001), this.lista) &&
t.checkExpect(this.listb.allBefore(2001), this.mtlist) &&
new ConsLoBook(this.lpp, new ConsLoBook(this.ll, this.mtlist))) &&
new ConsLoBook(this.ll, new ConsLoBook(this.lpp, this.mtlist)));
5.3 Sorting
The last method to design was defined in the interface ILoBook as:
// produce a list of all books in this list, sorted by their price ILoBook sortByPrice();
An empty list is sorted already, so in the class MtLoBook the method becomes:
// produce a list of all books in this list, sorted by their price public ILoBook sortByPrice() {
return this;
We do not need to create a new empty list, this one works perfectly well.
We need examples for the more complex cases. We recall our sample data:
//Books Book htdp = new Book("HtDP", "MF", 2001, 60);
Book lpp = new Book("LPP", "STX", 1942, 25);
Book ll = new Book("LL", "FF", 1986, 10);
// lists of Books ILoBook mtlist = new MtLoBook();
ILoBook lista = new ConsLoBook(this.lpp, this.mtlist);
ILoBook listb = new ConsLoBook(this.htdp, this.mtlist);
ILoBook listc = new ConsLoBook(this.lpp,
new ConsLoBook(this.ll, this.listb));
ILoBook listd = new ConsLoBook(this.ll,
new ConsLoBook(this.lpp,
new ConsLoBook(this.htdp, this.mtlist)));
ILoBook listdUnsorted =
new ConsLoBook(this.lpp,
new ConsLoBook(this.htdp,
new ConsLoBook(this.ll, this.mtlist)));
and our tests will be:
// test the method sort for the lists of books boolean testSort(Tester t) {
t.checkExpect(this.listc.sort(), this.listd) &&
t.checkExpect(this.listdUnsorted.sort(), this.listd);
Next we look at the template that is relevant for this question:
/* TEMPLATE: --------- Fields: ... this.first ... -- Book ... this.rest ... -- ILoBook Methods: ... this.sort() ... -- ILoBook Methods for Fields: ... this.rest.sort() ... -- ILoBook */
Reading the purpose statement for sort carefully, we see that (like allBefore above) this.rest.sort() does almost all the work for us — it produces a sorted rest of this list. What makes this method
more challenging than allBefore is that we aren’t simply prepending to the beginning of that resulting list; we need to insert the first element of the list into its appropriate place in the sorted
rest of the list. This sounds like a job for a helper method, so we add it to our wish list and move on.
When we do get to it, where should this helper method be defined?
Implementing sort for ConsLoBook is now straightforward: we just translate the English sentence above into Java.
// In ConsLoBook // produces a list of the books in this non-empty list, sorted by price public ILoBook sortByPrice() {
return this.rest.sortByPrice() // sort the rest of the list... .insert(this.first); // and insert the first book into that result }
Now we need to finish off the items on our wish list. We need insert to produce a list whose contents are the same as the contents of this (already sorted!) list, but with the given Book inserted
into its proper place. To do this, we’ll certainly need to compare whether one book is cheaper than another, so we’ll add that to our wish list and move on.
Implement insert for ConsLoBook. Pay careful attention to the use of the template to guide your recursive calls.
If we define insert as a method in ConsLoBook...
// in ConsLoBook // insert the given book into this list of books // already sorted by price public ILoBook insert(Book b) {
if (this.first.cheaperThan(b)) {
return new ConsLoBook(this.first, this.rest.insert(b));
else {
return new ConsLoBook(b, this);
...Java complains.
Why? What did we forget? (Hint: if you try this code in Eclipse, where does it indicate there are errors?)
Yes, we haven’t implemented cheaperThan yet. Let’s fix that right now:
// in Book // is the price of this book cheaper than the price of the given book? boolean cheaperThan(Book that) {
return this.price < that.price;
But still we have a problem. We’ve defined insert on ConsLoBook, but in the third line, we write this.rest.insert(b) — and we do not know anything about this.rest except that it’s an ILoBook, and
ILoBook says nothing about an insert method!
Ok let’s add the method to our interface:
// in ILoBook // insert the given book into this list of books // already sorted by price ILoBook insert(Book b);
And now we have fixed our problem here, only to create a new problem elsewhere.
This is another, subtle example of the benefits of writing down our types explicitly. In DrRacket, if we tried to define a function over a union type, and forgot a case, the only way we’d
find out is if a test caught the lapse. Here, Java can immedately warn us that we’ve forgotten something
Now that we’ve promised that all ILoBooks must implement insert, we need to implement it on MtLoBook too. How can we insert a Book into its proper place in an empty list? By building a list with only
the given book in it:
// in MtLoBook // insert the given book into this empty list of books // already sorted by price public ILoBook insert(Book b) {
return new ConsLoBook(b, this);
And now we are finally done!
This sorting order is called “lexicographic”, and is the generalization of sorting alphabetically to account for digits, punctuation, other alphabets, and all the other characters allowed
in strings.
Suppose we wanted to sort the books by title, instead of by price. We cannot use the < operator to compare Strings. Instead, Strings have a method compareTo(String) that returns:
□ -1 if this String is lexicographically before the given String
□ 0 if the strings are lexicographically equal
□ 1 if this String is lexicographically after the given String
Use this method to define a method titleBefore on Books, analogous to cheaperThan, and revise sort and/or insert to use it.
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2 cars travel towards each other from points 500km apart. The 2 cars meet in 4 hrs. What is the ave speed of each car if one travels 15kph faster than the other?
Let's denote the speed of the slower car as "x" km/h. Since the faster car is traveling 15 km/h faster, its speed would be "x + 15" km/h.
The formula for average speed is:
Average speed = Total distance / Total time
In this case, the total distance is 500 km and the total time is 4 hours.
For the slower car:
Average speed = 500 km / 4 hours = 125 km/h
For the faster car:
Average speed = 500 km / 4 hours = 125 km/h
Therefore, the average speed of each car is 125 km/h.
Added 7/17/2023 12:47:13 AM
This answer has been confirmed as correct and helpful.
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Linear Algebra And Its Applications 4th Edition Solutions.rar - bjarnabloggur
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NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 2 Relations and Functions PDF
NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise
NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions includes solutions to all Miscellaneous Exercise problems. The NCERT Solutions for Maths Class 11 Miscellaneous Exercises are
based on the ideas presented in Maths Chapter 2. This activity is crucial for both the CBSE Board examinations and competitive tests.
1. NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise
2. Access NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions
2.1Miscellaneous Exercise
3. Class 11 Maths Chapter 2: Exercises Breakdown
4. CBSE Class 11 Maths Chapter 2 Other Study Materials
5. Chapter-Specific NCERT Solutions for Class 11 Maths
Focus on the different types of relations and functions covered in the chapter. This understanding is important for solving problems effectively and improving your math skills. To do well in exams,
download the CBSE Class 11 Maths Syllabus in PDF format and practice them offline regularly.
FAQs on NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 2 - Relations and Functions
1. What is the key focus of the NCERT Solutions of Miscellaneous Exercise Class 11 Chapter 2?
The NCERT Solutions of Maths Miscellaneous Exercise Class 11 Chapter 2 focuses on applying the concepts of relations and functions. It covers different types of relations, functions, and their
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There are 12 questions in the Miscellaneous Exercise Class 11 Chapter 2. These questions cover a wide range of topics within relations and functions. They include different types of problems to test
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Exams often ask questions that prove the properties of functions. You might need to identify different types of relations. Solving complex problems involving sets and mappings is common. These
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Swap numbers without using third variable in C | Dremendo
Swap numbers without using third variable in C
scanf() - Question 10
In this question, we will see how to input two integers into two variables x and y in C programming using the scanf() function and swap their values without using a third variable. To know more about
scanf() function click on the scanf() function lesson.
Q10) Write a program in C to input two integers into two variables x and y and swap their values without using a third variable.
x = 5
y = 3
After Swap
x = 3
y = 5
#include <stdio.h>
#include <conio.h>
int main()
int x,y;
printf("Enter value of x ");
printf("Enter value of y ");
printf("After Swap\n");
return 0;
Enter value of x 10
Enter value of y 20
After Swap
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Solution Manual for Beginning and Intermediate Algebra, 5th Edition, Julie Miller, Molly O’Neill Nancy Hyde
This is completed downloadable of Solution Manual for Beginning and Intermediate Algebra, 5th Edition, Julie Miller, Molly O’Neill Nancy Hyde
Product Details:
• ISBN-10 : 1259616754
• ISBN-13 : 978-1259616754
• Author: Julie Miller, Molly O’Neill, Nancy Hyde
The Miller/O’Neill/Hyde author team continues to offer an enlightened approach grounded in the fundamentals of classroom experience in Beginning and Intermediate Algebra 5e. The text reflects the
compassion and insight of its experienced author team with features developed to address the specific needs of developmental level students. Throughout the text, the authors communicate to students
the very points their instructors are likely to make during lecture, and this helps to reinforce the concepts and provide instruction that leads students to mastery and success. Also included are
Problem Recognition Exercises, designed to help students recognize which solution strategies are most appropriate for a given exercise. These types of exercises, along with the number of practice
problems and group activities available, permit instructors to choose from a wealth of problems, allowing ample opportunity for students to practice what they learn in lecture to hone their skills.
In this way, the book perfectly complements any learning platform, whether traditional lecture or distance-learning; its instruction is so reflective of what comes from lecture, that students will
feel as comfortable outside of class as they do inside class with their instructor.
Table of Content:
Chapter 1: Set of Real Numbers
1.1 Fractions
1.2 Introduction to Algebra and the Set of Real Numbers
1.3 Exponents, Square Roots, and Order of Operations
1.4 Addition of Real Numbers
1.5 Subtraction of Real Numbers
Problem Recognition Exercises—Addition and Subtraction of Real Numbers
1.6 Multiplication and Division of Real Numbers
Problem Recognition Exercises—Adding, Subtracting, Multiplying and Dividing Real Numbers
1.7 Properties of Real Numbers and Simplifying Expressions
Chapter 2: Linear Equations and Inequalities
2.1 Addition, Subtraction, Multiplication, and Division Properties of Equality
2.2 Solving Linear Equations
2.3 Linear Equations: Clearing Fractions and Decimals
Problem Recognition Exercises—Equations vs.Expressions
2.4 Applications of Linear Equations: Introduction to Problem Solving
2.5 Applications Involving Percents
2.6 Formulas and Applications of Geometry
2.7 Mixture Applications and Uniform Motion
2.8 Linear Inequalities
Chapter 3: Graphing Linear Equations in Two Variables
3.1 Rectangular Coordinate System
3.2 Linear Equations in Two Variables
3.3 Slope of a Line and Rate of Change
3.4 Slope-Intercept Form of a Linear Equation
Problem Recognition Exercises-Linear Equations in Two Variables
3.5 Point-Slope Formula
3.6 Applications of Linear Equations
Chapter 4: Systems of Linear Equations in Two Variables
4.1 Solving Systems of Equations by the Graphing Method
4.2 Solving Systems of Equations by the Substitution Method
4.3 Solving Systems of Equations by the Addition Method
Problem Recognition Exercises: Systems of Equations
4.4 Applications of Linear Equations in Two Variables
4.5 Systems of Linear Equations in Three Variables
4.6 Applications of Systems of Linear Equations in Three Variables
Chapter 5: Polynomials and Properties of Exponents
5.1 Multiplying and Dividing Expressions with Common Bases
5.2 More Properties of Exponents
5.3 Definitions of b^0 and b^-n
Problem Recognition Exercises-Properties of Exponents
5.4 Scientific Notation
5.5 Addition and Subtraction of Polynomials
5.6 Multiplication of Polynomials and Special Products
5.7 Division of Polynomials
Problem Recognition Exercises-Operations on Polynomials
Chapter 6: Factoring Polynomials
6.1 Greatest Common Factor and Factoring by Grouping
6.2 Factoring Trinomials of the Form x^2 + bx + c
6.3 Factoring Trinomials: Trial-and-Error Method
6.4 Factoring Trinomials: AC-Method
6.5 Difference of Squares and Perfect Square Trinomials
6.6 Sum and Difference of Cubes
Problem Recognition Exercises-Factoring Strategy
6.7 Solving Equations Using the Zero Product Rule
Problem Recognition Exercises-Polynomial Expressions Versus Polynomial Equations
6.8 Applications of Quadratic Equations
Chapter 7: Rational Expressions and Equations
7.1 Introduction to Rational Expressions
7.2 Multiplication and Division of Rational Expressions
7.3 Least Common Denominator
7.4 Addition and Subtraction of Rational Expressions
Problem Recognition Exercises-Operations of Rational Expressions
7.5 Complex Fractions
7.6 Rational Equations
Problem Recognition Exercises-Comparing Rational Equations and Rational Expressions
7.7 Applications of Rational Equations and Proportions
Chapter 8: Relations and Functions
8.1 Introduction of Relations
8.2 Introduction of Functions
8.3 Graphs of Functions
Problem Recognition Exercises: Characteristics of Relations
8.4 Alebra of Functions and Composition
8.5 Variation
Chapter 9: More Equations and Inequalities
9.1 Compound Inequalities
9.2 Polynomial and Rational Enequalities
9.3 Absolute Value Equations
9.4 Absolute Value Inequalities
Problem Recognition Exercises: Equations and Inequalities
9.5 Linear Inequalities and Systems of Linear Inequalities in Two Variables
Chapter 10: Radicals and Complex Numbers
10.1 Definition of an nth Root
10.2 Rational Exponents
10.3 Simplifying Radical Expressions
10.4 Addition and Subtraction of Radicals
10.5 Multiplication of Radicals
Problem Recognition Exercises: Simplifying Radical Expressions
10.6 Division of Radicals and Rationalization
10.7 Solving Radical Equations
10.8 Complex Numbers
Chapter 11: Quadratic Equations and Functions
11.1 Square Root Property and Completing the Square
11.2 Quadratic Formula
11.3 Equations in Quadratic Form
Problem Recognition Exercises: Quadratic and Quadratic Type Equations
11.4 Graphs of Quadratic Functions
11.5 Vertex of a Parabola: Applications and Modeling
Chapter 12: Exponential and Logarithmic Functions and Applications
12.1 Inverse Functions
12.2 Exponential Functions
12.3 Logarithmic Functions
Problem Recognition Exercises: Identifying Graphs of Functions
12.4 Properties of Logarithms
12.5 The Irrational Number e and change of Base
Problem Recognition Exercises: Logarithmic and Exponential Forms
12.6 Logarithmic and Exponential Equations and Applications
Chapter 13: Conic Sections
13.1 Distance Formula, Midpoint Formula, and Circles
13.2 More on the Parabola
13.3 The Ellipse and Hyperbola
Problem Recognition Exercises: Formulas and Conic Sections
13.4 Nonlinear Systems of Equation in Two Variables
13.5 Nonlinear Inequalities and Systems of Inequalities in Two Variables
Chapter 14: Binomial Expansions, Sequences, and Series
14.1 Binomial Expansions
14.2 Sequences and Series
14.3 Arithmetic Sequences and Series
14.4 Geometric Sequences and Series
Problem Recognition Exercises: Identifying Arithmetic and Geometric Series
Chapter 15 (Online): Transformations, Piecewise-Defined Functions, and Probability
15.1 Transformations of Graphs and Piecewise-Defined Functions
15.2 Fundamentals of Counting
15.3 Introduction to Probability
Additional Topics Appendix
A.1 Additional Factoring
A.2 Mean, Median, and Mode
A.3 Introduction to Geometry
A.4 Solving Systems of Linear Equations Using Matrices
A.5 Determinants and Cramer’s Rule
Online Appendix B
B.1 Review of the Set of Real Numbers
B.2 Review of Linear Equations and Linear Inequalities
B.3 Review of Graphing
B.4 Review of Systems of Linear Equations in Two Variables
B.5 Review of Polynomials and Properties of Exponents
B.6 Review of Factoring Polynomials and Solving Quadratic Equations
B.7 Review of Rational Expressions
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Negative result brings project's current phase to an end » Bergsonian.org
Negative result brings project’s current phase to an end
We have disproved a certain conjecture that had taken a central place in the Bergsonian axioms project. This brings the current phase of the project to a close. It also provides a good moment to
briefly take stock of the project and to suggest what a future phase might look like.
Bergson’s key philosophical insight is that the world is the continual creation of new possibilities, rather than the successive realization of pre-existing possibilities. We developed the Bergsonian
axioms to formalize this idea — or, more precisely, the idea of a universe of mathematical possibilities constructing itself over time. There is a natural recipe for candidate structures to satisfy
the axioms; it takes as inputs a boolean algebra $B$ and a function $F$ from the set of $B$’s subalgebras into $B$. The most commonly-used boolean algebras do not work in the recipe because they have
too many symmetries. What is needed is a $B$ with “densely-nested” subalgebras, but without the “flexible homogeneity” property that prevents satisfaction of the axioms. We know of no such $B$, but
several years ago we came across an example of rigidly-nested von Neumann algebras (“simple subfactors”) and we were later advised by their inventor R. Longo that they can be densely nested as well.
The task then became to derive boolean algebras from these von Neumann algebras, in the hope that they too will prove, if not rigidly nested, then at least free of the flexible homogeneity property.
The cleanest method is to take the boolean completions of the algebras’ projection lattices. We proved that if “new possibilities” were obtained this way, they would be random quantum states, which
raised the question whether the physical world could be a model of the Bergsonian axioms. We investigated the advantages of an affirmative answer (see the paper “Bergson’s not-even-wrong theory, now
with extra math!”). However, in the last week or two we have proved that these boolean completions of projection lattices are flexibly homogeneous. This closes our main avenue of research. We will
certainly continue to maintain this site; millions have found Bergson’s underlying intuition to be compelling, and will continue to do so; in a world of eight billion people, at least a handful will
surely be drawn to the idea of formalizing this intuition; and we would like to spare them at least some wheel-reinvention. We would also suggest to them the investigation of substructures of the
projection lattices we have focused on; it is common in the theory of forcing to pass to a particular sub-poset of one’s forcing poset, thereby obtaining a forcing notion with very different
properties. It must be said, though, that direct use of the projection lattices would have made for an elegant solution (if one may speak counterfactually about facts that hold a priori!), and that
poking around for sub-posets here would have a somewhat ad-hoc character.
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Use Goal Seek to find the result you want by adjusting an input value
If you know the result that you want from a formula, but are not sure what input value the formula needs to get that result, use the Goal Seek feature. For example, suppose that you need to borrow
some money. You know how much money you want, how long you want to take to pay off the loan, and how much you can afford to pay each month. You can use Goal Seek to determine what interest rate you
will need to secure in order to meet your loan goal.
If you know the result that you want from a formula, but are not sure what input value the formula needs to get that result, use the Goal Seek feature. For example, suppose that you need to borrow
some money. You know how much money you want, how long you want to take to pay off the loan, and how much you can afford to pay each month. You can use Goal Seek to determine what interest rate you
will need to secure in order to meet your loan goal.
Note: Goal Seek works only with one variable input value. If you want to accept more than one input value; for example, both the loan amount and the monthly payment amount for a loan, you use the
Solver add-in. For more information, see Define and solve a problem by using Solver.
Step-by-step with an example
Let's look at the preceding example, step-by-step.
Because you want to calculate the loan interest rate needed to meet your goal, you use the PMT function. The PMT function calculates a monthly payment amount. In this example, the monthly payment
amount is the goal that you seek.
Prepare the worksheet
1. Open a new, blank worksheet.
2. First, add some labels in the first column to make it easier to read the worksheet.
a. In cell A1, type Loan Amount.
b. In cell A2, type Term in Months.
c. In cell A3, type Interest Rate.
d. In cell A4, type Payment.
3. Next, add the values that you know.
a. In cell B1, type 100000. This is the amount that you want to borrow.
b. In cell B2, type 180. This is the number of months that you want to pay off the loan.
Note: Although you know the payment amount that you want, you do not enter it as a value, because the payment amount is a result of the formula. Instead, you add the formula to the worksheet
and specify the payment value at a later step, when you use Goal Seek.
4. Next, add the formula for which you have a goal. For the example, use the PMT function:
a. In cell B4, type =PMT(B3/12,B2,B1). This formula calculates the payment amount. In this example, you want to pay $900 each month. You don't enter that amount here, because you want to use
Goal Seek to determine the interest rate, and Goal Seek requires that you start with a formula.
The formula refers to cells B1 and B2, which contain values that you specified in preceding steps. The formula also refers to cell B3, which is where you will specify that Goal Seek put the
interest rate. The formula divides the value in B3 by 12 because you specified a monthly payment, and the PMT function assumes an annual interest rate.
Because there is no value in cell B3, Excel assumes a 0% interest rate and, using the values in the example, returns a payment of $555.56. You can ignore that value for now.
Use Goal Seek to determine the interest rate
1. On the Data tab, in the Data Tools group, click What-If Analysis, and then click Goal Seek.
2. In the Set cell box, enter the reference for the cell that contains the formula that you want to resolve. In the example, this reference is cell B4.
3. In the To value box, type the formula result that you want. In the example, this is -900. Note that this number is negative because it represents a payment.
4. In the By changing cell box, enter the reference for the cell that contains the value that you want to adjust. In the example, this reference is cell B3.
Note: The cell that Goal Seek changes must be referenced by the formula in the cell that you specified in the Set cell box.
5. Click OK. Goal Seek runs and produces a result, as shown in the following illustration.
Cells B1, B2, and B3 are the values for the loan amount, term length, and interest rate.
Cell B4 displays the result of the formula =PMT(B3/12,B2,B1).
6. Finally, format the target cell (B3) so that it displays the result as a percentage.
a. On the Home tab, in the Number group, click Percentage.
b. Click Increase Decimal or Decrease Decimal to set the number of decimal places.
If you know the result that you want from a formula, but are not sure what input value the formula needs to get that result, use the Goal Seek feature. For example, suppose that you need to borrow
some money. You know how much money you want, how long you want to take to pay off the loan, and how much you can afford to pay each month. You can use Goal Seek to determine what interest rate you
will need to secure in order to meet your loan goal.
Note: Goal Seek works only with one variable input value. If you want to accept more than one input value, for example, both the loan amount and the monthly payment amount for a loan, use the Solver
add-in. For more information, see Define and solve a problem by using Solver.
Step-by-step with an example
Let's look at the preceding example, step-by-step.
Because you want to calculate the loan interest rate needed to meet your goal, you use the PMT function. The PMT function calculates a monthly payment amount. In this example, the monthly payment
amount is the goal that you seek.
Prepare the worksheet
1. Open a new, blank worksheet.
2. First, add some labels in the first column to make it easier to read the worksheet.
a. In cell A1, type Loan Amount.
b. In cell A2, type Term in Months.
c. In cell A3, type Interest Rate.
d. In cell A4, type Payment.
3. Next, add the values that you know.
a. In cell B1, type 100000. This is the amount that you want to borrow.
b. In cell B2, type 180. This is the number of months that you want to pay off the loan.
Note: Although you know the payment amount that you want, you do not enter it as a value, because the payment amount is a result of the formula. Instead, you add the formula to the worksheet
and specify the payment value at a later step, when you use Goal Seek.
4. Next, add the formula for which you have a goal. For the example, use the PMT function:
a. In cell B4, type =PMT(B3/12,B2,B1). This formula calculates the payment amount. In this example, you want to pay $900 each month. You don't enter that amount here, because you want to use
Goal Seek to determine the interest rate, and Goal Seek requires that you start with a formula.
The formula refers to cells B1 and B2, which contain values that you specified in preceding steps. The formula also refers to cell B3, which is where you will specify that Goal Seek put the
interest rate. The formula divides the value in B3 by 12 because you specified a monthly payment, and the PMT function assumes an annual interest rate.
Because there is no value in cell B3, Excel assumes a 0% interest rate and, using the values in the example, returns a payment of $555.56. You can ignore that value for now.
Use Goal Seek to determine the interest rate
1. On the Data tab, click What-If Analysis, and then click Goal Seek.
2. In the Set cell box, enter the reference for the cell that contains the formula that you want to resolve. In the example, this reference is cell B4.
3. In the To value box, type the formula result that you want. In the example, this is -900. Note that this number is negative because it represents a payment.
4. In the By changing cell box, enter the reference for the cell that contains the value that you want to adjust. In the example, this reference is cell B3.
Note: The cell that Goal Seek changes must be referenced by the formula in the cell that you specified in the Set cell box.
5. Click OK. Goal Seek runs and produces a result, as shown in the following illustration.
6. Finally, format the target cell (B3) so that it displays the result as a percentage. On the Home tab, click Increase Decimal Decrease Decimal
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Topic outline
• 1.1 Count, read and write whole numbers from 0 up to 200
Activity 1.1.1
Activity 1.1.2
Activity 1.1.3
Activity 1.1.4
Activity 1.1.5
Activity 1.1.6
Activity 1.1.7
Activity 1.1.8
Activity 1.1.9
Activity 1.1.10
Activity 1.1.11
Activity 1.1.13
1.2 Place value of each digit for numbers from 0 up to 200
Activity 1.2.1
Activity 1.2.2
Activity 1.2.3
591.3. Comparison of numbers from 0 up to 200
Activity 1.3.1
Activity 1.3.2
Activity 1.3.3
Activity 1.3.4
1.4 Arranging numbers within 200 in ascending and descending order
1.4.1 Arranging numbers from smallest to the largest.
Activity 1.4.1
Activity 1.4.2
1.4.2 Arranging numbers in descending order
Activity 1.4.3
Activity 1.4.4
Activity 1.4.5
1.5 Addition of numbers whose sum does not exceed 200
1.5.1 Mental calculation
Activity 1.5.1
Activity 1.5.2
Activity 1.5.3
1.5.2 Addition without carrying
Activity 1.5.4
Activity 1.5.5
Activity 1.5.6
Activity 1.5.7
1.5.2 Addition with carrying
Activity 1.5.8
1.6 Word problems involving numbers whose sum does not exceed 200
Activity 1.6.1
In the first week our school enrolled 123 new pupils. In the second week the school received other 54 new pupils. Find the total number of new pupils in this two weeks.
In the first week: 123
In the second week: 54
Question: The total or the sum
Answer: 123 + 54 = 177.
The total number of new pupils in this two weeks is 177.
Note: To solve a problem, make up a number sentence from a given number story.
123 + 54 = 177 is a number sentence.
Solve the following problems:
1.7 Subtraction within the range of 200
1.7.1 Mental work
Activity 1.7.1
261.7.2 Subtraction without Borrowing
Activity 1.7.2
Activity 1.7.3
Activity 1.7.4
Activity 1.7.5
1.7.3 Subtraction with Borrowing
Activity 1.7.6
Activity 1.7.7Hundreds
1.8 Solve problems involving subtraction in real life situations
Activity 1.8.1
Solve the following problems:
1. Our school has 200 cocks. If the headmaster sells 50 cocks, how many cocks will remain?
2. Uwera had 170 eggs. In this morning uwera sold 60 eggs. How many eggs left?
3. In the exam, Mugisha scored 156. If the pass mark is 200 marks. How many marks does he need to get the pass mark?
4. Shimwa produced 166 sacks of rice. Shema produced 187 sacks of rice. Find the difference between their sacks.
5. The family of Keza bought 178 cobs of maize. At the evening, they gave 69 cobs of maize to their visitors. How many cobs of
maize were left?
6. Kayiranga took 195 pineapples to the market. People bought 139 pineapples only. How many pineapples did he bring back
7. Our village has 187 families. 149 families have cows, How many families do not have cows in our village?
8. Muhizi had 187 sacks of cement. If 39 sacks will be used during the construction of the walls of his house, how many sacks of
cement will remain?
9. Bwiza Village has 172 families. If only 148 families of this village have health insurance , How many families of Bwiza village are
not insured?
1.9 Multiplication of whole numbers by 2 and the multiples of 2
Activity 1.9.1
Activity 1.9.2
1. Complete the multiplication table by 2.
2. Fill in the missing number in the multiplication table
1.10 Multiply a two-digit number by 2
Activity 1.10 .1
I have learnt that:
1.11 Word problems involving the multiplication by 2
Activity 1.11
The number of people : 42 x 2 = 84
The number of people is 84.
1.12 Division without a Remainder of a two or three-digit number by 2
Activity 1.12.1
Activity 1.12.2
I have learnt that:
Activity 1.12.3
I have learnt:
1.13 Word problems involving the division of a number by 2
If the sector shares 148 books between 2 schools equally. How many books will each school get?
The number of books for each school: 148 : 2 = 74
The number of books for each school is 74.
Then solve the following problens:
1.14 Multiplication of whole numbers by 3 and the multiples of 3
Activity 1.14.1
Activity 1.14.2
1.15 Multiply a two-digit number by 3
Activity 1.15 .1
I have learnt:
Activity 1.15 .2
31.16 Word problems involving the multiplication by 3
Activity 1.16
Work out the following problems:
1.17 Division without a Remainder of a two or three-digit number by 3
Activity 1.17.1
Activity 1.17.2
Activity 1.17.3
1.18 Word problems involving the division of a number by 3
Activity 1.18
Laptops to be shared to each school: 189 : 3 = 63
The number of Laptops to be shared to each school is 63.
Solve the following problens:
END UNIT ASSESSMENT 1
Files: 3URLs: 6
• 2.1 Count, read and write whole numbers from 0 up to 500
Activity 2.1.1
Study the picture and tell your friend the numbers you have see on it
Activity 2.1.2
Read loudly numbers on a), b), c) and d) using number names
Activity 2.1.3
Read numbers you see on the sign posts
Activity 2.1.4
Study the following numbers and read them in a loud voice
Activity 2.1.5
Count in hundreds and complete the following number line
Activity 2.1.6
Fill in the missing numbers
Activity 2.1.7
Fill in the missing numbers
Activity 2.1.8
2.1.7You have a container with number cards for the following numbers: 242, 318, 425, 499 and 384.
Pick randomly one number card from the container and tell your collegue the number you have picked.
Activity 2.1.9
Take 10 number cards with successive numbers between 200 and 500.
Arrange them from the smallest to largest.
Activity 2.1.1
Study the following pictures. What do you see ?
Using your own number cards, arrange numbers from 200 up to 500.
Activity 2.1.11
Fill in the missing numbers on the following number lines:
Activity 2.1.12
Fill in the missing numbers on the following number lines.
Activity 2.1.13
Write numbers in words:
Activity 2.1.14
Activity 2.1.15
Read and write these numbers in words
Activity 2.1.16
2.2 Place values of numbers from 0 up to 500
Activity 2.2.1
Use the example and write the numbers that follow in the table of place values
Activity 2.2.2
Use the table of place values to group numbers into hundreds (H), tens (T) and ones (O).
Activity 2.2.3
2.3 Comparing numbers from 0 up to 500
Activity 2.3.1
Activity 2.3.2
Put them on a table and use cards symbol <, > or = to compare the numbers.
Activity 2.3.3
Activity 2.3.4
The number of carrots produced by each class is given in this table:
Compare the harvest for the following classes:
2.4 Arrange numbers within 500 in ascending or descending order
2.4.1 Arrange numbers from the smallest to the largest.
Activity 2.4.1
Activity 2.4.2
Activity 2.4.3
2.4.2 Arranging numbers from the largest to the smallest.
Activity 2.4.3
Activity 2.4.4
Do the same and arrange your number cards from the largest to the smallest number.
Arrange the following numbers from the largest to the smallest number
2.5 Addition of numbers whose sum does not exceed 500
2.5.1 Mental calculation
Activity 2.5.1
Activity 2.5.2
2.5.2 Addition without carrying
Activity 2.5.3
Tell the activity taking place in the pictures below
Activity 2.5.4
Activity 2.5.6
Activity 2.5.7
Activity 2.5.8
2.6 Word problems involving the addition of numbers whose highest sum is 500
The total marks for Nahimana: 225 + 215 = 440
The total marks for Nahimana is 440.
1. Today the school leader buys 265 books for Mathematics and 19 books for Kinyarwanda. How many books does he
buy altogether?
2. Kanyinya Village planted 312 trees during Umuganda.
Kinyinya Village also planted 188 trees. How many trees were planted altogether?
2.7 Subtraction of numbers within the range of 500
2.7.1 Mental calculation
Activity 2.7.1
2.7. 2 Subtraction without borrowing
Activity 2.7.3
Activity 2.7.4
Activity 2.7.5
Then, 496 - 223 = 273
Work out :
2.7.3 Subtraction with Borrowing
Activity 2.7.6
2.8 Solve problems involving subtraction in real life situations
Activity 2.8
Our school has 378 as the total number of pupils. However, 132 pupils are in P6. How many pupils will remain after the departure of P6 pupils ?
There will remain: 378 - 132 = 246
1. Tito has got 170 eggs. In this morning 87 were broken. How many eggs are left?
2. Makuza has 466 sacks of beans. His Sister has 387 sacks of beans.
a) Who has more beans ?
b) What is the difference between the sacks of the two people ?
2.9 Multiplication of whole numbers by 4 and the multiples of 4
Activity 2.9.1
The multiplication by 4 looks like the repeated addition of fours.
Activity 2.9.2
Use the multiplication by 4 to complete the missing number
2.10 Multiply a two-digit number by 4
Activity 2.10 .1
Activity 2.10 .2
2.11 Word problems involving the multiplication of a number by 4
Activity 2.11
Study the worked out example below:
We are 42 pupils in the classroom. Every pupil has 4 books.
Find the total number of books we have in our classroom.
Total number of books: 42 × 4 = 168
The total number of books is 168
Solve the following problems:
1. At our school we are 82 pupils. We are going to plant trees so that every pupil plants 4 trees. How many trees
shall we plant altogether?
2. In the morning assemble the P3 pupils stand in rows in front of their classroom. If there are 22 pupils on each
row, find the total number of pupils in the assembly.
2.12 Division of a number by 4
Activity 2.12.1
2.13 Division of a two or three-digit numbers by 4 without a Remainder
Activity 2.13.1
Activity 2.13.2
2.14 Word problems involving the division of a number by 4
Activity 2.14
The head teacher bought 488 books. These books were shared equally to 4 classes.
How many books did each class get?
Each class received: 488 : 4 = 122
Each class got 122 books.
Solve the following problems:
1. We are 4 children at home. Our Mum wants us to share 144 notebooks equally. How many notebooks does each child get?
2. There are 368 people in the conference hall. People sit in 4 equal columns. How many people are in each column?
2.15 Multiplication of whole numbers by 5 and the multiples of 5
Activity 2.15.1
Activity 1.15.2
Fill in the missing number in the multiplication table by 5
2.16 Multiply a two-digit number by 5
Activity 2.16 .1
Then, 21 x 5 = 105
Find the answer by using the table of place values:
Activity 2.16.2
2.17 Word problems involving the multiplication by 5
The number of all chairs: 91 x 5 = 455
The number of all chairs is 455
Solve the following word problems:
1. During the distribution of mosquito nets, each family received 5 mosquito nets. How many nets were given to 81 families ?
2. If there are 5 cups on each tray, how many cups are there on 41 trays ?
3. There are 61 benches in the conference hall. How manypeople can sit in the conference hall if only 5 can sit on each
bench ?
4. One family has 5 people. How many people are in 31 families ?
5. There are 40 bottles of water in each box. How many bottles of water are in 5 boxes ?
2.18 Division of a two or three-digit number by 5 without a remainder
Activity 2.18.1
Activity 2.18.2
Activity 2.18.3
2.19 Word problems involving the division of a two or 3 digit number by 5
Activity 2.19
You have 65 oranges. If you share them equally among 5
pupils, how many oranges will each pupil get?
One pupil can get: 65 : 5 = 13
One pupil can get 13 oranges.
Then solve the following problems:
1. The cooperative of 5 farmers has 495 cows. If they share their cows equally, how many cows will each farmer get?
2. The health Center has 385 mosquito nets to be distributed equally to 5 villages in our Cell. Find the number of mosquito nets for each village.
END UNIT ASSESSMENT 2
1. Write in words or in figures
(a) 497
(b) Three hundred eighty six.
2. Underline the correct answer
(a) 3Ones 6Tens 4Hundreds = 1) 364 2) 463 3) 346
(b) 3Hundreds 2Ones 4Tens = 1) 324 2) 423 3) 342
3. Write the expanded number
(a) (4 × 100) + (8 × 10) + (7 × 1) =
(b) 300 + 70 + 6 =
4. Write each number in the table of place values
(a) 268 (b) 475 (c) 473 (d) 352
6. Arrange the following numbers in ascending order
(from the smallest to the largest)
439, 349, 493, 394, 387, 479
7. Arrange the following numbers in descending order
(from the largest to the smallest)
293, 239, 387, 470, 389, 499
8. Work out the following:
(a) 234 + 253 = (c) 378 + 114 =
(b) 257 + 208 = (d) 369 + 128 =
Find the difference:
(a) 459 – 327 = (c) 367 – 236 =
(b) 453 – 345 = (d) 381 – 274 =
10. Complete the following multiplication or division table:
11. Work out the following product:
12. Find the missing numbers in the following multiplication table:
13. Work out the following division by using the standard written form.
14. Word problems
a) Our Village planted 256 trees. The neighboring Village also planted 239 trees. Find the total number of trees planted
by the two villages.
b) Our school has 489 pupils. The number of boys is 297. Find the number of girls.
c) Head Mistress gave 4 books to every pupil. How many books did she give to 72 pupils?
d) Shared 496 books equally among 4 classrooms. How many books can each classroom get?
e) Chose the right answer:
Gisa shared equally 450 pineapples to 5 shops. Each shop got:
(i) 450 – 5 = 445 pineapples
(ii) 450 + 5 = 455 pineapples
(iii) 450 : 5 = 90 pineapples
f) Muhoza has 196 sweets. He wants to share them equally
among his 5friends . How many sweets will one each get?
• 3.1 Count, read and write whole numbers from 0 up to 1000
Activity 3.1.1
Activity 3.1.2
Activity 3.1.3
Activity 3.1.4
Activity 3.1.5
Activity 3.1.6
Activity 3.1.7
You have a container with number cards.
Pick randomly one number card from the container and tell your friend the number in words
Activity 3.1.8
Go to the classrooms of P1, P2 and P3. Ask them the number of pupils who are in each classroom.
Write these numbers and go back to your classroom. Read to your friend the numbers you wrote.
Activity 3.1.9
Study the pictures carefully and arrange numbers from 500 up to 1000.
Activity 3.1.10
Activity 3.1.11
Activity 3.1.12
Activity 3.1.13
Activity 3.1.15
3.2 Place value of each digit of numbers from 0 up to 999
Activity 3.2.1
Activity 3.2.2
Activity 3.2.3
3.3 Comparing numbers from 0 up to 999
Activity 3.3.1
Activity 3.3.3
Activity 3.3.4
The number of sugar canes harvested by every class is given in this table:
3.4 Arranging numbers within 999 in ascending or descending order
3.4.1 Arranging numbers in ascending order (from the smallest to the largest)
Activity 3.4.1
Activity 3.4.2
Activity 2.4.3
3.4.2 Arranging numbers in descending order (from the largest to the smallest)
Activity 3.4.4
Activity 3.4.5
Do the same and arrange your number cards from the largest to the smallest number.
Activity 3.4.6
3.5 Addition of numbers whose sum does not exceed 999
3.5.1 Mental work .
Activity 3.5.1
Activity 3.5.2
3.5.2 Addition without carrying
Activity 3.5.3
Activity 3.5.4
Activity 3.5.5
3.5.3 Addition with carrying
Activity 3.5.6
3.6 Word problems involving the addition of numbers with the highest sum of 999
Activity 3.6
1. During exams, pupils used 534 sheets of paper in mathematics and 365 in Kinyarwanda. Find the total number of paper used.
2. On Saturday party we served 450 mangoes. On Sunday we used 539 mangoes. How many mangoes did we serve altogether?
3. In the morning there were 723 people in the market and 276 more people came in the afternoon.
How many people came in the market altogether?
3.7 Subtraction of numbers within the range of 999
3.7.1 Mental work
3.7. 2 Subtraction without Borrowing
Activity 3.7.2
Activity 3.7.3
Use them to do the task below:
Activity 3.7.5
3.7.3 Subtraction with Borrowing
Activity 3.7.6
3.8 Solve problems involving subtraction in real life situations
Activity 3.8
Study this example carefully :
There were 850 reading books in the library. If 615 were taken to the classroom, How many books remained in the library?
The library remained with: 850 - 615 = 235
The library remained with 235 books
Solve the following problems:
1. Our teacher bought 500 pens. She gave us 342 pens. How many pens did she remain with ?
2. Butera harvested 646 sacks of sweet potatoes. His sister harvested 837 sacks
a) Who had more sacks of sweet potatoes?
b) Find the difference between Butera and his sister’s harvest.
3. Last year Zigama had 954 shirts in his shop. He sold 719 of them. How many shirts remained?
4. Our Sector bought 960 bottles of soda for a party.
Only 756 people attended the party and every person took one bottle of soda. How many bottles remained?
5. The government bought 942 cars. If 749 cars are small, how many big cars did the government buy?
3.9 Multiplication of whole numbers by 6 and the multiples.
Activity 3.9.1
Form different groups of 6 counters. Count the number of groups and the number of counters for those groups.
Do it in the following way:
1 group, 2 groups, 3 groups, 4 groups, 5 groups, 6 groups, 7 groups 8 groups, 9 groups and 10 groups.
Write the number sentences of the following: The number of counters for 5 groups is ...,
The number of counters for 9 groups is..., etc.
The multiplication by 6 looks like the repeated addition of sixes.
Activity 3.9.2
Use the multiplication by 6 and complete the missing number
3.10 Multiply a two or three-digit number by 6
Activity 3.10 .1
Activity 3.10 .2
70 x 6 =
3.11 Word problems involving the multiplication of a number by 6
Activity 3.11
During Umuganda for last month every person planted 6 trees. How many trees were planted by 91 people
Solve the following word problems:
1. In the church, 6 people sit on one bench. How many people can sit on 51 benches?
2. Every pupil has 6 notebooks. Find the total number of notebooks for 41 pupils.
3. A flat building in Kigali city center has 31 floors. If each floor has 6 rooms, find the total number of rooms in flat building.
4. In the morning assemble P5 pupils stood in 6 rows. If there are 61 pupils on each row, find the total number of pupils
who were in the assembly.
5. Chairs for the conference hall are arranged in 6 columns. If every column has 95 chairs, find the total number of
chairs in the conference hall.
6. A Carpenter has 6 big trees. If he cuts 50 pieces of timber from each tree.
Find the total number of pieces of timber he can cut from his trees.
3.12 Division of a number by 6
Activity 3.12
3.13 Division of a two or three-digit numbers by 6 without a Remainder
Activity 3.13
3.14 Word problems involving the division of a number by 6
Activity 3.14
The District shared 984 books equally among 6 schools. How many books does each school get?
Solve the following problems:
1. Share 246 notebooks equally among 6 pupils. What does each pupil get?
2. Musoni’s cows produce 486 liters of milk in 6 days. If the daily production is the same,
find the number of litters they produce in one day.
3. Share 864 balls equally among 6 schools. How many balls does each school get?
3.15 Multiplication of whole numbers by 10 or by 100
Activity 3.15.1
The multiplication by 10 looks like the repeated addition of tens.
Activity 3.15.2
1) Complete the multiplication by 10 or by 100
2) Complete this multiplication table
3) Work out the multiplication
END UNIT ASSESSMENT 3
1. Write in words or in figures
(a) 976 :
(b) Eight hundred thirty five
2. Underline the correct number
3. Write the expanded number
(a) (8 × 100) + (7 × 10) + (9 × 1) =
(b) 900 + 90 + 9 =
4. Write these numbers in a place value table
(a) 896 (b) 759 (c) 837 (d) 925
5. Use <, > and = to compare numbers
6. Arrange the following numbers from the smallest to the largest.
793, 947, 986, 969, 678, 789
7. Arrange the following numbers from the largest to the smallest.
972, 984, 837, 749, 839, 949
8. Carry out the addition
(a) 534 + 453 = (b) 738 + 241 =
(c) 572 + 418 = (d) 693 + 289 =
9. Carry out the subtraction
(a) 857 – 727 = (b) 967 – 856 =
(c) 935 – 798 = (d) 618 – 579 =
10. Complete the following multiplication or division table
11. Carry out the multiplication
12. Complete the multiplication by 10 or by 100
13. Complete the missing numbers in the following division or multiplication table
14. Divide the following numbers by 6
(a) 966 : 6 = (f) 870 : 6 =
(b) 684 : 6 = (g) 774 : 6 =
(c) 564 : 6 =
(d) 624 : 6 = (i) 978 : 6 =
(e) 864 : 6 = (j) 786 : 6 =
15. Word problems
(a) Shema had 78 cows. This morning he sold 568 cows. How many cows remained?
(b) What number can you add to 567 to get 999?
(c) There were 967 books in the library. If students borrowed 765 books,
how many books were left in the library?
(d) What number can you subtract from 987 to get 556?
(e) Which number can you add to 568 to get 879?
(f) Bumanzi Village has 235 men, 262 women and 302 children.
How many people are there altogether in Bumanzi village?
(g) Share 864 mosquito nets equally among 6 Villages. How many mosquito nets does each village get?
(i) Ntwari has 186 bottles of water. He wants to park these bottles equally in 6 boxes.
How many bottles of water will be in one box?
• Activity 4.1.1
Activity 4.1.2
Activity 4.1.3
Activity 4.1.3
(b) Drawing and shading one half of an object
Activity 4.1.4
Activity 4.1.5
Activity 4.2.1
Activity 4.2.2
Activity 4.2.3
Activity 4.2.4
Activity 4.2.5
Activity 4.2.6
Activity 4.3.1
Activity 4.3.2
Activity 4.3.3
Activity 4.3.4
Activity 4.3.5
Activity 4.3.6
4.4 Parts of a fraction
Activity 4.4
4.5 Comparing fractions
Activity 4.5.1
Compare using < > =
Activity 4.5.2
Activity 4.5.3
4.6 Putting fractions together to make a whole
Activity 4.6
4.7 Importance of fractions
Activity 4.7
END UNIT ASSESSMENT 4
1. Write in words and in figures the fraction related to the shaded parts
2. Draw a circle, divide it into fractions and shade the part equivalent to:
4. Use >,< or = to compare the following fractions
5. Answer by “Yes” or “No”
• 5. 0 Preliminary activities
Activity 5.0.1
Activity 5.0.2
Activity 5.0.3
Activity 5.0.4
Activity 5.0.5
Solve word problems
1. The chalkboard of our classroom has the length of 8m.
The chalkboard of the neighboring classroom has the length of 6m. Find the total length of the two chalkboards.
2. Kaneza’s garden has a length of 20 m. The garden of Mitari measures 18m. What is the total length for the two gardens?3. On Monday, Mariza bought 14m of pieces of clothe. On Tuesday, she
bought 13 m of the same cloth.
The next day she bought 12m. Find the total length for the pieces of clothes she bought.
4. Mayira has a rope of 10m. His brother’s rope has 19 dm. What is the total length for the two ropes?
5. Nshuti made a mat of 20 dm. Her sister Mutesi made a mat of 17 dm. What is the difference in the length of the two mats?
6. I made a rope of 72m. My father cut 12m from it to tie the banana plant and protect it against strong wind.
What is the length of the remaing rope?
7. Munezero has a timber of 12m. Kagabo’s timber measures 8 m. What is the total length for the two timbers?
5.1 Measuring the length of objects using a meter ruler
Activity 5.1
Do the following activity in groups :
1. Use a meter ruler and measure:
(a) The length of your desk
(b) The length of teacher’s table
2. Use a meter ruler and measure:
(a) The width of the teacher’s cupboard
(b) The width or the height of your blackboard
3. Use a meter ruler and measure: the perimeter of your classroom.
4. Use a meter ruler and measure:
(a) The width of your classroom door .
(b) The total length of two sides (length) of your classroom
5. Use a 30cm ruler and measure the length of notebooks and books, and other objects in your classrooms.
5.2 Dividing a meter into 10 equal parts
Activity 5.2
Do the following activities:
1. Get sugar cane of 1m long. Divide this cane in 10 equal parts.
2. Get a rope of 1m long. Cut it in 10 equal parts.
3. Get a thread of 1m long. Divide it in 10 parts of the same length.
4. Get a cloth measuring 1m long. Cut it in 10 equal parts.
5.3 Dividing a decimeter into 10 equal parts
Activity 5.3
In your groups do the following activities:
1. Take a rope of 1dm. Cut it in 10 equal parts.
2. Take a small tree of 1m. Divide it in 10 parts of the same length.
5.4 Conversion of Units of length
Activity 5.4.1
5.5 Comparing lengths
Activity 5.5
5.6 Measuring the length round objects
Activity 5.6
1. Use a meter ruler and measure the total length round your classroom.
2. Measure the length of 10 m in the playground.
3. Use a meter ruler and measure the length round a garden
4. Use a rope of 10 m to measure the length round the football pitch.
5.7 Arranging lengths of objects
Activity 5.7.1
Activity 5.7.2
5.8 Addition of lengths
Activity 5.8
5.9 Subtraction of units of lengths
Activity 5.9
5.10 Multiplication of units of length per a whole number
Activity 5. 10
5.11 Division of length by a whole number
Activity 5. 11
5.12 Word problems involving units of length
Activity 5. 12
Study this example on the word problem:
The distance between our classroom and the office of Headteacher is 45 dm. The distance between the office and
the play ground is 55dm. Find the total distance in meters between our classroom and the playground.
Distance between our classroom and the office of Headteacher: 45 dm
Distance between the office and the play ground : 55 dm
The distance between our classroom and the play ground is 10 m.
Solve problems:
1. Last year I planted a tree with 50 dm of height. Today, the tree has 80dm.
What is the difference in the height of this tree?
2. A carpenter bought a piece of timber measuring 100cm. He cut it into 5 equal parts. How long is each part?
3. Gatari bought a rope of 60 m. He wants to cut it in 3 equal ropes. What would be the length of each part.
4. Gatera had a field of 89m of length. Munezero’s field had 97 m of length.
(a) Between them, who had a longer field?
(b) Find the difference between their fields?
5. The distance from our home to school is 420 dm. Convert this distance in m.
5.13 The uses of units of length
Activity 5.13.1
Activity 5.13.2
Activity 5.13.3
5. 14 END UNIT ASSESSMENT 5
1. Comment by Yes or No
(a) The length for my class table is 100 cm ….....…...
(b) The meter is the standard unit of length measurement….........................
(c) We use the tape meter to measure the length of a cloth. ……………………….
(d) Units of length help us to find the measurement of length for objects………………..
(e) I use a meter ruler to measure the length for my notebook. ……………………..
(f) The units of length vary from one the next in the multiple of ten…………………..
2 Use a conversion table to convert
3. Use <, > or = to compare lengths
4. Arrange the lengths for objects from the shortest to the longest: 9 m, 75 dm, 8 m, 85 dm.
5. Arrange the lengths for objects from the longest to the shortest: 756 cm, 87 dm, 967 cm, 68 dm.
6.Work out:
7.Word problems
(a) Gisa walks on foot to go to visit his friend. He covers a distance of 45m. Convert this distance in dm.
(b) Keza bought a long cloth of 79m. She sold 70 dm from it. How long is the remaing piece of cloth cloth?
(c) Mucuruzi bought a cloth of 75m. He divided it in 5 equal parts. Find the length for each part.
(d) During the running race, the competitor Gwiza made 100m in 6 consecutive periods.
Find the total length covered by Gwiza.
• 6.1 The litre as a measuring tool
Activity 6.1
6.2 Measuring liquids
Activity 6.2 1
Activity 6.2 2
Activity 6.2 3
6.3 Comparing containers of liquids
Activity 6.3.1
Activity 6.3.2
Activity 6.3.3
Activity 6.3.4
6.4 Addition of capacities in litres
Activity 6.4.1
6.5 Word problems involving the addition of capacity measurements
Activity 6.5
Solve the following problems:
6.6 Subtraction or difference of capacities in litres
Activity 6. 6.1
Solve the following problems:
6.9 Word problems involving multiplication of capacities per a number of times
Activity 6. 9
Solve the following problems:
6.10 Division of capacity measurements by a whole number
Activity 6.10
6.11 Word problems involving the division of capacity measurements by a whole number
Activity 6. 11
Solve the following problems:
6.12 Importance of capacity measurements
Activity 6.12.1
List and explain where liters are used in real life.
Activity 6.12.2
Activity 6.12.3
END UNIT ASSESSMENT 6
61. Comment by Yes or No
(a) Liter is the standard unit of measuring the capacity of liquids…..............
(b) We use the liter to measure the length of a field…..
(c) Liter is used to measure the quantity of liquids such as water……..
2. Use <, > or = to compare
3. Arrange the capacity of measurements for objects from the lightest to the heaviest
4. Arrange the capacity measurements for objects from the heaviest to the lightest.
5. Find the answer
6. Problems
• 7.1 The Kilogram as the standard unit of mass
Activity 7.1
We measure mass in kilograms (Kg).
Give another way of measuring mass.
7.2 Balances and their types
Activity 7.2
7.3 Measuring masses of objects in Kg
Activity 7.3.1
Activity 7.3.2
Do the same and read the mass of different objects in Kilograms and record masses on a balance.
Activity 7.3.3
Activity 7.3.4
7.4 Importance of Kilogram (Kg)
Activity 7.4.1
7.4.1Activity 7.4.2
Activity 7.4.3
7.5 Comparing masses of objects
Activity 7.5.1
Activity 7.5.2
Activity 7.5.3
Activity 7.5.4
Activity 7.5.5
7.6 Addition of masses in kilogram
Activity 7. 6
Example: 205 kg + 414 kg =
7.7 Word problems involving the addition of mass measurements
Activity 7. 7
Look at the worked out example below:
I weigh 32Kg; my brother weighs 46Kg. Find our total weight?
Solve the following problems:
1. Last month Kamanzi kept 12Kg of cassava in the store.
His brother kept15 Kg of cassava. How much cassava did they save altogether?
2. One day, Rukundo sold 50Kg of rice in the morning. In the afternoon, he sold 25Kg of rice.
How much rice did Rukundo sell on the same day?
3. At home we cook 5Kg of bananas in the morning. In the evening we cooked 4 Kg of bananas.
Find the mass of bananas we cook per day.
4. Every day Mbabazi sells 15Kg of sugar and 25Kg of sorghum flour. Find the total number of Kg Mbabazi sells
per day.
7.8 Subtraction of units of mass in Kg
Activity 7. 8
Example: 475 kg - 364 kg =
7.9 Word problems on the subtraction of units of mass
Activity 7. 9
Study the worked out example below:
I poured 28Kg of rice in a sack that requires 59 Kg to be filled. How many Kg are needed to fill the sack?
Solve the following problems:
1. A businessman had 150Kg of beans. He sold 75 Kg from them.
How many kilograms of beans did he remain with?
2. Gisa harvested 247Kg of rice. He gave his neighbors 130 Kg of rice.
How many kilogram of rice did he remain with?
7.10 Multiplication of mass measurements by a whole number
Activity 7. 10
Example: 82 kg x 4 =
7.11 Word problems involving multiplication of mass measurements by a whole number
Activity 7. 11
Study the example below:
My parents harvested 6 sacks of beans. Each sack weighs 71Kg. How many kilograms of beans did my they harvest?
Solve the following problems:
1) At home we cook 6 Kg of potatoes. How much potatoes do we cook in 3 days?
2) Mugabo carries 61 Kg of bananas on the wheelbarrow. How many kilograms will he have if he caries bananas 3 times?
3) When preparing breads, Muhizi uses 31Kg of millet flour per day. How many kilogram of millet flour will he use in 10 days?
7.12 Division of mass measurements by a whole number
Activity 7. 12
7.13 Word problems involving the division of mass mea-surements by a whole number
Activity 7. 13
Discuss the example below:
Solve the following problems:
1. Share 450 Kg of rice equally among 5 people. How many kilograms will each person get?
2. Four people bought 328 Kg of sugar to be shared equally among them. Find the share for each person?
3. There are 284 Kg of beans to be shared equally in 4 sacks. What is the mass for each sack?
4. During the harvesting of beans, a mother got 48Kg. She equally shared this harvest among 4 children.
What was the share of each child?
5. At home we use 30Kg of potatoes in 5 days. How many kilograms of potatoes do we use in one day?
END UNIT ASSESSMENT 7
1. Comment by Yes or No
(a) Kg is the standard unit of mass measurements........
(b) Kg is used to measure the capacity of objects……
2. Give 3 types of balances.
3. Use <, > or = to compare masses =
4. Arrange the mass measurements for objects from the lightest to the heaviest mass
478 kg, 874 kg, 487 kg, 784 kg, 847 kg, 748 kg
5. Arrange the mass measurements for objects from the heaviest to the lightest mass
836 kg, 368 kg, 638 kg, 863 kg, 386 kg, 683 kg
6. Find the answer
7. Solve word problems
(a) Abatoni bought 6 sacks of cement. If one sack weighs 50Kg, Find the number of Kg she bought.
(b) During the beginning of season B of Agriculture, Rwema shared 85Kg equally to his 5 children.
Find the quantity for each child.
(c) In the first season of farming we got a harvest of 356 kg of rice.
In the second season we got 278 Kg and we got 319 Kg in the third season.
Find the total harvest we got in these three seasons.
(d) The store of our school had 895Kg of beans.
If the school used 547 Kg of beans for students’ meal, find the quantity of beans which remained in the store.
(e) Last year I got 21Kg of rice as a harvest. In this year I got 185 Kg of rice.
Find my total harvest for these two years.
(f) Share 472 Kg of sugar equally to 4 families; How much sugar will each family get?
(g) Kamana weighs 45Kg. His sister weighs 55Kg. Find the total weight for Kamana and his sister.
8.0 Preliminary activities
Activity 8.0.1
Activity 8.0.2
Activity 8.0.3
Activity 8.0.4
Activity 8.0.5
1. Kariza had a coin of 100Frw. She bought a sweet at 50 Frw. What was her balance?
2. Keza was given 80Frw by her parents. If she got 20Frw more. How much money did she get?
3. Kayitare was given 100Frw . He bought a pen at 50Frw and a banana at 40Frw. How much money did he remain with?
4. Peter bought a pencil at 20Frw and a mango at 50Frw. How much money did he use altogether?
5. Mutesi had 100Frw. She bought a pen and paid 50Frw. How much money was left?
8.1 Features of Rwandan currency from 1Frw to 1000Frw
Get different coins and notes (denominations of the Rwandan currency), group them according to their colors and values.
Say the denominations of the Rwandan currency from the smallest to the largest.
Activity 8.1.2
8.2 Importance of money
Activity 8.2.1
Activity 8.2.2
1) When you have 100Frw, what can you buy?
2) When you have 500Frw, what can you buy?
3) When you have 1000 Frw, can you buy a house?
Activity 8.2.3
8.3 Sources of money
Activity 8.3.1
Activity 8.3.2
Activity 8.3.3
8.4 Buying and selling
Activity 8.4.1
a) Mutoni wants to buy an orange and a mango. How much money will she pay?
b) Gisa bought a bottle of juice and one cob of maize. How much money did she pay?
c) Kangabe sent Uwase to buy one toilet paper, a banana and a loaf of bread. How much money did she pay altogether?
d) Mahame asked Butera to buy one cob of maize and a loaf of bread. How much money will he Pay altogether?
Activity 8.4.2
a) Muhizi has 750 Frw. If he buys a notebook and a bar of soap, what will be his balance?
b) Ingabire has a note of 500Frw. If she buys one pawpaw and a sweet, how much money will she remain with?
8.5 Exchange of Rwandan currency from 1Frw up to 1000Frw
Activity 8.5.1
Activity 8.5.2
8.6 List down items needed before buying
Activity 8.6.1
Activity 8.6.2
Activity 8.6.3
Find the sum of money he will pay for the items.
1. Onions = 200 Frw 3. Ground nuts = 200 Frw
2. Soap = 200 Frw 4. Irish potatoes= 300 Frw
8.7 Good use and management of money
Activity 8.7. 1
Activity 8.7.2
Tell what these people are doing?
Why do you think they are do so?
How can we keep money safely?
8.8 The habit of saving money
Activity 8.1
Is it good to save money in order to use in the future?
8.9 Starting a small income generating projects
Activity 8.9
Do you have an activity which can help you to get money? …......
8.10 Comparing the amount of money that does not exceed 1000Frw
Activity 8.10.1
Activity 8.10.2
Activity 8.10.3
8.11 Addition and subtraction of Rwandan currency with the sum not exceeding 1000Frw
Activity 8.11
8.12 Multiplication and division of an amount of money by a whole number
Activity 8.12
8.13 Word problems involving the addition or subtraction of money
Activity 8.13
Carefully study the example below:
Butera has 750Frw. He wants to buy a book which costs 950Frw.
How much more money will he need to buy that book?
Solve the following problems:
1. Mahoro bought a notebook at 350frw and pens that cost 200Frw. How much money did Mahoro pay?
2. Shema had a note of 500Frw. He went to buy a bottle of water at 300Frw. What was the balance.
3. Manirakiza was paid 900Frw. He bought juice and remained with 200Frw. How much money did he use to buy juice?
4. Gasore had 900Frw. He went to buy bread and he remained with 250Frw. How much money did he pay on the bread?
5. Uwamahoro bought bananas at 600Frw. She bought also one cabbage at 300Frw. How much money did she pay
8.14 Word problems involving the multiplication or division of money by a number
Activity 8.14
Study these example below:
One bottle of soda costs 400Frw. Tom is sent to the shop to buy two bottles of soda.
How much money will he pay?
Solve the following problems:
1 . Peter has 800Frw. If he shares it equally among 4 children. How much money will each child get?
2 . Share 900Frw equally among 3 pupils.
3 . One notebook costs 200Frw. If I buy 2 notebooks, how much money will I pay?
4 . One pizza costs 100Frw. How much money can I use if I buy 10 pizzas for my friends?
5 . Ishimwe wants to buy 6 books. If one book costs 100Frw, how much money will he pay?
END UNIT ASSESSMENT 8
1. Answer by Yes or Not
(a) Rwandan currency is made of different coins only...….….
(b) Rwandan currency is made of different notes only ……
(c) Rwandan currency is made of different coins and different notes...............…
(d) All Rwandan coins and notes have the coat of arm…….
2. Fill in correctly
3. Underline the source of money for your parents
Salary fishing art-craft farming commerce agriculture
4. Use >, < or = to compare amount of money
5. Arrange the following amount of money from the smallest to the largest
(a) 650Frw, 900Frw, 750Frw, 800Frw
(b) 400Frw, 700Frw, 650Frw, 300Frw
6. Arrange the following amount of money from the largest to the smallest
(a) 450Frw, 550Frw, 350Frw, 250Frw, 650Frw.
(b) F 850, F 250, F 500, F 950, F 400.
7. Write the number of coins or notes in the boxes:
8 Word problems
(a) Muhizi had 900Frw and he went to buy 1Kg of sugar. If the price of the sugar is 850Frw per Kg, how much money left?
(b) Keza bought the bread at 500Frw, eggs of 200Frw and one pizza of 200Frw. How much did she pay?
(c) Share 750Frw equally among 5 cyclists. How much money can each cyclist get?
(d) Masabo goes to school every day. If he pays 400Frw per day. How much money does he pay in 2 days?
(e) When I had 950Frw, I bought rice at 1 750Frw. How much money did I remain with?
• 9.1 Reading and Telling Time shown by a clock face
(a) Reading exact time: An hour o’clock
Activity 9.1.1
Activity 9.1.2
I have leant that:
(a) A clock face has two or three handsHour hand: It is the short hand of the clock, It tells time in hours. If it rotates once round the clock face, then the time taken is 12 hours
Minute hand: The long hand of the clock, it tells time in minutes. One full rotation equals 60 minutes
Second hand: The thinnest hand of the clock. it rotates the fastest. Its full rotation equals 60 seconds •
In the clock face we have:
- Numbers from 1 to 12;
- From one number to another there is 1 hour.
(b) Digital watch with numbers and a colon:
- The first number before the colon indicates hours;
- The number after the colon indicates minutes
- One hour is equivalent to 60minutes
- One day is equivalent to 24hours.
(c) A day
- A whole day has 2 main parts:
- A whole day has 2 main parts: Day and night
- Every part has 12 hours.
- The first part is divided in two: Before noon (morning) and after noon.
Activity 9.1.3
Activity 9.1.4
Reading and telling the time
Activity 9.1.5
I have learnt that:
On the watch with hands: When the hour hand reaches a number and the minute hand reaches the number 12,
it is a complete hour. Read the number followed by o’clock.
On the watch with numbers and a colon: When the first number is followed by two zeros after the colon,
it is a complete hour. Example: 7: 00 it is 7 o’clock.
Application activity
b) Half past an hour
Activity 9.1.6
I have leant that:
On the watch with hands: When the hour hand reaches the point half of the interval between two numbers and
the minute hand reaches the number 6, it is a half hour. Read “a half past ….(the previous number)”.
On the watch with numbers and a colon: When the first number is followed by 30 after the colon,
it is that hour past 30 or a half past that hour. Example: 9:30; it is “a half past nine”.
Activity 9.1.7
Application activity 9.1.7
9.2 The Calendar
Activity 9.2.1
a) How many days make a week?
b) What is the first day of the week?
c) What is the last day of the week?
d) How many working days does a week have?
e) How many weekend days does a week have?
I have learnt that:
7 days make a week.
The week starts at the first day (Sunday), it ends at the seventh day (Saturday).
1. How many days do you come to school in a week?
2. When do you go to the church with your family members?
3. On which day of the week do we do marriage parties?
4. Why do we have working days and weekend days?
Activity 9.2.2
(a) How many months are in a year?
(b) Do all months have the same number of days?
(c) List down of months which have 30 days.
(d) Which month the year has fewer days?
(e) How many weeks are in a month?
(f) How many weeks are in a year?
I have learnt that:
One year has 12 months.
- The second month “February” is the month with few
days. It has 28 or 29 days.
-One month has 4 weeks.
- One year has 52 weeks;
- A common year has 365 days. When the month of
February has 29 days, the year has 366 days.
Activity 9.2.3
9.3 Schools’ activities and timetable
Activity 9.3.1
I have learnt that:
An example of a time table showing school activities.
9.4 Preparing a weekly activity plan
Activity 9.4.1
I have learnt that:
- The weekly plan helps us to meet the deadline.
We decide to:
- To respect the timetable;
- Avoid being late at school;
- Meet the existing timeline for activities.
Activity 9.4.2
1. What is the time?
2. Complete the following sentence correctly
3. Write down a list of months with:
a) 31days b) 30 days.
END UNIT ASSESSMENT 9
1. Complete
2. Draw
(a) A clock face with hands showing “ten o’clock”.
(b) A clock face with hands showing “one o’clock”.
3. Complete the table below
• 10.0 Preliminary activities
Activity 10.0.1
Activity 10.0.3
Activity 10.0.1
10.1 Straight lines
(a) Straight and non closed lines
Activity 10.0.1
Activity 10.1.2
(a) Oblique straight line
(b) Horizontal line.
(c) Two vertical lines.
I have learnt that:
There are 4 types of lines:
• The horizontal straight line
• The vertical straight line
• Oblique straight line towards right
• Oblique straight line towards left.
Activity 10.1.3
(b) Closed lines
Activity 10.1.4
Activity 10.1.5
a) a zigzag closed line
b) a closed line
I have learnt that:
A closed line is a line which is not open.
(c) Non straight open lines
Activity 10.1.6
Activity 10.1.7
a) Left open line
b) Top open line
I have learnt:
An open line is a non closed line
Application activity 10.1
(d) Curved lines
Activity 10.1.8
Activity 10.1.9
a) A zigzag line
b)A curved down open line
Activity 10.1.10
I have learnt that:
– Curved lines are non straight lines.
– Zigzag lines are lines made by line segments of different directions.
10.2 Types of angles
(a) Right angle
Activity 10.2 1
Activity 10.2.2
I have learnt that:
A right angle is an angle formed by two intersecting straight lines: the horizontal and the vertical lines
Activity 10.2.3
(b) Acute angle
Activity 10.2.4
I have learnt that:
An acute angle is an angle made by two intersecting straight lines; one of them is oblique and this angle is less than the right angle.
Activity 10.2.5
(c) Obtuse angle
Activity 10. 2.6
Activity 10. 2.7
a) Two oblique lines
b) Horizontal lines and an oblique line.
I have learnt that:
An obtuse angle is greater than a right angle, it is made of:
- Two oblique lines or
- A vertical line and an oblique line
- A horizontal line and an oblique line
Activity 10. 2.8
Application activities 10.2
(d) Comparing right angle, obtuse angle and the acute angle
Activity 10.2.9
I have learnt that:
– A right angle is greater than an acute angle
– An obtuse angle is greater than a right angle
– An obtuse angle is greater than an acute angle
– An acute angle is less that a right angle
– An acute angle is less than an obtuse angle.
END UNIT ASSESSMENT 10
2. Answer by Yes or No
(a) An obtuse angle is greater than a right angle
(b) An obtuse angle is less than an acute angle
(c) A right angle is greater than an acute angle
3. Draw
(a) A right angle
(b) A closed line
(c) An oblique straight towards the right
(d) An obtuse angle
(e) A vertical straight line
(f) An acute angle
g) A horizontal straight line
• 11.0 Preliminary activities
Study this grid carefully.
2) Study this grid carefully.
11.1 Characteristics of a grid
Activity 11.1
I have learnt that:
A grid is formed by vertical and horizontal lines.
Vertical lines are called posts and horizontal lines are called crossing bars.
11.2 Construction of a grid
Activity 11.2
11.3 Putting a point on a grid
Activity 11.3
Activity 11.4
a) The point A is the intersecting point of the crossing bar number 2 and the post number 4 .
b) The point B is the intersecting point of the post number 5 and the crossing bar number 3.
11.4 Location of a point on a grid
Activity 11.5
– Count and number all posts from the first by using numbers: 1, 2, 3, 4, 5, 6.
– Count and number all crossing bars from the first by using numbers: 1, 2, 3, 4, 5, 6.
□ Show a point A at the intersection of post number 4 and crossing bar number 3.
□ Show a point B at the intersection of post number 5 and crossing bar number 6.
The answer is on this grid.
I have learnt that:
When locating a point on a grid, we start by the number of posts and then the number of crossing bars.
The point A is located at the intersection of post number 4 and the crossing bar number 3.
Activity 11.6
1 . Draw a grid with 5 posts and 5 crossing bars.
2. Put a point on:
a) The post number 3 and the crossing bar number 4
b) Post number 4 and the crossing bar number 5 c) Post number 2 and crossing bar number 3
3. Draw a grid with 7 posts and 7 crossing bars.
4. Draw a grid with 8 posts and 8 crossing bars.
Show the point A located at the post number 5 and the crossing bar number 4.
Put the point B at the post number 7 and the crossing bar number 6.
END UNIT ASSESSMENT 11
1. a. Construct a grid with 10 posts and 10 crossing bars.
b. Put the points on the grid at:
(a) Post number 3 and the crossing bar number 7.
(b) Post number 10 and the crossing bar number 8
(c) The crossing bar number 5 and the post number 9.
(d) Crossing bar number 7 and the post number 8
(e) Crossing bar number 4 and the post number 6
(f) Crossing bar number 6 and the post number 10.
2. Locate the position of each point in the given grids
• 12.1 The Square
(a) Properties of a square
Activity 12.1.1
I have learnt that:
Activity 12.1.2
Activity 12.1.3
(b) Perimeter of a square
Activity 12.1.4
Activity 12.1.5
I have learnt that:
Activity 12.1.6
12.2 The Rectangle
(a) Properties of a rectangle
Activity 12.2.1
I have learnt that:
Activity 12.1.6
Activity 12.2.3
(b) Perimeter of a rectangle
Activity 12.2.4
Activity 12.2.5
Activity 12.2.6
12.3 The Triangle
(a) Properties of a triangle
Activity 12.3.1
Activity 12.3.2
Activity 12.3.3
(b) Perimeter of a triangle
Activity 12.3.4
Activity 12.3.5
END UNIT ASSESSMENT 12
1. Name the following figures:
2. Answer by YES or NO
(a) A square has 4 equal sides………………
(b) The short sides of a rectangle are called length (L).….
(c) A rectangle has 4 right angles………………….……..
(d) A square has 4 acute angles………………………….
(e) A rectangle has 3 sides, for which 2 are parallel and equal……………..
(f) The long sides of a rectangle are called Width. …….….
(g) A triangle has 4 sides and 3 angles……………………..
3. Find the perimeter of:
(a) A square with the side of 12cm.
(b) A rectangle with the length of 12cm and the width of 8cm.
(c) A triangle which has: 7cm, 8cm and 9cm of sides.
4. Write 1 on a square, write 2 on a rectangle and write 3 on a triangle.
5. Find the perimeter of a flower gardens with the form of:
(a) A square of 80m of side.
(b) A rectangle with 54m of length and 40m of width.
(c) A triangle with 25m, 27m and 30m of sides.
6. Find the perimeter of the following figures:
• 13.1 Discover the unknown number by quick addition or subtraction
Activity 13.1.1
Activity 13.1.2
Activity 13.1.3
Activity 13.1.4
Activity 13.1.5
Activity 13.1.6
I have learnt that:
Activity 13.1.7
13.2 Finding the missing number in a number sentence with multiplication or division
Activity 13.2
I have learnt that:
13.3. Number pattern
(a) Finding the common difference in a number pattern
Activity 13.3.1
(b) Finding the missing number in the number pattern
Activity 13.3.2
END UNIT ASSESSMENT 13
• 14.1 Making groups of objects and showing them on a pictograph
Activity 14.1
14.2 Describing and interpreting various pictographs showing the number of objects.
Activity 14.2
1. Carefully study the following pictures of objects. Group them by putting together the similar objects.
Count the similar objects and tell their number.
2. Draw a pictograph with the following objects:
a) 6 pens
b) 9 bananas
c) 5 oranges
d) 3 trees.
END UNIT ASSESSMENT 14
1. Carefully study the following pictograph and answer to the following questions
a) How many flowers are missing in order to have a number of flowers that match with the number 4?
b) Which number that matches with the pineapples?
c) How many tomatoes are on the pictograph?
2. Draw a pictograph with the following pictures: 1 notebook,
5 balls, 3 cups, 2 flowers, and 6 leaves.
END OF YEAR ASSESSMENT
1. Write in figures or in words
(a) Four hundred ninety five.
(b) 979:
(c) Five hundred seventy nine
(d) 793:
2. Partition these numbers in hundreds, Tens and ones.
(a) 395: … (b) 921: …
3. Complete with the required number
(a) 6H 9O 4T = (b) 9O 9H 7T = (c) 3O 5T 9H =
4. Use <, > or = to compare the following numbers:
5. Arrange these numbers from the smallest to the biggest number.
(a) 251, 125, 215, 152 (b) 309, 930, 390, 903
6. Arrange these numbers from the biggest to the smallest number.
(a) 571, 175, 517,157 (b) 923, 293, 932, 239
7. Add and write the answer
(a) 123 + 456 = (b) 799 + 102 =
(c) 345 + 567 = (d) 524 + 415 =
8. Subtract and write the answer
(a) 997 – 654 = (c) 934 – 912 =
(b) 756 – 699 = (d) 543 – 497 =
9. Multiply and write the answer
10. Divide these numbers
(a) 996 : 2 = (c) 975 : 5 =
(b) 792 : 3 = (d) 648 : 4 =
11. Complete by 10 or 100
12. Complete the missing numbers
13. Complete the following multiplication tables
14. Find the perimeter of the following geometric figures
15. Name the following angles
16. Work out the following
17. Use >, < or = to compare the following measurements
18. Study the calendar and answer the following questions:
(a) How many days are in this month?
(b) How many Mondays are in this month?
(c) How many Tuesdays are in this month?
(d) How many weekends does this month have?
(e) What is the last day on this month?
19. Read and tell the time?
20. Word problems
(a) The total number of pupils of our school is 985. If 512 of them are girls, find the number of boys.
(b) Last year Karisa planted 432trees. This year he planted 515 trees. Find the total number of trees planted.
(c) Kayiranga has 1000Frw. If he buys 1Kg of sugar at 800Frw, how much money will he remain with?
(d) Butera has 500Frw. He needs to buy a book costing 900Frw. How much more money does he need to buy the book?
(e) Last year, Uwamahoro bought 492hens. In this year he bought 508 more hens.
Find the total number of hens bought by Uwamahoro in two years.
(f) There are 5 rows of chairs in the church. If each row has 101 chairs, find the number of chairs in the church.
(g) Gato paid 800Frw to buy sugar and 100Frw for the bread. How much money did he pay?
Find the quantity of rice I remained with.
(i) We have a tank containing 550 of water. If we use 350 to wash clothes, how much water can we remain with?
(j) Carefully study the picture below showing my way from home to school.
(1) Find the distance from home to school.
(2) Find the distance from home to the market.
(3) Find the distance from the market to school.
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Visualizing a Theory of Everything!
This is constructed interactively using the mouse GUI (or manually through keyboard entry of the node / line tables) from VisibLie_Dynkin found on TheoryOfEverything.org.
The Lie group names are calculated based on the Dynkin topology. The geometric permutation names are calculated (to rank 8) based on the binary pattern of empty and filled nodes. The node and line
colors can be used as indicators in Coxeter projections and/or Hasse diagrams.
This tool can be used to drive the visualizations in VisibLie_E8.
More SciAm images
An emulation (w/particle labels) of an E8 embedding from Lisi’s Dec. 2010 Scientific American article. There are 2 rings of 12 groups of 8 fermions (with 2 overlapping in the center) and 4 rings of
12 bosons.
A few new projections in honor of Garrett’s second SciAm Article this year.
Another article in SciAm
I created an emulation of the article pic (bottom)
and added a well known Lisi projection (middle),
and one of my own clearly showing 2 rings of 12 groups of 8 Fermions and 4 Boson rings of 12 :
My E8 image in SciAm
I finally made it into a notable publication (or at least my image has 😉
Sep. 2010 Scientific American Article “Rummaging for a Final Theory”
Dual 600 Cell (H4+H4/Phi) Petrie Projection to Square,Penta & Hexagon
I have published my ToE.pdf to vixra.org http://vixra.org/abs/1006.0063 . While looking around vixra, I found a paper by Tony Smith on E8 http://vixra.org/pdf/0907.0006v1.pdf. It has much of the
work on his website including references to Lisi and Bathsheba on their visualizations of E8. Some of the references are based on the 600 Cell set H4 & H4/Phi (shown by Richter to be isomorphic to
E8). I can show the E8 Dynkin diagram can indeed be folded to H4.
The H4 and its 120 vertices make up the 4D 600 Cell (which is made up of 96 vertices of the snub-24 Cell and the 24 vertices of the 24 Cell=[16 vertex Tesseract=8-Cell and the 8 vertices of the
4-Orthoplex=16 Cell]). It can be generated from the 240 split real even E8 vertices using a 4×8 rotation matrix:
x = (1, φ, 0, -1, φ, 0, 0, 0)
y = (φ, 0, 1, φ, 0, -1, 0, 0)
z = (0, 1, φ, 0, -1, φ, 0, 0)
w = (0, 0, 0, 0, 0, 0, φ^2, 1/φ)
where φ=Golden Ratio=(1+Sqrt(5))/2
I find in folding from 8D to 4D, that the 6720 edge counts split into two sets of 3360 from E8’s 6720 length Sqrt(2).
I recreated here a E8 Petrie projection which is isomorphic to 2D split real even E8 Petrie projection (except it only produces half of the 6720 “shortest edge” counts, of different length). All of
these use a single set of 3 projection basis vectors to produce a single 3D projected object with 3 unique faces. On one face, it has the famous E8 Petrie projection:
600 Cell E8 Petrie Projection
On another face it has a pentagonal projection:
Cell 600 dual pentagonal projection face
On the third unique face is the hexagonal projection:
Dual 600 cell Hexagonal face
I created a movie (similar to those of Lisi) of the H4+H4/Phi 600 Cells that rotates the 2D H-V E8 face projection to the H-Z Pentagonal face (shown in the middle frames of the movie). This is an
interesting rotation because this rotates through a square projection at around Pi/8.
Cell 600 square projection
A lower quality YouTube is here: 600 Cells E8 to Pentagonal. Download and play the higher quality .avi: cell600E8pent
I also created a 3D perspective spin of that projection (two sets of mutually orthogonal cubic faces). On YouTube:600 Cells 3D Spin. Download the higher quality .avi: cell6003Dspin2
Similar to the work of other 3D artists like Bathsheba and Wizzy in Second Life (SL), this object can be created in Real Life (RL) as 3D laser etched crystal or rezzed as a 3000+ prim object in SL.
The 3 (H,V,Z) basis projection vectors are:
600 Cells H, V, Z Projection Basis Vectors
While these are 8D vectors for the 600 cells – they could just as well be 4D (setting the last 4 to zero, as they are simply negatives of the first 4).
Orthogonally, the Zome like model from VisibLie_E8 shows:
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Gray Codes for A-Free Strings
Gray Codes for A-Free Strings
For any $q \geq 2$, let $\Sigma_{q}=\{0,\ldots,q\!-\!1\}$, and fix a string $A$ over $\Sigma_{q}$. The $A$-free strings of length $n$ are the strings in $\Sigma_{q}^n$ which do not contain $A$ as a
contiguous substring. In this paper, we investigate the possibility of listing the $A$-free strings of length $n$ so that successive strings differ in only one position, and by $\pm 1$ in that
position. Such a listing is a Gray code for the $A$-free strings of length $n$.
We identify those $q$ and $A$ such that, for infinitely many $n \geq 0$, a Gray code for the $A$-free strings of length $n$ is prohibited by a parity problem. Our parity argument uses techniques
similar to those of Guibas and Odlyzko (Journal of Combinatorial Theory A 30 (1981) pp. 183–208) who enumerated the $A$-free strings of length $n$. When $q$ is even, we also give the complementary
positive result: for those $A$ for which an infinite number of parity problems do not exist, we construct a Gray code for the $A$-free strings of length $n$ for all $n \geq 0$.
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Q1. Solve for x :
x2+5x−(a2+a−6)=0... | Filo
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Question Text Q1. Solve for :
Updated On Mar 19, 2023
Topic All topics
Subject Mathematics
Class Grade 12
Answer Type Video solution: 1
Upvotes 65
Avg. Video Duration 12 min
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Algebraic Geometry Seminar
Graph Formulas for Tautological Cycles
Speaker: Renzo Cavalieri, Colorado State University
Location: Warren Weaver Hall 201
Date: Tuesday, March 7, 2017, 3:30 p.m.
The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of
curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed
by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago,
Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the
tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the
combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence,
and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.
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If the dice are thrown once more, what is the probability of getting a sum 3? Two dice are thrown simultaneously 500 times. Each time the sum of two numbers appearing on their
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
If the dice are thrown once more, what is the probability of getting a sum 3? Two dice are thrown simultaneously 500 times. Each time the sum of two numbers appearing on their tops is noted and
recorded as given in the following table:
Sum Frequency
Given, two dice are thrown simultaneously 500 times.
Each time the sum of the two numbers appearing on their tops is noted and recorded.
The table represents the sum of the numbers on the top of the dice and their frequencies
We have to determine the probability of getting a sum of 3 if the dice are thrown once more.
Probability of an event = Number of trials in which the event has happened / Total number of trials
From the given table,
Number of time we get a sum of 3 = 30
So, number of trials in which the event has happened = 30
Total number trails = 500
Probability of getting a sum of 3 = 30/500
= 3/50
= 0.06
Therefore, the required probability is 0.06
✦ Try This: Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is 0.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 14
NCERT Exemplar Class 9 Maths Exercise 14.3 Problem 17(i)
If the dice are thrown once more, what is the probability of getting a sum 3? Two dice are thrown simultaneously 500 times. Each time the sum of two numbers appearing on their tops is noted and
recorded as given in the following table: Sum Frequency 2 14 3 30 4 42 5 55 6 72 7 75 8 70 9 53 10 46 11 28 12 15
Two dice are thrown simultaneously 500 times. Each time the sum of two numbers appearing on their tops is noted and recorded as given in the table. If the dice are thrown once more, the probability
of getting a sum 3 is 0.06
☛ Related Questions:
Math worksheets and
visual curriculum
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IML how to calculate new matrix based on values of two matrices
I am looking to generate a matrix that's based on two others (letters are column names and numbers are row numbers to visualize😞
X = {u1 h1 a1, u2 h2 a2, u3 h3 a3};
Y = {b1 f1 r1, b2 f2 r2};
what I am trying to come up with is this:
Z = {(b1-u1)+(f1-h1)+(r1-a1) (b2-u1)+(f2-h1)+(r2-a1)
, (b1-u2)+(f1-h2)+(r1-a2) (b2-u2)+(f2-h2)+(r2-a2)
, (b1-u3)+(f1-h3)+(r1-a3) (b2-u3)+(f2-h3)+(r2-a3)
, (b1-u4)+(f1-h4)+(r1-a4) (b2-u4)+(f2-h4)+(r2-a4)}
what's the most efficient way to go about this? I can only think of do-loops for each expression within parenthesis to create submatrices and then add them together? has any one done something
similar. please point me in the right direction. many thanks in advance!
05-08-2020 09:02 PM
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Subtracting Mixed Numbers With Regrouping Worksheets Pdf
Subtracting Mixed Numbers With Regrouping Worksheets Pdf act as foundational devices in the world of mathematics, offering a structured yet versatile platform for students to explore and understand
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Subtracting Mixed Numbers With Regrouping Worksheets Pdf
Subtracting Mixed Numbers With Regrouping Worksheets Pdf -
Transcript To subtract mixed numbers first align the whole numbers and fractions so they can be subtracted separately If the fractions have different denominators find a common denominator and
convert them accordingly If the fraction on the bottom is larger regroup by borrowing from the whole number on top
Our subtracting mixed numbers worksheets give 4th grade and 5th grade students a head start learning subtraction Subtract the whole number and the fractional parts separately to solve this pdf
exercise Easy Moderate Difficult Subtracting Mixed Numbers with Like Denominators Vertical
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Subtracting Mixed Numbers With Unlike Denominators With Regrouping Worksheet
Subtracting Mixed Numbers With Unlike Denominators With Regrouping Worksheet
1 Change the mixed numbers to improper fractions 2 3 4 11 4 and 11 2 3 2 2 Subtract using the new improper fractions using one of the strategies from above 11 4 3 2 22 12 8 10 8 1 2 8 11 4 The same
strategy used to subtract two mixed numbers is used for adding mixed numbers as well Strategy 2
Case 3 Adding and Subtracting Mixed Numbers Method 1 Step 1 Convert all mixed numbers into improper fractions Step 2 Check Do they have a common denominator If not find a common denominator Step 3
When necessary create equivalent fractions Step 4 Add or subtract the numerators and keep the denominator the same
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Mixed Addition And Subtraction With Regrouping Worksheets
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Subtracting Fractions FREE Subtract pairs of fractions with the same denominator No simplifying Vertical problems example 3 5 1 5 2 5 3rd through 5th Grades View PDF
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Addition And Subtraction Worksheets With Regrouping
Subtracting Mixed Numbers With Borrowing Worksheet
Check more of Subtracting Mixed Numbers With Regrouping Worksheets Pdf below
Adding And Subtracting Mixed Numbers With Regrouping Worksheet Printable Word Searches
Subtracting Fractions With Regrouping Worksheet With Answer Key Printable Pdf Download
Subtracting Fractions Worksheets
Subtracting Mixed Numbers With Regrouping Worksheets Pdf 2023 NumbersWorksheets
16 Adding And Subtracting Mixed Worksheets Worksheeto
Free Printable Subtraction With Regrouping
Subtracting Mixed Numbers Worksheets Math Worksheets 4 Kids
Our subtracting mixed numbers worksheets give 4th grade and 5th grade students a head start learning subtraction Subtract the whole number and the fractional parts separately to solve this pdf
exercise Easy Moderate Difficult Subtracting Mixed Numbers with Like Denominators Vertical
Subtract Mixed Numbers unlike Denominators K5 Learning
Fractions worksheets Subtracting mixed numbers with unlike denominators Below are six versions of our grade 5 math worksheet on subtracting mixed numbers from mixed numbers where the fractional parts
have different denominators These worksheets are
Our subtracting mixed numbers worksheets give 4th grade and 5th grade students a head start learning subtraction Subtract the whole number and the fractional parts separately to solve this pdf
exercise Easy Moderate Difficult Subtracting Mixed Numbers with Like Denominators Vertical
Fractions worksheets Subtracting mixed numbers with unlike denominators Below are six versions of our grade 5 math worksheet on subtracting mixed numbers from mixed numbers where the fractional parts
have different denominators These worksheets are
Subtracting Mixed Numbers With Regrouping Worksheets Pdf 2023 NumbersWorksheets
Subtracting Fractions With Regrouping Worksheet With Answer Key Printable Pdf Download
16 Adding And Subtracting Mixed Worksheets Worksheeto
Free Printable Subtraction With Regrouping
Free Printable 3 Digit Subtraction Worksheets With Regrouping
Subtraction Worksheet With Regrouping 2 Digit
Subtraction Worksheet With Regrouping 2 Digit
Subtracting Mixed Fractions With Borrowing
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EC Dispenser Series
Outsmarting Bugs
EC Dispenser
Level: 6
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Sellback: 72 NGs before 24 Hours, 20 NGs after 24 hours
Location: EbilCorp Weapons Shop
Equip Slot: Head
Damage Type: None
Damage: 19-31
Hits: 4
Energy: 9
Cooldown: 4
Bonuses: None
Special Effects:
1st Hit has a chance for "Corrosive Strike - Armor Dissolves: -X to Defense", lowers opponent's Defense by X for 3 turns, where X is a random number between 20 and 40.
2nd Hit has a chance for "Sugar Coated Weapons: -X% to Damage", lowers opponent's Boost by X for 3 turns, where X is a random number between 20 and 40.
3rd Hit has a chance for "Sugar Rush: -X% Attack Bonus", lowers opponent's Bonus by X for 3 turns, where X is a random number between 20 and 40.
4th Hit has a chance for "Sugar Crash: Losing Energy", causes 3-6 EP DoT for 5 turns.
Combos: None
Description: This insane head weapon dispenses EvilCorp's latest snack sensation, Exploding Candy!
Image: EC Dispenser
< Message edited by Peachii -- 4/4/2011 2:40:17 >
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OpenStax College Physics for AP® Courses, Chapter 30, Problem 49 (Problems & Exercises)
Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)?
a. $1s^1$
b. $1d^3$
c. $4s^2$
d. $3p^7$
e. $6h^{20}$
Question by
is licensed under
CC BY 4.0
Final Answer
(b) and (d) violate constraints on possible quantum numbers.
Solution video
OpenStax College Physics for AP® Courses, Chapter 30, Problem 49 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. We have here some of the rules about atomic quantum numbers and here are some spectroscopic notations for electron states and we have to see which
ones are impossible. So this means that the principal quantum number is 1 so the spectroscopic notation has the first big number in front is the letter n, principal quantum number and then after that
is a letter representing angular momentum quantum number so that's l and then the superscript here is the number of electrons that are in that state. So or in that subshell would be another way to
say that so number of electrons in this l-orbital. So this shell is 1 and the orbital is the s-orbital which corresponds to l equals 0 and there's 1 electron in that state and sure it can be 1
electron in any state so definitely this is a possible spectroscopic notation. Part (b) it says that the principal quantum number is 1 and that there is a subshell d, or an orbital d, with three
electrons. Now three electrons do fit in the d-orbital but the d-orbital cannot exist with a principal quantum number of 1 because it's constrained by this constraint here where the maximum value of
l is 1 less than n. Now with n being 1, that makes the maximum value of l, 0 and so it's at the most l max because n minus 1 which is 1 minus 1 in this case which is 0. And so you can't have a d
-orbital which corresponds to an l equals to 2 so this is not a possible state and that's why we have an 'X' right here. OK. Then we are looking at part (c) and we have an s orbital at principal
quantum number 4 and it contains 2 electrons and that's fine. So I mean s corresponds to l equal to 0 which is certainly possible because n is well l equals 0 is always possible and the s-orbital can
contain 2 electrons; 1 being spin up and other being spin down because with a angular momentum quantum number of 0, the angular momentum projection is going to have to be 0 and in this state of, you
know, n l, m l and m s—these are the four quantum numbers— you can have four, zero, zero and this can be either positive a half spin up or it can be negative a half spin down which corresponds to two
different electrons and so 2 is OK for that subscript; had this been a 3, we would have said no or more. OK. Part (d) says that the principal quantum number is 3 and the orbital is p which
corresponds to l equal to 2 and you cannot have 7 electrons in a p-orbital. Here's a formula for suborbital I should say; here's a formula telling you how many electrons will fit in a suborbital and
it is 2 times 2 times the angular momentum quantum number plus 1 so oh I'm sorry, this is l equal to 1 by the way for p yeah so in the spectroscopic notation, it's l and then the symbol 0 is s, 1 is
p, 2 is d and 3 is f. OK. So this is 6 is the maximum number of electrons you can have in a p-orbital. So not possible because there's a 7 there. And then in (e), h corresponds to l equal to 5 and
with this formula telling you the suborbital capacity, we have 2 times 5 is 10 plus 1 is 11 times 2 makes 22 and so there could be 22 electrons in this suborbital and it's saying 20 there so that's
OK and then we have to check to make sure this number 5 is acceptable given principal quantum number 6 and it is acceptable because l at its most can be 1 less than n and n being 6 makes l being 5 is
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Logistic regression model
This page shows the addition of a logistic regression formula on Evidencio. The standard formula for a logistic regression is:
β[0] is the intercept and Xβ is the model linear predictor. Estimated by the summation of the intercept term and all β values multiplied by that variable. E.g.: β[0] + β[var1] ⋅ var1 + β[var2] ⋅
Numerous logistic regression based models can be found in published research. However, not all models describe the coeficients necessary to add a model on Evidencio. Most models do show the results
of the multivariate analysis as odds ratios. The beta coefficients can be calculated by taking the natural logarithm of the odds ratio. Besides, on Evidencio it is possible to enter odds ratios
instead of beta coefficients. The intercept, however, is really necessary to add the model on Evidencio. Without the intercept, the calculations will be faulty.
Adding a logistic regression model
A step-by-step guide to add a logistic regression model is seen below. A novel Briganti model is the example used in this case, described by Gandaglia et al. (2017). Original article can be found
The coefficients used in this model were added as supplementary data, the model added in this guide concerns model 1, found in the table below.
Deriving model coefficients
The example above shows a model where all the coefficients were given. This is of course very nice, but is unfortunately not always described very well. Several models display the odds ratios and a
nomogram, but the intercept is missing. The example above of the Novel Briganti model also used a nomogram in their published work. The nomogram is displayed below:
So what if we only have the odds ratios and this nomogram for this model. We can actually derive the intercept based on the input in a nomogram.
• We know the beta coefficients if we have the odds ratios. In the nomogram above we can see that PSA is a predictor where the PSA level of 50 ng/ml gets 100 points assigned. So the increase of 1
ng/ml PSA adds 2 points to the calculation.
• We know that the beta coefficient of PSA is 0.0826 (odds ratio 1.086). So every point in the nomogram is worth 0.0413 beta.(0.0826/2)
• We can see in the nomogram that 80 points corresponds with a risk of 7% and a risk of 60% corresponds with approximately 152 points.
□ ∑βX for 80 points is equal to 80 · 0.0413 = 3.304.
□ ∑βX for 152 points is equal to 150 · 0.0413 = 6.195.
• Now we have all the parts of the equation except for the intercept for two different examples. We can enter the formula without the intercept and solve the equation.
□ For the 80 points = 7% risk example we get an intercept of -5.89069
□ For the 152 points = 60% risk example we get an intercept of -5.78953
□ On average our intercept is -5.84011.
□ The intercept of the example model is given, so if we compare our derivation with the given intercept we see that the intercept should be -5.8717.
• With this derivation we can estimate the intercept, but bear in mind that this estimation might not be completely accurate! Still using the formula instead of the nomogram improves accuracy
measures and makes it more efficient to calculate a risk.
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Due Date Calculator - HEALTH - Calculator Cafe | Calculate Online for Free
Wondering and asking yourself "when is my due date?" or "how many weeks pregnant am I? If so, the due date calculator here can help you with that for free. As a pregnancy week calculator, it will
give you information about the duration of pregnancy, weeks to the due date, days to the due date, estimated due date, and even the baby's horoscope.
Due Date
The estimated due date (EDD) is the date when labor is expected to start spontaneously. The due date may be predicted by subtracting 280 days ( 9 months and 7 days) to the beginning day of the last
menstrual period (LMP). "Pregnancy wheels" employ this technique. The consistency of the EDD calculated using this approach is dependent on the mother's exact recollection, normal 28-day cycles, and
the assumption that ovulation and conception occur on day 14 of the cycle.
Pregnancy Calculator
To find out how far along you are in your pregnancy, follow these three basic steps:
1. Determine whether you conceived on the first day of your last menstruation period or on the date of conception.
2. Enter the required dates.
3. Then hit "Calculate" to get an estimate.
Remember that each pregnancy is different, therefore the outcome of the due date calculator will be an estimate rather than a specific date.
First Day of Your Last Menstrual Period (LMP)
Counting 40 weeks from the first day of your last menstrual cycle is the most frequent method for determining your pregnancy due date. That's how the majority of healthcare practitioners go about it.
If your menstrual cycle is the typical length (28 days), it began around two weeks before you conceived. This explains why, instead of 38 weeks, pregnancies are supposed to last 40 weeks.
This approach disregards the length of your menstrual cycle or the time you believe you may have conceived. Women, on average, ovulate around two weeks after their menstrual cycle begins.
Conception Date
Should you know exactly when you conceived – for example, if you used an ovulation predictor kit or kept track of your ovulation symptoms – you may use that information to calculate your pregnancy
due date. Simply select the appropriate calculating technique from the drop-down menu above and enter your date.
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Why is x= _3C_9 impossible to evaluate? | Socratic
Why is #x= _3C_9# impossible to evaluate?
1 Answer
It's not impossible to evaluate: it's just 0.
The best way to think of ${\setminus}_{n} {C}_{r}$ is as "n choose r", or "how many ways can I choose r things from n things?"
In your case, that would mean "how many ways can I choose 9 things from 3 things?" If I only have 3 things, there is no way I can choose 9 things. Hence, there are 0 possible ways to do it.
If you wanted to consider ${\setminus}_{9} {C}_{3}$, we can easily calculate that:
#\ _9C_3 = (9!)/(3!6!) = (9*8*7)/(3*2*1) = 3 * 4 * 7 = 84 #
Impact of this question
4424 views around the world
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Happy PI Day - MANGO Math Group
March 14 (3.14) is just a few days away. Here are some creative activities to celebrate the day and reinforce some important math skills. Thank you to Mrs. Burke, who has put together some wonderful
Pi Day activities, some of which we have linked to below.
• Hold a Pi Day Scavenger Hunt: students have to find quantities of items that align with the numbers of pi
• Have student write a Pi-em, a poem with corresponding number of syllables, words, or letters in each line (3 in the first line, 1 in the second, 4 in the third, etc.)
• Set up the Amazing Pi Race in your classroom or school, students “travel” to places to solve pi-related math problems
• Play Jeopardy with questions about Pi
• Create a paper chain of Pi numerals, with each paper color representing a number. Students can decorate their number. String them up down the hall and see how far you can go
• In shop class, make Pi symbols out of wood or engrave Pi into a box or piece of furniture
• Use actual pie or cakes of different dimensions to do calculations to ensure each student receives the same volume of pie
• Have students bring round things from home (hulla hoops, bike wheels, pizza pan)
• and take measurements of the circumference and diameter (it helps to lay the measurements out flat to visualize the proportion of circumference to diameter)
• Do reports or have discussions about the history of Pi
• Act out a play about scholars using and discussing Pi in ancient history
• Look at the Guinness Book of World Records to learn about who has recited the most digits of Pi. Figure out how long it took and how many digits he recited per minute or per hour.
• See who can memorize the most digits of Pi
• Read “Sir Cumference and the Dragon of Pi” (Cindy Neuschwander), “Piece of Pi” (Naila Bokhari), or “The Joy of Pi” (David Blatner)
• Write your own book about Pi
• Compose a song about the digits of Pi or how the number is used
• Of course, be sure to eat some Pi Pie!
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Waveform Design for a Dual-Function MIMO RadCom System
This example shows how to design a set of waveforms for a dual-function multiple-input-multiple-output (MIMO) radar-communication (RadCom) system. It then uses the designed waveforms to simulate a
RadCom system that can detect targets in range and azimuth, as well as transmit data to multiple downlink users.
Joint Radar-Communication
Radar systems require large bandwidth to achieve high resolution while communication systems require large bandwidth to achieve high data rates. With a growing number of connected devices, the
frequency spectrum becomes increasingly congested. To make the frequency spectrum usage more efficient the radar and the communication communities have recently proposed a host of approaches for
radar-communication coexistence. These approaches range from non-cooperative spectrum sharing that rely on spectrum sensing and dynamic spectrum access to joint RadCom systems optimized for both
functions [1].
This example shows how to design a set of dual-function waveforms for a joint RadCom system. This RadCom system uses a phased array to facilitate MIMO communication. It also takes advantage of
waveform diversity to achieve good radar performance. Assume the carrier frequency is 6GHz, the peak transmit power is 0.5 MW, and the same uniform linear array (ULA) is used for transmitting and
receiving. Create a ULA with $\mathit{N}$ = 16 isotropic antenna elements spaced half of a wavelength apart.
rng('default'); % Set random number generator for reproducibility
fc = 6e9; % Carrier frequency (Hz)
Pt = 0.5e6; % Peak transmit power (W)
lambda = freq2wavelen(fc); % Wavelength (m)
N = 16; % Number of array elements
d = 0.5*lambda; % Array element spacing (m)
% Use isotropic antenna elements
element = phased.IsotropicAntennaElement('BackBaffled', true);
array = phased.ULA('Element', element, 'NumElements', N, 'ElementSpacing', d, 'ArrayAxis', 'y');
Since the objectives of the radar and the communication components of the RadCom system are fundamentally different, each component imposes different requirements on the transmitted waveforms.
Objective of the Radar Component
The goal of the Rad component is to form multiple transmit beams in the directions of targets of interest (for example, the tracker subsystem can provide predicted target positions).
Create a desired transmit beam pattern with 10 degree beams pointing at each target.
% Three targets of interest
tgtAz = [-60 10 45]; % Azimuths of the targets of interest
tgtRng = [5.31e3, 6.23e3, 5.7e3]; % Ranges of the targets of interest
ang = linspace(-90, 90, 200); % Grid of azimuth angles
beamwidth = 10; % Desired beamwidth
% Desired beam pattern
idx = false(size(ang));
for i = 1:numel(tgtAz)
idx = idx | ang >= tgtAz(i)-beamwidth/2 & ang <= tgtAz(i)+beamwidth/2;
Bdes = zeros(size(ang));
Bdes(idx) = 1;
plot(ang, Bdes, 'LineWidth', 2)
xlabel('Azimuth (deg)')
title('Desired Beam Pattern')
grid on
ylim([0 1.1])
Objective of the Communication Component
The goal of the Com component is to transmit communication symbols to downlink users. Let there be K=4 single-antenna users and a communication message consisting of M=30 symbols. Create a K-by-M
matrix S of QPSK symbols that represents the desired communication message.
K = 4; % Number of communication users
M = 30; % Number of communication symbols
Q = 4;
data = randi([0 Q-1], K, M); % Binary data
S = pskmod(data, Q, pi/Q); % QPSK symbols
Create a K-by-N matrix H that models a scattering channel between N RadCom antenna elements and K downlink users. This matrix is typically assumed to be known to the RadCom system prior to the
waveform design.
% User locations are random
txpos = [rand(1, K)*1.5e3; rand(1, K)*2.4e3 - 1.2e3; zeros(1, K)];
% Create a scattering channel matrix assuming 100 independent scatterers
numscat = 100;
rxpos = array.getElementPosition();
H = scatteringchanmtx(rxpos, txpos, numscat).'
H = 4×16 complex
6.9220 - 1.1247i 7.1017 - 0.9594i 7.1987 - 0.8061i 7.2116 - 0.6666i 7.1400 - 0.5421i 6.9853 - 0.4333i 6.7501 - 0.3404i 6.4382 - 0.2632i 6.0549 - 0.2010i 5.6067 - 0.1526i 5.1011 - 0.1165i 4.5465 - 0.0908i 3.9525 - 0.0733i 3.3289 - 0.0617i 2.6864 - 0.0536i 2.0357 - 0.0465i
-10.6335 - 8.1281i -9.9214 - 8.4327i -9.1425 - 8.7545i -8.3116 - 9.0850i -7.4454 - 9.4151i -6.5620 - 9.7352i -5.6804 -10.0353i -4.8206 -10.3055i -4.0023 -10.5361i -3.2454 -10.7180i -2.5688 -10.8428i -1.9903 -10.9031i -1.5263 -10.8926i -1.1909 -10.8064i -0.9961 -10.6411i -0.9511 -10.3950i
4.5973 + 3.1755i 4.6600 + 3.4353i 4.7256 + 3.6782i 4.8029 + 3.9025i 4.9008 + 4.1061i 5.0280 + 4.2865i 5.1927 + 4.4408i 5.4020 + 4.5658i 5.6621 + 4.6581i 5.9779 + 4.7141i 6.3528 + 4.7302i 6.7885 + 4.7027i 7.2849 + 4.6283i 7.8401 + 4.5038i 8.4503 + 4.3264i 9.1098 + 4.0940i
-3.2160 - 5.8685i -3.3480 - 5.4857i -3.5441 - 5.0363i -3.8054 - 4.5215i -4.1313 - 3.9437i -4.5198 - 3.3060i -4.9672 - 2.6122i -5.4682 - 1.8670i -6.0158 - 1.0757i -6.6016 - 0.2443i -7.2158 + 0.6208i -7.8475 + 1.5129i -8.4848 + 2.4247i -9.1149 + 3.3491i -9.7246 + 4.2785i -10.3003 + 5.2058i
Next, the example will show how to create a set of radar waveforms that satisfy the radar objective and form the desired transmit beam pattern. It will then show how to embed the desired
communication message into the synthesized waveforms. Finally, the example will model a transmission from the RadCom system. It will plot the resulting range-angle response and compute the symbol
error rate to verify that the system is capable of performing both functions.
MIMO Radar Waveform Synthesis
The Rad component of the RadCom system is a MIMO radar. Unlike a traditional phased array radar that transmits a scaled version of the same waveform from each antenna element, a MIMO radar utilizes
waveform diversity and transmits $N$ different waveforms. These waveforms combine to form the transmit beam pattern. By adjusting the transmitted waveforms, a MIMO radar can dynamically adjust its
transmit beam pattern.
Let an $N$-by-$M$ matrix $X$ represent the transmitted waveforms such that the rows of $X$ correspond to the $N$ antenna elements and the columns of $X$ to $M$ subpulses. The resulting transmit beam
pattern is [2]
$B\left(\theta \right)=\frac{1}{4\pi }{a}^{H}\left(\theta \right)\cdot R\cdot a\left(\theta \right)$
where $a\left(\theta \right)$ is the array steering vector, $\theta$ is the azimuth angle, and $R=\frac{1}{M}X{X}^{H}$ is the waveform covariance matrix. This relationship shows that design of a
transmit beam pattern for a MIMO radar is equivalent to designing the waveform covariance matrix. Therefore, the waveform design process can be broken into two steps. First, you find a covariance
matrix that produces the desired beam pattern, and then synthesize the waveforms based on the found covariance matrix. Designing the covariance matrix first is much easier compared to selecting the
waveforms directly, since typically $M\gg N$, and thus $R$ has much fewer unknowns than $X$.
From Beam Pattern to Covariance Matrix
Given a desired beam pattern ${B}_{des}\left(\theta \right)$, a waveform covariance matrix $R$ can be found by minimizing the squared error between the desired and the actual beam pattern [2]
$\underset{{}_{R}}{\mathrm{min}}{\int }_{\theta }{|{B}_{des}\left(\theta \right)-\frac{1}{4\pi }{a}^{H}\left(\theta \right)\cdot R\cdot a\left(\theta \right)|}^{2}cos\left(\theta \right)d\theta$
s.t. $diag\left(R\right)=\frac{{P}_{t}}{N}1$
$R\in {\mathbb{S}}_{+}^{N}$
where ${P}_{t}$ is the total transmit power. The first constraint in this optimization problem requires that the diagonal elements of $R$ are equal to $\frac{{P}_{t}}{N}$. This guarantees that each
antenna element transmits equal power, and the total transmitted power is equal to ${P}_{t}$. The second constraint restricts $R$ to the set of positive semidefinite matrices, which is a necessary
requirement for a valid covariance matrix. Use the helperMMSECovariance helper function to solve this optimization problem for the desired beam pattern in Bdes. The solution is the optimal covariance
matrix Rmmse that results in a beam pattern closest to the desired beam pattern in the minimum mean squared error (MMSE) sense.
% Normalize the antenna element positions by the wavelength
normalizedPos = rxpos/lambda;
% Solve the optimization problem to find the covariance matrix
Rmmse = helperMMSECovariance(normalizedPos, Bdes, ang);
% helperMMSECovariance returns a covariance matrix with 1s along the
% diagonal such that the total transmit power is equal to N. Renormalize
% the covariance matrix to make the total transmit power equal to Pt.
Rmmse = Rmmse*Pt/N;
% Matrix of steering vectors corresponding to the angles in the grid ang
A = steervec(normalizedPos, [ang; zeros(size(ang))]);
% Compute the resulting beam pattern given the found covariance matrix
Bmmse = abs(diag(A'*Rmmse*A))/(4*pi);
Compare the beam pattern obtained by optimization of the minimum squared error with the desired beam pattern.
hold on
plot(ang, pow2db(Bdes + eps), 'LineWidth', 2)
plot(ang, pow2db(Bmmse/max(Bmmse)), 'LineWidth', 2)
grid on
xlabel('Azimuth (deg)')
legend('Desired', 'MMSE Covariance', 'Location', 'southoutside', 'Orientation', 'horizontal')
ylim([-30 1])
title('Transmit Beam Pattern')
The beam pattern obtained from the covariance matrix in Rmmse has three main lobes that match the main lobes of the desired beam pattern and correspond to the directions of the targets of interest.
The resulting sidelobe level depends on the available aperture size.
From Covariance Matrix to Waveform
Once the covariance matrix is known, the problem of synthesizing radar waveforms that produce the desired beam pattern reduces to the problem of finding a set of waveforms with a given covariance $R$
. But to be useful in practice these waveforms also must meet a number of constraints. For power-efficient transmission both radar and communication systems typically require the waveforms to have a
constant modulus. Since, it might be numerically difficult to find a set of constant modulus waveforms with a given covariance matrix, the waveforms can be constrained in terms of the
peak-to-average-power ratio (PAR) instead. The PAR for the $n$th transmitted waveform is
where ${P}_{n}=\frac{1}{M}\sum _{m=1}^{M}|X\left(n,m\right){|}^{2}$ is the average power, $n=1,2,\dots .N$, and $m=1,2,\dots ,M$. Notice that the diagonal elements of the waveform covariance matrix
$R\left(n,n\right)$ are equal to the average powers of the waveforms ${P}_{n}$. If each antenna element transmits equal power, and the total transmit power is ${P}_{t}$, then ${P}_{n}=\frac{{P}_{t}}
{N}$. For a set of $N$ waveforms, the following low PAR constraint can be set
$\left\{\begin{array}{c}\begin{array}{ccc}\frac{1}{M}\sum _{m=1}^{M}|X\left(n,m\right){|}^{2}=\frac{{P}_{t}}{N}& & n=1,2,\dots ,N\\ |X\left(n,m\right){|}^{2}\le \eta \frac{{P}_{t}}{N}& & \forall n,m\
text{}\text{and}\text{}\eta \in \left[1,M\right]\end{array}\end{array}$
This constraint requires that the average power of each waveform in the set is equal to the total power divided by the number of antenna elements and that the power for each subpulse does not exceed
the average power by more than a factor of $\eta$. If $\eta$ is set to 1, the low PAR constraint becomes the constant modulus constraint.
In [3] the authors propose a cyclic algorithm (CA) for synthesizing waveforms given a desired covariance matrix such that the PAR constraint is satisfied. The helper function
helperCAWaveformSynthesis implements a version of this algorithm described in [4] that also minimizes the autocorrelation sidelobes for each waveform. Set the parameter $\eta$ to 1.1 and use
helperCAWaveformSynthesis to find a set of waveforms that have a covariance matrix equal to the optimal covariance matrix in Rmmse. Set the number of subpulses in the generated waveform to be equal
to the number of symbols in the desired communication symbol matrix S.
eta = 1.1; % Parameter that controls low PAR constraint
% Find a set of waveform with the covariance equal to Rmmse using the
% cyclic algorithm. The length of each waveform is M.
Xca = helperCAWaveformSynthesis(Rmmse, M, eta);
% Covariance matrix of the computed waveforms
Rca = Xca*Xca'/M;
% The resulting beam pattern
Bca = abs(diag(A'*Rca*A))/(4*pi);
Compare the beam pattern formed by the synthesized waveforms Xca with the beam pattern produced by Rmmse and the desired beam pattern Bdes.
hold on
plot(ang, pow2db(Bdes + eps), 'LineWidth', 2)
plot(ang, pow2db(Bmmse/max(Bmmse)), 'LineWidth', 2)
plot(ang, pow2db(Bca/max(Bca)), 'LineWidth', 2)
grid on
xlabel('Azimuth (deg)')
legend('Desired', 'MMSE Covariance', 'Radar Waveforms', 'Location', 'southoutside', 'Orientation', 'horizontal')
ylim([-30 1])
title('Transmit Beam Pattern')
The waveforms synthesized by CA form a beam pattern close to the beam pattern produced by Rmmse. Some distortion is present, but the main lobes are mostly undistorted. Since the diagonal elements of
Rmmse were constraint to be equal to $\frac{{P}_{t}}{N}$, the synthesized waveforms Xca must have average power equal to 0.5e6/16=31250 watt. Verify that the defined low PAR constraint is satisfied
for all synthesized waveforms.
% Average power
Pn = diag(Rca);
% Peak-to-average power ratio
PARn = max(abs(Xca).^2, [], 2)./Pn;
array2table([Pn.'; PARn.'], 'VariableNames', compose('%d', 1:N), 'RowNames', {'Average Power', 'Peak-to-Average Power Ratio'})
ans=2×16 table
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
Average Power 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250 31250
Peak-to-Average Power Ratio 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1
To verify that the generated waveforms have low range sidelobes (an important requirement for a good radar waveform), plot the autocorrelation function for one of the waveforms in Xca. Use the
sidelobelevel function to compute the peak-to-sidelobe level (PSL).
% Plot the autocorrelation function for the nth waveform in Xca
n = 1;
[acmag_ca, delay] = ambgfun(Xca(n, :), 1, 1/M, "Cut", "Doppler");
acmag_ca_db = mag2db(acmag_ca);
psl_ca = sidelobelevel(acmag_ca_db);
ax = gca;
colors = ax.ColorOrder;
plot(delay, acmag_ca_db, 'Color', colors(3, :), 'Linewidth', 2)
yline(psl_ca,'Label',sprintf('Sidelobe Level (%.2f dB)', psl_ca))
xlabel('Lag (subpulses)')
title(sprintf('Autocorrelation Function for Waveform #%d', n))
grid on;
ylim([-45 0]);
The PSL for the generated waveform is -13.85 dB, which is slightly lower than the PSL of a rectangular LFM pulse.
Radar-Communication Waveform Synthesis
The RadCom system uses the same waveforms for joint radar and communication. If $H$ is a known channel matrix and $X$ is a matrix containing transmitted waveforms, the signal received by the downlink
users is a product $HX$. One way to design a set of dual-function waveforms is to embed the communication symbols into the radar waveforms such that the received signal $HX$ is as close as possible
to the set of the desired communication symbols $S$. In other words, such that the difference $HX-S$ is minimized. This difference is known as the multi-user interference (MUI).
A set of suitable waveforms is a solution to the following trade-off objective function with two terms: the first term is the MUI, while the second term represents the mismatch between the desired
and the resulting beam pattern
$\underset{X}{\mathrm{min}}\rho ‖HX-S{‖}_{F}^{2}+\left(1-\rho \right)‖X-{X}_{0}{‖}_{F}^{2}$
s.t. $\frac{1}{M}diag\left({XX}^{H}\right)=\frac{{P}_{t}}{N}1$
here ${X}_{0}$ is the initial set of radar waveforms that form the desired beam pattern and $\rho$ is a parameter controlling the trade-off between the radar and the communication function. Again,
the constraint in this optimization problem guarantees that all antenna elements transmit equal power, and the total power is equal to ${P}_{t}$.
If $\rho$ is close to zero, the obtained waveforms will be as close as possible to the initial radar waveforms guaranteeing a good radar performance. However, the MUI in this case can be high
resulting in a poor communication performance. When $\rho$ is close to one, the MUI minimization is prioritized resulting in a good communication performance, but possibly poor radar performance. In
this case the resulting waveforms might diverge significantly from the initial radar waveforms and therefore the resulting beam pattern might be significantly different from the desired beam pattern.
Authors in [5] use the Riemannian Conjugate Gradient (RCG) method to solve the trade-off optimization and find a set of dual-function waveforms. Use the helperRadComWaveform helper function that
implements the RCG method to embed the 4-by-30 set of QPSK symbols S into the set of synthesized radar waveforms Xca using the generated 4-by-16 channel matrix H.
% Radar-communication trade-off parameter
rho = 0.4;
% Use RCG to embed the communication symbols in S into the waveforms in Xca
% assuming known channel matrix H.
Xradcom = helperRadComWaveform(H, S, Xca, Pt, rho);
% Compute the corresponding waveform covariance matrix and the transmit beam pattern
Rradcom = Xradcom*Xradcom'/M;
Bradcom = abs(diag(A'*Rradcom*A))/(4*pi);
Compare the beam pattern formed by the obtained waveforms Xradcom against the desired beam pattern Pdes, the beam pattern obtained using the optimal covariance matrix Rmmse, and the beam pattern
formed by the radar waveforms Xca generated by CA.
hold on;
plot(ang, pow2db(Bdes + eps), 'LineWidth', 2);
plot(ang, pow2db(Bmmse/max(Bmmse)), 'LineWidth', 2);
plot(ang, pow2db(Bca/max(Bca)), 'LineWidth', 2);
plot(ang, pow2db(Bradcom/max(Bradcom)), 'LineWidth', 2);
grid on;
xlabel('Azimuth (deg)');
legend('Desired', 'MMSE Covariance', 'Radar Waveforms', 'RadCom Waveforms',...
'Location', 'southoutside', 'Orientation', 'horizontal', 'NumColumns', 2)
ylim([-30 1])
title('Transmit Beam Pattern')
Embedding communication data into the radar waveform causes some transmit beam pattern distortion, however the main lobes are not affected in a significant way. To show the effect of MUI minimization
compare the autocorrelation functions of a dual-function waveform in Xradcom and its radar counterpart from Xca. Use the sidelobelevel function to compute the PSL of the dual-function waveform.
[acmag_rc, delay] = ambgfun(Xradcom(n, :), 1, 1/M, "Cut", "Doppler");
acmag_rc_db = mag2db(acmag_rc);
psl_rc = sidelobelevel(acmag_rc_db);
ax = gca;
colors = ax.ColorOrder;
hold on;
plot(delay, acmag_ca_db, 'Color', colors(3, :), 'LineWidth', 2)
plot(delay, acmag_rc_db, 'Color', colors(4, :), 'LineWidth', 2)
yline(psl_rc,'Label',sprintf('Sidelobe Level (%.2f dB)', psl_rc))
xlabel('Lag (subpulses)')
title(sprintf('Autocorrelation Function for Waveform #%d', n))
grid on;
ylim([-30 0]);
legend('Radar Waveform', 'RadCom Waveform');
It is evident that after embedding the communication information into the radar waveform the close range sidelobes have increased. However, the PSL has not decreased significantly and is still below
-13 dB. The behavior of the sidelobes depends on the value of the parameter $\rho$. In general, increasing $\rho$ results in increased range sidelobes, since more priority is given to the
communication function and less to the radar.
Joint Radar-Communication Simulation
You will use the designed dual-function waveforms to compute a range-angle response and a symbol error rate. Assume that the RadCom system is located at the origin. Let the duration of each subpulse
be $\tau$=0.25$\mu$s. This means that the bandwidth of the system is $B=1/\tau$=4 MHz. Let the pulse repetition frequency be equal to 15 KHz.
tau = 5e-7; % Subpulse duration
B = 1/tau; % Bandwidth
prf = 15e3; % PRF
fs = B; % Sample rate
% The RadCom system is located at the origin and is not moving
radcompos = [0; 0; 0];
radcomvel = [0; 0; 0];
From each antenna element of the transmit array transmit a pulse that consists of the corresponding dual-function waveform.
t = 0:1/fs:(1/prf);
waveform = zeros(numel(t), N);
waveform(1:M, :) = Xradcom' / sqrt(Pt/N);
transmitter = phased.Transmitter('Gain', 0, 'PeakPower', Pt/N, 'InUseOutputPort', true);
txsig = zeros(size(waveform));
for n = 1:N
txsig(:, n) = transmitter(waveform(:, n));
Radiate the transmitted pulses and propagate them towards the targets through a free space channel.
% Positions of the targets of interest in Cartesian coordinates
[x, y, z] = sph2cart(deg2rad(tgtAz), zeros(size(tgtAz)), tgtRng);
tgtpos = [x; y; z];
% Assume the targets are static
tgtvel = zeros(3, numel(tgtAz));
% Calculate the target angles as seen from the transmit array
[tgtRng, tgtang] = rangeangle(tgtpos, radcompos);
radiator = phased.Radiator('Sensor', array, 'OperatingFrequency', fc, 'CombineRadiatedSignals', true);
radtxsig = radiator(txsig, tgtang);
channel = phased.FreeSpace('SampleRate', fs, 'TwoWayPropagation', true, 'OperatingFrequency',fc);
radtxsig = channel(radtxsig, radcompos, tgtpos, radcomvel, tgtvel);
Reflect the transmitted pulses off the targets and receive the reflected echoes by the receive array.
% Target radar cross sections
tgtRCS = [2.7 3.1 4.3];
% Reflect pulse off targets
target = phased.RadarTarget('Model', 'Nonfluctuating', 'MeanRCS', tgtRCS, 'OperatingFrequency', fc);
tgtsig = target(radtxsig);
% Receive target returns at the receive array
collector = phased.Collector('Sensor', array, 'OperatingFrequency', fc);
rxsig = collector(tgtsig, tgtang);
receiver = phased.ReceiverPreamp('Gain', 0, 'NoiseFigure', 2.7, 'SampleRate', fs);
rxsig = receiver(rxsig);
Process the received pulses by computing the range-angle response.
% Compute range-angle response
rngangresp = phased.RangeAngleResponse(...
resp = zeros(numel(t), rngangresp.NumAngleSamples, N);
% Apply matched filter N times. One time for each waveform. Then integrate
% the results.
for i = 1:N
[resp(:, :, i), rng_grid, ang_grid] = rngangresp(rxsig, flipud(Xradcom(i, :).'));
% Plot the range angle response
resp_int = sum(abs(resp).^2, 3);
resp_max = max(resp_int, [], 'all');
imagesc(ang_grid, rng_grid, pow2db(resp_int/resp_max))
clim([-20 1])
axis xy
xlabel('Azimuth (deg)')
ylabel('Range (m)')
title('Range-Angle Response')
cbar = colorbar;
cbar.Label.String = '(dB)';
All three targets of interest are visible on the range-angle response. Now, transmit the dual-function waveforms through the MIMO scattering channel H and compute the resulting error rate.
rd = pskdemod(H*Xradcom, Q, pi/Q);
[numErr, errRatio] = symerr(data, rd)
The error rate is close to 4.2%. Note that the resulting performance heavily depends on the trade-off parameter $\rho$. To show this for different values of $\rho$ compute the symbol error rate, the
PSL of the range response (averaged over N waveforms), and the squared error between the desired beam pattern and the beam pattern formed by the dual-function waveforms.
% Vary the radar-communication trade-off parameter from 0 to 1
rho = 0.0:0.1:1;
er = zeros(size(rho));
psl_ca = zeros(size(rho));
bpse = zeros(size(rho));
for i = 1:numel(rho)
Xrc = helperRadComWaveform(H, S, Xca, Pt, rho(i));
% Transmit through the communication channel and compute the error
% rate
rd = pskdemod(H*Xrc, Q, pi/Q);
[~, er(i)] = symerr(data, rd);
% Compute the peak to sidelobe level
psl_ca(i) = helperAveragePSL(Xrc);
% Compute the beam pattern
Rrc = Xrc*Xrc'/M;
Brc = abs(diag(A'*Rrc*A))/(4*pi);
% Squared error between the desired beam pattern and the beam pattern produced by the RadCom waveforms Xrc
bpse(i) = trapz(deg2rad(ang), (Bdes.' - Brc/max(Brc)).^2.*cosd(ang).');
tiledlayout(3, 1);
ax = nexttile;
semilogy(rho, er, 'LineWidth', 2);
xlim([0 1]);
ylim([1e-3 1]);
grid on;
title('Symbol Error Rate');
plot(rho, psl_ca, 'LineWidth', 2);
grid on;
title({'Peak to Sidelobe Level', '(averaged over 16 waveforms)'});
plot(rho, pow2db(bpse), 'LineWidth', 2);
grid on;
title('Squared Error Between the Desired and the RadCom Beam Patterns');
Increasing $\rho$ decreases the symbol error rate. When $\rho$ is equal to 0.8 or larger the symbol error rate becomes zero. This is due to a relatively small number of transmitted symbols used in
this simulation. On the other hand, increasing $\rho$ results in the higher sidelobes and a large error between the desired and the resulting transmit beam pattern. By carefully selecting $\rho$ a
balance between the radar and the communication performance can be achieved such that the symbol error rate as well as the sidelobes of the range response and the beam pattern distortion are low.
This example shows how to design a set of dual-function waveforms for a joint MIMO RadCom system. The resulting waveforms must form a desired beam pattern to steer transmit beams towards multiple
radar targets of interest and transmit a desired symbol matrix to the downlink users over a known MIMO channel. First, the example shows how to synthesize a set of radar waveforms that produce the
desired transmit beam pattern. Then, the communication symbols are embedded into the synthesized radar waveforms such that the multi-user interference and the distortions in the transmit beam pattern
are minimized. The example then simulates a transmission of the designed waveforms and computes the resulting range-angle response and symbol error rate.
1. Martone, Anthony, and Moeness Amin. "A view on radar and communication systems coexistence and dual functionality in the era of spectrum sensing." Digital Signal Processing 119 (2021): 103135.
2. Fuhrmann, Daniel R., and Geoffrey San Antonio. "Transmit beamforming for MIMO radar systems using signal cross-correlation." IEEE Transactions on Aerospace and Electronic Systems 44, no. 1
(2008): 171-186.
3. Stoica, Petre, Jian Li, and Xumin Zhu. "Waveform synthesis for diversity-based transmit beampattern design." IEEE Transactions on Signal Processing 56, no. 6 (2008): 2593-2598.
4. He, Hao, Jian Li, and Petre Stoica. Waveform design for active sensing systems: a computational approach. Cambridge University Press, 2012.
5. Liu, Fan, Longfei Zhou, Christos Masouros, Ang Li, Wu Luo, and Athina Petropulu. "Toward dual-functional radar-communication systems: Optimal waveform design." IEEE Transactions on Signal
Processing 66, no. 16 (2018): 4264-4279.
Supporting Functions
type 'helperMMSECovariance.m'
function R = helperMMSECovariance(elPos, Pdesired, ang)
% This function computes a waveform covariance matrix that generates a
% desired transmit beam pattern. The computation is based on the squared
% error optimization described in
% Fuhrmann, Daniel R., and Geoffrey San Antonio. "Transmit beamforming for
% MIMO radar systems using signal cross-correlation." IEEE Transactions on
% Aerospace and Electronic Systems 44, no. 1 (2008): 171-186.
% elPos is a 3-by-N matrix of array element positions normalized by the
% wavelength. Pdesired is the desired beam pattern evaluated at the angles
% specified in ang.
N = size(elPos, 2);
% Initial covariance is random. x_ is a vector of all elements that are
% in the upper triangular part of the matrix above the main diagonal.
x_ = initialCovariance(N);
% Normalized the desired beam pattern such that the total transmit power is
% equal to the number of array elements N.
Pdesired = N * Pdesired / (2*pi*trapz(deg2rad(ang), Pdesired.*cosd(ang)));
Pdesired = Pdesired * 4 * pi;
% Matrix of steering vectors corresponding to angles in ang
A = steervec(elPos, [ang; zeros(size(ang))]);
% Parameters of the barrier method
mu = 4;
% The barrier term is weighted by 1/t. At each iteration t is multiplied
% by mu to decrease the contribution of the barrier function.
t = 0.02;
epsilon = 1e-1;
stopCriteriaMet = false;
J_ = squaredErrorObjective(x_, t, A, Pdesired, ang);
while ~stopCriteriaMet
% Run Newton optimization using x_ as a starting point
x = newtonOptimization(x_, t, A, Pdesired, ang);
J = squaredErrorObjective(x, t, A, Pdesired, ang);
if abs(J) < abs(J_)
stopCriteriaMet = abs(J - J_) < epsilon;
x_ = x;
J_ = J;
% Increased t by too much, step back a little.
t = t / mu;
mu = max(mu * 0.8, 1.01);
t = t * mu;
R = constrainedCovariance(x, N);
function x = newtonOptimization(x_, t, A, Pdesired, ang)
epsilon = 1e-3;
% Parameters for Armijo rule
s = 2;
beta = 0.5;
sigma = 0.1;
stopCriteriaMet = false;
J_ = squaredErrorObjective(x_, t, A, Pdesired, ang);
while ~stopCriteriaMet
[g, H] = gradientAndHessian(x_, t, A, Pdesired, ang);
% Descent direction
d = -(H\g);
% Compute step size and the new value x using the Armijo rule
m = 0;
gamma = g'*d;
while true
mu = (beta^m)*s;
x = x_ + mu*d;
J = squaredErrorObjective(x, t, A, Pdesired, ang);
if abs(J_) - abs(J) >= (-sigma*mu*gamma)
x_ = x;
stopCriteriaMet = abs(J - J_) < epsilon;
J_ = J;
m = m + 1;
function [G, H] = gradientAndHessian(x, t, A, Pdesired, ang)
N = size(A, 1);
M = N*(N-1);
F = constrainedCovariance(x, N);
numAng = numel(ang);
FinvFi = zeros(N, N, M);
Alpha = zeros(M, numAng);
idxs = find(triu(ones(N, N), 1));
[r, c] = ind2sub([N N], idxs);
for i = 1:M/2
[Fi_re, Fi_im] = basisMatrix(N, r(i), c(i));
% Matrix inverses used in Eq. (26) and (27)
FinvFi(:, :, i) = F\Fi_re;
FinvFi(:, :, i + M/2) = F\Fi_im;
% Eq. (29)
Alpha(i, :) = 2*real(conj(A(r(i), :)) .* A(c(i), :));
Alpha(i + M/2, :) = -2*imag(conj(A(r(i), :)) .* A(c(i), :));
G = zeros(M, 1);
H = zeros(M, M);
D = (real(diag(A'*F*A).') - Pdesired) .* cosd(ang);
ang_rad = deg2rad(ang);
for i = 1:M
% Eq. (33a)
G(i) = -trace(squeeze(FinvFi(:, :, i))) * (1/t) + 2*trapz(ang_rad, Alpha(i, :) .* D);
for j = i:M
% Eq. (33b)
H(i, j) = trace(squeeze(FinvFi(:, :, i))*squeeze(FinvFi(:, :, j))) * (1/t) ...
+ 2*trapz(ang_rad, Alpha(i, :).*Alpha(j, :).*cosd(ang));
H = H + triu(H, 1)';
function [Fi_re, Fi_im] = basisMatrix(N, i, j)
Fi_re = zeros(N, N);
Fi_re(i, j) = 1;
Fi_re(j, i) = 1;
Fi_im = zeros(N, N);
Fi_im(i, j) = 1i;
Fi_im(j, i) = -1i;
function J = squaredErrorObjective(x, t, A, Pdesired, ang)
% Squared error between the desired beam pattern in Pdesired and the
% beam pattern formed by a covariance matrix defined by the vector x
% containing the above diagonal elements
N = size(A, 1);
% Beam patter defined by x
F = constrainedCovariance(x, N);
P_ = real(diag(A'*F*A).');
% Squared error weighted by angle
E = abs(Pdesired - P_).^2 .* cosd(ang);
% Total error over all angles
J = trapz(deg2rad(ang), E);
% Barrier function
d = eig(F);
if all(d >= 0)
phi = -log(prod(d));
phi = Inf;
J = J + (1/t)*phi;
function F = constrainedCovariance(x, N)
% Reconstruct the covariance matrix from a vector x of above diagonal
% values. The diagonal elements are all equal to 1.
Re = zeros(N, N);
Im = zeros(N, N);
M = numel(x);
idxs = triu(ones(N, N), 1) == 1;
Re(idxs) = x(1:M/2);
Im(idxs) = x(M/2+1:end);
F = eye(N, N);
F = F + Re + Re.' + 1i*Im - 1i*Im.';
function x = initialCovariance(N)
% Create a random covariance matrix
X = randn(N, 10*N) + 1i*randn(N, 10*N);
L = sum(conj(X) .* X, 2);
X = X./sqrt(L);
R = X*X';
M = N*(N-1);
x = zeros(M, 1);
% Select the elements that are above the main diagonal
idxs = triu(ones(N, N), 1) == 1;
x(1:M/2) = real(R(idxs));
x(M/2+1:end) = imag(R(idxs));
type 'helperCAWaveformSynthesis.m'
function X = helperCAWaveformSynthesis(R, M, rho)
% This function generates a set of waveforms that form the desired
% covariance matrix R. M is the number of waveform samples and rho is a
% parameter controlling the resulting peak-to-average power ratio (PAR).
% This algorithm is described in the chapter 14 of
% He, Hao, Jian Li, and Petre Stoica. Waveform design for active sensing
% systems: a computational approach. Cambridge University Press, 2012.
% and in
% Stoica, Petre, Jian Li, and Xumin Zhu. "Waveform synthesis for
% diversity-based transmit beampattern design." IEEE Transactions on Signal
% Processing 56, no. 6 (2008): 2593-2598.
N = size(R, 1);
epsilon = 0.1;
% P can control the autocorrelation level for each waveform. If P = M the
% optimization is trying to minimize sidelobes for across all M-1 lags.
P = M;
% Arbitrary semi-unitary matrix
U = (randn(P + M - 1, N*P) + 1i*randn(P + M - 1, N*P))/sqrt(2);
% Expended covariance is used to minimize the autocorrelation sidelobes. See
% eq. (14.10) and (14.11)
Rexp = kron(R, eye(P));
% Square root of the expended covariance
[L, D] = ldl(Rexp);
Rexp_sr = sqrt(D)*L';
% Cyclic algorithm
numIter = 0;
maxNumIter = 1000;
while true
% Z has waveforms that form R but do not satisfy the PAR constraint
% Eq. (14.6)
Z = sqrt(M) * U * Rexp_sr;
X = zeros(M, N);
Xexp = zeros(P + M - 1, N*P);
% Go through each waveform
for n = 1:N
% Retrieve the nth waveform from the expended matrix Z
zn = getzn(Z, M, P, n);
gamma = R(n, n);
% Enforces PAR constraint using nearest vector algorithm
xn = nearestVector(zn, gamma, rho);
X(:, n) = xn;
% Compute new expended waveform matrix
for p = 1:P
Xexp(p:p+M-1, (n-1)*P + p) = xn;
% Recompute U. Eq. (5) and (11) in the paper
[Ubar, ~, Utilde] = svd(sqrt(M)*Rexp_sr*Xexp', 'econ');
U_ = Utilde * Ubar';
numIter = numIter + 1;
if norm(U-U_) < epsilon || numIter > maxNumIter
U = U_;
X = X';
function zn = getzn(Z, M, P, n)
zn = zeros(M, P);
for i = 1:P
zn(:, i) = Z(i:i+M-1, (n-1)*P + i);
zn = mean(zn, 2);
% Nearest vector algorithm, pages 70-71
function z = nearestVector(z, gamma, rho)
M = numel(z);
S = z'*z;
z = sqrt(M*gamma/S)*z;
beta = gamma * rho;
if all(abs(z).^2 <= beta)
ind = true(M, 1);
for i = 1:M
[~, j] = max(abs(z).^2);
z(j) = sqrt(beta)*exp(1i*angle(z(j)));
ind(j) = false;
S = z(ind)'*z(ind);
z(ind) = sqrt((M-i*rho)*gamma/S)*z(ind);
if all(abs(z(ind)).^2 <= (beta+eps))
type 'helperRadComWaveform.m'
function X = helperRadComWaveform(H, S, X0, Pt, rho)
% This function implements Riemannian Conjugate Gradient (RCG) algorithm
% described in
% Liu, Fan, Longfei Zhou, Christos Masouros, Ang Li, Wu Luo, and Athina
% Petropulu. "Toward dual-functional radar-communication systems: Optimal
% waveform design." IEEE Transactions on Signal Processing 66, no. 16
% (2018): 4264-4279.
% H is a K-by-N channel matrix where K is the number of down-link users
% and N is the number of antennas at the RadCom base station. S is a K-by-L
% symbol matrix, where L is the number of symbols. X0 is a N-by-L matrix of
% waveforms that generate the desired radar beam pattern. Pt is total
% transmit power, and rho is a trade-off parameter to balance the radar and
% communication performance.
delta = 0.5e-6;
maxNumIterations = 600;
[N, M] = size(X0);
S = S * sqrt(Pt/size(S, 1));
A = [sqrt(rho)*H; sqrt(1-rho)*eye(N)];
B = [sqrt(rho)*S; sqrt(1-rho)*X0];
X = (randn(N, M) + 1i*randn(N, M))/sqrt(2);
X = Rx(X, 0, Pt);
% Euclidean gradient
dF = 2 * A' * (A * X - B);
% Projection of the gradient onto the tangent space at X
gradF = Px(X, dF, Pt);
% Descent direction
G = -gradF;
% Armijo rule parameters
s = 1;
beta = 0.5;
sigma = 0.01;
k = 1;
while true
fx = norm(A*X-B, 'fro').^2;
% Iterate until the maximum number of iterations is reached or the norm
% of the gradient is less than delta
if k > maxNumIterations || norm(gradF, 'fro') < delta
% If the inner product of the gradient and the descent direction is
% positive reset the descent direction and make it equal to the
% negative gradient
gamma = real(trace(gradF*G'));
if gamma > 0
G = -gradF;
gamma = real(trace(gradF*G'));
% Step size and new X computation using Armijo rule
m = 0;
mu = 1;
while (-sigma*mu*gamma) > 0
mu = (beta^m)*s;
X_ = Rx(X, mu*G, Pt);
fx_ = norm(A*X_ - B, 'fro').^2;
if (fx(1) - fx_(1)) >= (-sigma*mu*gamma)
X = X_;
m = m + 1;
% Previous projected gradient
gradF_ = gradF;
% Previous descent direction
G_ = G;
% Euclidean gradient at X
dF = 2 * A' * (A * X - B);
% Projection of the gradient onto the tangent space at X
gradF = Px(X, dF, Pt);
% Polak-Ribiere combination coefficient
tau = real(trace(gradF'* (gradF - Px(X, gradF_, Pt)))) / real(trace(gradF_'*gradF_));
% New descent direction is a combination of the current negative
% gradient and the previous gradient translated into the current
% tangent space
G = -gradF + tau * Px(X, G_, Pt);
k = k + 1;
% Tangent space projector
function PxZ = Px(X, Z, Pt)
[N, M] = size(X);
d = diag(Z*X');
PxZ = Z - diag(d)*X*(N/(M*Pt));
% Retractor mapping
function RxZ = Rx(X, Z, Pt)
[N, M] = size(X);
Y = X + Z;
d = diag(Y*Y').^(-1/2);
RxZ = sqrt(M * Pt / N)*diag(d) * Y;
type 'helperAveragePSL.m'
function avpsl = helperAveragePSL(X)
[n, m] = size(X);
psl = zeros(n, 1);
for i = 1:n
acmag = ambgfun(X(i, :), 1, 1/m, "Cut", "Doppler");
psl(i) = acmag(m)/max([acmag(1:m-1) acmag(m+1:end)]);
avpsl = mag2db(mean(psl));
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In Signals Notebook, it only shows one decimal number and rounds it up to this, which is very inaccurate. Could you please explain why?
Date Posted:
Product: Signals Notebook
In Signals Notebook, it only shows one decimal number and rounds it up to this, which is very inaccurate. Could you please explain why?
It seems the number of decimal digits you receive at the end will depend on the number of significant digits on the values the user enters.
For example:
They have a reagent with a FM of 232.06. If they add 0.6 mol (1sf) that equates to 0.6x232.06 g = 139.236g. In kg that is 0.13926, which to 1sf is 0.1 kg.
In reverse, if they specify 0.1kg then that is 100g/232.06 = 0.4309 mol = 0.4 mol to 1 sf
The system is doing what is mathematically appropriate, which is reflecting the same level of accuracy as their least accurate user entered value. If they changed the accuracy of their entered value
as follows they would see a different accuracy:
They have a reagent with a FM of 232.06. If they add 0.60 mol (2sf) that equates to 0.6x232.06 g = 139.236g. In kg that is 0.13926, which to 2sf is 0.14 kg.
In reverse, if they specify 0.10 kg then that is 100g/232.06 = 0.4309 mol = 0.43 mol to 2 sf
They have a reagent with a FM of 232.06. If they add 0.600 mol (3sf) that equates to 0.6x232.06 g = 139.236g. In kg that is 0.13926, which to 3sf is 0.139 kg.
In reverse, if they specify 0.100 kg then that is 100g/232.06 = 0.4309 mol = 0.431 mol to 3 sf
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Article is closed for comments.
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2.4: Lab 4 - Introduction to Folds
Last updated
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Map techniques for folds
Both contouring and stereographic projection are powerful techniques for the analysis of folds. When contouring folded surfaces, you may have to try out more than one possible solution. Map patterns
in folded rocks may show V-shapes that represent true fold hinges, and other V-shapes that represent the effect of valleys and ridges intersecting with planar surfaces. Try to identify any V-shapes
that cannot be attributed to topography; these are likely to be outcrops of fold hinges. Fold hinge outcrops lie on fold axial traces. These are lines on the map that divide the map into different
fold limbs. Try to work on one limb at a time when contouring. Remember, there may be more than one possible solution, but the simplest solution, geologically, is usually the best.
If a fold is cylindrical, then all the planes in the folded surface contain the same line. In principle, if we find the intersection of any two planes, we can define the fold axis. The plane
perpendicular to the fold axis is the profile plane. We can also use the stereographic projection to find the inter-limb angle, if we have measurements of the orientation of the folded surface on the
limbs at the inflection points.
Do the questions in any order, so as to use the various materials when they are available. (Samples for question 1 may not be available outside the lab hours.)
1.* Look at the samples of folded rocks displayed in the lab. Choose one sample in each group, and make a labelled diagram of the fold in profile view. Make your diagram large (fill a whole page).
Your diagram should be a scientific illustration, not a work of art! Use simple clear lines to show boundaries of layers within the sample. If there are too many layers to show precisely, use a
dashed ornament to show the form for the layer traces. Label as many invariant features as you can.
2.* Why would you not label variant features of these folds?
3. Construct a topographic profile and cross-section through Map 1. Proceed as follows. Mark with colours the various surfaces that separate the units. Look for places where V-shapes in the traces
cannot be explained by valleys and ridges in the topography, and lightly circle possible locations for fold hinge points. Draw structure contours for each surface, on each fold limb, using lead
pencil. Use coloured numbers on the contours, to show which contour corresponds to which surface. Use these to construct the geology on the cross-section.
4. Describe the folds as completely as you can using the terms in the previous sections of this manual. (Note that the units are shown in their correct stratigraphic order in the legend.)
5. Examine the map of the Great Cavern Petroleum prospect.
a) Draw structure contours on the fault surface.
b) Contour the top surface of the Great Cavern Limestone. In the southeast of the map, use the intersections of the outcrop trace with the contours. Elsewhere, use the elevations marked against each
of the 26 dry oil wells. Make your contours as smooth as possible, consistent with the data provided. (Make the contours as smooth as possible in 3 dimensions too: this means that the spacing of
contours should be as even as possible on each fold limb.) Mark the position of any fold hinges, and draw hinge lines. Remember that top limestone structure contours will be cut off by fault
structure contours of the same elevation.
c) *Calculate the plunge and trend of the easternmost fold in the area. Keep a note of your answer as you will need it in lab 5.
d) *Construct cross-section A-B
e) Construct cross-section C-D
f) On the map, and on cross-section C-D, show the potential maximum size of an oil reservoir that might have been missed by the existing wells. (Note that in porous units like the Great Cavern
Limestone, oil usually rises to the highest point in the reservoir rock; the base of an oil reservoir is typically a horizontal oil-water contact.) Draw the maximum extent of the potential reservoir
on the map and the cross section. Suggest a spot for well 27 on the map and cross section which would maximize the chance for an oil discovery.
g) *Make an estimate of the maximum potential volume of the reservoir, if the Great Cavern Limestone has 10% porosity. To do this, estimate the maximum possible area of the reservoir in square
metres. Also calculate its maximum vertical thickness. Then, by approximating the shape of the reservoir as a cone, use the equation for the volume of a cone (one third base area times height) to
figure out the approximate volume of the reservoir, and from this, a rough estimate of the maximum volume of petroleum it might contain.
(For a more realistic estimate of the potential resource it would be necessary to take into account the potential presence of other fluids in the reservoir, the effects of pressure, and the
proportion of the fluids that could be economically extracted.)
h) *Research: find out the conversion factor between cubic metres and barrels, and express your answer in barrels!
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Adding And Subtracting Decimals Worksheets 5th Grade
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Can you solve it?
If have login problems remove cookies and clear browser cache.
05-21-2014, 06:36 PM
Post: #129
Leo9ardo Posts: 143
Noob Joined: Dec 2013
RE: Can you solve it?
(05-21-2014 05:54 PM)Donholy28 Wrote:
(05-21-2014 05:11 PM)Leo9ardo Wrote:
(05-21-2014 12:24 PM)Donholy28 Wrote: What is the end of everything?
its g
you cheated didn't u?
is it the right answer ? I just guess . I am not a cheater
05-22-2014, 12:13 AM
Post: #130
Donholy28 Posts: 890
Legend Joined: Nov 2013
RE: Can you solve it?
I am not alive but i need air to grow
05-22-2014, 04:48 AM
Post: #131
Leo9ardo Posts: 143
Noob Joined: Dec 2013
RE: Can you solve it?
(05-22-2014 12:13 AM)Donholy28 Wrote: I am not alive but i need air to grow
Ballon or Plant
05-22-2014, 10:33 PM
Post: #132
Donholy28 Posts: 890
Legend Joined: Nov 2013
RE: Can you solve it?
05-23-2014, 05:52 AM
Post: #133
Leo9ardo Posts: 143
Noob Joined: Dec 2013
RE: Can you solve it?
(05-22-2014 10:33 PM)Donholy28 Wrote: Wrong
05-23-2014, 06:05 AM
Post: #134
Sneha Posts: 232
Noob Joined: Dec 2013
RE: Can you solve it?
(05-23-2014 05:52 AM)Leo9ardo Wrote:
(05-22-2014 10:33 PM)Donholy28 Wrote: Wrong
my answer Fire
05-23-2014, 09:35 AM
Post: #135
Sneha Posts: 232
Noob Joined: Dec 2013
RE: Can you solve it?
Three men in a cafe order a meal the
total cost of which is $15. They each
contribute $5. The waiter takes the
money to the chef who recognizes the
three as friends and asks the waiter to
return $5 to the men.
The waiter is not only poor at
mathematics but dishonest and
instead of going to the trouble of
splitting the $5 between the three he
simply gives them $1 each and pockets
the remaining $2 for himself.
Now, each of the men effectively paid
$4, the total paid is therefore $12. Add
the $2 in the waiters pocket and this
comes to $14.....where has the other $1
gone from the original $15?
05-23-2014, 11:23 AM
Post: #136
Agent P.A.I.N Posts: 464
Noob Joined: Mar 2014
RE: Can you solve it?
If they paid $15 for meal and chef $5 as per you the waiter distributed $1 for three men(which is totally $3 in $5) and pocketed remaining $2,then the whole bucks paid for the meal is $13.How can it
be $12?
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MathFiction: The Theory of Death (Faye Kellerman)
The apparent suicides of a math student and math professor at Kneed Loft College are investigated by a detective, his wife, and a former detective now studying law. It was sufficiently engrossing and
kept me both reading and liking the characters through to the end, which is all one generally asks of a murder mystery.
This novel, one of many in the Decker/Lazarus series, does discuss math and stereotypes of math frequently. Through her characters, the author tries several times to explain the basic ideas behind
Fourier analysis, Fourier transforms and eigenvectors, the tools of the trade for the victims and the prime suspects. Some of the ideas are conveyed accurately, but others (like the concept of an
eigenvector, which is incorrectly defined repeatedly) are not. In any case, although these ideas are "part of the scenery" in this mystery, they do not play an otherwise important role.
The interpersonal dynamics and general "goings on" in this math department are more relevant to the plot. The professors are shown to be working on interesting research projects with the students
(including some applied projects involving investment banking and the mapping of debris in low Earth orbits). Perhaps it is understandable that for the sake of the plot, the conflict between the
faculty (especially over who gets to work with which students) and the anti-social nature of people in math are both exaggerated, but I did occasionally find it annoying. (It was also sometimes funny
to me when the author would get something wrong, such as the way the students kept talking about working in different professors' "labs" and theoretical mathematicians spoke of their "data". That
just isn't the terminology we use.)
The character of Mordecai Gold, a Harvard math professor, who appeared in an earlier Decker/Lazarus novel to help with a decryption problem, plays a small role in this novel as well.
In summary, I can recommend this book to fans of the mystery genre who enjoy a mathematical backdrop for the action. However, one should not expect the representation of math and mathematicians to be
either accurate or enlightening.
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In the figure, BD bisects ∠ABC , point D resides on BD such that
In the figure, BD bisects ∠ABC , point D resides on BD such that ∠A+∠C=180° . Prove that AD=DC
Drop altitudes from D to the two legs of the angle \angle ABC
Since BD bisects \angle ABC ,
In the right triangle \triangle ADE
\angle ADE+ \angle DAE = 90 \degree
\angle ADE+ (180\degree -\angle A ) = 90 \degree
\angle A -\angle ADE =90\degree
In the right triangle \triangle DCF
\angle C+\angle FDC = 90\degree
Addition of (2) and (3) gives
\angle A+\angle C+\angle FDC -\angle ADE=180\degree
Substituting the given condition ∠A+∠C=180° yields
\angle FDC =\angle ADE
According to (1) and (4), \triangle ADE and \triangle DCF are congruent triangles.
\triangle ADE \cong \triangle DCF AD and DC are their corresponding sides. Hence,
AD = DC
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Sudoku Online - 120 new Sudoku puzzles every day!
Our online Sudoku can be easily operated with the mouse or a touch screen. If you click in an empty field, 9 small help numbers appear, which you can toggle and enter as a solution in the field. In
addition, there are 2 hints if you get stuck. An entered number can also be deleted by simply clicking on it, in which case the help numbers appear again.
The help numbers are available in 2 colors, whether you use both colors and how you use them is up to you. For example, you can use the 2nd color to mark 2 possible positions of a number within a
column/row/quadrant if they are already blocking other positions for this number.
To make the online Sudoku as comfortable as possible, we have integrated 2 game modes, a mode for the mouse on the PC and a mode for the touch screen. The Online Sudoku tries to determine the correct
mode automatically, but the game mode can also be switched manually.
In Sudoku mode for the mouse on the PC, the small auxiliary numbers are switched by a simple mouse click and a number is entered as the solution by double-clicking on the corresponding auxiliary
Operation with the mouse is very direct and intuitive, no unnecessary additional clicks.
In touchscreen mode, additional buttons are displayed to switch between the auxiliary numbers and to enter a number as a solution. Any Sudoku field is then activated by tapping it and these buttons
then affect this field.
Direct switching of the auxiliary numbers is deactivated in touchscreen mode, especially as it is difficult to hit the small numbers correctly with your finger. Double-clicking on the touch screen
zooms in and out, so it cannot be used for operation either.
We offer 2 tools for Sudoku Online: scanning of all rows, columns and quadrants, including display of all unique solutions found, and the option of entering a correct number in a field.
These tools are ideal for beginners, i.e. get help and then see for yourself why this solution is correct. But of course you can also use them if you get stuck - the focus should be on fun, not
frustration because you are stuck.
The “Scan everything” button uncovers the auxiliary numbers in all fields and starts a scan process in all rows, columns and quadrants, checks in which fields which number is blocked and also whether
the respective field within the row, column or quadrant must have a certain number. Unique solutions are then highlighted in color.
All numbers entered are taken into account and there is no access to the solution, i.e. if an incorrect number was previously entered in a field, the scanning process can also display incorrect
solution numbers.
The scanning process for Sudoku Online also only takes into account entered numbers and, in some cases, clear solutions that have already been determined. It may therefore be necessary to consider
“ambiguous” positions for further solution numbers, e.g. if a number must be in one of 2 fields within a column/row or quadrant and other positions in other columns/rows/quadrants are already blocked
as a result. This can lead to the scanning process not finding any further solutions.
The " Uncover" button enters the correct number in the Sudoku field. If a field is already selected in touchscreen mode, the correct number is entered here, otherwise a field must be selected.
The Sudoku statistics only record solutions where the solution has been checked once at the end, i.e. all solutions with aids or intermediate checks are not taken into account. There is also a time
window, if a Sudoku was solved impossibly quickly or the solution time is too long, e.g. due to absence in the meantime, it is not included in the statistics. The update of the statistics can also be
Once you have solved and checked a Sudoku, you have the option of sharing your result, including a link, on social media. If a friend now clicks on this link, he will see the given times and actions
in [] in front of the timer and on the buttons and can try to be better and can then also share his result.
In addition to the light and dark designs, there is also the option of displaying the website and the Sudoku in high contrast, i.e. pure black and white. In the Sudoku, the given numbers are then
displayed with a solid underline and incorrectly entered numbers with a dotted underline. This representation can be used to compensate for all forms of color blindness.
A Sudoku consists of 9 columns and 9 rows, which together make 81 Sudoku squares, which in turn are divided into 3x3 square quadrants. The numbers 1-9 appear once in each row/column/quadrant. In a
real Sudoku there is only one solution.
( 878 Words = 5196 Characters)
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ISIS Application Documentation
ratio Printer Friendly View | TOC | Home
Divide two cubes
Overview Parameters Example 1 Example 2
This program divides two cubes. It operates in one of two manners: 1) the denominator cube must have the same number of bands as the numerator cube or 2) the denominator cube must have exactly one
band. In the former case, corresponding bands are divided, for example, band one is divided by band one, band two by band two, and so on. In the later case, all bands in the numerator cube are
divided by the single band in the denominator cube. Special pixels values are handled identically in both cases. Whenever a special pixel occurs, in either the numerator or denominator, the output
pixel is set to NULL. Likewise, if the denominator is zero, the output is set to NULL.
• Math and Statistics
Related Applications to Previous Versions of ISIS
This program replaces the following applications existing in previous versions of ISIS:
• ratio
• cosi
Eric Eliason 1990-09-01 Original version
Jeff Anderson 2002-06-17 Converted to Isis 3.0
Kim Sides 2003-05-13 Added application test
Stuart Sides 2003-05-16 Modified schema location from astogeology... to isis.astrogeology..."
Stuart Sides 2003-07-22 Changed NUM, DEN and To from type filename to type cube
Stuart Sides 2003-07-29 Modified filename parameters to be cube parameters where necessary
Files: NUMERATOR
The numerator cube. All pixels in this cube will be divided by corresponding pixels in the denominator cube.
Type cube
File Mode input
Filter *.cub
Files: DENOMINATOR
The denominator cube. The number of bands in this cube must match the numerator cube or be exactly one. In the former case, a band-by-band division occurs. In the later, each numerator band is
divided by the single denominator band.
Type cube
File Mode input
Filter *.cub
Files: TO
The output cube containing the results of the ratio. A NULL pixel will be output when either of the numerator or denominator pixels is special. Similarly, a NULL will be output if the denominator
pixel is zero.
Type cube
File Mode output
Pixel Type real
Filter *.cub
Example 1
Dividing one band by one band
This example presents dividing one band in a cube by a single band. In this case, we divide band 5 by band 4 in peaks.cub and generate a single band output cube. Note that the file (peaks.cub) does
not have to be the same name for NUMERATOR and DENOMINATOR so long as the spatial size (samples and lines) match between the given files.
Command Line
ratio numerator=peaks.cub:5 denominator=peaks.cub:4 to=ratio.cub
Example 2
Dividing multiple bands by one band
This example presents dividing all bands in a cube by an individual band. In this case, the seven bands in peaks.cub are divided by band 4 and the results output to ratio.cub.
Command Line
ratio numerator=peaks.cub denominator=peaks.cub:4 to=ratio.cub
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Determining the Dynamic Coefficient of Friction Using Newton's Laws
Determining the Dynamic Coefficient of Friction Using Newton's Laws
Core Concepts
The dynamic coefficient of friction is less than the static coefficient of friction, and this relationship can be demonstrated through a theoretical-experimental approach using Newton's laws.
The article presents a theoretical-experimental methodology to model the dynamics of a mass-inclined plane system and determine the relationship between the static and dynamic coefficients of
Key highlights:
1. The static coefficient of friction can be determined from the critical angle at which the object on the inclined plane is at the limit of translational equilibrium.
2. For the dynamic case, the acceleration of the object depends on the angle of inclination and the static coefficient of friction. This can be used to derive an expression for the time it takes the
object to traverse the inclined plane.
3. By comparing the theoretical time using the static coefficient of friction with the experimental time, the dynamic coefficient of friction can be determined.
4. The experimental results show that the dynamic coefficient of friction is indeed less than the static coefficient, and the proposed model accurately describes the system's dynamics.
5. The article also discusses the importance of this experiment in reinforcing the understanding of Newton's laws and the significance of experimental validation of theoretical concepts.
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Obtaining the dynamic coefficient of friction from the application of Newton's laws
The time it takes the object to traverse the inclined plane is given by the expression: t = sqrt(2L / (g * cos(θ) * (1 - μs * tan(θ)))) where: t is the time L is the length of the inclined plane g is
the acceleration due to gravity θ is the angle of inclination μs is the static coefficient of friction
"The dynamic coefficient of friction is less than the static one." "The objective of this didactic proposal is for the student to have a theoretical-experimental design to model and interpret the
dynamics of a physical system in the presence of friction, and be able to give a quantitative answer to the questions that were raised as motivation."
Deeper Inquiries
How would the results change if the materials of the inclined plane and the object were different?
The results of the experiment would likely vary significantly if different materials were used for the inclined plane and the object. The coefficient of friction, both static and dynamic, is highly
dependent on the surface characteristics of the materials in contact. For instance, if a rubber block were placed on a wooden inclined plane, the static and dynamic coefficients of friction would
generally be higher compared to a plastic block on an acrylic surface. This is due to the increased interlocking of surface irregularities and greater adhesive forces between materials with higher
friction coefficients. Consequently, the angle at which the object begins to slide (the critical angle) would be greater for materials with higher friction, leading to a different relationship
between the angle of inclination and the time taken to traverse the ramp. Additionally, the overall dynamics of the system, including acceleration and the time of descent, would be affected,
potentially resulting in a more pronounced difference between the static and dynamic coefficients of friction.
What are the potential sources of error in the experimental measurements, and how could the experimental design be improved to minimize them?
Several potential sources of error could affect the accuracy of the experimental measurements in determining the dynamic coefficient of friction. These include: Measurement Errors: Inaccuracies in
timing due to human reaction time when starting and stopping the timer can lead to significant discrepancies. Using a video camera to record the motion and analyzing the footage frame-by-frame can
help mitigate this error. Surface Conditions: Variations in the surface texture of the inclined plane and the object, such as dirt or wear, can alter the frictional properties. Regular cleaning and
maintenance of the surfaces can help ensure consistent conditions. Angle Measurement: Errors in measuring the angle of inclination can lead to incorrect calculations of the forces involved. Using a
protractor with higher precision or digital inclinometers can improve accuracy. Environmental Factors: Changes in temperature and humidity can affect the materials' properties. Conducting experiments
in a controlled environment can help minimize these effects. Reproducibility: Variability in the experimental setup, such as the placement of the inclined plane or the object, can lead to
inconsistent results. Standardizing the setup and ensuring that all measurements are taken under the same conditions can enhance reproducibility. To improve the experimental design, incorporating
automated timing systems, using high-precision measuring instruments, and conducting multiple trials to average out anomalies would enhance the reliability of the results. Additionally, implementing
a systematic approach to data collection and analysis, including statistical methods to assess the uncertainty, would provide a more robust understanding of the relationship between static and
dynamic friction.
What other physical systems could this theoretical-experimental approach be applied to in order to study the relationship between static and dynamic friction?
The theoretical-experimental approach outlined in the study can be applied to various physical systems to explore the relationship between static and dynamic friction. Some potential applications
include: Inclined Planes with Different Materials: Similar experiments can be conducted using various combinations of materials for both the inclined plane and the object, allowing for a
comprehensive analysis of how different surface interactions affect friction coefficients. Rolling Objects: Investigating the frictional forces involved in rolling objects, such as balls or
cylinders, on inclined surfaces can provide insights into the differences between static and dynamic friction in rolling motion. Sliding Blocks on Different Surfaces: A setup where blocks slide down
different surfaces (e.g., metal, wood, rubber) can help quantify how surface roughness and material properties influence friction. Friction in Mechanical Systems: The approach can be adapted to study
friction in mechanical systems, such as gears or bearings, where understanding the transition from static to dynamic friction is crucial for performance optimization. Automotive Applications: The
principles can be applied to analyze tire friction on various road surfaces, which is vital for vehicle safety and performance, particularly in understanding how tires behave under different
conditions. Sports Equipment: The study of friction in sports, such as the interaction between a ball and a playing surface (e.g., tennis balls on grass vs. clay), can provide valuable insights into
performance and equipment design. By applying this theoretical-experimental methodology to these diverse systems, researchers can deepen their understanding of frictional forces and their
implications in real-world applications.
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Alexander Pruss's Blog
In an earlier post, I said that an account that insists that all fundamental causation is simultaneous but secures the diachronic aspects of causal series by means of divine conservation is “a close
cousin to occasionalism”. For a diachronic causal series on this theory has two kinds of links: creaturely causal links that function instantaneously and divine conservation links that preserve
objects “in between” the instants at which creaturely causation acts. This sounds like occasionalism, in that the temporal extension of the series is entirely due to God working alone, without any
contribution from creatures.
I now think there is an interesting way to blunt the force of this objection by giving another role to creatures using a probabilistic trick that I used in my previous post. This trick allows created
reality to control how long diachronic causal series take, even though all creaturely causation is simultaneous. And if created reality were to control how long diachronic causal series take, a
significant aspect of the diachronicity of diachronic causal series would involve creatures, and hence the whole thing would look rather less occasionalist.
Let me explain the trick again. Suppose time is discrete, being divided into lots of equally-spaced moments. Now imagine an event A[1] that has a probability 1/2 of producing an event A[2] during any
instant that A[1] exists in, as long as A[1] hasn’t already produced A[2]. Suppose A[1] is conserved for as long as it takes to produce A[2]. Then the probability that it will take n units of time
for A[2] to be produced is (1/2)^n+1. Consequently, the expected wait time for A[2] to happen is:
• (1/2)⋅0+(1/4)⋅1+(1/8)⋅2+(1/16)⋅3+...=1.
We can then similarly set things up so that A[2] causes A[3] on average in one unit of time, and A[3] on causes A[4] on average in one unit of time, and so on. If n is large enough, then by the
Central Limit Theorem, it is likely that the lag time between A[1] and A[n] will be approximately n units of time (plus or minus an error on the order of n^1/2 units), and if the units of time are
short enough, we can get arbitrarily good precision in the lag time with arbitrarily high precision.
If the probability of each event triggering the next at an instant is made smaller than 1/2, then the expected lag time from A[1] to A[n] will be less than n, and if the probaility is bigger than 1/
2, the expected lag time will be bigger than n. Thus the creaturely trigger probability parameter, which we can think of as measuring the “strength” of the causal power, controls how long it takes to
get to A[n] through the “magic” of probabilistic causation and the Central Limit Theorem. Thus, the diachronic time scale is controlled precisely by creaturely causation—even though divine
conservation is responsible for A[i] persisting until it can cause A[i+1]. This is a more significant creaturely input than I thought before, and hence it is one that makes for rather less in the
way of occasionalism.
This looks like a pretty cool theory to me. I don’t believe it to be true, because I don’t buy the idea of all causation being simultaneous, but I think it gives a really nice.
Consider the Causal Simultaneity Thesis (CST) that all causation is simultaneous. Assume that simultaneity is absolute (rather than relative). Assume there is change. Here is a consequence I will
argue for: determinism is false. In fact, more strongly, there are no diachronic deterministic causal series. What is surprising is that we get this consequence without any considerations of free
will or quantum mechanics.
Since there is a very plausible argument from presentism to CST (a non-simultaneous fundamental causal relation could never obtain between two existent things given presentism), we get an argument
from presentism to indeterminism.
Personally, I am inclined to think of this argument as a bit of evidence against CST and hence against presentism, because it seems to me that there could be a deterministic world, even though there
isn’t. But tastes differ.
Now the argument for the central thesis. The idea is simple. On CST, as soon as the deterministic causes of an effect are in place, their effect is in place. Any delay in the effect would mean a
violation of the determinism. There can be nothing in the deterministic causes to explain how much delay happens, because all the causes work simultaneously. And so if determinism is true—i.e., if
everything has a deterministic cause—then all the effects happen all at once, and everything is already in the final state at the first moment of time. Thus there is no change if we have determinism
and CST.
The point becomes clearer when we think about how it is an adherent of CST explains diachronic causal series. We have an item A that starts existing at time t[1], persists through time t[2] (kept in
existence not by its own causal power, as that would require a diachronic causal relation, but either by a conserver or a principle of existential inertia), then causes an item B, which then persists
through time t[3] and then causes an item C, and so on. While any two successive items in the causal series A,B,C,... must overlap temporally (i.e., there must be a time at which they both exist),
we need not have temporal overlap between A and C, say. We can thus have things perishing and new things coming into being after them.
But if the causation is deterministic, then as soon as A exists, it will cause B, which will cause C, and so on, thereby forcing the whole series to exist at once, and destroying change.
In an earlier post, I thought this made for a serious objection to CST. I asked: “Why does A ‘wait’ until t[2] to cause B?” But once we realize that the issue above has to do with determinism, we see
that an answer is available. All we need to do is to suppose there is probabilistic causation.
For simplicity (and because this is what fits best with causal finitism) suppose time is discrete. Then we may suppose that at each moment of time at which A exists it has a certain low probability p
[AB] of causing B if B does not already exist. Then the probability that A will cause B precisely after n units of time is (1−p[AB])^np[AB]. It follows mathematically that “on average” it will
cause B after p[AB]/(1−p[AB]) fundamental units of time.
It follows that for any desired average time delay, a designer of the universe can design a cause that has that delay. Let’s say that we want B to come into existence on average u fundamental units
of time after A has come into existence. Then the designer can give A a causal power of producing B at any given moment of time at which B does not already exist with probability p[AB]=1/(1+u).
The resulting setup will be indeterministic, and in particular we can expect significant random variation in how long it takes to get B from A. But if the designer wants more precise timing, that can
be arranged as well. Let’s say that our designer wants B to happen very close to precisely one second after A. The designer can then ensure that, say, there are a million instants of time in a
second, and that A has the power to produce an event A[1] with a probability at any given instant such that the expected wait time will be 0.0001 seconds (i.e., 100 fundamental units of time), and A
[1] the power to produce A[2] with the same probability, and so on, with A[10000]=B. Then by the Central Limit Theorem, the average wait time between A and B can be expected to be fairly close to
10000×0.0001=1 seconds, and the designer can get arbitrarily high confidence of an arbitrarily high precision of delay by inserting more instants in each second, and more intermediate causes
between A and B, with each intermediate cause having an average delay time of 100 fundamental units (say). (This uses the fact that the geometric distribution has a finite third moment and the
Barry-Esseen version of the Central Limit Theorem.)
Thus, a designer of the universe can make an arbitrarily precise and reliable near-deterministic changing universe despite CST. And that really blunts the force of my anti-deterministic observation
as a consideration against CST.
Here’s a tempting principle:
1. If x and y ground z, then the fusion of x and y grounds z.
In other words, we don’t need proper pluralities for grounding—their fusions do the job just as well.
But the principle is false. For the principle is only plausible if any two things have a fusion. But if x and y do not overlap, then x and y ground their fusion. And then (1) would say that the
fusion grounds itself, which is absurd.
This makes it very plausible to think that plural objectual grounding does not reduce to singular objectual grounding.
Start with the concept of “narrowly physical” for facts about the arrangement of physical entities and first-order physical properties such as “charge” and “mass”.
Here are two observations I have not seen made:
1. On Lewis-Ramsey accounts of laws, laws of nature concerning narrowly physical facts do not supervene on narrowly physical facts.
2. On Lewis’s account of causation, causal facts about narrowly physical events do not supervene on narrowly physical facts.
This means that in a Lewisian system we have at least four things we could mean by “physical”:
3. narrowly physical
4. grounded in the laws of narrowly physical facts and/or the narrowly physical facts themselves
5. grounded in the causal facts about narrowly physical events and/or the narrowly physical facts themeselves
6. grounded in the causal facts about narrowly physical events, the laws concerning narrowly physical facts and/or the narrowly physical facts themselves.
Here’s a corollary for the philosophy of mind:
7. On a Lewisian system, we should not even expect the mental properties of purely narrowly physical beings to supervene on narrowly physical facts.
Argument for (1): The laws are the optimal systematization of particular facts. But now imagine a possible world where there is just a coin that is tossed a trillion times, and with no discernible
pattern lands heads about half the time. In the best systematization, we attribute a chance of 1/2 to the coin landing heads. But now imagine a possible world with the same narrowly physical facts,
but where there is an angel that thought about ℵ[3] about a million times—each time, with a good prior mental explanation of the train of thought—and each of these times was a time just before the
coin landed heads. Then the best systematization of the coin tosses will no longer make them simply have a chance of 1/2 of landing heads. Rather, they will have a chance 1/2 of landing heads when
the angel didn’t just think about ℵ[3].
Argument for (2): Add to the world in the above argument some cats and suppose that on any day when the fattest cat in the world eats n mice, that leads the angel to think about ℵ[n], though there
are other things that can get the angel to think about ℵ[n]. We can set things up so that the fattest cat’s eating three mice in a day causes the coin to land heads on the Lewisian counterfactual
account of causation, but if we subtract the angel from the story, this will no longer be the case.
Endicott has observed that functionalism in the philosophy of mind contradicts the widely accepted supervenience of the mental on the physical, because you can have worlds where the functional
features are realized by non-physical processes.
My own view is that a functionalist physicalist shouldn’t worry about this much. It seems to be a strength of a functionalist view that it makes it possible to have non-physical minds, and the
physicalist should only hold that in the actual world all the minds are physical (call this “actual-world physicalism”).
But here is something that might worry a physicalist a little bit more.
• If functionalism and actual-world physicalism are true, there is a possible world which is physically exactly like ours but where there is no pain.
Here is why. On functionalism, pain is constituted by some functional roles. No doubt an essential part of that role is the detection of damage and the production of aversive behavior. Let’s suppose
for simplicity that this role is realized in C-fiber firing in all beings capable of pain (the argument generalizes straightforwardly if there are multiple realizers). Now imagine a possible world
physically just like this one, but with two modifications: there are lots of blissful non-physical angels, and all C-fiber equipped brains have an additional non-physical causal power to trigger
C-fiber firing whenever an angel thinks about that brain. It is no longer true that the functional trigger for C-fiber firing is damage. Now, the functional trigger for C-fiber firing is the
disjunction of damage and being thought about by an angel, and hence C-fiber firing no longer fulfills the functional role of pain. But now add that the angels never actually think about a rain while
that brain is alive, though they easily could. Then the world is physically just like ours, but nobody feels any pain.
One might object that a functional role of a detector is unchanged by adding a disjunct to what is being detected. But that is mistaken. After all, imagine that we modify the hookups in a brain so
that C-fiber firing is triggered by damage and lack of damage. Then clearly we’ve changed the functional role of C-fiber firing—now, the C-fibers are triggered 100% of the time, no matter what—even
though we’ve just added a disjunct.
We can also set up a story where it is the aversive behavior side of the causal role that is removed. For instance, we may suppose that there is a magical non-physical aura normally present
everywhere in the universe, and C-fiber firing interacts with this aura to magically move human beings in the opposite direction to the one their muscles are moving them to. The aura does nothing
else. Thus, if the aura is present and you receive a painful stimulus, you now move closer to the stimulus; if the aura is absent, you move further away. It is no longer the case that C-fibers have
the function of producing aversive behavior. However, we may further imagine that at times random abnormal holes appear in the aura, perhaps due to a sport played by non-physical pain-free imps, and
completely coincidentally a hole has always appeared around any animal while its C-fibers were firing. Thus, the physical aspects of that world can be exactly the same as in ours, but there is no
The arguments generalize to show that functionalists are committed to zombies: beings physically just like us but without any conscious states. Interestingly, these are implemented as the reverse of
the zombies dualists think up. The dualist’s zombies lack non-physical properties that the dualist (rightly) thinks we have, and this lack makes them not be conscious. But my new zombies are
non-conscious precisely because they have additional non-physical properties.
Note that the arguments assume the standard physicalist-based functionalism, rather than Koons-Pruss Aristotelian functionalism.
Taking the privation theory literally, evil is constituted by the non-existence of something that should exist. This leads to a lot of puzzling questions of what that “something” is in cases such as
error and pain.
But I am now wondering whether one couldn’t have a privation theory of evil on which evil is a lack of something, but not of an entity. What do I mean? Well, imagine you’re a thoroughgoing
nominalist, believing in neither tropes nor universals. Then you think that there is no such thing as red, but of course you can say that sometimes a red sign fades to gray. It is natural to say that
the faded sign is lacking the due color red, and the nominalist should be able to say this, too.
Suppose that in addition to being a thoroughgoing nominalist, you are a classical theist. Then you will want to say this: the sign used to participate in God by being red, but now it no longer thusly
participates in God (though it still otherwise participates in God). Even though you can’t be a literal privation theorist, and hold that some entity has perished from the sign, you can be a
privation theorist of sorts, by saying that the sign has in one respect stopped participating in God.
A lot of what I said in the previous two paragraphs is fishy. The “thusly” seems to refer to redness, and “one respect” seems to involve a quantification over respects. But presumably nominalists say
stuff like that in contexts other than God and evil. So they probably think they have a story to tell about such statements. Why not here, then?
Furthermore, imagine that instead of a nominalist we have a Platonist who does not believe in tropes (not even the trope of participating). Then the problems of the “thusly” and “one respect” and the
like can be solved. But it is still the case that there is no entity missing from the sign. Yet we still recognizably have a privation theory.
This makes me wonder: could it be that a privation theory that wasn’t committed to missing entities solve some of the problems that more literal privation theories face?
Aquinas believes that it follows from omnipotence that:
1. Any being that depends on creatures can be created by God without its depending on creatures.
But, plausibly:
2. If x and y are a couple z, then z depends on x and y.
3. If x and y are a couple z, then necessarily if z exists, z depends on x and y.
4. Jill and Joe Biden are a couple.
5. Jill and Joe Biden are creatures.
But this leads to a contradiction. By (4), we have a couple, call it “the Bidens”, consisting by Jill and Joe Biden, and by (2) that couple depends on Jill and Joe Biden. By (1) and (5), God can
create the Bidens without either Jill or Joe Biden. But that contradicts (3).
So, Aquinas’ principle (1) implies that there are no couples. More generally, it implies that there are no beings that necessarily depend on other creatures. All our artifacts would be like that:
they would depend on parts. Thus, Aquinas’ principle implies there are no artifacts.
Thomists are sometimes tempted to say that artifacts, heaps and the like are accidental beings. But the above argument shows that that won’t do. God’s power extends to all being, and whatever being
creatures can bestow, God can bestow absent the creatures. If the accidental beings are beings, God can create them without their parts. But a universe with a heap and yet nothing heaped is absurd.
So, I think, we need to deny the existence of accidental beings.
If we lean on (1) further, we get an argument for survivalism. Either Socrates depends on his body or not. If Socrates does not depend on his body, he can surely survive without his body after death.
But if Socrates does depend on his body, then by (1) God can create Socrates disembodied, since Socrates’ body is a creature. But if God can create Socrates disembodied, surely God can sustain
Socrates disembodied, and so Socrates can survive without his body. In fact, the argument does not apply merely to humans but to every embodied being: bacteria, trees and wolves can all survive death
if God so pleases.
Things get even stranger once we get to the compositional structure of substances. Socrates presumably depends on his act of being. But Socrates’ act of being is itself a creature. Thus, by (1), God
could create Socrates without creating Socrates’ act of being. Then Socrates would exist without having any existence.
I like the sound of (1), but the last conclusion seems disastrous. Perhaps, though, the lesson we get from this is that the esse of Socrates isn’t an entity? Or perhaps we need to reject (1)?
It is tempting to say that I value a wager W at x provided that I would be willing to pay any amount up to x for W and unwilling to pay an amount larger than x. But that’s not quite right. For often
the fact that a wager is being offered to me would itself be relevant information that would affect how I value the wager.
Let’s say that you tossed a fair coin. Then I value a wager that pays ten dollars on heads at five dollars. But if you were to try to sell me that wager for a dollar, I wouldn’t buy it, because your
offering it to me at that price would be strong evidence that you saw the coin landing tails.
Thus, if we want to define how much I value a wager at in terms of what I would be willing to pay for it, we have to talk about what I would be willing to pay for it were the fact that the wager is
being offered statistically independent of the events in the wager.
But sometimes this conditional does not help. Imagine a wager W that pays $123.45 if p is true, where p is the proposition that at some point in my life I get offered a wager that pays $123.45 on
some eventuality. My probability of p is quite low: it is unlikely anybody will offer me such a wager. Consequently, it is right to say that I value the wager at some small amount, maybe a few
Now consider the question of what I would be willing to pay for W were the fact that the wager is being offered statistically independent of the events in the wager, i.e., independent of p. Since my
being offered W entails p, the only way we can have the statistical independence is if my being offered W has credence zero or p has credence one. It is reasonable to say that the closest possible
world where one of these two scenarios holds is a world where p has credence one because some wager involving a $123.45 has already been offered to me. In that world, however, I am willing to pay up
to $123.45 for W. Yet that is not what I value W at.
Maybe when we ask what we would be willing to pay for a wager, we mean: what we would be willing to pay provided that our credences stayed unchanged despite the offer. But a scenario where our
credences stay unchanged despite the offer is a very weird one. Obviously, when an offer is made, your credence that the offer is made goes up, unless you’re seriously irrational. So this new
counterfactual question asks us what we would decide in worlds where we are seriously irrational. And that’s not relevant to the question of how we value the wager.
Maybe instead of asking about the prices at which I would accept an offer, I should instead ask about the prices at which I would make an offer. But that doesn't help either. Go back to the fair coin
case. I value a wager that pays you ten dollars on heads at negative five dollars. But I might not offer it to you for eight dollars, because it is likely that you would pay eight dollars for this
wager only if you actually saw that the coin turned out heads, in which case this would be a losing proposition for me.
The upshot is, I think, that the question of what one values a wager at is not to be defined in terms of simple behavioral tendencies or even simple counterfactualized behavioral tendencies. Perhaps
we can do better with a holistic best-fit analysis.
We might call the following three statements "the Paradox of Charity":
1. In charity, we love our neighbor primarily because of our neighbor’s relation to God.
2. In the best kind of love, we love our neighbor primarily because of our neighbor’s intrinsic properties.
3. Charity is the best kind of love.
I think this paradox discloses something very deep.
Note that the above three statements do not by themselves constitute a strictly logical contradiction. To get a strictly logical contradiction we need a premise like:
4. No intrinsic property of our neighbor is a relation to God.
Now, let’s think (2) through. I think our best reason for accepting (2) is not abstract considerations of intrinsicness, but particular cases of properties. In the best kind of love, perhaps, we love
our neighbor because our neighbor is a human being, is a finite person, has a potential for human flourishing, etc. We may think that these features are intrinsic to our neighbor, but we directly see
them as apt reasons for the best kind of love, without depending on their intrinsicness.
But suppose ontological investigation of such paradigm properties for which one loves one’s neighbor with the best kind of love showed that these properties are actually relational rather than
intrinsic. Would that make us doubt that these properties are a fit reason for the best kind of love? Not at all! Rather, if we were to learn that, we would simply deny (2). (And notice that plenty
of continentally-inclined philosophers do think that personhood is relational.)
And that is my solution. I think (1), (3) and (4) are true. I also think that the best kind of neighbor love is motivated by reasons such as that our neighbor is a human being, or a person, or has a
potential for human flourishing. I conclude from (1), (3) and (4) that these properties are relations to God.
But how could these be relations to God? Well, all the reality in a finite being is a participation in God. Thus, being human, being a finite person and having a potential for human flourishing are
all ways of participating in God, and hence are relations to God. Indeed, I think:
5. Every property of every creature is a relation to God.
It follows that no creature has any intrinsic property. The closest we come to having intrinsic properties are what one might call “almost intrinsic properties”—properties that are relational to God
We can now come back to the original argument. Once we have seen that all creaturely properties are participations in God, we have no reason to affirm (2). But we can still affirm, if we like:
6. In the best kind of love, we love our neighbor primarily because of our neighbor’s almost intrinsic properties, i.e., our neighbor’s relations only to God.
And there is no tension with (1) any more.
Suppose I take a nasty fall while biking. But I remain conscious. Here is the obvious first thing for a formal epistemologist to do: increase my credence in the effectiveness of this brand of
helmets. But by how much?
In an ordinary case of evidence gathering, I simply conditionalize on my evidence. But this is not an ordinary case, because if things had gone otherwise—namely, if I did not remain conscious—I
wouldn’t be able to update or think in any way. It seems like I am now subject to a survivorship bias. What should I do about that? Should I simply dismiss the evidence entirely, and leave unchanged
my credence in the effectiveness of helmets? No! For I cannot deny that I am still conscious—my credence for that is now forced to be one. If I leave all my other credences unchanged, my credences
will become inconsistent, assuming they were consistent before, and so I have to do something to my other credences to maintain consistency.
It is tempting to think that perhaps I need to compensate for survivorship bias in some way, perhaps updating my credence in the effectiveness of the helmet to be bigger than my priors but smaller
than the posteriors of a bystander who had the same priors as I did but got to observe my continued consciousness without a similar bias, since they would have been able to continue to think even
were I to become unconscious.
But, no. What I should do is simply update on my consciousness (and on the impact, but if I am a perfect Bayesian agent, I have already done that as soon as it was evident that I would hit the
ground), and not worry about the fact that if I weren’t conscious, I wouldn’t be around to update on it. In other words, there is no such problem as survivorship bias in the first person, or at least
not in cases like this.
To see this, let’s generalize the case. We have a situation where the probability space is partitioned into outcomes E[1],...,E[n], each with non-zero prior credence. I will call an outcome E[i]
normal if on that outcome you would know for sure that E[i] has happened, you would have no memory loss, and would be able to maintain rationality. But some of the outcomes may be abnormal. I will
have a bit more to say about the kinds of abnormality my argument can handle in a moment.
We can now approach the problem as follows: Prior to the experiment—i.e., prior to the potentially incapacitating observation—you decide rationally what kind of evidence update procedures to adopt.
On the normal outcomes, you get to stick to these procedures. On the abnormal ones, you won’t be able to—you will lose rationality, and in particular your update will be statistically independent of
the procedure you rationally adopted. This independence assumption is pretty restrictive, but it plausibly applies in the bike crash case. For in that case, if you become unconscious, your credences
become fixed at the point of impact or become scrambled in some random way, and you have no evidence of any connection between the type of scrambling and the rational update procedure you adopted. My
story can even handle cases where on some of the abnormal outcomes you don’t have any credences, say because your brain is completely wiped or you cease to exist, again assuming that this is
independent of the update procedure you adopted for the normal outcomes.
It turns out to be a theorem that under conditions like this, given some additional technical assumptions, you maximize expected epistemic utility by conditionalizing when you can, i.e., whenever a
normal outcome occurs. And epistemic utility arguments are formally interchangeable with pragmatic arguments (because rational decisions about wager adoption yield a proper epistemic utility), so we
also get a pragmatic argument. The theorem will be given at the end of this post.
This result means we don’t have to worry in firing squad cases that you wouldn’t be there if you weren’t hit: you can just happily update your credences (say, regarding the number of empty guns, the
accuracy of the shooters, etc.) on your not being hit. Similarly, you can update on your not getting Alzheimer’s (which is, e.g., evidence against your siblings getting it), on your not having fallen
asleep yet (which may be evidence that a sleeping pill isn’t effective), etc., much as a third party who would have been able to observe you on both outcomes should. Whether this applies to cases
where you wouldn’t have existed in the first place on one of the items in the partition—i.e., whether you can update on your existence, as in fine-tuning cases—is a more difficult question, but the
result makes some progress towards a positive answer. (Of course, it woudn’t surprise me if all this were known. It’s more fun to prove things oneself than to search the literature.)
Here is the promised result.
Theorem. Assume a finite probability space. Let I be the set of i such that E[i] is normal. Suppose that epistemic utility is measured by a proper accuracy scoring rule s[i] when E[i] happens for i
∈I, so that the epistemic utility of a credence assignment c[i] is s[i](c[i]) on E[i]. Suppose that epistemic utility is measured by a random variable U[i] on E[i] (not dependent on the choice of
the c[j] for j∈I) for i not in I. Let U(c)=∑[i∈I] 1[E[i]]⋅s[i](c[i])+∑[i∉I] 1[E[i]]⋅U[i]. Assume you have consistent priors p that assign non-zero credence to each normal E[i], and the
expectation E^p of the second sum with respect to these priors is well defined. Then the expected value of U(c) with respect to p is maximized when c[i](A)=p(A∣E[i]) for i∈I. If additionally the
scoring rules are strictly proper, and the p-expectation of the second sum is finite, then the expected value of U(c) is uniquely maximized by that choice of c[i].
This is one of those theorems that are shorter to prove than to state, because they are pretty obvious once fairly clearly formulated.
Normally, all the s[i] will be the same. It's worth thinking if any useful generalization is gained by allowing them to be different. Perhaps there is. We could imagine situtions where depending on
what happens to you, your epistemic priorities rightly change. Thus, if an accident leaves you with some medical condition, knowing more about that medical condition will be valuable, while if you
don't get that medical condition, the value of knowing more about it will be low. Taking that into account with a single scoring rule is apt to make the scoring rule improper. But in the case where
you are conditioning on that medical condition itself, the use of different but individually proper scoring rules when the condition eventuates and when it does not can model the situation rather
Proof of Theorem: Let c[i] be the result of conditionalizing p on E[i]. Then the expectation of s[i](c[i]′) with respect to c[i] is maximized when (and only when, if the conditions of the last
sentence of the theorem hold) c[i]′=c[i] by propriety of s[i]. But the expectation of s[i](c[i]′) with respect to c[i] equals 1/p(E[i]) times the expectation of 1[E[i]]⋅s[i](c[i]′) with respect to
p. So the latter expectation is maximized when (and only when, given the additional conditions) c[i]′=c[i].
Functionalism holds that two (deterministic) minds think the same thoughts when they engage in the same computation and have the same inputs. What does it mean for them to engage in the same
This is a hard question. Suppose two computers run programs that sort a series of names in alphabetical order, but they use different sorting algorithms. Given the same inputs, are the two computers
engaging in the same computation?
If we say “no”, then functionalism doesn’t have the degree of multiple realizability that we thought it did. We have no guarantee that aliens who behave very much like us think very much like us, or
even think at all, since the alien brains may have evolved to compute using different algorithms from us.
If we say “yes”, then it seems we are much better off with respect to multiple realizability. However, there is a tricky issue here: What counts as the inputs and outputs? We just said that the
computers using different sorting algorithms engage in the same computation. But the computer using a quicksort typically returns an answer sooner than a computer using a bubble sort, and heats up
less. In some cases, the time at which an output is produced itself counts as an output (think of a game where timing is everything). And heat is a kind of output, too.
In my toy sorting algorithm example, presumably we didn’t count the timing and the heat as features of the outputs because we assumed that to the human designers and/or users of the computers the
timing and heat have no semantic value, but are merely matters of convenience (sooner and cooler are better). But when we don’t have a designer or user to define the outputs, as in the case where
functionalism is applied to randomly evolved brains, things are much more difficult.
So, in practice, even if we answered “yes” in the toy sorting algorithm case, in a real-life case where we have evolved brains, it is far from clear what counts as an output, and hence far from clear
what counts as “engaging in the same computation”. As a result, the degree to which functionalism yields multiple realizability is much less clear.
One of the well-known challenges in accounting for killing in a just war is the thought that even soldiers fighting on a side without justice think they have justice on their side, hence are
subjectively innocent, and thus it seems wrong to kill them.
But I wonder if there isn’t an opposite problem. As is well-known, human beings have a very strong visceral opposition to killing. Even those who kill with justice on their side are apt to feel
guilty, and it wouldn’t be surprising if often they not only feel guilty but judge themselves to have done wrong. Thus, it could well be that soldiers who kill on both sides of a war have a tendency
to be subjectively guilty, even if one of the sides is waging a just war.
Or perhaps things work out this way: Soldiers who kill tend to be subjectively guilty unless they are waging a clearly just war. If so, then those who are on a side without justice are indeed apt to
be subjectively guilty, since rarely does a side without justice appear manifestly just. And those who are on a side with justice are may very well also be subjectively guilty, unless the war is one
of those where justice is manifest (as was the case for the Allies in World War II).
I doubt that things work out all that neatly.
In any case, the above considerations do show that a side with justice has very strong moral reason to make that justice as manifest as possible to the soldiers. And when that is not possible, those
in charge should be persons of such evident integrity that it is easy to trust their judgment.
Consider this argument:
1. An action is intrinsically evil if and only if it is wrong to do no matter what.
2. In doing anything wrong, one does something (at least) prima facie bad with insufficient moral reason.
3. No matter what, it is wrong to do something prima facie bad with insufficient moral reason.
4. So in doing anything wrong, one performs an intrinsically evil action.
This conclusion seems mistaken. Lightly slapping a stranger on a bus in the face is wrong, but not intrinsically wrong, because if a malefactor was going to kill everyone on the bus who wasn’t
slapped by you, then you should go and slap everybody. Yet the argument would imply that in lightly slapping a stranger on a bus you do something intrinsically wrong, namely slap a stranger with
insufficient moral reason. But it seems mistaken to think that in slapping a stranger lightly you perform an intrinsically evil action.
The above argument threatens to eviscerate the traditional Christian distinction between intrinsic and extrinsic evil. What should we say?
Here is a suggestion. Perhaps we should abandon (1) and instead distinguish between reasons why an action is wrong. Intrinsically evil actions are wrong for reasons that do not depend on
consideration of consequences and extrinsically evil actions are wrong but not for any reasons that do not depend on consideration of consequences.
Thus, lightly slapping a stranger with insufficient moral reason is extrinsically evil because any reason that makes it wrong is a reason that depends on consideration of consequences. On the other
hand, one can completely explain what makes an act of murder wrong without adverting to consequences.
But isn’t the death of the victim a crucial part of the wrongness of murder, and yet a consequence? After all, if the cause of death is murder, then the death is a consequence of the murder.
Fortunately we can solve this: the act is no less wrong if the victim does not die. It is the intention of death, not the actuality of death, that is a part of the reasons for wrongness.
So, when we distinguish between acts made wrong by consequences and and wrong acts not made wrong by consequences, by “consequences” we do not mean intended consequences, but only actual or foreseen
or risked consequences.
But what if Alice slaps Bob with the intention of producing an on-balance bad outcome? That act is wrong for reasons that have nothing to do with actual, foreseen or risked consequences, but only
with her intention. Here I think we can bite the bullet: to slap an innocent stranger with the intention of producing an on-balance bad outcome is intrinsically wrong, just as it is intrinsically
wrong to slap an innocent stranger with the intention of causing death.
Note that this would show that an intrinsically evil action need not be very evil. A light slap with the intention of producing an on-balance slightly bad outcome is wrong, but not very wrong.
(Similarly, the Christian tradition holds that every lie is intrinsically evil, but some lies are only slight wrongs.)
Here is another advantage of running the distinction in this way, given the Jewish and Christian tradition. If an intrinsically evil action is one that is evil independently of consequences, it could
be that such an action could still be turned into a permissible one on the basis of circumstantial factors not based in consequences. And God’s commands can be such circumstantial factors. Thus, when
God commands Abraham to kill Isaac, the killing of Isaac becomes right not because of any new consequences, but because of the circumstance of God commanding the killing.
Could we maybe narrow down the scope of intrinsically evil actions even more, by saying that not just consequences, but circumstances in general, aren’t supposed to be among the reasons for
wrongness? But if we do that, then most paradigm cases of intrinsically evil actions will fail: for instance, that the victim of a murder is innocent is a circumstance (it is not a part of the
agent’s intention).
To avoid scepticism, we need to trust that human epistemic practices and reality match up. This trust is clearly at least a part of a central epistemic virtue.
Now, trusting persons is a virtue, the virtue of faith. But trusting in general, apart from trusting persons, is not. Theism can thus neatly account for how the trusting that is at the heart of human
epistemic practices is virtuous: it is an implicit trust in our creator.
The metaphysical problem of evil consists in the contradiction between:
1. Everything that exists is God or is created by God.
2. God is not an evil.
3. God does not create anything that is an evil.
4. There exists an evil.
The classic Augustinian response is to deny (4) by saying that evil “is” just a lack of a due good. This has serious problems with evil positive actions, errors, pains, etc.
Here is a different way out. Say that a non-fundamental object x is an object x such that the proposition that x exists is wholly grounded in some proposition that makes no reference to x. Now we
deny (3) and replace it with:
5. God does not create anything fundamental that is an evil.
How could God create something non-fundamental that is an evil? By a combination of creative acts and refrainings from creative acts whose joint outcome grounds the existence of the non-fundamental
evil, while foreseeing without intending the non-fundamental evil. Of course, this requires the kind of story about intention that the Principle of Double Effect uses.
Thus, consider George Shaw’s erroneous (initial) error that there are no platypuses. God creates George Shaw. He creates Shaw’s belief. He creates platypuses. The belief isn’t an evil. The platypuses
aren’t an evil. The combination of the belief and the platypuses is an error. But the combination of the two is not a fundamental entity (even if the belief and the platypuses are). God can intend
the belief to exist and the platypuses to exist without intending the combination to exist.
I’ve been thinking a bit about the virtue ethical claim that the right (i.e., obligatory) action is one that a virtuous person would do and the wrong one is one that a virtuous person wouldn’t do.
I’ve argued in my previous posts that this is a problematic claim, since given either naturalism or the Hebrew Scriptures, it is possible for a virtuous person to do something wrong.
Maybe instead of focusing on the person, the virtue ethicist can focus on the virtues. Here is an option:
1. An action is wrong if and only if it could not properly (non-aberrantly) flow from the relevant virtues.
This principle is compatible with a virtuous person doing something wrong, as long as that wrong thing doesn’t flow from virtue.
The “properly” in (1) is an “in the right way” condition. Once we have allowed, as I think we should, that a virtuous person can do the wrong thing, we should also allow that a wrong action can flow
from virtue in some aberrant way. For instance, we can imagine a wholly virtuous person falling prey to a temptation to brag about being wholly virtuous (and instantly losing the virtue, of course).
The bragging flows from the virtue—but aberrantly.
A down-side of (1) is that it is a pretty strong condition on permissibility. One might think that there are some permissible morally neutral actions which can be done by a perfectly virtuous person
but which do not flow from their virtue. If we accept (1), then in effect we are saying that there are no morally neutral actions. I think that is the right thing to say.
The big problem with (1) is the “properly”.
Virtue ethics is committed to this claim:
1. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A.
But (1) implies this generalization:
2. A person who has the relevant virtues explanatorily prior to a choice never chooses wrongly.
In my previous post I argued that Aristotelian Jews and Christians should deny (2), and hence (1).
Additionally, I think naturalists should deny (1). For we live in a fundamentally indeterministic world given quantum mechanics. If a virtuous person were placed in a position of choosing between
aiding and insulting a stranger, there will always be a tiny probability of their choosing to insult the stranger. We shouldn’t say that they wouldn’t insult the stranger, only that they would be
very unlikely to do so (this is inspired by Alan Hajek’s argument against counterfactuals).
And (2) itself is dubious, unless we have such a high standard of virtue that very few people have virtues. For in our messy chaotic world, very little is at 100%. Rare exceptions should be expected
when human behavior is involved.
(Perhaps a dualist virtue ethicist who does not accept the Hebrew Scriptures could accept (1) and (2), holding that a virtuous soul makes the choices and is not subject to the indeterminacy of
quantum mechanics and the chaos of the world.)
There is a natural way out of the above arguments, and that it so to change (1) to a probabilistic claim:
3. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would be very unlikely to have chosen A.
But (3) is false. Suppose that Alice is a virtuous person who has a choice to help exactly one of a million strangers. Whichever stranger she chooses to help, she does no wrong. But it is
mathematically guaranteed that there is at least one stranger such that her chance of helping them is at most one in a million (for if p[n] is her chance of helping stranger number n, then p[1]
+...+p[1000000]≤1, since she cannot help more than one; given that 0≤p[n] for all n, it follows mathematically that for some n we have p[n]≤1/1000000). So her helping a particular such
stranger is very unlikely to be chosen, but isn’t wrong.
Or for a less weighty case, suppose I say something perfectly morally innocent to start off a conversation. Yet it is very unlikely that a virtuous person would have said so. Why? Because there are
so very many perfectly morally innocent ways to start off a conversation, it is very unlikely that they would have chosen the same one I did.
If virtue ethics is correct:
1. An choice is wrong if and only if a person with the relevant virtues and in these circumstances wouldn’t have made that choice. (Premise)
If Aristotelian virtue ethics is correct:
2. An adult lacking a virtue is defective. (Premise)
3. Humans became defective because of the choice of Adam and Eve to eat the forbidden fruit. (Premise)
And it seems that:
4. Adam and Eve were adult humans when they chose to eat the forbidden fruit. (Premise)
Thus it seems:
5. When Adam and Eve chose to eat the forbidden fruit, they were not lacking relevant virtues. (By 2–4)
6. Thus, persons (namely Adam and Eve!) with the relevant virtues and in their circumstances did choose to eat the forbidden fruit. (By 5)
7. Thus, their choice to eat the forbidden fruit wasn’t wrong. (1 and 6)
8. But their choice was wrong. (Premise)
9. Contradiction!
Here is one thing the classic virtue ethicist can question about this argument: the derivation of (5) depends on how we read premise (1). We could read (1) as:
10. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A
or as:
11. A choice of A is wrong if and only if a person who had the relevant virtues while having chosen A and was in these circumstances would not have chosen A.
If we opt for (10), the derivation of (5) works, and the argument stands. But if we opt for (11) then we can say that as soon as Adam and Eve chose to eat the fruit, they no longer counted as
Could the virtue ethicist thus opt for (11) in place of (10)? I don’t think so. It seems central to virtue ethics that the right choices are ones that result from virtue. And that is what (10)
captures. To a great extent (11) would trivialize virtue ethics, in that obviously in doing a bad thing one isn’t virtuous.
Somehow it hasn’t occurred to me until yesterday that quantum multiverse theories (without the traveling minds tweak) undercut half of ethics, just as Lewis’s extreme modal realism does.
For whatever we do, total reality is the same, and hence no suffering is relieved, no joy is added, etc. The part of ethics where consequences matter is all destroyed. There is no point to preventing
any evil, since doing so just shifts which branch of the multiverse one inhabits.
At most what is left of ethics is agent-centered stuff, like deontology. But that’s only about half of ethics.
Moreover, even the agent-centered stuff may be seriously damaged, depending on how one interprets personal identity in the quantum multiverse.
Consider three theories.
On the first, I go to all the outgoing branches, with a split consciousness. On this view, no matter what, there will be branches where I act well and branches where I act badly. So much or all of
the agent-centered parts of ethics will be destroyed.
On the second, whenever branching happens, the persons in the branches are new persons. If so, then there are no agent-centered outcomes—if I am deliberating between insulting or comforting a
suffering person, no matter what, I will do neither, but instead a descendant of me will insult and another descendant will comfort. Again, it’s hard to fit this with the agent-centered parts of
The third is the infinitely many minds theory on which there are infinitely many minds inhabiting my body, and whenever a branching happens, infinitely many move into each branch. In particular, I
will move into one particular branch. On this theory, if somehow I can control which branch I go down (which is not clear), there is room for agent-centered outcomes. But this is not the most
prominent of the multiverse theories.
A standard dualist theory of perception goes like this:
1. You sense stuff physically, the data goes to the brain, the brain processes the data, and out of the processed data produces qualia.
There is a lot of discussion of the “causal closure” of the physical. What people generally mean by this is that the physical is causally backwards-closed: the cause of a physical thing is itself
physical. This is a controversial doctrine, not least because it seems to imply that some physical things are uncaused. But what doesn’t get discussed much is a more plausible doctrine we might call
the forwards causal closure of the physical: physical causes only have physical effects. Forwards causal closure of the physical is, I think, a very plausible candidate for a conceptual truth. The
physical isn’t spooky—and it is spooky to have the power of producing something spooky. (One could leave this at this conceptual argument, or one could add the scholastic maxim that one cannot cause
what one does not in some sense have.)
By forwards closure, on the standard dualist theory, the brain is not a physical thing. This is a problem. It is supposed to be one of the advantages of the standard dualist theory that it is
compatible with property dualism on which people are physical but have non-physical properties. But if the brain is not physical, there is no hope for people to be physical! Personally, I don’t mind
losing property dualism, but it sure sounds absurd to hold that the brain is not physical.
Recently, I have been thinking about a non-causal dualist theory that goes like this:
2. You sense stuff physically, the data goes to the brain, the brain processes the data, and the soul “observes” the brain’s processed data. (Or, perhaps more precisely, the person "feels" the
neural data through the soul.)
To expand on this, what makes one feel pain is not the existence of a pain quale, but a sui generis “observation” relation between the soul and the brain’s processed data. This observation relation
is not caused by the data, but takes place whether there is data there or not (if there isn’t, we have a perceptual blank slate). The soul is not changed intrinsically by the data: the “observation”
of a particular datum—say, a datum representing a sharp pain in a toe—is an extrinsic feature of the soul. Note that unlike the standard theory, this up-front requires substance dualism of some sort,
since the observing entity is not physical given the sui generis nature of the “observation” relation.
The non-causal dualist theory allows one to maintain forwards closure of the physical and the physicality of the brain. For the brain doesn’t cause a non-physical effect. The brain simply gets
It is however possible that the soul causes an effect in the brain—for instance, the “observation” relation may trigger quantum collapse. Thus, the theory may violate backwards closure. And that’s
fine by me. Backwards closure does not follow conceptually from the concept of the physical—a physical thing doesn’t become spooky for having a spooky cause.
There is a difficulty here, however. Suppose that the soul acts on the “observed” data, say by causing one to say “You stepped on my foot.” Wouldn’t we want to say that the brain data correlated with
the pain caused one to say “You stepped on my foot”?
I think this temptation is resistable. Ridiculously oversimplifying, we can imagine that the soul has a conditional causal power to cause an utterance of “You stepped on my foot” under the condition
of “observing” a certain kind of pain-correlated neural state. And while it is tempting to say that the satisfied conditions of a conditional causal power cause the causal power to go off, we need
not say that. We can, simply, say that the causal power goes off, and the cause is not the condition, but the thing that has the causal power, in this case the soul.
On this story, if you step on my foot, you don’t cause me to say “You stepped on my foot”, though you do cause the condition of my conditional causal power to say so. We might say that in an extended
sense there is a “causal explanation” of my utterance in terms of your stepping, and your stepping is “causally prior” to my utterance, even though this causal explanation is not itself an instance
of causation simpliciter. If so, then all the stuff I say in my infinity book on causation should get translated into the language of causal explanation or causal priority. Or we can just say that
there is a broad and a narrow sense of “cause”, and in the broad sense you cause me to speak and in the narrow you do not.
I think there is a very good theological reason to think this makes sense. For we shouldn’t say that our actions cause God to act. The idea of causing God to do anything seems directly contrary to
divine transcendence. God is beyond our causal scope! Just as by forwards closure a physical thing cannot cause a spiritual effect, so too by transcendence a created thing cannot cause a divine
effect. Yet, of course, our actions explain God’s actions. God answers prayers, rewards the just and punishes the unrepentant wicked. There is, thus, some sort of quasi-causal explanatory relation
here that can be used just as much for non-causal dualist perception.
Thursday November 11, 2021, at 4 pm Eastern (3 pm Central), the Rutgers Center for Philosophy of Religion and the Princeton Project in Philosophy of Religion present a joint colloquium: Alex Pruss
(Baylor), "A Norm-Based Design Argument".
The location will be https://rutgers.zoom.us/s/95159158918
Some people don’t like Cantorian ways of comparing the sizes of sets because they want to have a “whole is bigger than the (proper) part” principle, denying which they consider to be
Suppose that there is a relation ≤ which provides a way of comparing the sizes of sets of real numbers (or just the sizes of countable sets of real numbers) such that:
1. the comparison satisfies the “the whole is bigger than the part” principle, so that if A is a proper subset of B, then A<B
2. there are no incommensurable sets: given any A and B, at least one of A≤B and B≤A holds
3. the relation ≤ is transitive and reflexive.
Then the Banach-Tarski paradox follows from (a)–(c) without any use of the Axiom of Choice: there is a way to decompose a ball into a finite number of pieces and move them around to form two balls of
the same size as the original. And Banach-Tarski feels like a direct violation of the "whole is bigger" principle!
Thus, intuitive as the “whole is bigger” principle is, the price of being able to compare the sizes of sets of real numbers in conformity with the principle is quite high. I suspect that most people
who think that denying the “whole is bigger” principle also think Banach-Tarski is super problematic.
For our next observation, let’s add one more highly plausible condition:
4. the relation ≤ is weakly invariant under reflections of the real line: for any reflection ρ, we have A≤B if and only if ρA≤ρB.
Proposition: Conditions (a)–(d) are contradictory.
So, I think we should deny that, in the context of comparing the number of elements of a set, the whole is bigger than the proper part.
Proof of Proposition: Write A∼B iff A≤B and B≤A. Then I claim we have A∼ρA for any reflection ρ. For otherwise we either have A<ρA or ρA<A by (b). If we have A<ρA, then we also have ρA
<ρ^2A by (d), and since ρ^2A=A, we have ρA<A, a contradiction. If we have ρA<A, then we have ρ^2A<ρA by (d), and hence A<ρA, again a contradiction.
Since any translation τ can be made out of two reflections, it follows that A∼τA as well. Let τ be translation by one unit to the right. Then {0,1,2,...} ∼ τ{0,1,2,...} = {1,2,3,...},
which contradicts (a).
Most paradoxes of actual infinities, such as Hilbert’s Hotel, depend on the intuition that:
1. A collection is bigger than any proper subcollection.
A Dedekind infinite set is one that has the property that it is the same cardinality as some proper subset. In other words, a Dedekind infinite set is precisely one that violates (1).
In Zermelo-Fraenkel (ZF) set theory, it is easy to prove that any Dedekind infinite set is infinite. More interestingly, assuming the consistency of ZF, there are models of ZF with infinite sets that
are Dedekind finite.
It is easy to check that if A is a Dedekind finite set, then A and every subset of A satisfies (1). Thus an infinite but Dedekind finite set escapes most if not all the standard paradoxes of
infinity. Perhaps enemies of actual infinity, should thus only object to Dedekind infinities, not all infinities?
However, infinite Dedekind finite sets are paradoxical in their own special way: they have no countably infinite subsets—no subsets that can be put into one-to-one correspondence with the natural
numbers. You might think this is absurd: shouldn’t you be able to take one element of an infinite Dedekind finite set, then another, then another, and since you’ll never run out of elements (if you
did, the set wouldn’t be finite), you’d form a countably infinite sequence of elements? But, no: the problem is that repeating the “taking” requires the Axiom of Choice, and infinite Dedekind finite
sets only live in set-theoretic universes without the Axiom of Choice.
In fact, I think infinite Dedekind finite sets are much more paradoxical than a run-of-the-mill Dedekind infinite sets.
Do we learn anything philosophical here? I am not sure, but perhaps. If infinite Dedekind finite sets are extremely paradoxical, then by the same token (1) seems an unreasonable condition in the
infinite case. For Dedekind finitude is precisely defined by (1).
Van Inwagen distinguishes the General Composition Question:
• (GCQ) What are the nonmereological necessary and sufficient conditions for the xs to compose y?
from the Special Composition Question:
• (SCQ) What are the nonmereological necessary and sufficient conditions for the xs to compose something?
He thinks that the GCQ is probably unanswerable, but attempts to give an answer to the SCQ. Note that an answer to the GCQ immediately yields an answer to the SCQ by existential quantification over y
There are two main families of mereological theories:
• Bottom-Up: The proper parts explain the whole.
• Top-Down: The whole explains the proper parts.
Van Inwagen generally eschews talk of explanation, but the spirit of his work is in the bottom-up camp.
It’s interesting to ask how the GCQ and SCQ look to theorists in the top-down camp. On top-down theories, the xs that compose y are explained by or identical to y. It seems unlikely to suppose that
in all cases there would be some relation among the xs that does not involve y which marks the xs out as all parts of one whole. That would be like thinking there is a necessary and sufficient
condition for Alice, Bob and Carl to be siblings that makes no reference to a parent. Therefore, it is likely that any top-down answer to the SCQ must make reference to the whole that is composed of
the xs. But if we can give such an answer, then it is very likely that we can also give an answer to the GCQ.
If my plausible reasoning is right, then on top-down theories either:
1. An answer can be given to the GCQ, or
2. No answer can be given to the SCQ.
Suppose mereological universalism is true, and that I make a pizza and an alien a long time ago in galaxy far, far away (or even in another universe) makes a sandwich. Then I and the alien have
engaged in an amazing collaboration spanning time and space, and maybe even across universes, and produced a fusion of a pizza and a sandwich. Surely I cannot so very easily collaborate in the
production of things with beings so far off!
Some philosophers hold that all fundamental instances of causal relations are simultaneous. Many of these philosophers are Aristotelians, though presentism provides a plausible route to this
simultaneity doctrine independently of (other) Aristotelian considerations. I think the phenomenon of substantial change shows that this simultaneity doctrine is false.
When a horse changes into a carcass (or an electron-positron pair changes into a pair of photons) we have substantial change. Clearly, in such a case, the horse is the cause of the carcass.
Notice, however, that there is never a time at which both the horse and the carcass exists. This means that substantial change involves properly diachronic rather than simultaneous causation. And
while there is a way of building a diachronic causal explanation out of simultaneous causation and persistence, I don’t see any way of doing that here. The trick in that way was to use the
persistence of one or both of the relata of simultaneous causation to extend the relationship temporally. But here adding the earlier persistence of the horse or the later persistence of the carcass
does not help, because it is still not the case that the horse and carcass have any moment of co-existence.
I think that this is where an Aristotelian will try to bring in matter. The horse has matter. When the horse perishes, its matter persists and comes to make up a carcass. We have a simultaneous
relation between the horse and its matter, and then we have a simultaneous relation between the matter and the carcass.
But this doesn’t solve the problem. For the horse doesn’t just cause a heap of matter—it causes a carcass of a particular sort, made up of substances other than the substance of the horse. What
persists in substantial change, on a classic Aristotelian view, is at most the prime matter. And the prime matter does not explain the form had by the carcass (or the parts of the carcass, if the
carcass counts as a heap of substances).
We can even see the problem at the level of the accidents. Take the horse’s shape S[h] and the carcass’s very similar shape S[c]. Then, clearly, S[h] causes S[c]. One can see this empirically: if one
rearranges the legs of the dying horse, the carcass’s shape changes correspondingly. But the only relevant thing that on the Aristotelian story persists across the change from horse to carcass is the
matter: S[h] does not persist, nor does anything that grounds S[h].
On reflection, the last line of thought shows that there could be a problem even for accidental change. For it seems likely to the case that a substance has an accident A which partially causes
itself to be replaced by an accident B incompatible with itself. For instance, consider my current shape S[1]. In a moment, my body will shift into a new shape S[2]. The shape S[1] partially causes
the shape S[2]. Yet there is never a moment where I have both shapes. Indeed, at any time where I have shape S[1], that’s the only shape I have. So, the shape S[1] cannot cause any different future
shape of me, assuming causation is always simultaneous.
This problem may be less serious than for substantial change, however, because one might say that S[1] does not cause S[2], but there is some deeper persisting accident that first causes me to have S
[1] and then causes me to have S[2], so that there is no more a causal relationship between S[1] and S[2] than between a shadow of a moving person first appearing in one place and then in another. I
think it is implausible to think that all cases where an accident A partially causes its immediate replacement by an accident B can be accounted by positing A and B to be mere epiphenomena, but I am
not sure I have as good an argument against this as I do against the substantial change case.
I conclude from all this that while simultaneous causation is possible, it is not the case that all diachronic causation reduces to simultaneous causation.
Xenophanes famously lambasted Greek religion for its anthropomorphism:
if cattle or lions had hands, so as to paint with their hands and produce works of art as men do, they would paint their gods and give them bodies in form like their own-horses like horses,
cattle like cattle.
Two and a half millenia later, accusations of anthropomorphism continue to be made against monotheistic religions, typically by naturalists.
I was thinking about this, and had an odd thought. According to monotheism, the root of all explanation is the activity of God. According to standard naturalism, the root of all explanation is the
activity of the fundamental physical entities, either particles or fields. But humans are more like fundamental physical entities than like the God of the monotheistic religions. The difference
between us and the fundamental physical entities is merely finite. The difference between us and God is infinite. Thus, in an important sense, it is standard naturalism that is more anthropomorphic
in its fundamental explanatory agents than monotheism.
If we do not feel this—if we feel ourselves more God-like than electron-like—then we are infinitely elevating ourselves or infinitely demoting God or both.
That said, the three Western monotheistic religions do think that the physical universe is made for us. Thus, while the religions are not anthropomorphic, they do have an anthropocentric view of our
physical universe. Interestingly, though, to some (albeit lesser) extent so does the most plausible current naturalist view, namely a multiverse theory together with the weak anthropic principle.
Here are two theories of divine conservation, tendentiously labeled:
• Occasionalist conservation: That a creature that previously existed continues to exist is solely explained by God’s power.
• Concurrentist conservation: That a creature that previously existed continues to exist is explained by God’s power concurring with creaturely causal powers (typically, the creature’s power to
continue to exist).
It is usual in classical theism to say that divine conservation is very similar to divine creation. This comparison might seen to favor occasionalist conservation. However, that is not so clear once
we realize that classical theism holds that all finite things are created by God, and hence creation itself comes in two varieties:
• Creation ex nihilo: God creates something by the sole exercise of his power.
• Concurrentist creation: God creates things by concurring with a creaturely cause.
Most of the objects familiar to us are the product of concurrentist creation. Thus, an acorn is produced by God in concurrence with an oak tree, and a car inconcurrence with a factory. (The human
soul is an exception according to Catholic tradition.)
Because of this, even if we opt for concurrentist conservation, we can still save the comparison between conservation and creation, as long as we remember that often creation is concurrentist
Which of the two theories of conservation should we prefer?
On general principles, I think we have some reason to prefer concurrentist conservation, simply because it preserves the explanatory connections within the natural world better.
However, if we insist on presentism, then we may be stuck with occasionalist conservation, because presentism makes cross-time causal relations problematic.
[Edited Nov. 4 2020 to replace "cooperation" with the more usual term "concurrence".]
According to the Principle of Sufficient Reason (PSR), every contingent fact has a sufficient reason. What does “sufficient” mean here? A natural thought is that it means that the reason is logically
sufficient for the fact. My own work on the PSR rejects this natural thought. I say that a sufficient reason is one that suffices to explain the fact, not necessarily one that suffices for the fact
to be true. I occasionally worry that this is too wimpy a take on the PSR, indeed a kind of bait-and-switch.
When I worry about this, it helps me to come back to Leibniz, whom nobody considers a wimp with respect to the PSR. How does Leibniz understand “sufficient”?
In the Principles of Nature and Grace, Leibniz talks of the
grand principe … qui porte que rien ne se fait sans raison suffisante; c’est-à-dire que rien n’arrive sans qu’il soit possible à celui qui connaîtrait assez les choses de rendre une raison qui
suffise pour déterminer pourquoi il en est ainsi, et non pas autrement [great principle … which holds that nothing happens without sufficient reason; that is to say, that nothing happens without
its being possible for someone who knows enough about how things are to give a reason that suffices to determine why it is so and not otherwise]. (my italics)
Leibniz does not say that the reason is sufficient to determine the fact. Rather, Leibniz carefully says that the reason is sufficient to determine why the fact occurred. You can read off the
explanation, the answer to the why question, from the reason, but no claim is made that you can read the explained fact off from it.
Indeed, the only necessitation in the paragraph is hypothetical:
De plus, supposé que des choses doivent exister, il faut qu’on puisse rendre raison pourquoi elles doivent exister ainsi, et non autrement. [Further, supposing things must exist, it has to be
possible to give a reason why they must exist so and not otherwise.] (my italics)
I wish Leibniz had this weaker picture of sufficient reason consistently. Sadly for me, he does not. In a 1716 letter to Bourguet he writes:
Mr. Clark … n’a pas bien compris la force de cette maxime, que rien n’arrive sans une raison suffisante pour le determiner. [Mr. Clark … has not understood well the force of the maxim that
nothing happens without a reason sufficing to determine it.]
Oh well.
I comfort myself, however, that my philosophical hero does, after all, have two kinds of necessity, and hopefully the determination in the PSR involves the weaker one.
As a four-dimensionalist, I have been puzzled both by the arguments that divine conservation is necessary secure the persistence of substances and the idea of existential inertia as a metaphysical
Temporal extent seems little different metaphysically to me from spatial thickness, the “problem of persistence” seems to me to be a pseudo-problem, and both solutions to this pseudo-problem seem to
me to be confused.
On the existential inertia side, a metaphysical principle that objects continue to exist unless their existence is interrupted by some other cause seems as ridiculous to me as a principle that
objects are maximally thick (and long and deep) unless and until their thickness (or length or depth) is stopped by other causes. And divine action is needed to secure persistence only to the extent
that it is needed to secure thickness (and length or depth). That said, I do think divine action is needed to secure thickness, as well as all other accidents of a thing, because substances are in
some sense causes of their accidents, but all creaturely causation requires divine cooperation. But that, I think, is a slightly different line of argument from the arguments for persistence of
substances (in particular, I don’t have a good argument for it that doesn’t already presuppose theism, while the arguments for conservation are supposed to provide reasons for accepting theism).
However, I now see how it is that presentism yields a real problem of persistence. Here’s the line of thought. First, note that contrary to the protestations of some presentists, it is very plausible
1. Presentism implies that all causation is simultaneous.
For something that exists, at least at the time at which it is caused, cannot have as its cause something that doesn’t exist. But given presentism, only something present exists. So at a time at
which E is caused, if the cause of E did not exist, we would have the exercise of a non-existent causal power, which is absurd.
But even if all causation is simultaneous, nonetheless:
2. There is diachronic causal explanation.
Setting the alarm at night explains why it goes off in the morning, even if by the simultaneity thesis (1), setting the alarm cannot be the cause of the alarm going off. Diachronic causal explanation
cannot simply be causation. So what is it? Here is the best presentist story I know (and it’s not original to me).
First, we can get some temporal extension by the following trick. Imagine a thing A persists over an interval of time from t[1] to t[2]. At t[2] is causes a thing B that persists over an interval of
time from t[2] to t[3]. The existence of A at t[1] then causally explains the existence of B at t[3]. Note, however, that the existence of A at t[1] does not cause the existence of B at t[3].
Causation happens at t[2] (or perhaps over an interval of times—thus, A might persist until some time t[2.5]<t[3], and be causing B over all of the interval from t[2] to t[2.5]), but not at any
earlier time, since at earlier times A doesn’t exist. Thus, by supplementing the simultaneous causal relation between A and B at t[2] with the persistence of A before t[2] and/or the persistence of B
after t[2], we have, we can extend the relation into what one might call a fundamental instance of diachronic causal explanation.
Thus, a fundamental link in diachronic causal explanation consists of an instance of causation preceded and/or followed by an instance of persistence of the causing thing and/or the caused thing
respectively. And a non-fundamental instance of diachronic causal explanation is a chain of fundamental links of diachronic causal explanations. (It may be that these diachronic causal explanations
are very close to what Aquinas calls per accidens causal sequences.)
But for this to be genuine explanation, the persistence of the cause and/or effect needs to have an explanation. Divine conservation provides a very neat explanation: God necessarily exists
eternally, and is simultaneous with everything (there may be some complications, though, with a timeless being given presentism), so God can cause A to persist from t[1] to t[2] and B to persist from
t[2] to t[3]. Thus, fundamental links in diachronic causal explanations depend on divine conservation.
An existential inertia view also gives a solution, but a far inferior one. For existential inertia requires the earlier existence of A, together with the metaphysical principle of existential
inertia, to explain the later existence of A. But such a cross-time explanatory relation seems too much like the already rejected idea of cross-time causation. For it’s looking like A qua existing at
t[1] explains A existing at t[2]. But at t[2], according to presentism A qua existing at t[1] is in the unreal past, and it is absurd to suppose that what is in the unreal past can explain something
real now.
In summary, given presentism, all fundamental explanatory relations need to be simultaneous. But it is an evident fact that there are diachronic causal explanatory relations. The only way to build
those out of simultaneous explanatory relations is by supposing a being that can be simultaneous with things that exist at more than one time—a timelessly eternal being—whose causal efficacy provides
the diachronic aspects of the explanatory linkage.
That said, I think there are two serious weaknesses in this story. The first is that it’s a close cousin of occasionalism. For there is no purely non-divine explanatory chain from the setting of the
alarm at night to the alarm going off in the morning—divine action explains the persistences that make the chain diachronic.
A second problem is the puzzle of what explains why A causes B at t[2] rather than as soon as A comes into existence. Why does A “wait” until t[2] to cause B? Crucial to the story is that A is the
whole cause, which then persists from t[1] to t[2]. But why doesn’t it cause B right away, with B then causing whatever effect it has right away, and with everything in the whole causal history of
the universe happening at once? Again, one might give this an occasionalist solution—A causes B only because God cooperates with creaturely causation, and God might hold off his cooperation until t
[2]. But this makes the story even more occasionalist, by making God involved in the timing of causation.
Imagine there is an infinite stack of cards labeled with the natural numbers (so each card has a different number, and every natural number is the number of some card). In the year 2021−n, you
perfectly shuffled the bottom n cards in the stack.
Now you draw the bottom card from the deck. Whatever card you see, you are nearly certain that the next card will have a bigger number. Why? Well, let’s say that the card you drew has the number N on
it. Next consider the next M cards in the deck for some number M much bigger than N. At most N−1 of these have numbers smaller than N on them. Since these bottom M cards were perfectly shuffled
during the year 2021−(M+1), the probability that the number you draw is bigger than N is at most (N−1)/M. And since M can be made arbitrarily large, it follows that the probability that the
number you draw is bigger than N is infinitesimal. And the same reasoning applies to the next card and so on. Thus, after each card you draw, you are nearly certain that the next card will have a
bigger number.
And, yet, here’s something you can be pretty confident of: The bottom 100 cards are not in ascending order, since they got perfectly shuffled in 1921, and after that you’ve shuffled smaller subsets
of the bottom 100 cards, which would not make the bottom 100 cards any less than perfectly shuffled. So you can be quite confident that your reasoning in the previous paragraph will fail. Indeed,
intuitively, you expect it to fail about half the time. And yet you can’t rationally resist engaging in this reasoning!
The best explanation of what went wrong is, I think, causal finitism: you cannot have a causal process that has infinitely many causal antecedents.
Occasionally, people have thought that one can refute determinism as follows:
1. If determinism is true, then all our thinking is determined.
2. If our thinking is determined, then it is irrational to trust its conclusions.
3. It is not irrational to trust the conclusions of our thinking.
4. So, determinism is not true.
But now notice that, plausibly, even if we have indeterministic free will, other animals don’t. And yet it seems at least as reasonable to trust a dog’s epistemic judgment—say, as to the presence of
an intruder—as a human’s. Nor would learning that a dog’s thinking is determined or not determined make any difference to our trust in its reliability.
One might respond that things are different in a first-person case. But I don’t see why.
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Eagen, L. Li. E. 2022. Bulletproofs++. [Cryptology ePrint Archive, Report 2022/510].
Added by: Plowsof (5/5/22, 1:11 PM) Last edited by: Rucknium (4/5/24, 10:13 PM)
Resource type: Miscellaneous Categories: Monero-focused
BibTeX citation key: Eagen2022 Keywords: Bulletproofs, Monero, Whitepaper Views: 204/6473
View all bibliographic details Creators: Eagen
Building on Bulletproofs [1] and Bulletproofs+ [2], I describe several new range proofs that achieve both shorter proof sizes and witness lengths as well as a new confidential transaction protocol
for multiple types of currency. The first section describes how to modify the (weighted) inner product protocol to prove a norm relation, i.e. self inner product, while only committing to the vector
once. In the second section, this is used to construct a binary digit range proof of half the witness length of Bulletproofs(+). Using a novel permutation argument, which is essentially the
logarithmic derivative of [3], and the norm argument, I then construct a family of range proofs for arbitrary bases. In the case of 64 bit range proofs, using 16 hexadecimal digits, the reciprocal
range proof achieves a proof size of 10 curve points and 3 scalars, 416 bytes in Curve25519 and 418 in SECP256k1, and witness length of 23 scalars. This proof size is approximately 27% smaller than
Bulletproofs+ and 38% smaller than Bulletproofs. The witness length, which is proportional to verification complexity, is reduced by a factor of roughly 6, which asymptotically approaches 8 as the
number of ranges increases. Finally, I use the permutation argument to construct a zero knowledge confidential transaction protocol for multiple types of currency. This uses one multiplication per
input and per output and supports multiparty proving, substantially improving on both ring signature [12] and Bulletproof [4] based confidential transactions.
CCS for Peer Review: https://ccs.getmonero.org/proposals/bulletproofs-pp-peer-review.html
Peer review completed: https://moneroresearch.info/index.php?action=resource_RESOURCEVIEW_CORE&id=217
A rust implementation of BP++ https://github.com/sanket1729/rust-bulletproofs-pp
Added by: Rucknium Last edited by: Rucknium
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Mathematical model of complex control of the vibratory transportation and technological process
The vibratory transportation and technological process is a dynamically sensitive operation which includes physically different components: vibro-exciter, elastic system, working member (absolutely
rigid or of finite rigidity) and various friable loads. Interaction of these components predetermines the behavior of the friable material on the surface of the working member (WM). At the same time,
existing simple models or physical experiments cannot provide sufficient precision to adequately research the mentioned complex process. Therefore, it is necessary to develop a more precise
mathematical model ensuring the study and revelation of the still hidden factors influencing the vibratory process. A new generalized dynamical spatial model of the loaded vibratory technologic
machine (vibro-exciter, working member, load) developed on the basis of the systemic approach is presented in the work and a system of interconnected equations of movements of the constituent masses
considering dynamical, geometrical and physical parameters, is obtained. The change of parameters is reflected on the variation of dynamical characteristics of the system that allows a thorough study
of the technological process with the help of mathematical modeling. Using the presented model, it is possible to find the physical parameters and their combinations, realization of which will
promote the improvement of the technological process. Some results of the modeling are presented. A new design of the vibro-exciter developed on the basis of the results of modeling is presented as
1. Introduction
The vibratory transportation and technologic machines are widely used in various spheres of industry for transportation of friable materials and individual parts, measured feeding and sorting and
also for carrying out various technologic operations on them [1-7].
The vibratory conveyors are especially effective for transportation of powdery, explosive, chemically aggressive, abrasive, heated and other friable materials, for the displacement of which, other
types of conveyors are less suitable [1, 8].
The advantages of these machines are:
– Possibility of hermetization of the transportation and technologic processes that decrease the pollution of the environment;
– Elimination of losses and dirtying of the materials to be transported;
– Minimal wear of the working member;
– Compatibility of the process of transportation with scattering, heating, cooling, washing, mixing and other technologic operations.
In spite of the great number of researches in the sphere of the vibratory transportation and technologic machines, many unsolved problems remain in the direction of theory, design and fabrication.
For example, the reasons of generation of the working member parasitic (non-working) vibrations [9] and their influence on the technologic process are studied insufficiently; such reasons may be
incorrect transfer of the exciting force, features of the elastic system (springs) or other constructional errors (Fig. 1) [10].
In Fig. 1 is presented a two-mass system “vibro-drive – working member” and are shown spatial deviations of the working member caused by various possible errors: I – nominal (designed) position of
the working member; II – position of the working member (${M}_{1}$) considering the errors including the eccentricities ${e}_{x}$, ${e}_{y}$, ${e}_{z}$ of displacement of the center of gravity from
position ${O}_{1}$ to point ${O}_{1}^{"}$; ${O}_{2}{O}_{1}^{"}$ – possible deviation of the exciting force; $\delta$ – possible deflection of the elastic element; ${\theta }_{0}$, ${\psi }_{0}$, ${\
phi }_{0}$ – turnings of the coordinate axes caused by the assembly errors of the vibratory machine and transition from position ${O}_{1}{x}_{1}{y}_{1}{z}_{1}$ to position ${O}_{1}^{"}{x}_{1}^{\text
{'}}{y}_{1}^{\text{'}}{z}_{1}^{\text{'}}$. 1 – basic elastic system of the vibro-machine; 2 – suspensions of the vibro-machine; $Q\left(t\right)$ – exciting force; the vibro-exciter (${M}_{2}$ – ${O}
_{2}{x}_{2}{y}_{2}{z}_{2}$) can also have the similar changes that are not shown in the drawing.
Fig. 1Deviations of the working member from the designed position stipulated by the vibro-machine design errors and spring specificity
The axes ${x}_{1}$ and ${x}_{2}$ of the active and reactive parts (${M}_{1}$[]and ${M}_{2}$) in this figure coincide in the initial position. The main interaction of the mentioned masses is realized
along $x$ (${x}_{1}$, ${x}_{2}$) axis that is reflected in the Eq. (7). The influence of vibrations of mass ${M}_{2}$[]on mass ${M}_{1}$ and respectively on mass ${M}_{3}$ in other directions is
small in contrast to the influence of mass ${M}_{1}$[]on mass ${M}_{3}$, because in addition to elastic forces the interaction of inertial forces takes place here.
The aim of the work is to draw up the generalized mathematical model allowing to study the influence of the parameters and characteristics of masses of the system on the vibratory technologic process
[11-16]. In this regard, it is expedient to consider spatial, relative-translatory and absolute movement of the three-mass dynamic system; it is analogous to the three-mass vibrational technological
system, including the following parts: vibro-exciter, working member, load to be processed (to be transported).
A dynamical model of the mentioned machine (Fig. 1(b)) is shown in Fig. 1(a), where to each mass the corresponding coordinate systems ${O}_{1}{x}_{1}{y}_{1}{z}_{1}$, ${O}_{2}{x}_{2}{y}_{2}{z}_{2}$, $
{O}_{3}{x}_{3}{y}_{3}{z}_{3}$ are connected; $Oxyz$ – immobile (inertial) coordinate system; 1 – basic elastic system connecting a vibro-exciter to the working member, 2 – suspension of the vibratory
machine, 3 – conventional elastic system connecting a friable load to the working member surface; $Q\left(t\right)$ – exciting vibratory force.
Though these masses are integrated into the common system, each of them is characterized by the proper physical and mechanical properties significantly different from each other; they should be taken
into account at drawing up a generalized mathematical model of their movement.
For the inclusion of the technologic load (mass ${M}_{3}$) in the common spatial system (Fig. 4) and imparting it a generalized character, we present it as a rigid body connected to the WM (${M}_{1}$
) by the conventional elastic system 3 (Fig. 2), describing elastic and damping properties of the friable material.
At a fixed moment of time, elastic system 3 (as well as 1 and 2 in Fig. 2 and 3) is decomposed into three components, describing elastic properties of the material in space.
Fig. 2Vibratory technologic machine: a) generalized dynamical model; b) physical prototype with a bunker
A distinctive feature of the elastic system 3 is a non-retaining character of its connection with the WM in dynamics. With the help of the elastic-damping elements are described characteristics
between the layers of the technologic friable materials, interaction between the layers and between the lower layer and WM. In contrast to the existing models [1, 2, 14, 16] all degrees of freedom
are considered in it, i.e. it can be enclosed in the model of the general spatial system (Fig. 2, 3) and depending on the concrete problem, it can be reduced to the simpler form (plane, linear,
Fig. 3A conventional elastic and damping system of the technologic load
Presentation of the technologic load (TL) by the rigid body (at drawing up expressions of kinetic energy) is stipulated by the necessity of obtaining the equation of motion in the more generalized
form not only for translational (in this case TL could be considered as a material point) but also for rotary movements.
The deformation of the friable TL layer is modeled at movement by the elastic elements with coefficients of elasticity ${k}_{{x}_{3}}$, ${k}_{{y}_{3}}$, ${k}_{{z}_{3}}$, ${k}_{{\theta }_{3}}$, ${k}_
{{\psi }_{3}}$, ${k}_{{\phi }_{3}}$ (Fig. 3). The dissipation of energy at the deformation of the layer is considered by dampers with coefficients of resistance ${C}_{{x}_{3}}$, ${C}_{{y}_{3}}$, ${C}
_{{z}_{3}}$, ${C}_{{\theta }_{3}}$, ${C}_{{\psi }_{3}}$, ${C}_{{\phi }_{3}}$ (not shown in Figure). Therefore, direct contact of the TL with WM is replaced by elastic-frictional connections.
2. Drawing up the mathematical model
For the deduction of the equations of movement, consider a spatial dynamical model of the system (Fig. 4).
The masses are considered in two positions (Fig. 4(b)): I – ideal-immobile, determined according to the design drawings, II – dynamical displacement under the action of the exciting force determined
by the direction cosines in accordance with Tables 1 and 2, where $i=$ 1, 2, 3 (mass numbers), $m=$ I, II (positions of masses). Direction cosines in Table 2 are presented in the form of the angles
of Euler-Krylov [17], where, because of small values of the rotary displacements, products of the angles no greater than second degree are taken into account at the expansion. Table 2 shows direction
cosines for mass ${M}_{1}$[](working member) only. For other masses, forms of expressions of the angles will be similar.
Table 1Direction cosines
${x}_{i}$ ${y}_{i}$ ${z}_{i}$
${x}_{i}^{m}$ ${\left({\alpha }_{i}^{m}\right)}_{11}$ ${\left({\alpha }_{i}^{m}\right)}_{12}$ ${\left({\alpha }_{i}^{m}\right)}_{13}$
${y}_{i}^{m}$ ${\left({\alpha }_{i}^{m}\right)}_{21}$ ${\left({\alpha }_{i}^{m}\right)}_{22}$ ${\left({\alpha }_{i}^{m}\right)}_{23}$
${z}_{i}^{m}$ ${\left({\alpha }_{i}^{m}\right)}_{31}$ ${\left({\alpha }_{i}^{m}\right)}_{32}$ ${\left({\alpha }_{i}^{m}\right)}_{33}$
Table 2Direction cosines in angles of Euler-Krylov
${x}_{1}^{II}$ ${y}_{1}^{II}$ ${z}_{1}^{II}$
${x}_{1}^{I}$ $1-{\psi }_{1}^{2}/2-{\varphi }_{1}^{2}/2$ ${\varphi }_{1}+{\psi }_{1}{\theta }_{1}$ ${\psi }_{1}$
${y}_{1}^{I}$ ${\varphi }_{1}$ $1-{\theta }_{1}^{2}/2-{\varphi }_{1}^{2}/2$ $-{\theta }_{1}$
${z}_{1}^{I}$ $-{\psi }_{1}-{\varphi }_{1}{\theta }_{1}$ $-{\varphi }_{1}{\psi }_{1}+{\theta }_{1}$ $1-{\psi }_{1}^{2}/2-{\theta }_{1}^{2}/2$
After dynamical displacement the positions of the coordinate systems will be: ${O}_{1}^{\mathrm{"}}{x}_{1}^{\mathrm{"}}{y}_{1}^{\mathrm{"}}{z}_{1}^{\mathrm{"}}$, ${O}_{2}^{\mathrm{"}}{x}_{2}^{\mathrm
{"}}{y}_{2}^{\mathrm{"}}{z}_{2}^{\mathrm{"}}$, ${O}_{3}^{\mathrm{"}}{x}_{3}^{\mathrm{"}}{y}_{3}^{\mathrm{"}}{z}_{3}^{\mathrm{"}}$ (Fig. 4(b)).
Fig. 4A generalized model of the spatial movement of the vibratory technologic machine with a load
Fig. 4 shows the following indications: ${A}_{i}$, ${B}_{i}$, ${C}_{i}$ – free points of masses, ${R}_{1}$, ${R}_{2}$, ${R}_{3}$ – radius-vectors between the coordinate origin points (centers of
masses), ${R}_{1i}^{\mathrm{"}}$, ${R}_{2i}^{\mathrm{"}}$, ${R}_{3i}^{\mathrm{"}}$ – ${A}_{i}$, ${B}_{i}$, ${C}_{i}$ – radius-vectors of the points relative to the origin of the coordinate axes of
the corresponding masses, ${r}_{1i}^{\mathrm{"}}$, ${r}_{2i}^{\mathrm{"}}$, ${r}_{3i}^{\mathrm{"}}$ – ${A}_{i}$, ${B}_{i}$, ${C}_{i}$ – radius-vectors of the points relative to the origin of
coordinate axes of own masses.
With the help of Fig. 2, 4 and methods of deduction of equations of motion of the material points and rigid bodies [5, 17], we obtain equations of spatial movement of masses ${M}_{1}$ and ${M}_{3}$.
The following ideas were taken into account in the establishment of the differential equations of motion:
– Spatial motion equations were established for the interconnected motion masses of the working body (${M}_{1}$) and the technological load (${M}_{3}$), as relative-translatory in relation to each
– Because ${M}_{2}$ is connected to mass ${M}_{1}$ by the potential and damping forces (${Q}_{q}^{\mathrm{"}}$= $f\left({q}_{i},\stackrel{.}{{q}_{i}}\right)$, $i=$ 1, 2; $q={x}_{1},{x}_{2},\dots ,{\
phi }_{2}$) whose influence on the motion of the TL is insignificant, its spatial movement is described by linear differential Eq. (2); at the same time, ${M}_{2}$ as an electromagnetic vibro-drive
realizes vibratory motion through angle $\beta$ with respect to mass ${M}_{1}$ (to the axis ${O}_{1}{x}_{1}$ of WM) by means of main elastic system 1 (Fig. 1 and 4).
– The influence of the mass of the load (${M}_{3}$) on the working body (${M}_{1}$) is indicated by coefficient $\mu$, the value of which could vary between 0-1 depending on a regime of the load
movement (sporadic or with working body).
– In Eqs. (1) and (3), the multiplication type 2nd order nonlinear members will be considered mostly in the high amplitude (resonance) vibrations.
Working member motion equations will be as follows:
$\left({M}_{1}+\mu {M}_{3}\right){\stackrel{¨}{x}}_{1}+\mu {M}_{3}\left[\left({\stackrel{¨}{\psi }}_{1}{z}_{3}+2{\stackrel{˙}{\psi }}_{1}{z}_{3}+{\stackrel{¨}{\varphi }}_{1}{y}_{3}-2{\stackrel{˙}{\
varphi }}_{1}{y}_{3}-{\stackrel{¨}{y}}_{3}{\varphi }_{1}+{\stackrel{¨}{z}}_{3}{\psi }_{1}\right)\mathrm{cos}{\alpha }_{1}$$+\mathrm{cos}{\alpha }_{1}{\stackrel{¨}{x}}_{3}+{\stackrel{¨}{\theta }}_{1}
{y}_{3}+{\stackrel{¨}{x}}_{3}{\psi }_{1}-2{\stackrel{˙}{\theta }}_{1}{\stackrel{˙}{y}}_{3}-2{\stackrel{˙}{x}}_{3}{\stackrel{˙}{\psi }}_{1}-{\stackrel{¨}{\psi }}_{1}{x}_{3}+{\stackrel{¨}{y}}_{3}{\
theta }_{1}\right)\mathrm{sin}{\alpha }_{1}\right]={Q}_{x1}+{Q}_{x1}^{"},$$\left({M}_{1}+\mu {M}_{3}\right){\stackrel{¨}{y}}_{1}+\mu {M}_{3}\left({\stackrel{¨}{\varphi }}_{1}{x}_{3}-{\stackrel{¨}{\
theta }}_{1}{z}_{3}-2{\stackrel{˙}{\theta }}_{1}{\stackrel{˙}{z}}_{3}-2{\stackrel{˙}{\varphi }}_{1}{\stackrel{˙}{x}}_{3}+$$+{\stackrel{¨}{x}}_{3}{\varphi }_{1}+{\stackrel{¨}{y}}_{3}-{\stackrel{¨}{z}}
_{3}{\theta }_{1}\right)={Q}_{y1}+{Q}_{y1}^{"},$$\left({M}_{1}+\mu {M}_{3}\right){\stackrel{¨}{z}}_{1}+\mu {M}_{3}\left[\left({\stackrel{¨}{\theta }}_{1}{y}_{3}+2{\stackrel{˙}{\theta }}_{1}{\stackrel
{˙}{y}}_{3}-{\stackrel{¨}{\psi }}_{1}{x}_{3}-2{\stackrel{˙}{\psi }}_{1}{\stackrel{¨}{x}}_{3}-{\stackrel{¨}{x}}_{3}{\psi }_{1}+{\stackrel{¨}{y}}_{3}{\theta }_{1}\right)\mathrm{cos}{\alpha }_{1}$$-\
mathrm{sin}{\alpha }_{1}{\stackrel{¨}{x}}_{3}+\mathrm{cos}{\alpha }_{1}{\stackrel{¨}{z}}_{3}+{\stackrel{¨}{\varphi }}_{1}{y}_{3}-{\stackrel{¨}{\psi }}_{1}{z}_{3}-2{\stackrel{˙}{\psi }}_{1}{\stackrel
{˙}{z}}_{3}$$+2{\stackrel{˙}{y}}_{3}{\stackrel{˙}{\varphi }}_{1}+{\stackrel{¨}{y}}_{3}{\varphi }_{1}-{\stackrel{¨}{y}}_{3}{\psi }_{1}\right)\mathrm{sin}{\alpha }_{1}\right]={Q}_{z1}+{Q}_{z1}^{"},$$
{A}_{1\theta }{\stackrel{¨}{\theta }}_{1}+{A}_{2\theta }{\stackrel{¨}{\psi }}_{1}{\varphi }_{1}+{A}_{3\theta }{\stackrel{˙}{\varphi }}_{1}{\stackrel{˙}{\psi }}_{1}+\mu {M}_{3}\left({\stackrel{¨}{z}}_
{1}{y}_{3}\mathrm{cos}{\alpha }_{1}+{\stackrel{¨}{x}}_{1}{y}_{3}-{\stackrel{¨}{y}}_{1}{z}_{3}+{\stackrel{¨}{z}}_{3}{y}_{3}-{\stackrel{¨}{y}}_{3}{z}_{3}\right)$$+{A}_{4\theta }{\stackrel{¨}{\psi }}_
{3}{\varphi }_{1}+{A}_{5\theta }{\stackrel{˙}{\varphi }}_{3}{\stackrel{˙}{\psi }}_{3}+{A}_{6\theta }{\stackrel{¨}{\varphi }}_{3}{\psi }_{1}+{A}_{7\theta }{\stackrel{˙}{\psi }}_{3}{\stackrel{˙}{\
varphi }}_{1}+{A}_{8\theta }\left(\stackrel{¨}{{\theta }_{3}}+{\stackrel{¨}{\phi }}_{3}{\stackrel{¨}{\psi }}_{3}-{\stackrel{¨}{\varphi }}_{3}\psi \right)={Q}_{\theta 1}+{Q}_{\theta 1}^{"},$
where ${\alpha }_{1}=\alpha +\beta$, $\alpha$ – the angle of inclination of the working member, $\beta$ – the angle of vibrations, ${Q}_{{x}_{1}}$, ${Q}_{{y}_{1}}$, ${Q}_{{z}_{1}}$ – components of
the exciting force; ${Q}_{{x}_{1}}^{"}$, ${Q}_{{y}_{1}}^{"}$, ${Q}_{{z}_{1}}^{"}$ – elastic and damping forces from elastic systems 1 and 3 and weight of the TL; ${A}_{1\theta }$, ${A}_{2\theta }$, …
– sum of the moments of inertia of masses relative corresponding axes, e.g.:
${A}_{1\theta }={J}_{{x}_{1}}+{J}_{{x}_{3}},{A}_{2\theta }={J}_{{x}_{1}}-{J}_{{y}_{1}}+{J}_{{z}_{3}}-{J}_{{y}_{3}},$
${A}_{3\theta }={J}_{{x}_{1}}-{J}_{{y}_{1}}+{J}_{{z}_{3}}-{J}_{{y}_{3}}+{J}_{{z}_{1}}+{J}_{{z}_{3}}.$
In the equation systems Eqs. (1) and (3), not all rotational movement equations are shown (in ${\psi }_{1}$, ${\varphi }_{1}$, ${\psi }_{3}$, ${\varphi }_{3}$ directions).
Spatial vibratory movements of mass ${M}_{2}$ are described by the equations:
${M}_{2}{\stackrel{¨}{x}}_{2}={Q}_{x2}+{Q}_{x2}^{"},{M}_{2}{\stackrel{¨}{y}}_{2}={Q}_{y2}+{Q}_{y2}^{"},{M}_{2}{\stackrel{¨}{z}}_{2}={Q}_{z2}+{Q}_{z2}^{"},$${C}_{\theta }{\stackrel{¨}{\theta }}_{2}=
{Q}_{\theta 2}+{Q}_{\theta 2}^{"},{C}_{\psi }{\stackrel{¨}{\psi }}_{2}={Q}_{\psi 2}+{Q}_{\psi 2}^{"},{C}_{\varphi }{\stackrel{¨}{\varphi }}_{2}={Q}_{\varphi 2}+{Q}_{\varphi 2}^{"},$
where ${C}_{\theta }$, ${C}_{\psi }$, ${C}_{\varphi }$ are principal moments of inertia; ${Q}_{{x}_{2}}$, ${Q}_{{y}_{2}}$, ${Q}_{{z}_{2}}$ – components of the exciting force; ${Q}_{{x}_{2}}^{"}$, $
{Q}_{{y}_{2}}^{"}$, ${Q}_{{z}_{2}}^{"}$ – elastic and damping forces from elastic systems 1 (they are functions of ${q}_{1,2}$, ${\stackrel{˙}{q}}_{1,2}$, where $q$ takes the values $x$, $y$, $z$, $\
theta$, $\psi$, $\varphi$.
For technological load (${M}_{3}$):
${M}_{3}{\stackrel{¨}{x}}_{3}+{M}_{3}\left({\stackrel{¨}{x}}_{1}\mathrm{cos}{\alpha }_{1}-{\stackrel{¨}{z}}_{1}\mathrm{sin}{\alpha }_{1}\right)+{M}_{3}\left({\stackrel{¨}{\psi }}_{1}{z}_{3}-{\
stackrel{¨}{z}}_{1}{\psi }_{1}\mathrm{cos}{\alpha }_{1}-{\stackrel{˙}{x}}_{1}{\psi }_{1}\mathrm{sin}{\alpha }_{1}$$-{\stackrel{¨}{y}}_{1}{\varphi }_{1}+2{\stackrel{˙}{\psi }}_{1}{\stackrel{˙}{z}}_{3}
-{\stackrel{¨}{\varphi }}_{1}{y}_{3}-2{\stackrel{˙}{\varphi }}_{1}{\stackrel{˙}{y}}_{3}\right)+{c}_{{x}_{1}}\left({\stackrel{˙}{x}}_{1}\mathrm{cos}{\alpha }_{1}-{\stackrel{˙}{z}}_{1}\mathrm{sin}{\
alpha }_{1}+{\stackrel{˙}{x}}_{3}\right)$$+{c}_{{x}_{3}}{\stackrel{˙}{x}}_{3}-{M}_{3}g\left(\mathrm{sin}\alpha -{\psi }_{1}\mathrm{cos}\alpha \right)=-\left({f}_{x}{N}_{z}+{f}_{y}{N}_{y}\right)\
mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\stackrel{˙}{x}}_{3}\right),$${M}_{3}{\stackrel{¨}{y}}_{3}+{M}_{3}{\stackrel{¨}{y}}_{1}+{M}_{3}\left[\left({\stackrel{¨}{z}}_{1}{\theta }_{1}-{\stackrel
{¨}{x}}_{1}{\phi }_{1}\right){\mathrm{cos}\alpha }_{1}+\left({\stackrel{¨}{x}}_{1}{\theta }_{1}+{\stackrel{¨}{z}}_{1}{\phi }_{1}\right){\mathrm{sin}\alpha }_{1}-{\stackrel{¨}{\theta }}_{1}{z}_{3}$$-
{2\stackrel{˙}{\theta }}_{1}{\stackrel{˙}{z}}_{3}+{2\stackrel{˙}{\phi }}_{1}{\stackrel{˙}{x}}_{3}\right]+{c}_{{y}_{1}}\left({\stackrel{˙}{y}}_{1}+{\stackrel{˙}{y}}_{3}\right)+{c}_{{y}_{3}}{\stackrel
{˙}{y}}_{3}+{k}_{{y}_{3}}{y}_{3}$$+{M}_{3}g\left({\phi }_{1}\mathrm{sin}\alpha -{\theta }_{1}\mathrm{cos}\alpha \right)=-{f}_{z}{N}_{z}sign\left({\stackrel{˙}{y}}_{3}\right),$${M}_{3}{\stackrel{¨}
{z}}_{3}+{M}_{3}\left({\stackrel{¨}{z}}_{1}\mathrm{cos}{\alpha }_{1}-{\stackrel{¨}{x}}_{1}\mathrm{sin}{\alpha }_{1}\right)+{M}_{3}\left({\stackrel{¨}{x}}_{1}{\psi }_{1}\mathrm{cos}{\alpha }_{1}-{\
stackrel{¨}{z}}_{3}{\psi }_{1}\mathrm{sin}{\alpha }_{1}-{\stackrel{¨}{y}}_{1}{\theta }_{1}$$+{\stackrel{¨}{\theta }}_{1}{y}_{3}+2{\stackrel{˙}{\theta }}_{1}{\stackrel{˙}{y}}_{3}-2{\stackrel{˙}{x}}_
{3}{\stackrel{˙}{\psi }}_{1}\right)+{c}_{{z}_{1}}\left({\stackrel{˙}{z}}_{1}\mathrm{cos}{\alpha }_{1}+{\stackrel{˙}{x}}_{1}\mathrm{sin}{\alpha }_{1}+{\stackrel{˙}{z}}_{3}\right)+{c}_{{z}_{3}}{\
stackrel{˙}{z}}_{3}$$+{k}_{{z}_{3}}{z}_{3}+{M}_{3}g\left(\mathrm{cos}\alpha -{\psi }_{1}\mathrm{sin}\alpha \right)=-{f}_{y}{N}_{y}sign\left({\stackrel{˙}{z}}_{3}\right),$${B}_{1\theta }\left({\
stackrel{¨}{\theta }}_{3}+{\stackrel{¨}{\theta }}_{1}\right)+{B}_{2\theta }{\stackrel{¨}{\psi }}_{3}{\varphi }_{1}+{B}_{3\theta }{\stackrel{˙}{\varphi }}_{1}{\stackrel{˙}{\psi }}_{3}-{B}_{4\theta }{\
stackrel{¨}{\varphi }}_{3}{\psi }_{3}+{B}_{5\theta }{\stackrel{˙}{\varphi }}_{3}{\stackrel{˙}{\psi }}_{3}+{B}_{6\theta }{\stackrel{¨}{\varphi }}_{3}{\psi }_{1}$$+{B}_{7\theta }{\stackrel{˙}{\psi }}_
{1}{\stackrel{˙}{\varphi }}_{3}+{B}_{8\theta }{\stackrel{¨}{\psi }}_{1}{\varphi }_{1}+{B}_{9\theta }{\stackrel{¨}{\varphi }}_{1}{\psi }_{1}+{B}_{10\theta }\left({\stackrel{¨}{\varphi }}_{1}{\psi }_
{1}-{\stackrel{¨}{\varphi }}_{1}{\psi }_{3}\right)=\left({F}_{fr}{\right)}_{y}{r}_{y}sign\left({\stackrel{˙}{\theta }}_{3}\right),$
where ${c}_{{x}_{1}}$, ${c}_{{y}_{1}}$, ${c}_{{z}_{1}}$ – coefficients of resistance of WM elastic system; ${B}_{1\theta }$, ${B}_{2\theta }$, ... – functions of the sum of the inertia moments of
masses relative corresponding axes; The coefficients ${c}_{{x}_{1}}$, ${c}_{{y}_{1}}$, ${c}_{{z}_{1}}$, ${c}_{{x}_{3}}$, ${c}_{{y}_{3}}$, ${c}_{{z}_{3}}$, ${k}_{{y}_{3}}$, ${k}_{{z}_{3}}$ (Fig. 3) in
Eq. (3) change according to the movement of the material in relation to the working body (${z}_{3}$) (the coefficient values reduce within the half period from the moment of detachment of the
material from the surface [3, 5]).
The numerical values of stated (empirical) coefficients [3, 5] correspond to the ordinary coal with 5 % humidity, or ore with maximum fragment size no greater than 100 mm and with 2-3 % humidity.
For the description of the friction force between the WM and TL of the friable type various approaches [3, 18-20] are used, the essence of which is that reaction of the TL on WM is proportional to
the velocity and deformation of TL (Fig. 3):
that will be written in the expanded form as follows:
where $q$ takes the values ${x}_{3}$, ${y}_{3}$, ${z}_{3}$; consequently, friction forces of the TL on the surfaces of WM (e.g. of the tray-shape form) and their moments will have the form:
where $f$ – coefficient of friction of the TL on the surface of WM; ${r}_{q}$ – distance between the friction surface and center of mass of the TL along the coordinate $q$.
Since the rotary movement of the material is relatively small than the linear one, and only the material’s longitudinal displacement is taken into consideration, only linear coordinates are provided
for in the material friction forces Eqs. (5), (6).
Decomposing expression ${\left({F}_{fr}\right)}_{q}$ along the surfaces of the WM, we obtain:
where ${f}_{x}$, ${f}_{y}$, ${f}_{z}$ – coefficients of friction along ${x}_{1}$, ${y}_{1}$, ${z}_{1}$; ${N}_{y}$, ${N}_{z}$ – normal reactions on the surfaces of the WM with tray-shape form; “$sign$
” – non-linear function depending on sign of velocities ${\stackrel{˙}{x}}_{3}$, ${\stackrel{˙}{y}}_{3}$, ${\stackrel{˙}{z}}_{3}$: $sign$ = 1, at ${\stackrel{˙}{x}}_{3}{\left(\stackrel{˙}{y}}_{3},{\
stackrel{˙}{z}}_{3}\right)<0$; $sign$ = –1, at ${\stackrel{˙}{x}}_{3}{\left(\stackrel{˙}{y}}_{3},{\stackrel{˙}{z}}_{3}\right)>0$. Moments of the friction forces will be determined by putting
projections Eq. (6) into expressions of ${M}_{q}$ Eq. (5).
As it was mentioned, the TL (friable material) is unilaterally connected to WM (Fig. 3). It is periodically compressed and released depending on the action of the working member, or the pressure on
the WM is increased and decreased (Fig. 2). The coefficients ${c}_{{q}_{3}}$ and ${k}_{{q}_{3}}$, change respectively in the expression ${N}_{q}={c}_{{q}_{3}}{\stackrel{˙}{q}}_{3}+{k}_{{q}_{3}}{q}_
{3}$ as well as in Eq. (3) in the process of modeling depending on ${z}_{3}$; When ${z}_{3}$ ≤ 0 (TL is displaced together with WM) values of ${c}_{{q}_{3}}$ and ${k}_{{q}_{3}}$ are significantly
greater than when ${z}_{3}$ > 0 (TL loses contact partially or fully with WM) and these changes as well as the coefficients are different depending on the TL properties (dry, humid etc.) [3].
For illustration, in Fig. 5 is shown the trajectory of the friable material layer (${z}_{3}$) relative to the working member vibration (${x}_{1}$) [21].
Analysis of the oscillograms shows that the vibratory movement trajectory of the friable material layer, at various amplitudes of the working member vibration, is different: a) the material moves on
the working surface with slipping without losing the contact with it; b) with increase of the amplitude the material loses the contact partially with the surface; c) with further increase of the
amplitude a single touch of the material with the surface takes place during one period of the vibration followed by its jump (vibration velocity at losing the contact ${\stackrel{˙}{x}}_{1}=max$);
d) with increase of the vibration amplitude (with increase of the velocity) the length of the material jump increases but since the vibration velocity is small at the moment of the material falling
it cannot lose the contact until the velocity reaches the maximum again (the oscillogram of the vibration velocity ${\stackrel{˙}{x}}_{1}$ is not given in the Figure and it is meant that it is
shifted from the displacement ${x}_{1}$ by the phase 90°).
The modeling was carried out according to the parameters of the resonant vibro-feeder with 500 W electro-magnetic vibro-exciter [5, 23-26]. At that, equations of movement of mass ${M}_{2}$ are
replaced by the equation of variation of the electromagnetic flow $Ф$ [27]:
$\frac{dФ}{dt}=\frac{{U}_{0}}{W}\mathrm{s}\mathrm{i}\mathrm{n}\omega t-\frac{\left(\delta -x\right)r}{{\mu }_{0}S{W}^{2}}Ф,$
depending on which the exciting force is determined:
$Q=\frac{0,051}{{\mu }_{0}S}{\mathrm{\Phi }}^{2},$
where $W$ – number of windings of the coil, $\delta$ – initial clearance between the magnet pole and anchor; ${\mu }_{0}$ – magnet penetration of the air; $S$ – area of the electromagnet core
section; $r$ – total resistance; $x$ – displacement of the reduced mass of the active and reactive parts of the vibratory feeder.
The following assumptions were considered during the modeling:
– Movement of the material is presented at modeling by system Eq. (3) and reaction of the TL on WM is not provided for in Eq. (1) of the working member or $\mu$ = 0, which we often find in the
researches [1-4, 7] and it does not change the qualitative characteristics of the results (in general, $\mu$ varies between 0-1 and its value depends on the type of friable material and vibration
regime: that is the next stage of our research and the subject of our next publication).
– Only linear coordinates are provided for in the material friction forces Eqs. (5), (6), which is explained above.
Fig. 6 shows some oscillograms of movement of the vibro-feeder with the load: vibrations of the working member – ${x}_{1}$, displacement of the friable material – ${x}_{3}$ and variation of the
electromagnetic flow $\mathrm{Ф}$.
The mentioned oscillograms are obtained from the solutions of Eqs. (1), (3), (7) and (8) equations, when special non-working vibrations ${y}_{1}$, ${z}_{1}$, ${\theta }_{1}$, ${\psi }_{1}$, ${\varphi
}_{1}$ are far from the working resonance (50 Hz).
Fig. 51 – trajectory of a grain movement (z3), 2 – amplitude of the working member (x1), a) displacement of a grain without losing touch with the working member, b), c), d) –displacement of a grain
with losing touch with the working member
The process of the vibro-feeder’s entering the resonance, when amplitude ${x}_{1}\mathrm{}$of the working member and respectively angle of rise of the material displacement ${x}_{3}$ increase, is
depicted in the picture; It should be noted in this connection that the equation of the material longitudinal displacement (first equation of (3)) does not include elastic deformation ${x}_{3}$ of
the material (Fig. 3) because displacement of the material is not restricted in this direction (the working member groove is open in the longitudinal direction) in contrast to directions ${y}_{3}$
and ${z}_{3}$, when the working member bottom and walls restrict the material displacement and therefore, their equations include ${y}_{3}$ and ${z}_{3}$; Consequently, the material vibratory
displacement ${x}_{3}$ has the form [3], shown in Fig. 6 and with increase of the velocity, angle of its rise increases.
Fig. 6Oscillograms of vibrations of the working member x1, variation of the electromagnetic flow Ф and displacements of the technologic load x3
In view of the fact that the exciting force in Eq. (1) presents in the non-linear form Eq. (8), there are two resonant positions at variation of the frequency – basic (50 Hz), sub-harmonic (25 Hz)
and super-harmonic (100 Hz) that is reflected in the obtained results (Fig. 7, 8). The pictures show the dependence of the velocity of the longitudinal displacement of a friable material (${V}_{x}$)
on the variation of the rotary (${\psi }_{1}$), and vertical (${z}_{1}$) vibration frequencies (${\omega }_{{\psi }_{\mathrm{}}}$ and ${\omega }_{z}$) of the working body (${M}_{1}$), during their
passage in resonance areas (25, 50, 100 Hz).
The influence of partial vibrations (${z}_{1}\text{,}$${\psi }_{1}$ etc.) on the process (velocity) of material displacement is realized by means of the non-linear terms of Eq. (3), where they appear
(e.g.$\mathrm{}{\stackrel{¨}{\psi }}_{1}{z}_{3}$, ${\stackrel{¨}{z}}_{1}{\psi }_{1}$ etc.).
The mathematical model was adjusted to reach the identicality with the physical experiment [3, 5, 15] to achieve the reliability of the modeling results: a vibratory displacement velocity of the
friable material (ground coal) was measured on the real vibro-feeder. Then a simple numerical experiment was carried out by variation of the material model coefficients (Fig. 3) and were selected
those values of the coefficients at which the results (${V}_{x}$) of the physical and numerical experiments were identical.
At modeling the working member operates constantly in the basic (working) resonant mode (${\omega }_{exc}={\omega }_{x}=$ 50 Hz). Besides, the sub- and super-harmonic resonant vibrations with 25 (1/2
${\omega }_{exc}$) and 100 Hz (2 ${\omega }_{exc}$) are also generated in the electromagnetic vibrator feeded from the 50 Hz (${\omega }_{exc}$) power supply network, by corresponding change of the
rigidity. If we vary coefficients of rigidity i.e. eigenfrequencies ${\omega }_{y}$, ${\omega }_{z}$, ${\omega }_{\theta }$, ${\omega }_{\psi }$, ${\omega }_{\phi }$ in the equations of the working
spatial vibrations in the range of 0-110 Hz, we obtain resonant vibrations of 25, 50, and 100 Hz in these directions. With such approach it becomes possible to enhance nonworking spatial vibrations
and study their influence on the material velocity in combination with basic working resonant vibration (${\omega }_{x}=$ 50 Hz = const).
The graphs shown on Fig. 7 are obtained from the solutions of Eqs. (1), (3), (7) and (8), when the vibration of only ${z}_{1}$ direction enters into resonance and other non-working vibrations ${y}_
{1}$, ${\theta }_{1}$, ${\psi }_{1}$, ${\phi }_{1}$ are far from the working resonance regime ($A=$ 2.8 mm, ${\omega }_{x}=$ 50 Hz), their influence is insignificant and the change of the
longitudinal displacement velocity (${V}_{x}$) depends on only non-working (“parasite”) change of ${z}_{1}$.
Fig. 7Dependence of the velocity of the longitudinal displacement Vx of the load M3 on the frequency of the WM resonant vibrations along axis O1z1; ωx= 50 Hz, A= 2.8 mm
The graphs on Fig. 8 are also obtained from the solutions of Eqs. (1), (3), (7) and (8), when the vibration of only ${\psi }_{1}$ direction enters into the resonance, whereas the rest of non-working
vibrations ${y}_{1}$, ${z}_{1}$, ${\theta }_{1}$, ${\phi }_{1}$ are far from the continuous working resonance regime ($A=$ 2 mm, ${\omega }_{x}=$ 50 Hz), i.e. they show the change of the longitudinal
displacement velocity (${V}_{x}$) depending only on the change of ${\psi }_{1}$. The change of ${V}_{x}$ on Fig. 7 and Fig. 8 begins when ${z}_{1}$ and ${\psi }_{1}$ non-working vibrations begin to
enter the resonance and end when they exit the resonance, i.e. in the following limits of ${\omega }_{z}$ and ${\omega }_{\psi }$ frequencies 10 $>{\omega }_{z}>\mathrm{}$55 Hz, 35 $>{\omega }_{\psi
}>$ 105 Hz. Outside of those frequencies, the value of ${V}_{x}$ is constant: on Fig. 7 ${V}_{x}\approx$ 0,12 m/s, on Fig. 8 ${V}_{x}\approx$ 0,08 m/s and they correspond to the nominal
(experimental) values when the vibro-feeder mainly operates in the resonance regime and no outside factors affect the process of material displacement.
The graphs on Fig. 9 are obtained from the solutions of Eqs. (1), (3), (7) and (8), when all of the non-working vibrations ${y}_{1}$, ${z}_{1}$, ${\theta }_{1}$, ${\psi }_{1}$, ${\phi }_{1}$ are far
from the working resonance regime ($A=$ 4 mm, ${\omega }_{x}=$ 50 Hz) and their influence on the process is insignificant; the graphs show the dependence of the friable material’s velocity on the
angle of vibration ($\beta$) at fixed values of the angle of inclination of the working member ($\alpha$). As it is seen from the graph, the best result is reached at angles: $\alpha \approx$ 10° and
$\beta \approx$ 10°.
Fig. 8Dependence of the velocity of the longitudinal displacement Vx of the load M3 on the frequency of the WM rotary vibrations ψ1; ωx= 50 Hz, A= 2 mm
Fig. 9Dependence of velocity Vx of the friable material on the variation of the angle of vibrations β at fixed angles of the inclination of the working member α; ωx= 50 Hz, A= 4 mm
The working resonance frequency of the WM during the modeling process is constant: ${\omega }_{x}=$ 50 Hz = const.
Fig. 10Design of the vibro-feeder with a controllable trajectory of vibrations of the working member
The results of the modeling have shown that some combinations of the spatial (non-working) vibrations and basic working vibrations may have a positive influence on the transportation and technologic
process of the material (increase both, transportation velocity of the friable material and the intensity of movement).
Fig. 10 shows a design of the vibro-feeder with a new vibro-exciter. It is developed on the basis of the results of the modeling and it allows generation of various forms (I, II, III) of the working
vibrations with the help of the variation of the angle between electromagnetic and elastic forces and thereby control the vibratory technologic process of the friable material.
3. Conclusions
1) A new mathematical model of spatial movement of the three-mass system – analog of the vibratory (Fig. 4) machine with technological load, is developed. A system of differential equations describes
the interconnected movement of component parts of the loaded machine and allows many-sided research into vibratory transportation and the technologic process of the friable materials.
2) The presented graphs of the results show that mathematical modeling may reveal new nuances promoting improvement of both, the technologic process and the machine design.
3) It was, for example, ascertained that some spatial non-working (parasitic) vibrations in combination with basic (working) vibrations can significantly increase velocity of the material’s vibratory
displacement (see Figs. 7, 8).
4) A new design of the electromagnetic vibratory feeder was developed on the base of the modeling results that allows obtaining various configurations of the working vibrations and increasing
velocity of the material transportation by the change of the elastic force direction. The obtained design (Fig. 10) confirms reliability of the developed mathematical model because the influence of
the vertical partial (non-working) vibration is qualitatively identical for both – the modeling (Fig. 7 – ${z}_{1}$) and the design (III-${z}_{1}^{\mathrm{*}}$).
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About this article
Vibration generation and control
vibratory technological process
spatial vibrations
vibratory transportation of the friable material
generalized mathematical model
This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [ N FR17_292, “Mathematical Modeling of the Vibratory Technologic Processes and Design of the New, Highly
Effective Machines”.
Copyright © 2020 V. Zviadauri, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Problem solving with functions and proof | Oak National Academy
Hello, Mr. Robson here.
Welcome to Maths.
Great decision to join me today, especially seeing as we're problem solving with functions and proofs.
I love problem solving, you love problem solving, so let's solve some problems. Our learning outcome is I'll be able to use our knowledge of functions and proof to solve problems. Lots of keywords
you're gonna hear throughout today's lesson.
Two of which are function and expression.
A function is a mathematical relationship that uniquely maps values of one set to the values of another set.
An expression contains one or more terms where each term is separated by an operator.
Two parts of our learning today, we're gonna begin by looking at problems with functions.
If f of x equals five x minus four and g of x equals x minus 11, we can write expressions for composite functions like g of f of x and f of g of x.
G of f of x would look like so, that's the function f of x going into the function g of x and simplifying of course.
F of g of x looks like so.
That's the function g of x being input into the function f of x giving us five x minus 59.
We can use the same skills to write expressions for all sorts of manipulations of the function f of x, such as two f of x, f of two x, f of x plus two and f of x plus two.
Let's have a closer look at what those four things mean.
Two f of x is two lots of the function f of x.
f of x plus two is two being added to the function f of x.
F of two X is when two x is input into our function.
So would make logical sense that f of x plus two is when x plus two is input into the function f of x.
So in the case of two lots of f of x, that's two lots of f of x or two lots of five x minus four.
We'll leave it like that rather than expanding those brackets.
That's a nice concise expression.
f of x plus two, well that's f of x and then we'll add two to that.
Let's simplify five x minus two.
F of two x input into two x into that function f of x would look like that.
Let's simplify, 10 x minus four.
f of x plus two.
Well, when we input x plus two into our function f of x, we get that.
Let's expand that bracket, add the like terms, simplify and we get five x plus six.
Quick check.
You can do that.
If f of x equals seven x plus 11 and g of x equals three lots of five minus x, can you write expressions for these six things? Pause and give it a go now.
Welcome back.
Let's see how we did.
Three lots of f of x would look like that.
That's seven x plus 11 being multiplied by three.
f of x minus 15.
Well that's our function f of x and then subtract 15.
Eight lots of g of x would look like that.
Again, you're welcome to leave that expression in its brackets or expand it out, either is fine.
g of x plus five, well that's our function g of x.
When we add five to it, we will expand that bracket and simplify those like terms for that one.
F of seven x, well that's seven x going into our function f of x giving us 49 x plus 11.
g of x plus eight.
That's x plus eight going into our function g of x which expands and simplifies to negative three x minus nine.
Andeep and Izzy are manipulating the function f of x equals x plus four.
Andeep says "I'm great at algebra.
I know that three times a times b equals b times a times three.
So I predict that three f of x equals F of three x." Izzy says "it's good to make predictions, but it's even better when we test them." Well done Izzy, good point.
Can you test Andeep's prediction? Is he right? Pause.
Have a play with this bit of mathematics.
I'll be back with a big reveal in a moment.
Welcome back.
Hopefully you noticed three lots of f of x gives us three lots of x plus four which expands to three x plus 12, whereas f of three x gives us three x plus four so it wasn't true.
Three f of x is not equal to f of three x.
Andeep says "I was wrong.
But I don't mind making errors in maths Izzy.
Every error is an opportunity to learn." Izzy says "I totally agree Andeep, I agree too.
Three f of x and f of three x are function notation so three f of x does not equal f of three x." But is that always true? Could there be a function f of x such that three f of x equals f of three x?
Andeep says, "hold on Izzy, I found some cases whereby three f of x equals F of three x.
Can you think of any?" Is Andeep right? Is it ever possible that this is true? Pause, have a think.
See if you can come up with some examples.
Welcome back.
I wonder if you found any.
Andeep found some and he presents them to Izzy.
Izzy says "well done.
It's true for these two.
There will be some cases where three f of x equals F of three x." Quick check you've got there.
For a function f of x, it is true that a f of x equals F of axe where a is an integer.
Is that always true, sometimes true or never true? Pause and have a think.
Welcome back.
Hopefully you said sometimes.
In most cases a F of a will not equal F of axe, but there will be some cases where they are equal.
Sometimes we're given limited information and have to work backwards to find the expression of the original function or the manipulation that has happened to a function.
f of x equals kx minus seven.
Ooh, we're not given the actual f of x function here, but we are told F of three equals two and then challenged to find k where k is a constant.
Lots of ways we could do this.
If we input three into that f of x function, we know k multiplied by three minus seven will give us that output two because we're told F of three was equal to two.
There, we can set up this equation.
Three k minus seven equals two, so three k must equal nine and k must equal three.
Another example, g of x equals nine x plus one.
We have the expression of g of x, but we haven't got the manipulation in this case.
If kg of negative one equals negative 40, can you find k where k is a constant? Well, let's look at an expression for kg of x, that would be k lots of nine x plus one.
We know an input of negative one will give us an output of negative 40, so let's express that.
From there, we can do some simplifying and find that k multiplied by negative eight equals negative 40, so k must be equal to five.
Some examples like this might include composite functions.
We know f of x but we're not given all the information of g of x.
We're told it's kx minus four and then we're told that f of g two equals 56 and challenged to find k where k is a constant.
We could start with an expression for f of g of x, f of g of x will be the output of g of x kx minus four going into the function f of x.
We're gonna do some simplifying from there and we're happy at that point.
We're now gonna put our input two and reflect the fact that we know it gives us an output of 56.
Some simplifying and tidying from there and we find that 2k minus two equals eight, so 2k equals 10k equals five.
We can check that that's true.
We think that g of x is five x minus four, so let's find out if that's true.
G of two would therefore be five lots of two minus four, that's six.
When that goes into f of x, it should give us 56.
It does.
We know we're right.
We could have solved this same problem by considering inputs and outputs.
F of what makes 56? Well that what would be six.
Why do we care about that? Well, that's because six is what we need to input into the function f of x to get an output of 56 therefore g of two equals six.
G of two equals six, then k multiplied by two minus four equals six, k equals five.
Absolutely no surprise to arrive at the exact same answer.
In fact, it's reassuring if you can do the same problem in two different ways and you get the same answer, you are reassured that you are right.
Quick check.
You can do similar problems to those.
There's three problems here for you to think about and in each case we're told that k is a constant.
See if you can find the solutions to these three problems now.
Pause, see you in a moment for the solutions.
Welcome back.
For the first problem.
If f of three equals negative three, find k.
Let's reflect that input of three going into the expression and gives us an output of negative three.
If that's true, then three k plus 12 equals negative three.
Therefore k equals negative five.
For the second problem, an expression for kg of x helps.
We then reflect the fact that an input of nine gives us an output of negative 296.
Let's simplify that.
We find that k equals negative four.
For the final problem, g of x plus k.
We can express like so.
Let's reflect that input and output and we find that k equals 13.
Let's try a similar problem but with a composite function.
If g of f of five equals 311, can you find k where k is a constant in this example.
Pause, give this problem a go, see you in a moment so you can compare your work to mine.
Welcome back.
Lots of ways we could have done this.
One way is to ask ourself the question g of what makes 311? Well, g of what makes 311, that input needs to be 105.
Why do we care about that? Because f of five must equal 105 if g of f of five equals 311.
So if f of five equals 105, then 15k equals 105, k equals seven.
We can check that that's true.
We think f of x equals seven lots of x plus 10, so let's evaluate f of five and then input that output into the function g of x and we get to 311.
Our composite function mapped five to 311, so k must be equal to seven.
An alternative method was to write an expression for g of f of x.
Either method is perfectly valid.
Practise time now.
We're given two functions, f of x and g of x and we're asked to write expressions for these six things.
Pause and write those now.
For question two, we're given a function f of x and g of x and we're asked to show that f of x plus five minus two f of x equals 16 minus three x.
For part B, we're asked to show that f of four x is not equal to four f of x.
For part C, we're asked to show that g of x plus three minus g of x is equal to three lots of two x plus one.
Pause and see if you can prove those three facts now.
Question three, we're given the fact that f of x equals k, lots of X minus seven and g of x equals x squared plus two x minus 15.
for A, B and C, we're asked to find k where k is a constant and we're given an inputs and outputs for all three of those things.
Pause, see if you can solve these problems now.
Feedback time.
Let's see how we did.
Question one, part A, write an expression for eight f of x, we should have got 64 x minus 24.
For B, we should have got eight x plus five by the time you've simplified.
For C, we should've got four x plus 27.
For D, we should've got four lots of x plus six.
You might leave that expression bracketed.
You might expand those brackets.
Either is fine.
Part E, f of g of x plus five.
It's useful to write an expression for f of g of x first 32 x plus 221.
Now when we input x plus five into that expression, we get 32 x plus 381.
Part F was a little bit complicated.
We can undo some of that complication by writing an expression for g of f of x, which we four lots of eight x plus four.
Now we're gonna input two x and multiply the output by five.
That'll be reflected like so that simplifies to 320 x plus 80.
For question two, we are trying to show that a few things are true.
For part A, it helps to express f of x plus five and two f of x.
Once we've got those two expressions, we can show that f of x plus five minus two f of x does indeed equal 16 minus three x.
Part B, f of four x is 12 x minus one, whereas four f of x is 12 x minus four and they are not equal.
Part C, lets write an expression for g of x plus three.
We're gonna expand and simplify there and get the expression x squared plus four x plus eight.
When we take that and we subtract g of x, we end up with that which we simplify to six x plus three which factorises to three lots of two x plus one.
Question three, we are finding k, the constant in each example.
For part A, we'll input 10 and show that gives us an output of negative 21.
Therefore k must be equal to negative seven.
For part B, we can input seven express the fact that it gives us an output of negative 96 has a lot of simplifying scoring in that bracket there.
Once we've done that, we find that 48k equals negative 96, k is equal to negative two.
For part C, an expression for h of j of x would be three lots of kx squared minus four.
Let's show the fact that an input of five gives an output of 221.
That takes us to the point that 75k equals 225, so k must equal three.
Onto the second part of our learning now, problems with proof.
Sometimes in maths we come across proofs, which we know can't be right.
Sofia says, "did you know that two equals one Lucas?" And Lucas says, "no, it doesn't.
There's no way that's true." Sofia says 'it is true.
I can prove it." Well, Sofia, I'm looking forward to this.
Sofia writes, if x equals y, then x squared equals x multiplied by y.
Okay, add negative y squared to both sides.
Factorise, I'm with you.
Divide both sides by x minus y.
Okay, and then x is equal to y, so we can rewrite that.
Therefore, two y equals y divide both sides of the equation by y and two is equal to one.
Sofia, I'm stunned.
Two does equal one.
Sofia says, "I told you, I'm a genius.
I have broken maths." Or has she? Can you see any fault in Sofia's proof or is she right? Pause and have a good look at that work there.
Welcome back.
I wonder if you spotted it.
If we see a proof that we know cannot be true, we need to investigate.
In fact, there are some proofs in maths that are still being checked by the finest mathematicians on the planet.
We're gonna check this one because we know that two doesn't equal one.
The problem with this proof comes here in this step.
Can you see why? Well spotted if you can spot this.
The step is to divide both sides by x minus y.
At first glance, this looks fine, but when you remember that x equals y, the problem becomes clearer.
If x equals y, then x minus y equals zero and dividing by zero is undefined.
That means this is not a valid step in the proof.
Substituting in a value makes this error more obvious.
Rather than starting with x equals y, let's start with five equals five.
If five equals five, then five squared equals five times five.
It does.
Five squared minus five squared equals five times five minus five squared.
That's true, and we can factorised those and that remains true, but here when we've divided through by five minus five on both sides, we're now at a position of having a false statement.
Five plus five is not equal to five, so we can't say that two lots of five equals five and we can't say that two equals one.
Quick check that you can spot the error in proof.
Spot the error in this proof.
This proof proves that three equals two.
Alarm bells should be ringing.
Something must be wrong somewhere.
Pause and see if you can spot it.
Welcome back.
Did you spot it? At first glance, it appears perfectly valid, but if you solve this equation, you'd find that x equals three, therefore two x minus six equals zero.
Dividing by zero is not a valid step.
In order for it to be a proof, all steps need to be valid.
Sometimes an invalid step can convince us to believe something is true when it is not.
26 over 65 cancels down to two fifths.
It does.
Here's a proof.
In 26, we'll just scratch out the ones digit and in 65 we'll scratch out the tens digit.
That's why it's equal to two over five.
Here's another example.
16 over 64 equals a quarter.
Here's my proof.
Let's just scratch out the ones digit in the numerator.
The tens digit in the denominator, 16 over 64 equals a quarter.
Here's another proof.
I'm just gonna scratch out the ones digit in the numerator.
The tens digit in the denominator, 19 over 95 cancels down to one fifth.
What I'd like you to do is come up with a counterexample to show that this is an invalid step.
I can't just do this.
Scratch out the ones in the numerator, the tens in the denominator.
Can you come up with a counterexample to show that that is not a valid proof? Welcome back.
Lots of examples you could have used here.
You might have shown 12 over 24 when you scratch out the ones digit in the 12 and the tens digit in the 24, we show that 12 over 24 is equal to one quarter, which is not an equivalent fraction.
That's not true.
This is not a valid proof.
Many other counterexamples were acceptable.
In order for it to be a proof, all steps need to be valid.
This was a case of some very invalid steps.
This proof is also clearly wrong.
We're proving that pi equals three.
It doesn't, but can you see why it's wrong? Have a good look at that proof.
See if you can spot where it's gone wrong.
Welcome back.
Let's have a close look.
The problem with this proof comes here.
We have to be very careful when taking the square root of both sides of an equation.
X squared equals 25 does indeed have two solutions.
X equals positive five and x equals negative five, but if A squared equals B squared, we might say that A equals positive or negative B, but it's not necessarily the case that both are true.
For example, if A equals two and B equals negative two.
It is true that A squared equals B squared, but when we work backwards, it's not the case that A equals positive B or negative B.
This proof assumes that the positive root is correct, but we can see from the result that this was not the case because pi is not equal to three.
If we take the negative root at this step, we get back to the start, three plus pi equals two x.
We have to be very careful when taking the square root of both sides of an equation.
Sometimes we get two truths, sometimes we get one truth and one falsehood, so be careful.
It's not just in algebra that we may see suspicious proofs.
When these four shapes are placed together, they form an eight by eight square with an area of 64 square units.
When we arrange the four pieces to make a rectangle, they form a 13 by five rectangle with an area of 65 square units.
My key question for you, where has the extra square come from? Pause and see if you can spot it.
Welcome back.
I wonder if you noticed.
Your eyes have been deceived by what we call a dissection fallacy.
The problem is that the triangles are not identical.
In the red triangle on the left hand side of the screen, the gradient of the hypotenuse is three over eight.
This is a straight line.
When we look at the red triangle or is it a triangle? We see that it starts with a gradient of two over five and then changes to have a gradient of one over three.
This is not a straight line, so that shape there is in fact not a triangle.
It's a quadrilateral.
Your eyes have been deceived by a dissection fallacy.
If we insert congruent triangles, you can see the space that created the extra one, that little gap in there.
Quick check you've got that.
In geometrical proofs, we can be deceived by a bisection fallacy, dissection fallacy or dissecting falsely.
Pause and answer this one.
I hope you said B, dissection fallacy.
To avoid falling for a false proof, pay careful attention to the detail in every shape in the dissection.
Practise time now.
Question one, A equals zero, B equals two and C equals four and there's a proof.
A proof that shows that four is equal to zero.
Alarm bells are ringing, something wrong In that proof.
What I'd like you to do is find the error in the proof and write a sentence explaining why it's an invalid step.
Pause and do that now.
Question two.
If A equals B equals C, then look at that proof, three is equal to one.
Well, of course it isn't, which means there's an error somewhere.
I'd like you to find the error in this proof and write a sentence explaining why it's an invalid step.
Pause and do that now.
Question three.
I'd like you to write your own proof to show that zero equals 100.
My hint for you is you could start with A equals 50, B equals 100 and C equals zero.
I enjoyed doing this problem myself.
I hope you will too.
Pause and give it a go.
It might take you more than one attempt.
Question four, I'd like you to explain why in the second diagram there's an extra square.
Pause and have a good study of this screen.
Feedback time now, finding the error in this proof and writing a sentence explaining why it's an invalid step.
The problem is here.
The positive root of each term was taken.
It is wrong to assume that the positive roots are always the correct ones.
Question two.
Again, finding the error and the error was here.
If B equals C, then B minus C equals zero.
Dividing by zero is not a valid step.
In order for it to be a proof, all steps need to be valid.
Question three, I challenged you to write your own proof to show that zero equals 100.
You might have written if A equals 50, B equals 100 and C equals zero, then C plus B equals two A and so on and so on and so on and so on and so on and so on and so on, and therefore zero equals 100.
That is an example of a proof of zero equaling 100, but we know of course that taking only the positive roots here has invalidated the proof.
However, we could use this to convince a less vigilant mathematician that 100 equals zero and wouldn't that be fun? Question four, I asked you to explain why in the second diagram there's an extra
We're being fooled by a dissection fallacy.
This is not a triangle.
It is a quadrilateral.
The gradient of the red triangle is three over eight, the orange triangle, two over five.
That is not a straight line.
When we reverse the triangles and put the one with the greater gradient first we create space for the extra square.
If we overlay the respective gradients from the second shape, you can see more clearly the extra space created.
Well, we're at the end of the lesson now sadly, but what have we learned? We've learned that larger functions and proof can be used to solve a wide variety of problems. If we know f of x, then we can
write expressions for manipulation such as four f of x and f of four x and know that they are not necessarily the same.
Invalid steps in algebraic proofs can be identified and incorrect steps such as dividing by zero or making assumptions around the square roots of an equation can be spotted.
Hope you've enjoyed today's lesson as much as I've enjoyed it? And I look forward to seeing you again soon for more maths.
Goodbye for now.
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Everyday Calculator - Online Calculator
Everyday Calculator
Use this Percent to Goal Calculator to measure your progess in percentage towards a certain goal.
Everyday Calculator
Use this Audiobook Percentage Calculator to find how much percent of the audiobook is left to listen.
Everyday Calculator
Use this typing converter tool to convert your typing speed from Words Per Minutes (WPM) to Characters Per Minute (CPM).
Everyday Calculator
Date of Birth to be 21 Today Calculator as the name suggests is an online calculator which tells you the date you need to be born in order to be 21 or older today. Just enter today’s date in the
input field, and check whether you are 21 or not according to today.
Everyday Calculator
Plus Two Percentage Calculator is an online calculator that helps you to calculate the total marks and percentage of class 12th results. This tool is designed specifically for Kerala plus two (+2)
students. But you can still use this calculator to find your plus two percentage, given the total marks of each subject is 200.
Everyday Calculator
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Linear Systems
13.2.2 Linear Systems
Now that the phase space has been defined as a special kind of state space that can handle dynamics, it is convenient to classify the kinds of differential models that can be defined based on their
mathematical form. The class of linear systems has been most widely studied, particularly in the context of control theory. The reason is that many powerful techniques from linear algebra can be
applied to yield good control laws [192]. The ideas can also be generalized to linear systems that involve optimality criteria [28,570], nature [95,564], or multiple players [59].
Let be a phase space, and let be an action space for . A linear system is a differential model for which the state transition equation can be expressed as
in which and are constant, real-valued matrices of dimensions and , respectively.
Example 13..5
(Linear System Example) For a simple example of (
), suppose , , and let
Performing the matrix multiplications reveals that all three equations are linear in the state and action variables. Compare this to the discrete-time linear Gaussian system shown in Example
Recall from Section 13.1.1 that linear constraints restrict the velocity to an -dimensional hyperplane. The linear model in (13.37) is in parametric form, which means that each action variable may
allow an independent degree of freedom. In this case, . In the extreme case of , there are no actions, which results in . The phase velocity is fixed for every point . If , then at every a
one-dimensional set of velocities may be chosen using . Note that the direction is not fixed because is added to all components of . In general, the set of allowable velocities at a point is an
-dimensional hyperplane in the tangent space (if is nonsingular).
In spite of (13.37), it may still be possible to reach all of the state space from any initial state. It may be costly, however, to reach a nearby point because of the restriction on the tangent
space; it is impossible to command a velocity in some directions. For the case of nonlinear systems, it is sometimes possible to quickly reach any point in a small neighborhood of a state, while
remaining in a small region around the state. Such issues fall under the general topic of controllability, which will be covered in Sections 15.1.3 and 15.4.3.
Although not covered here, the observability of the system is an important topic in control [192,478]. In terms of the I-space concepts of Chapter 11, this means that a sensor of the form is defined,
and the task is to determine the current state, given the history I-state. If the system is observable, this means that the nondeterministic I-state is a single point. Otherwise, the system may only
be partially observable. In the case of linear systems, if the sensing model is also linear,
then simple matrix conditions can be used to determine whether the system is observable [192]. Nonlinear observability theory also exists [478].
As in the case of discrete planning problems, it is possible to define differential models that depend on time. In the discrete case, this involves a dependency on stages. For the continuous-stage
case, a time-varying linear system is defined as
In this case, the matrix entries are allowed to be functions of time. Many powerful control techniques can be easily adapted to this case, but it will not be considered here because most planning
problems are time-invariant (or stationary).
Steven M LaValle 2020-08-14
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Samacheer Kalvi 5th Maths Guide Term 2 Chapter 1 Geometry InText Questions
Students can download 5th Maths Term 2 Chapter 1 Geometry InText Questions and Answers, Notes, Samacheer Kalvi 5th Maths Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus,
helps students complete homework assignments and to score high marks in board exams.
Tamilnadu Samacheer Kalvi 5th Maths Solutions Term 2 Chapter 1 Geometry InText Questions
Try These (Text Book Page No.2)
Question 1.
Tick (✓) the correct alternative
The shortest distance between the points C and D is shown by the segment CD the curve CD
the segment CD
line PQ and line QP represent different lines / the same line
the same line
point C lies on the ray AB / ray BD
ray AB
Segment MN has infinite / finite length
finite length
Ray RT is a part / is not a part of the line TR
a part
Question 2.
Write the type of the angle
Right angle
Acute angle
Straight angle
Obtuse angle
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Time Settings – Factors
There are three factors that get applied to time slips and one that is used when budgeting time for phases.
Overtime Factor - this is generally 1.5. When a time slip is marked as overtime the labor cost at calculated at the employee base rate times this factor. If the employee base rate is $20 per hour
then the overtime rate at 1.5 would generate a labor cost of $30 per hour.
Overhead Factor - this is used to burden projects with overhead costs so that net profits can be calculated. Every firm has a different overhead factor and it is covered in a separate article. The
overhead factor is multiplied times the direct labor costs to determine the overhead costs to apply towards the phase and project associated with the time slip. We have clients with an overhead
factor down close to 1.0 and others pushing the higher end near 2.0. The average overhead factor is around 1.6.
Target Profit Percentage - this is the minimum ideal net profit that the organization would like to achieve each fiscal year. Using this value we calculate how much of the fee has been burned through
based upon direct labor + overhead and finally allocating funds for the bottom line.
Here is an example set of values:
Let's assume you have 1,000 dollars of direct labor with an overhead factor of 1.6 and a target profit of 18%. The total fee burned through would be (1000 + 1000 x 1.6) / (1 - .18) = 3,170.73. Now to
check the numbers. If revenue was 3,170.73 and direct costs plus overhead was 2,600 how much would be left for net profits? The answer is 570.73. Net profit percentage is net profit divided by
revenue which gives us 570.73 / 3170.73 = 0.18 or 18%.
Blended Billing Rate - this is the average billing rate charged to your time slips. You could also use a blended burden rate. Wait, what is a burden rate? The burden rate is direct rate + overhead +
profit. We use this to assist you in calculating how many hours you should budget for each phase of a project. We take the net fee and divide it by this rate to calculate how many production hours
you should spend on each phase. If you hit this value your less, you should obtain your target profit goals.
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Summarizing a fitted CLV model
summary.clv.fitted {CLVTools} R Documentation
Summarizing a fitted CLV model
Summary method for fitted CLV models that provides statistics about the estimated parameters and information about the optimization process. If multiple optimization methods were used (for example if
specified in parameter optimx.args), all information here refers to the last method/row of the resulting optimx object.
## S3 method for class 'clv.fitted'
summary(object, ...)
## S3 method for class 'clv.fitted.transactions.static.cov'
summary(object, ...)
## S3 method for class 'summary.clv.fitted'
digits = max(3L, getOption("digits") - 3L),
signif.stars = getOption("show.signif.stars"),
object A fitted CLV model
... Ignored for summary, forwarded to printCoefmat for print.
x an object of class "summary.clv.no.covariates", usually, a result of a call to summary.clv.no.covariates.
digits the number of significant digits to use when printing.
signif.stars logical. If TRUE, ‘significance stars’ are printed for each coefficient.
This function computes and returns a list of summary information of the fitted model given in object. It returns a list of class summary.clv.no.covariates that contains the following components:
name.model the name of the fitted model.
call The call used to fit the model.
tp.estimation.start Date or POSIXct indicating when the fitting period started.
tp.estimation.end Date or POSIXct indicating when the fitting period ended.
estimation.period.in.tu Length of fitting period in time.units.
time.unit Time unit that defines a single period.
coefficients a px4 matrix with columns for the estimated coefficients, its standard error, the t-statistic and corresponding (two-sided) p-value.
estimated.LL the value of the log-likelihood function at the found solution.
AIC Akaike's An Information Criterion for the fitted model.
BIC Schwarz' Bayesian Information Criterion for the fitted model.
KKT1 Karush-Kuhn-Tucker optimality conditions of the first order, as returned by optimx.
KKT2 Karush-Kuhn-Tucker optimality conditions of the second order, as returned by optimx.
fevals The number of calls to the log-likelihood function during optimization.
method The last method used to obtain the final solution.
A list of additional options used for model fitting.
additional.options Whether the correlation between the purchase and the attrition process was estimated.
Correlation coefficient measuring the correlation between the two processes, if used.
For models fits with static covariates, the list additionally is of class summary.clv.static.covariates and the list in additional.options contains the following elements:
Whether L2 regularization for parameters of contextual factors was used.
The regularization lambda used for the parameters of the Lifetime process, if used.
The regularization lambda used for the parameters of the Transaction process, if used.
Constraint covs
Whether any covariate parameters were forced to be the same for both processes.
Constraint params
Name of the covariate parameters which were constraint, if used.
See Also
The model fitting functions pnbd.
Function coef will extract the coefficients matrix including summary statistics and function vcov will extract the vcov from the returned summary object.
# Fit pnbd standard model, no covariates
clv.data.apparel <- clvdata(apparelTrans, time.unit="w",
estimation.split=40, date.format="ymd")
pnbd.apparel <- pnbd(clv.data.apparel)
# summary about model fit
# Add static covariate data
data.apparel.cov <-
data.cov.life = apparelStaticCov,
names.cov.life = "Gender",
data.cov.trans = apparelStaticCov,
names.cov.trans = "Gender",
name.id = "Id")
# fit model with covariates and regualization
pnbd.apparel.cov <- pnbd(data.apparel.cov,
reg.lambdas = c(life=2, trans=4))
# additional summary about covariate parameters
# and used regularization
version 0.10.0
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Square Numbers and Exponents in Python, With Examples
Home » Programming » Python » Python Exponent Square
Square Numbers and Exponents in Python, With Examples
This tutorial will show you how to use exponents (e.g., calculating the square root of a number) in Python, with some code examples.
This tutorial covers several ways to calculate exponents in Python 3.
Using ** (Power Operator)
The ** mathematical operator (known as the power operator) will calculate an exponent, raising the number on the left to the power of the number on the right of the operator:
4**5 # Will evaluate 4*4*4*4*4 and return the integer value 1024
Note that:
• You will receive a ZeroDivisionError if you try to raise 0 (zero) to a negative power
• Raising a negative number to a fractional power will result in a complex number
• If floating-point (non-integer) numbers are supplied, the result will be a floating-point number
Using pow(x, y)
The built-in pow() function will do the same – the result will be identical – it just uses a function to calculate the result rather than a mathematical operator:
pow(4, 5) # Will also return 1024, the same as above
Note that:
• Everything that applies to the ** operator applies to the pow() function
Using math.pow()
The math library also includes a pow() function which does the same thing – but with slightly different behavior and results:
import math
math.pow(4, 5) # Returns 1024 - but as a floating point number, not an integer
Note that:
• While the number returned is the same as the ** operator and the built-in pow() function, the number type is different. math.pow() converts the numbers passed to it to floating-point numbers –
and always returns its answer as a floating-point number
• Both math.pow(1.0, num) and math.pow(num, 0.0) always return 1.0 when using math.pow()
□ This occurs even when num is zero or NaN (Not a Number)
• math.pow() throws a ValueError exception when:
□ The first argument is negative
□ The second argument is not an integer
Square Numbers
‘Squaring’ a number is simply raising it to a power (exponent) of 2.
So, 5 squared would be calculated as follows using the above methods:
pow(5, 2)
math.pow(5, 2)
All of the above will return the number 25.
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I'm Brad, and I'm nearing 20 years of experience with Linux. I've worked in just about every IT role there is before taking the leap into software development. Currently, I'm building desktop and
web-based solutions with NodeJS and PHP hosted on Linux infrastructure. Visit my blog or find me on Twitter to see what I'm up to.
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BankRound Function
Returns the number rounded to the number of decimal places using the rules of bankers rounding.
Returns the number rounded to the number of decimal places using the rules of bankers rounding.
BankRound(number, digits)
number Numeric - The number to round.
digits Integer - The number of decimal digits to round the number to.
● With bankers rounding, note that .5 (.05, .005, etc) can round up sometimes and down sometimes. When the number to round is exactly halfway between two rounded values, the result is the rounded
value that has an even digit in the far right decimal position. So both 1.5 and 2.5 round to 2, and 3.5 and 4.5 both round to 4. This process is also known as rounding toward even, or Gaussian
● Example: BankRound(2.55, 1) returns 2.6.
● Example: BankRound(2.45, 1) returns 2.4.
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Weekly schedule
Week Dates Sections Topics Videos to view before class in WebAssign (click on “Resources” tab)
October 1.8,1.4 Introduction; Intermediate Value Theorem; Kazmierczak videos in precalc text sections 2.4 and 13.3
10 22-26 Tangent and Velocity; Final Exams
Math 224 2.1 The Derivative and Rates of Change
Begins 2.2 Derivative as a Function
October 2.3 Differentiation Formulas All future videos for course located in the 224 website
11 29-November 2.4 Derivatives of Trig Functions
2 2.5 Chain Rule
2.5 More Chain Rule
12 November 5-9 2.6 Differentiation of Implicitly Defined
2.7 Rates of Change in Science Basic Skills Test 1 (starts Monday, Nov 5: Students may take 3 attempts between Nov 5 and Nov 27, no more than one in the same
November 2.8 Related Rates day), on limits, continuity, and differentiation formulas
13 12-16 2.8 More Related Rates
S n o w D a y !!!
14 November 19 Review
Thanksgiving Break
In class Midterm Exam covering Sections 1.6
November through 2.8
15 26-November 2.9 and Linear Approximation (no differentials)
30 3.1 and Max/Min Values Basic Skills Test 2 (starts Monday, November 26: Students may take 3 attempts between Nov 26 and Dec 7, no more than two in the
3.3 How derivatives affect the shape of a same week, no more than one in the same day), on derivatives of trig functions, chain rule, and implicit differentiation
December 3.4 Limits at Infinity
16 3-December 7 3.5 Curve Sketching
Review Last Day for Skills Test 2
17 December 12 Final Wednesday, December 12, 12:50–2:50pm
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Best Stock Momentum Strategy Crash Indicator?
What indicator works best to mitigate stock momentum strategy crashes? In his March 2015 paper entitled “Momentum Crash Management”, Mahdi Heidari compares performances of seven indicators for
avoiding conventional stock momentum strategy crashes: (1) prior-month market return; (2) change in prior-month market return: (3) market volatility (standard deviation of 52 weekly returns); (4)
dispersion (variance) of daily returns across all stocks; (5) market illiquidity (aggregate impact of trading on price); (6) momentum volatility (standard deviation of momentum strategy returns the
past six months); and, (7) change in momentum volatility. The conventional strategy is each month long (short) the value-weighted tenth of stocks with the highest (lowest) returns from 12 months ago
to one month ago. For each of the competing indicators, he invests in the conventional momentum strategy (cash) when the indicator is below (within) the top 10% of its values over the past five
years. He uses portfolio turnover to compare implementation costs. Using data for a broad sample of relatively liquid U.S. stocks during January 1926 through December 2013, he finds that:
• Over the entire sample period, the conventional momentum strategy has:
□ Gross average monthly return 1.38%.
□ Gross three-factor (market, size, book-to-market) alpha 1.88%.
□ Gross annualized Sharpe ratio 0.59.
□ Very negative return skewness, with the most extreme crashes in 1932 (losing 92% in two months) and 2009 (losing 73% in three months).
• Regarding the seven potentially crash-avoiding indicators:
□ All relate negatively to next-month momentum return, both overall and during crashes.
□ Almost all of the predictive power across indicators comes from crash periods.
□ Most of the predictive power comes from loser (short) side of portfolios.
□ Change in momentum volatility has the highest predictive power, explaining about 12% of the variation in next-month momentum returns over the entire sample.
• Using the binary signal test strategy, all seven indicators boost the Sharpe ratio and dampen/eliminate the negative return skewness of the conventional strategy (see the chart below). Enhanced
gross annualized Sharpe ratios range from 0.68 for prior-month market return to 1.10 for change in momentum volatility.
• Turnovers of the enhanced momentum strategies are similar to that of the conventional strategy (84% to 104%). suggesting similar implementation costs.
• Combining any of the enhanced momentum strategies with a value strategy based on long (short) positions in the tenth of stocks with the highest (lowest) book-to-value ratios boosts Sharpe ratio
The following chart, taken from the paper, compares the log gross cumulative values of equal initial investments in the conventional stock momentum strategy (Mom) and the seven binary-signal risk
mitigation strategies specified above.
All seven variations improve performance. Change in momentum volatility (M-MomVolChg) produces the highest terminal value and the highest Sharpe ratio.
In summary, evidence from a strategy horse race indicates that monthly change in momentum strategy return volatility is the best (binary signal) crash protection indicator for a conventional stock
momentum strategy.
Cautions regarding findings include:
• The study does not address data snooping bias from trying many indicators on the same data, so the best-performing indicator overstates expectations. There may be additional hidden snooping bias
from unpublished experimentation with other indicators.
• While offering some subperiod testing, the study does not test robustness of the five-year look-back interval used to determine binary signals.
• Results do not explicitly quantify trading frictions from monthly portfolio reformation or costs of shorting. These costs would reduce performances of all strategies. Shorting may not be feasible
for some stocks.
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Convex and Nonsmooth Optimization
\(\) We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic
gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily
specialized to … Read more
Horoballs and the subgradient method
To explore convex optimization on Hadamard spaces, we consider an iteration in the style of a subgradient algorithm. Traditionally, such methods assume that the underlying spaces are manifolds and
that the objectives are geodesically convex: the methods are described using tangent spaces and exponential maps. By contrast, our iteration applies in a general Hadamard space, … Read more
Second-Order Strong Optimality and Second-Order Duality for Nonsmooth Constrained Multiobjective Fractional Programming Problems
\(\) This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and
second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke
directional derivative and the … Read more
Slow convergence of the moment-SOS hierarchy for an elementary polynomial optimization problem
We describe a parametric univariate quadratic optimization problem for which the moment-SOS hierarchy has finite but increasingly slow convergence when the parameter tends to its limit value. We
estimate the order of finite convergence as a function of the parameter. Article Download View Slow convergence of the moment-SOS hierarchy for an elementary polynomial optimization problem
Fast convergence of the primal-dual dynamical system and algorithms for a nonsmooth bilinearly coupled saddle point problem
\(\) This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretizations associated with a nonsmooth bilinearly coupled convex-concave saddle
point problem. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on the implicit discretization, we propose
… Read more
Scalable Projection-Free Optimization Methods via MultiRadial Duality Theory
Recent works have developed new projection-free first-order methods based on utilizing linesearches and normal vector computations to maintain feasibility. These oracles can be cheaper than
orthogonal projection or linear optimization subroutines but have the drawback of requiring a known strictly feasible point to do these linesearches with respect to. In this work, we develop new …
Read more
ε-Optimality in Reverse Optimization
The purpose of this paper is to completely characterize the global approximate optimality (ε-optimality) in reverse convex optimization under the general nonconvex constraint “h(x) ≥ 0”. The main
condition presented is obtained in terms of Fenchel’s ε-subdifferentials thanks to El Maghri’s ε-efficiency in difference vector optimization [J. Glob. Optim. 61 (2015) 803–812], after converting the
… Read more
The stochastic Ravine accelerated gradient method with general extrapolation coefficients
Abstract: In a real Hilbert space domain setting, we study the convergence properties of the stochastic Ravine accelerated gradient method for convex differentiable optimization. We consider the
general form of this algorithm where the extrapolation coefficients can vary with each iteration, and where the evaluation of the gradient is subject to random errors. This general … Read more
On Averaging and Extrapolation for Gradient Descent
\(\) This work considers the effect of averaging, and more generally extrapolation, of the iterates of gradient descent in smooth convex optimization. After running the method, rather than reporting
the final iterate, one can report either a convex combination of the iterates (averaging) or a generic combination of the iterates (extrapolation). For several common stepsize … Read more
The Maximum Singularity Degree for Linear and Semidefinite Programming
Article Download View The Maximum Singularity Degree for Linear and Semidefinite Programming
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Woo Algorithm
Amanatides-Woo Algorithm in Voxel Engines
The Amanatides-Woo algorithm is a popular method for fast voxel traversal in 3D grids, often used in ray-casting applications like voxel engines. Its ability to efficiently step through voxels along
a ray path makes it indispensable for various operations such as visibility determination, collision detection, and lighting computations. This article provides a detailed breakdown of the
algorithm's implementation, its significance in voxel engines, and common optimizations, pitfalls, and use cases.
Table of contents:
Overview of Amanatides-Woo Algorithm
The Amanatides-Woo algorithm, developed in 1987, is a grid traversal algorithm designed to efficiently step through the cells (voxels) that a ray intersects in a 3D grid. The algorithm works by
computing how far a ray needs to travel in each axis direction before crossing the next voxel boundary. This approach allows the algorithm to quickly identify which voxel the ray is in at each step,
making it ideal for voxelized environments where fast traversal is crucial.
In voxel engines, the Amanatides-Woo algorithm is typically used for tasks like ray-casting, where a ray needs to be traced through a voxel grid to determine which voxels are hit. It is particularly
useful for rendering, collision detection, and light propagation in voxel-based games or simulations. The algorithm is extremely efficient because it avoids unnecessary floating-point operations and
checks only the nearest voxel boundaries in each step.
The core advantage of the Amanatides-Woo algorithm is its efficiency, as it computes the exact traversal path through the voxel grid without visiting voxels that the ray does not intersect. This
makes it much faster than brute-force methods that would check every voxel along the ray’s path, particularly in large, sparse voxel grids. Its simplicity and speed make it an essential tool in
modern voxel engine design.
Algorithm Implementation
The implementation of the Amanatides-Woo algorithm starts by calculating the ray’s direction and determining the voxel grid cell in which the ray originates. Next, the algorithm calculates the step
size along each axis (x, y, z) and determines how far the ray must travel in each axis before it intersects the next voxel boundary. These distances are stored in variables called `tMaxX`, `tMaxY`,
and `tMaxZ` for each respective axis.
Each time the ray advances, the algorithm checks which of the three values (`tMaxX`, `tMaxY`, `tMaxZ`) is smallest, as this determines which voxel boundary the ray crosses first. The corresponding
tMax variable is then incremented by a pre-calculated `tDelta`, which is the distance between voxel boundaries along that axis. This process repeats until the ray either exits the grid or hits a
voxel of interest, such as a solid block.
The key to the algorithm’s efficiency lies in its constant time complexity per voxel step. Rather than iterating through every voxel along the ray’s path, it only checks the nearest voxel boundary
and updates the corresponding tMax value. This makes it particularly well-suited for voxel engines, where large numbers of voxels need to be checked quickly, and unnecessary computations can
significantly slow down performance.
Optimizing Voxel Traversal
While the basic Amanatides-Woo algorithm is highly efficient, there are several optimizations that can be applied to improve its performance in voxel engines. One common optimization involves using
integer arithmetic instead of floating-point calculations. Since voxel grids are inherently discrete, many of the calculations can be performed with integers, reducing the computational cost and
improving performance, especially on hardware with limited floating-point capabilities.
Another optimization is to implement early exit conditions. In many voxel engine applications, such as ray-casting for visibility or shadow determination, the ray often hits a solid voxel long before
it traverses the entire grid. By checking for collisions or hits at each step and exiting the loop early when a condition is met, unnecessary voxel checks can be avoided, further speeding up the
Additionally, voxel engines can benefit from spatial partitioning techniques such as octrees or grids to reduce the number of voxels that need to be traversed. By subdividing the voxel space into
hierarchies or bounding boxes, the algorithm can quickly discard regions of space that the ray cannot possibly intersect. This reduces the number of steps needed to complete the ray traversal and
improves overall performance in complex voxel scenes.
Use Cases in Voxel Engines
The Amanatides-Woo algorithm has several critical use cases in voxel engines, making it a versatile tool for various tasks. One of the most common uses is in ray-casting for visibility determination.
In voxel games, determining which voxels are visible to the player is essential for rendering optimization. By using the Amanatides-Woo algorithm to trace rays from the player’s viewpoint, the engine
can quickly determine which voxels need to be drawn and which can be skipped.
Another common use case is in collision detection. Voxel engines often need to detect when a player or object collides with solid voxels in the world. By tracing rays along the path of the moving
object and using the Amanatides-Woo algorithm to check for voxel intersections, the engine can efficiently detect collisions and respond accordingly. This is particularly important for physics-based
voxel games, where accurate collision detection is crucial for realistic gameplay.
The algorithm is also used in lighting and shadowing calculations. When computing shadows in a voxel world, rays must be traced from light sources to the scene to determine which voxels are in
shadow. The Amanatides-Woo algorithm can efficiently handle this task by stepping through the voxel grid and determining which voxels block the light. This allows the engine to generate accurate
shadows without having to check every voxel in the scene.
Common Pitfalls
Despite its efficiency, the Amanatides-Woo algorithm has some common pitfalls that developers should be aware of. One such pitfall is precision issues. When working with floating-point numbers,
rounding errors can occur, especially over long distances. This can cause the algorithm to step over voxel boundaries incorrectly, leading to missed intersections or inaccurate results. Using integer
arithmetic, where possible, can help mitigate this issue.
Another potential issue is when rays are nearly parallel to one of the coordinate axes. In this case, the algorithm may take many small steps along the other two axes, leading to inefficiencies.
Special handling for these edge cases, such as detecting when the ray direction is aligned with an axis and skipping unnecessary checks, can improve performance and accuracy in these situations.
Lastly, the algorithm may struggle in highly dense or irregular voxel grids. In voxel engines with complex geometries or non-uniform voxel sizes, the regular grid-based approach of the Amanatides-Woo
algorithm may not be as effective. In such cases, more sophisticated traversal techniques or adaptive grids may be necessary to maintain performance and accuracy.
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PhySU - Planning Your Math Courses
Planning Your Math Courses
Should I take harder math courses than what is required?
Many math classes you will take have multiple streams. One stream is intended for students completing a math specialist. These courses are very challenging, rigorous, and often have fewer
applications to physics. The other stream is more focused on the applications of math and are very useful in physics. First and second year calculus courses have other streams as well.
The math specialist stream gives a much stronger understanding of math, but most physics students will not find this extra knowledge to be useful in undergraduate physics courses. The more applied
math courses provide knowledge that is very useful in physics courses.
PhySU encourages students to take the math courses they are interested in and comfortable with. Otherwise, we suggest taking only the level of math courses that you need for your degree. Higher level
math courses are generally not helpful to your physics degree.
Which second-year (multivariate) calculus course should I take?
• MAT235: Calculus II
□ View the course information at: fas.calendar.utoronto.ca/course/mat235y1
□ You may benefit from taking this course if:
☆ You are more interested in the applications of math than in the theory
☆ You are not comfortable writing proofs
• MAT237: Multivariable Calculus
□ View the course information at: fas.calendar.utoronto.ca/course/mat237y1
□ You may benefit from taking this course if:
☆ You are interested in both the theory and applications of math
☆ You are enrolled in a Physics Specialist or Physics and Philosophy Specialist, or intend to switch into one of these programs
• MAT257: Analysis II
□ View the course information at: fas.calendar.utoronto.ca/course/mat257y1
□ You may benefit from taking this course if:
☆ You are enrolled in a Mathematics Specialist or Mathematics and Physics Specialist, or intend to switch into one of these programs
☆ You are more interested in the theory than in the applications of math
☆ You intend to take higher-level Math Specialist courses
When should I take ODEs? Which course should I take?
• MAT244: Introduction to Ordinary Differential Equations
• MAT267H1: Advanced Ordinary Differential Equations
□ View the course information at: fas.calendar.utoronto.ca/course/mat267h1
□ This course is intended for math specialists but it has less application in physics. Only take this course if your degree requires it or if you have interest in this course.
When should I take PDEs? Which course should I take?
• APM 346
□ View the course information at: Academic Calendar
□ This course is intended for Physics specialists.
• MAT 351
□ View the course information at: Academic Calendar
□ This course is intended for Mathematics and Physics specialists.
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Spacetime of General Relativity
General relativity is commonly represented in popular science as in the following extract of the Fabric of Cosmos, what is space? hosted by Brian Greene:
But a few years ago, I read somewhere that this usual visual representation of spacetime as a bent sheet of cloth was wrong. Worse than that, not only was it wrong, it was also deeply misleading.
Indeed, this representation works only if we consider that objects on the sheet are pulled down… by gravity! And yet, Einstein clearly stated that gravity was not a force!
This is troubling…
It is! So, lately, I’ve decided to dig a little bit deeper into the theory of general relativity to come up with new representations which would be less misleading… And this will take us into the
beautifully harmonious mathematics of general relativity!
I want to stress that I’m not criticizing other popular scientists here. On the contrary, they all do an amazing work and their usual visual representation really is awesome. In particular, I’m a
huge fan of Brian Greene! But, to go further, you need to acknowledge the wrongfulness of this simplified model.
Most introductions to general relativity first define the metric tensor, but I’ll use an alternative equivalent approach which I find more intuitive, based on affine connections.
The Spacetime Manifold
So, what is it that is so wrong about the usual visual representation?
In these representations, time and space are separated! Yet, Einstein taught us that space and time are rather mixed together in a fabric he called spacetime. And, just like it doesn’t make much
sense to talk separately about the rhum and the lime juice in a mojito, we should not talk about space and time separately. If you’re familiar with special relativity only, you might have thought of
this mixing as a commodity. I know I did. In general relativity though, this mixing is more than essential to have a clear insight in Einstein’s thoughts.
So space and time are mixed… What does that even mean?
This means that the Universe is made of a spacetime in which we follow trajectories.
A trajectory?
Yes. At any given time, we have some position in spacetime. But as time flows by… we are moving in time! Or, more accurately, in spacetime! By tracing all our successive positions, we are drawing our
trajectory in the spacetime of our Universe. How poetic is that?
I guess it is… But I have trouble visualizing that…
Our spacetime is in 4 dimensions, and that’s hard to visualize indeed. So, in this article, we’ll stick with a curved 2 dimension spacetime to illustrate Einstein’s general relativity, like the one
on the right, where I drew a possible trajectory in spacetime.
There’s actually a terrible misleading error in this figure… But I’ll get back to that later!
I’m not sure I see what you mean by “dimensions”…
That’s where we need a bit of differential geometry.
Waw! That sounds complicated!
Don’t be scared! It’s not! If you’ve read my article on differential calculus, you should expect me to associate the word differential with something that looks like a line when you zoom in. This is
precisely what differential geometry is.
So differential geometry is about spaces or spacetimes which look like lines when you zoom in?
Pretty much! For instance, the Earth is round. But when you zoom in on Google Earth, its roundness eventually disappears, and you end up with a flat two-dimensional surface. Similarly, Einstein’s
great insight was to imagine that our spacetime appeared flat to us, only because we are too small to notice its roundness. In some sense, as explained in Scott’s article on map-making, his
breakthrough is very similar to the Greek’s discovery of the roundness of the Earth!
I’m not sure I see what you mean…
Imagine that, on the curved 2D spacetime I showed earlier, small creatures were living and were too small to notice that their spacetime was in fact curved. They would feel like living in a flat 2D
spacetime, as illustrated below:
In mathematics, such objects which look flat when we zoom in are called manifolds. As Scott explained it in his article on the geometry of general relativity, what makes reasonings difficult is that
we are embedded in our 4D spacetime. In fact, the visual 2D spacetime example above is already a misleading simplification of what manifolds can be, as most 2D manifolds cannot be embedded in our 3D
space. But it’s the best visual example I can offer!
Hummm… This sounds complicated…
But we can sort it out! Using the zooming-in property I mentioned earlier, we can say that at each point in spacetime, spacetime seems like a flat vector spacetime called tangent spacetime. There is
a tangent spacetime associated to each point in spacetime! These tangent spacetimes are essential as they can be regarded as the building blocks of spacetime! As illustrated on the right.
People usually simply talk about tangent spaces rather than tangent spacetimes. But I prefer to stress the mixing of space and time. Also, the picture on the right only displays a finite number of
tangent spacetimes where as there actually is an infinite number of them: One for each point of the spacetime manifold.
But how can we use these tangent spacetimes to understand what happens in the actual spacetime?
Tangent spacetimes are simple. I mean, relatively simple. Indeed, they are vector spaces which can be described by special relativity. Motions in these tangent spaces of objects on which only gravity
acts are simply lines.
Don’t you mean vectors rather than lines?
We’ll see later that the description involving the metric tensor requires us to use vectors. But, by using affine connections, we can directly work with straight lines in tangent spacetimes! That’s
much simpler and more intuitive!
This explains how we move in the tangent spacetimes when only gravity acts… But not in spacetime!
Indeed. The tangent space is only valid for small scales. Thus, so far, we can only describe how motions work in all the small tangent spacetimes which fill spacetime, as displayed on the right. To
go further, we need to connect tangent spacetimes. This is done using an affine connection.
Affine Connection
So what are these affine connections you’ve already talked so much about?
An affine connection tells how a direction is moved from a tangent spacetime to a neighbour tangent spacetime. This deformation is similar to a spacetime deformation in special relativity, as I
explained it in my article on space deformation and group representation.
I’m not sure I see what you mean…
On the right is illustrated the affine connection of a tangent spacetime with its neighbour tangent spacetimes. In particular, I’ve represented how the orange and purple directions get transformed
when moved to neighbour tangent spacetimes. We say that the orange and purple directions undergo parallel transports to neighbour tangent spacetimes. Here’s an explanation I gave on my Youtube
But how are these transformations of directions described mathematically?
Through the concept of differentials! Basically, starting at an original tangent spacetime, for one step in any direction of spacetime, the affine connection is an operator which tells how directions
are moved into the new tangent spacetime. And because this deformation is proportional to the length of the step, it’s described by a differential. This differential describes how the motion is
moved. Or, equivalently, the motion of the motion!
So the affine connection is a differential?
Yes! More precisely, the affine connection is a differential of motions in tangent spacetimes. It maps a motion and the direction in which the motion is moved with variation of the motion. The affine
connection is thus formally a mapping of a tangent spacetime and 2 motions in it with 1 motion of the first of the two motions in the direction of the second (take your time to read this phrase!).
This is why we call it a tensor (1,2).
More precisely, a tensor (1,2) is a a linear operator that maps a point, a linear form field and two vector fields with a real scalar. The affine connection also has to satisfy usual properties of
derivatives, such as
Leibniz rule
. If you’re a bit lost here, don’t worry, these are just technical details…
But this doesn’t really describe motions in spacetime, does it?
It nearly does! The only concept you need is the simplest one: When only gravity acts upon an object, the object always moves in straight line in spacetime. Or rather, its trajectory never gets
curved in its own tangent spacetime, and it moves to its next tangent spacetime, while remaining parallel to itself. But, from the perspective of another observer, the successive tangent spacetimes
of the trajectory do get curved!
Are these what people call geodesics?
Yes, although they are usually rather defined differently! But I prefer to see geodesics as trajectories which are always going straight in tangent spacetimes. The picture on the right illustrates
two geodesics, constructed based on this idea.
I prefer this definition of geodesics because it is more intuitive. More often than not, and we’ll discuss it more later, it is rather defined as the shortest or longest path between two points. This
is both not totally correct (it only has to be a local extremum) and not intuitive at all (does the particle “choose” its ending point and then its trajectory?).
We are now ready to describe Einstein’s equation…
Einstein Equation
Really? That’s so exciting!
Hummm… Don’t set your hopes to high… To really describe Einstein equation, I’d need to introduce even more tensors like the metric tensor, the Riemann tensor, the Ricci tensor, the Einstein tensor
and the stress-energy tensor. And those are all very hard to visualize. At least for me.
But if you can write about these tensors, please do! I guess you’ll be allowed to get more technical, especially if you first refer your readers to this present article!
That’s a bummer…
Still, this won’t stop me from telling you the important message of Einstein equation: Energy is what bends neighbour tangent spacetimes!
I thought mass did it…
Einstein’s famous formula $E=mc^2$ actually says that mass is just a form of energy.
So energy deforms spacetime… What does that even mean?
Let’s take an example on a 2D spacetime. Let’s track the trajectories of a Sun and its Earth in this spacetime. From its perspective, its trajectory is a line. Now, let’s consider an Earth influenced
by the gravity of the Sun. Because no force acts on the Earth, it is moving straight in its tangent spacetime. However, its tangent spacetime gets deformed towards the Sun because of the gravity
caused by the mass of the Sun. This is displayed in the following figure:
Notice that this explains why the trajectory of the Earth does not depend on its mass. In other words, the equality of the inertial mass (the mass in $F=ma$) and the gravitational mass (the ones in
$F=Gm_1m_2/r^2$), which is a surprising assumption in Newton’s mechanics, is an obvious consequence of Einstein theory. This is known as the equivalence principle.
So what does the 2D spacetime manifold look like in this case?
I’m not sure. As I said, considering 2D spacetime manifolds as a sheet is nice to introduce the idea of curved spacetimes, but it’s misleading since most 2D manifolds can’t even be represented in our
3D world. That’s where you need to make a quantum leap by considering the spacetime manifold as something that locally looks like a vector space, but does not have an easily describable global
structure. This global structure is rather described by differential topology, and its study often poses difficult open problem such as the Poincaré conjecture.
So, to get back to the figure, at each step in time, tangent spacetimes get deformed?
Yes! In particular, each direction in tangent spacetimes is rotated towards the mass. So, if you think about our actual Earth, we are moving straight, and rather than a force pulling us towards the
Sun, there are gravitational waves coming from the Sun which, from its perspective, bend our tangent spacetime as we move in time. Now, to go even further in the understanding of general relativity,
I should talk a bit more about space and time.
Space and Time
Our spacetime has an additional structure which is inherited from special relativity and is due to the speed of light being a fundamental limit of spacetime. More precisely, tangent spacetimes are
ruled by special relativity.
Now, this means that the speed of light is a limit only in tangent spacetimes. But in the spacetime manifold, it’s possible for objects to move away or towards each other at speeds greater than the
speed of light. In fact, it’s going to be the case of distant galaxies. Even more surprising, this may mean that by creating the right gravity waves, it might be able to
travel in the spacetime manifold faster than the speed of light globally
, even though locally we never exceed that speed!
Now, if you have read my article on special relativity, you know that an essential element to describe special relativity is…
Dr. Sheldon Cooper?
No! Even us dummer people can figure it out! Provided we use…
The light cone?
Yes! This is an important feature of spacetime of general relativity! At each point in spacetime, spacetime around looks like a 4D vector space with a light cone defined. This light cone is essential
because it separates spacetime into 3 components: space, future and past. Needless to say these three components are very different. Thus, we distinct the motions in spacetime in four categories:
• Motions in space, called spacelike vectors: they point towards the outside of the light cone.
• Motions at light speed, called null vectors: they point along the borders of the light cone.
• Motions towards the future: they point towards the upper inside of the light cone.
• Motions towards the past: they point towards the lower inside of the lightcone.
The two last motions are motions in time, and they are called timelike vectors.
In fact, it doesn’t even seem that a distinction can be made between future and past. In a curved spacetime, Gödel showed that there could be path always going towards the future which end up back to
the initial point. In sci-fi words, Gödel proved that Einstein’s equations allowed for time-travel… As the story goes, Gödel presented his result to Einstein as a gift, but the German physicist did
not like it! For our purpose, let’s assume that future and past are here uniquely defined…
On the right is the image of the light cone in a 3D spacetime taken from wikipedia.
Wait… Wikipedia’s image is more complete. You’ve taken off the directions of space and time…
That’s because Wikipedia’s image is misleading! In fact, the mere idea of spacetime made of 1 direction of time and 3 of space is very misleading. What’s usually meant by that is that the most
natural and useful ways to decompose spacetime are by using 1 direction of time and 3 of space. But as explained by Henry Reich on Minute Physics, there is no “fourth” dimension. In fact, you could
generate spacetime with 4 timelike vectors! If you’re familiar with vector spaces, this shouldn’t be hard for you to see…
Is that why you said that your first image of the 2D spacetime with the arrow for time was wrong?
Yes! There’s no such thing as the arrow of time. Rather, there are a bunch of arrows of time at each point in spacetime.
Wait. You did say the most natural decompositions was with 1 time dimension and 3 space dimensions…
Yes. This is what makes spacetime a Lorentzian manifold of signature (1,3), and is equivalent to saying that there is a light cone decomposing spacetime in 3 components. Learn more with my article on
space deformation and group representation.
OK. So, in the spacetime of general relativity, we can either move in space or time…
Sorry but motions in space aren’t going to happen! Einstein’s postulate which says that nothing goes faster than light in tangent spacetimes implies that we can’t have motions in space. Light is just
fast enough to move along the light cone. But objects with mass like us can only move in time. And because motions are all relative, there is no way of saying that they move less in time and more in
space than others. But to better understand this relationship between space and time, we need to involve the metric tensor I have been avoiding so far.
The Metric Tensor
To understand the metric tensor, let’s restrict ourselves to motions in time. After all, only these are the ones which make sense…
So what does this metric tensor do with motions in time?
It compares them! More precisely, if two trajectories in spacetime (not necessarily geodesics) meet at a point, then the metric tensor at this point can tell how fast a trajectory is moving compared
to the other. Now, if you are familiar with special relativity, then you should know that, rather than the relative speed $v$, we rather equivalently work with the Lorentz factor $\gamma = (1-v^2/c^
2)^{-1/2}$. The metric tensor is precisely the mapping of any two trajectories and a meeting point with this value $\gamma$.
Well, that wasn’t that complicated!
That’s because I’ve simplified it. Technically, we need to map the trajectories at the meeting point with a vector which indicates the direction of the trajectories. Now, assuming that the future is
well-defined, we only need to find a normalization of this vector. In this setting, at each point in spacetime, the metric tensor has to be viewed as a bilinear form which, when applied twice to the
same vector gives a sort of square of the norm of the vector.
Linear and bilinear forms are essential objects of
linear algebra
. If you can, please write about them! It’s classical to match definite bilinear forms with the square of the norm, but more counter-intuitive when the bilinear form is of signature (1,3) as is the
case for the bilinear forms associated to our metric tensor. In this setting, for the norm to make sense, we need the vector to be a time vector. But that’s a given if trajectories are actual
trajectories in time.
The obtained norm of a vector can then be interpreted as the proper time of path passing by this vector.
The proper time?
Yes! The concept of time really depends on the observer. And the proper time of an observer is how the observer perceives his own time.
Wait. What’s an observer?
An observer is basically another object which will follow another path in spacetime. Now, if you are moving in a parallel direction from him in spacetime, then you’re not moving in space relatively
to the observer. Thus, you’ll be moving a lot in time. On the opposite, if you are moving in a very different direction in spacetime, then your time will be slowed from his perspective. This
perturbing aspect is illustrated in the following extract from the Fabric of the cosmos – the Illusion of Time by Brian Greene:
As explained by Brian Greene, time is not unique. In fact, each observer has its own time. More precisely, each trajectory has its own clock ticking. And the passing of time is measured by the norm
of the vector. This leads us to a natural normalization of vectors of motions in time, by fixing their proper times to be equal to 1. These vectors of motions in time with proper time 1 correspond to
a 3D hyperboloid. On the right is represented a 1D hyperboloid at the tangent spacetime where the trajectories meet (there are two sheets, one for the past, the other for the future).
So what does $\gamma$ have to do in all of that?
The coefficient $\gamma$ is then obtained by applying the bilinear form of the metric tensor to the vectors of motions of one unit of proper time. In other words, we first need to normalize the
vectors of motions before computing $\gamma$.
So how does now the metric tensor relate to everything we’ve mentioned so far?
Geodesics between two points which can be joined with a trajectory in time are trajectories which maximize the proper time! Visually, you can see that if we zigzag, the steps of one unit of proper
time are greater and we can get from a point in spacetime to another in little proper time. On the contrary, the more direct path takes more proper time.
More precisely, geodesics are local maxima: A trajectory between two points which is nearly a geodesics will have time flowing slower than someone following the geodesics.
I don’t see the link with affine connections…
Using the Euler-Lagrange equation which computes geodesics of points extremely close, this condition of proper time maximization can then be translated as a relationship between the affine connection
and the derivatives of the metric tensor. This implies that the knowledge of either the affine connection or the metric tensor uniquely defines the other. Overall, both are defined by the
distribution of masses and energies in the universe, accordingly to Einstein equation.
Is this related to the twin paradox?
Yes! I mentioned it in my article on special relativity, but, now that we have explained general relativity, we can explain it accurately!
Can you reexplain what the twin paradox is about?
Sure. The twin paradox says that if two twins are born at the same point in spacetime, take different paths in spacetime and then meet again, then both would be still in their inertial system. Thus,
both should have time flowing faster than the other’s.
The mere statement of this paradox seems wrong… What’s an inertial system?
You’re totally right! In the setting of general relativity, there is absolutely no paradox. And it’s quite easy to resolve the apparent paradox. If one of the twin stays on a planet while the other
is travelling around the universe, the settled brother is nearly following a geodesics and thus maximizes its proper time. Time will thus flow faster for him. And when they’ll meet again, he’ll be
How about GPS? I’ve heard general relativity was essential for that…
Indeed! Think about satellites. No force acts on them. Thus, they really follow geodesics in spacetime. Meanwhile, on Earth, we oppose gravity by standing on the surface of the Earth. This means that
we divert from geodesics by using the electromagnetic forces that prevent us from crashing towards the center of the Earth. Thus, our time isn’t maximized as is the time of satellites. Time thus
flows slower for us than for satellites. Taking this distortion of time into account is essental for GPS, as explained by Henry Reich on Minute Physics.
Let’s Conclude
We have got through the main ideas of general relativity! How awesome is that?
It’s pretty cool!
Frankly, when I started this article, I was quite scared, not only to be able to make it simple, but also simply of how well I understood general relativity. I’ve never had a course in general
relativity. All I did is read through this introduction in French. But I feel like I’m providing here a description which is very faithful to the mathematical one! Now, I’m absolutely no expert in
this field, and if you do master this beautiful theory, I’d love your feedbacks! Obviously, I skipped the difficulty of actually explaining Einstein equation, and the reason for that is that I myself
can’t really visualize it (especially the Ricci tensor!). But if you can, please do!
It’s hard to believe that this theory is right!
Well, in fact, when you really get into it and appreciate its amazing beauty, I’d say that it’s hard not to! It’s worth mentioning that it took several years to confirm this theory. In fact, general
relativity was first refuted twice! Today, it’s one of the two foundations of theoretical physics, and is impressively accurate for large scale observations. And I hope you’ll now have goose bumps
when you hear physicists talk about it and you’ll be thinking: I kind of get what they’re talking about!
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Calculate The Staircase Shuttering Quantity - Surveying & ArchitectsCalculate The Staircase Shuttering Quantity
How To Calculate The Staircase Shuttering Qty ( Quantity )
In this Article, I will show u the step-by-step process to calculate the required shuttering qty for the staircase.
The estimation process of the staircase shuttering materials is almost the same as calculation process of slab’s shuttering materials. Yet there is the little difference. However, let’s see…
Step – One :-
Calculate The Sheathing Materials.
Generally, wooden lumber is used as a sheathing material. The size of the lumber is normally one″ × Six″.
With that, let’s calculate the sheathing for the waist slab.
To do that, obtain the Staircase‘s design. It will resemble the picture below.
This is a prime example of a staircase layout. Get the stairwell section as well. It will seem as follows:
With that-
The Calculate inclined length of the waist slab using by the Pythagorean theorem.
= √{(5′)Two+(7½′)2} = Nine feet.
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Now, Calculate The sheathing materials for waist slab.
In my example, the width of waist slab is four feet.
But the sheathing should extend Six inches beyond each side of waist slab to support stringer bracing blocks. So, the sheathing width for waist slab will be,
= four′ + Six″ + Six″ = Five′ And, the area of the sheathing is,
= Width of the sheathing × Length of the sheathing
= five′ × nine′ [The length of the sheathing is equal to the inclined length of waist slab]
= Forty Five sq.ft. As we are using 1-inch thick lumber, the qty of the required wood will be,
= 45 sq.ft. × one″
= 3.73 cubic feet. [One″ = 0.083′] This is for 1 waist slab.
But we have 2 waist slabs. So the required wood for the both waist slabs is,
= Two × 3.73 = 7.46 cubic feet.
Now Next, Calculate the sheathing for the landing slab.
From the Photo –
The width of slab is four feet. If sheathing extends six inches beyond both sides of landing to support stringer bracing blocks, it will be,
= four′ + Six″ + Six″
= Five′
Similarly, the length will be:
= (8′-Six″) + Six″ + Six″
= 9½′ So, the sheathing area for landing is,
= Five′ × 9½′
= 47.5 sq.ft. &, for One″ thick lumber, the qty of the required wood is,
= 47.5 sq.ft. × One″
= 3.94 cubic feet. [One″ = 0.083′]
So, The total qty of the lumber for sheathing for our example staircase is,
= 7.46 + 3.94 = 11.40 cubic feet.
Say, twelve cubic feet.
Step-Two Calculate Joists :-
The Calculate Joists are used below the sheathing to support them. These also help to adjust the level of staircase.
Normally, Two″ × four″ wooden batten are used at 2 feet centers for this purpose.
With that, 1st, Calculate joists for waist slab.
The formula is,
= Number of the joists × Length of the joist
The number of the joists is,
= Width of the sheathing for the waist slab ÷ Distance of between joists
= five′ ÷ two′ = 2½
Say, three nos. And, the length of the joist is equal to the length of waist slab which is nine feet (see above).
So, the required qty of the joists are,
= three × nine′ = twenty seven running feet.
For 2 waist slabs, we need,
= two × 27 = Fifty four running feet.
Next, Calculate the joists for the landing slab.
The formula is, = Number of the joists × Length of the joist Here,
The number of the joists is,
= Length of landing ÷ distance of between joists
= 8½′ ÷ two′ = 4.25 Say, five nos. And, the length of the joist for the landing is somehow equal to the width of the landing which is four feet.
So, the required joists are,
= five × four′ = twenty running feet.
And, the total required of joists for our example staircase is,
= Joists for waist slab + joists for landing
= 54 + twenty = 74 running feet.
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Step Three Calculate Supports :-
Supports are used below joists at two feet centers. U can use bamboo or wooden post (Four″ × Four″) for this purpose.
The formula for the calculating supports is,
= (Length of the joists ÷ distance of between supports) × length of the post.
Here, The length of the joists is seventy four running feet (we calculated this above). Distance between supports is two feet. And, Length of the support varies as the height of the waist slab is not
That’s why, we will take half of floor height which is five feet (The floor height is normally Ten feet).
So, the required post for the supports is,
= (seventy four′ ÷ Two′) × Five′ = 185 Running feet.
Step Four Calculate Stringer :-
The Calculate Stringer is used on both sides of the steps.
The height of stringer is normally 1 foot.
So, the area of the stringers for waist slab is,
= Two × Length of the waist slab × Height of stringer
= Two × Nine′ × One′ [we calculated the length of the waist slab above]
= Eighteen Square feet. For both waist slabs,
= Two × 18 = Thirty six square feet.
Normally, One” thick lumber is used for the stringers. Therefore, the required qty of the wood for this is,
= 36 × One” [One” = 0.083′]
= 2.98 Say, Three cubic feet.
Step Five :-
Calculate Riser’s Face The area of the riser faces is,
= Number of the riser × Height of the riser × Width of the step
The number of the risers is,
= Floor height ÷ Riser height
= Ten’ ÷ Six” = twenty nos.
From the above formula,
= twenty × 6” × four’ = Forty square feet. [Six” = ½’]
Normally, One” thick lumber is used for the riser face.
Therefore, the lumber qty is,
= forty sq.ft × One” = 3.32 cubic feet. [One”=0.083’]
Say, Four feet.
These are somehow the required shuttering materials for the staircase.
Step Six :-
Summarize Shuttering Materials For The Staircase
One” × Six” lumber = Nineteen cubic feet (Sheathing = twelve cubic feet, Stringer = three cubic feet, Riser = four cubic feet).
Two” × Four” wooden batten = Seventy Four running feet
Four × 4” wooden batten = One Eighty Five running feet
Conclusion :-
There is no hard & fast rule for the calculating shuttering materials for the staircase. And there is no rule that which shuttering materials U will use for the staircase’s formwork. Most of time you
can use materials that are available in your project. I have just tried to give u an idea of how to calculate the shuttering materials for the staircase.
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To Get more information, Visit Our Official website
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Lens Maker’s Formula
Lens maker’s formula is used by the lens manufacturers, for manufacturing of the lenses having the desired focal length. For different optical instruments, lenses having different focal lengths are
used. The focal length of the lens is dependent on the radii of curvature and the refractive index of the lens. The lens maker’s equation is another formula and it gives the relationship between the
radii of curvature, refractive index, focal length, of the two spheres that are used in the lenses. The lens maker’s formula can be described as below.
Here f represents the focal length, n is the refractive index of the material that is used to make the lens, R1 is the radius of the curvature of the first sphere, and R2 is the radius of curvature
of sphere 2. But this equation can only be used for the thin lenses. The lenses, whose thickness is too less to be considered as negligible, as compared to the radius of curvature can be referred to
as thin lenses.
For the thin lenses, the power of the lens is approximately the sum of the surface powers. The radii of curvature can be measured according to the cartesian sign convention. The radius is positive
for the double convex lenses, and its measurement is done on the front surfaces. R2 radius is negative, as it is extending to the left side from the surface of the second one. Sometimes, it may
happen that the thickness of the lens is ignored, but sometimes, it is considered. The trouble of this concept can be resolved by the statement that when the ray of light is traveling from the medium
or air to the medium of the lens, then it is undergoing the refraction. It is possible to ignore the double refraction if the optic’s lenses are thin enough for making the assumption that the light
is refracted for only 1 time. This can be done, by making the ray optical calculation simpler, but at the first step, the constituents of thick and thin lenses should be identified.
Assumptions for Derivation of Lens Maker’s Formula
• The lens is thin and its distance measured from poles of surfaces can be considered as equal to the distance, from the optical center of the lens.
• Only the point objects are considered.
• The aperture of the lens is small.
• The refracted and incident rays make the small angles.
Limitations of Lens Maker’s Formula
The medium on both sides of lenses should be the same. The lenses should be thin. Due to this reason, the separation between two of the refracting surfaces will also be small.
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How to Solve Puzzles | Puzzazz | The best way to solve puzzles in the digital world
How to Solve KenKen® Puzzles
by Roy Leban
If you like Sudoku, there’s a good chance you’ll love KenKen. If you hate Sudoku, there’s a good chance you’ll love KenKen. Invented by Japanese mathematics teacher Tetsuya Miyamoto in 2004, KenKen
is an elegant and rich logic puzzle with a few easy-to-understand rules, which helps explain why New York Times Puzzle Editor Will Shortz called it “The most addictive puzzle since Sudoku.”
KenKen’s rules are straightforward:
1. Fill in each square cell in the puzzle with a number between 1 and the size of the grid. For example, in a 4×4 grid, use the numbers 1, 2, 3, & 4.
2. Use each number exactly once in each row and each column.
3. The numbers in each “Cage” (indicated by the heavy lines) must combine — in any order — to produce the cage’s target number using the indicated math operation. Numbers may be repeated within a
cage as long as rule 2 isn’t violated.
4. No guessing is required. Each puzzle can be solved completely using only logical deduction. Harder puzzles require more complex deductions.
That’s all you need to know. The rest is just logical deduction derived from those rules.
Solving Techniques
Here’s a sample puzzle that we’ll use to illustrate solving techniques:
Each cage in a KenKen contains a target number and most contain an operator. If you see a single-cell cage with just a number and no operator, it means that the value in that cell is the target
number. Such single-cell cages work like givens in Sudoku puzzles. You won’t see these in every puzzle, but when you do see one, you should start there. In this puzzle, we can immediately place a 4
in the upper right cell:
Whenever we place a number, this narrows down the possibilities for other cells, so we want to look for that. In this puzzle, we know that the 7+ cage in the third column must contain 3 & 4, since
that is the only possibility that adds to 7. Given the 4 that we just placed, combined with the rule that we must use each number exactly once in each row and each column, we can now tell which of
the cage’s cell contains a 3 and which contains a 4:
We can now tell that the two empty cells in the third column (in the 4× cage) contain 1 & 2, but we don’t know the order. However, given that information, we can place a 2 in the lower right cell to
make the cage’s product be 4:
Remember that a cage can repeat numbers in an irregularly shaped cage as long as no number is duplicated within a single row or column. In this puzzle, before we knew the two values in the 7+ cage,
we didn’t know if the 4× cage contained the numbers 1, 1, & 4 or 1, 2, & 2. Now that we know about the 2, we can immediately finish the 4× cage because we know the second 2 must be in the third row:
For more complex deductions, it can be useful to take notes in the puzzle, which you can do by using the right side of the on-screen keyboard, holding down the Shift key while typing on a physical
keyboard, or switching to Pencil input with TouchWrite. In the 2÷ cage in the upper left, there are only two possibilities — 1 & 2 or 2 & 4. The latter is excluded by the 3 & 4 already in the row, so
we can deduce that two cells contain 1 & 2 (in an unknown order). Since knowing this doesn’t let us place any additional numbers immediately, we can use notes to help us remember it for future use.
Next, we can look at the 1- cage in the first column. Without knowing any constraints, the cells can contain 1 & 2 or 2 & 3 or 3 & 4. But our notes show us that the first column will already contain
either a 1 or 2, which means the 1- cage cannot contain 1 & 2. This means it must have either 2 & 3 or 3 & 4. Whichever it is, it means the 1- cage will definitely contain a 3. And that means the
bottom left cell cannot be a 3, which means it must be a 4:
That also lets us place the 3 in the bottom row and then the 4 in the second row tells us how to place the 1 and the 4 above it:
Now we can place the 2 and then the 1 in the top row:
Next, we finish up the first column, and we can tell the order because the third row already has a 2 in it:
Finally, we wrap up the puzzle by placing the last two numbers in the fourth column:
Note that this is just one way to solve this puzzle. Because this is an Easy puzzle, there is more than one deductive path for solving. With harder puzzles, this is not always the case.
If you’re just getting started with KenKen puzzles, don’t forget you can get hints. Just tap on the to fill in a value or remove mistakes.
KenKen® is a registered trademark of KenKen Puzzle LLC. All rights reserved. www.kenkenpuzzle.com.
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