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[Solved] Statement I If a>0 and b2−4ac<0, then the value of the... | Filo
Statement I If and , then the value of the integral will be of the type , where are constants. Statement II If , then can be written as sum of two squares.
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If and , then where . which will have an answer of the type or Thus, choice (a) is correct.
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Question Text Statement I If and , then the value of the integral will be of the type , where are constants. Statement II If , then can be written as sum of two squares.
Topic Integrals
Subject Mathematics
Class Class 12
Answer Type Text solution:1
Upvotes 148 | {"url":"https://askfilo.com/math-question-answers/statement-i-if-a0-and-b2-4-a-c-then-the-value-of-the-integral-int-fracd-xa-x2b","timestamp":"2024-11-12T00:21:23Z","content_type":"text/html","content_length":"478335","record_id":"<urn:uuid:6865506a-5ee3-4061-955e-239d33fd3ed0>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00679.warc.gz"} |
Question ID - 54238 | SaraNextGen Top Answer
For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is
a) 2 b) c) d)
For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=54238","timestamp":"2024-11-07T20:30:27Z","content_type":"text/html","content_length":"16507","record_id":"<urn:uuid:e3f8e4b4-1c64-408a-89f2-fdb954fb9974>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00597.warc.gz"} |
A Mechanism for Overriding Ufuncs
A Mechanism for Overriding Ufuncs¶
Author: Blake Griffith
Contact: blake.g@utexas.edu
Date: 2013-07-10
Author: Pauli Virtanen
Author: Nathaniel Smith
Executive summary¶
NumPy’s universal functions (ufuncs) currently have some limited functionality for operating on user defined subclasses of ndarray using __array_prepare__ and __array_wrap__ [1], and there is little
to no support for arbitrary objects. e.g. SciPy’s sparse matrices [2] [3].
Here we propose adding a mechanism to override ufuncs based on the ufunc checking each of it’s arguments for a __numpy_ufunc__ method. On discovery of __numpy_ufunc__ the ufunc will hand off the
operation to the method.
This covers some of the same ground as Travis Oliphant’s proposal to retro-fit NumPy with multi-methods [4], which would solve the same problem. The mechanism here follows more closely the way Python
enables classes to override __mul__ and other binary operations.
[1] http://docs.scipy.org/doc/numpy/user/basics.subclassing.html
[2] https://github.com/scipy/scipy/issues/2123
[3] https://github.com/scipy/scipy/issues/1569
[4] http://technicaldiscovery.blogspot.com/2013/07/thoughts-after-scipy-2013-and-specific.html
The current machinery for dispatching Ufuncs is generally agreed to be insufficient. There have been lengthy discussions and other proposed solutions [5].
Using ufuncs with subclasses of ndarray is limited to __array_prepare__ and __array_wrap__ to prepare the arguments, but these don’t allow you to for example change the shape or the data of the
arguments. Trying to ufunc things that don’t subclass ndarray is even more difficult, as the input arguments tend to be cast to object arrays, which ends up producing surprising results.
Take this example of ufuncs interoperability with sparse matrices.:
In [1]: import numpy as np
import scipy.sparse as sp
a = np.random.randint(5, size=(3,3))
b = np.random.randint(5, size=(3,3))
asp = sp.csr_matrix(a)
bsp = sp.csr_matrix(b)
In [2]: a, b
Out[2]:(array([[0, 4, 4],
[1, 3, 2],
[1, 3, 1]]),
array([[0, 1, 0],
[0, 0, 1],
[4, 0, 1]]))
In [3]: np.multiply(a, b) # The right answer
Out[3]: array([[0, 4, 0],
[0, 0, 2],
[4, 0, 1]])
In [4]: np.multiply(asp, bsp).todense() # calls __mul__ which does matrix multi
Out[4]: matrix([[16, 0, 8],
[ 8, 1, 5],
[ 4, 1, 4]], dtype=int64)
In [5]: np.multiply(a, bsp) # Returns NotImplemented to user, bad!
Out[5]: NotImplemted
Returning NotImplemented to user should not happen. Moreover:
In [6]: np.multiply(asp, b)
Out[6]: array([[ <3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>],
[ <3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>],
[ <3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>,
<3x3 sparse matrix of type '<class 'numpy.int64'>'
with 8 stored elements in Compressed Sparse Row format>]], dtype=object)
Here, it appears that the sparse matrix was converted to a object array scalar, which was then multiplied with all elements of the b array. However, this behavior is more confusing than useful, and
having a TypeError would be preferable.
Adding the __numpy_ufunc__ functionality fixes this and would deprecate the other ufunc modifying functions.
[5] http://mail.scipy.org/pipermail/numpy-discussion/2011-June/056945.html
Proposed interface¶
Objects that want to override Ufuncs can define a __numpy_ufunc__ method. The method signature is:
def __numpy_ufunc__(self, ufunc, method, i, inputs, **kwargs)
• ufunc is the ufunc object that was called.
• method is a string indicating which Ufunc method was called (one of "__call__", "reduce", "reduceat", "accumulate", "outer", "inner").
• i is the index of self in inputs.
• inputs is a tuple of the input arguments to the ufunc
• kwargs are the keyword arguments passed to the function. The out arguments are always contained in kwargs, how positional variables are passed is discussed below.
The ufunc’s arguments are first normalized into a tuple of input data (inputs), and dict of keyword arguments. If there are output arguments they are handeled as follows:
• One positional output variable x is passed in the kwargs dict as out : x.
• Multiple positional output variables x0, x1, ... are passed as a tuple in the kwargs dict as out : (x0, x1, ...).
• Keyword output variables like out = x and out = (x0, x1, ...) are passed unchanged to the kwargs dict like out : x and out : (x0, x1, ...) respectively.
• Combinations of positional and keyword output variables are not supported.
The function dispatch proceeds as follows:
• If one of the input arguments implements __numpy_ufunc__ it is executed instead of the Ufunc.
• If more than one of the input arguments implements __numpy_ufunc__, they are tried in the following order: subclasses before superclasses, otherwise left to right. The first __numpy_ufunc__
method returning something else than NotImplemented determines the return value of the Ufunc.
• If all __numpy_ufunc__ methods of the input arguments return NotImplemented, a TypeError is raised.
• If a __numpy_ufunc__ method raises an error, the error is propagated immediately.
If none of the input arguments has a __numpy_ufunc__ method, the execution falls back on the default ufunc behaviour.
In combination with Python’s binary operations¶
The __numpy_ufunc__ mechanism is fully independent of Python’s standard operator override mechanism, and the two do not interact directly.
They however have indirect interactions, because Numpy’s ndarray type implements its binary operations via Ufuncs. Effectively, we have:
class ndarray(object):
def __mul__(self, other):
return np.multiply(self, other)
Suppose now we have a second class:
class MyObject(object):
def __numpy_ufunc__(self, *a, **kw):
return "ufunc"
def __mul__(self, other):
return 1234
def __rmul__(self, other):
return 4321
In this case, standard Python override rules combined with the above discussion imply:
a = MyObject()
b = np.array([0])
a * b # == 1234 OK
b * a # == "ufunc" surprising
This is not what would be naively expected, and is therefore somewhat undesirable behavior.
The reason why this occurs is: because MyObject is not an ndarray subclass, Python resolves the expression b * a by calling first b.__mul__. Since Numpy implements this via an Ufunc, the call is
forwarded to __numpy_ufunc__ and not to __rmul__. Note that if MyObject is a subclass of ndarray, Python calls a.__rmul__ first. The issue is therefore that __numpy_ufunc__ implements “virtual
subclassing” of ndarray behavior, without actual subclassing.
This issue can be resolved by a modification of the binary operation methods in Numpy:
class ndarray(object):
def __mul__(self, other):
if (not isinstance(other, self.__class__)
and hasattr(other, '__numpy_ufunc__')
and hasattr(other, '__rmul__')):
return NotImplemented
return np.multiply(self, other)
def __imul__(self, other):
if (other.__class__ is not self.__class__
and hasattr(other, '__numpy_ufunc__')
and hasattr(other, '__rmul__')):
return NotImplemented
return np.multiply(self, other, out=self)
b * a # == 4321 OK
The rationale here is the following: since the user class explicitly defines both __numpy_ufunc__ and __rmul__, the implementor has very likely made sure that the __rmul__ method can process
ndarrays. If not, the special case is simple to deal with (just call np.multiply).
The exclusion of subclasses of self can be made because Python itself calls the right-hand method first in this case. Moreover, it is desirable that ndarray subclasses are able to inherit the
right-hand binary operation methods from ndarray.
The same priority shuffling needs to be done also for the in-place operations, so that MyObject.__rmul__ is prioritized over ndarray.__imul__.
A pull request[6]_ has been made including the changes proposed in this NEP. Here is a demo highlighting the functionality.:
In [1]: import numpy as np;
In [2]: a = np.array([1])
In [3]: class B():
...: def __numpy_ufunc__(self, func, method, pos, inputs, **kwargs):
...: return "B"
In [4]: b = B()
In [5]: np.dot(a, b)
Out[5]: 'B'
In [6]: np.multiply(a, b)
Out[6]: 'B'
A simple __numpy_ufunc__ has been added to SciPy’s sparse matrices Currently this only handles np.dot and np.multiply because it was the two most common cases where users would attempt to use sparse
matrices with ufuncs. The method is defined below:
def __numpy_ufunc__(self, func, method, pos, inputs, **kwargs):
"""Method for compatibility with NumPy's ufuncs and dot
without_self = list(inputs)
del without_self[pos]
without_self = tuple(without_self)
if func == np.multiply:
return self.multiply(*without_self)
elif func == np.dot:
if pos == 0:
return self.__mul__(inputs[1])
if pos == 1:
return self.__rmul__(inputs[0])
return NotImplemented
So we now get the expected behavior when using ufuncs with sparse matrices.:
In [1]: import numpy as np; import scipy.sparse as sp
In [2]: a = np.random.randint(3, size=(3,3))
In [3]: b = np.random.randint(3, size=(3,3))
In [4]: asp = sp.csr_matrix(a); bsp = sp.csr_matrix(b)
In [5]: np.dot(a,b)
array([[2, 4, 8],
[2, 4, 8],
[2, 2, 3]])
In [6]: np.dot(asp,b)
array([[2, 4, 8],
[2, 4, 8],
[2, 2, 3]], dtype=int64)
In [7]: np.dot(asp, bsp).A
array([[2, 4, 8],
[2, 4, 8],
[2, 2, 3]], dtype=int64) | {"url":"https://docs.scipy.org/doc/numpy-1.10.1/neps/ufunc-overrides.html","timestamp":"2024-11-06T10:50:46Z","content_type":"text/html","content_length":"31242","record_id":"<urn:uuid:63e800d3-f8ec-4f40-ac88-184da0dace13>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00250.warc.gz"} |
How Delta-Sigma Works, part 2: The Anti-Aliasing Advantage
This post is part of a series on delta-sigma techniques: analog-to-digital and digital-to-analog converters, modulators, and more. A complete list of posts in the series are in the How Delta-Sigma
Works tutorial page.
Today, let’s take another look at delta-sigma conversion. The first part of this series showed how a one-bit, first-order delta-sigma modulator creates a bitstream, the average value of which equals
the input voltage. It turns out that how we find that average value makes a big difference in the performance of a delta-sigma analog-to-digital converter. In fact, if done right, not only does it
improve performance, but it greatly simplifies the analog circuitry preceding the A-to-D. Let’s take a look at why this works!
Aliasing Explained
One phenomenon that happens with any analog-to-digital conversion is known as aliasing. A-to-D is inherently a sampling process, in which an analog signal that is continuous in time is converted to
a digital signal that exists in discrete chunks, or samples. The rate at which those samples are taken is known as the sampling rate. The figure below shows three analog sine waves, in red, being
sampled at the times marked with the blue vertical lines. The top sine wave has a frequency of 1 Hz, and it is being sampled at 7 Hz. (The seventh sample is not obvious, because it is zero and the
left- hand or the right-hand edge of the graph. Which edge you choose is unimportant.)
The second and third plots in the figure show aliasing in action. When any signal with a frequency above one-half the sampling rate is sampled, the signal is <i>aliased</i> down to a frequency
between 0 Hz and one-half the sampling rate. The second line shows a 6 Hz sine wave in red, which is being sampled at 7 samples per second (sps). The blue lines show where the samples fall.
Because 6 Hz is above 3.5 sps (one half of 7 sps), the sine wave will be aliased. As you can see, the blue samples are identical to those you see in the top trace, except that they have the
opposite polarity. The dark blue trace connects them and shows that a 6 Hz sine wave is indistinguishable from a 1 Hz sine wave when both are sampled at 7 sps. That is aliasing in action.
The same process happens for any input frequency above one-half the sampling rate. The third plot in the figure, for example, shows an 8 Hz sine wave. Again, it is sampled at 7 Hz. This time the
samples are identical to those in the top line. The sampled waveform (in dark blue once again) is indistinguishable from the sampled version of our original 1 Hz sine wave.
Aliasing is so important that one-half the sampling rate has become known as the “Nyquist frequency” for a system. You may see people refer to signals as being “above Nyquist” or “below Nyquist”.
Aliasing happens in a repeated pattern as the frequency rises, and each repetiion of that pattern is called a “Nyquist zone”. All of this is in memory of Harry Nyquist (1889-1976), who with Claude
Shannon discovered much of the mathematics behind sampling.
Because of aliasing, analog-to-digital converters are usually preceded by an anti-aliasing filter. This filter removes the frequency content outside of the desired Nyquist zone, so that noise and
interfering signals do not alias into the passband of the ADC. Most often, a low-pass filter is used, so that the selected Nyquist zone runs from DC (0 Hz) to 1/2 the sampling frequency. There are
some exotic RF applications in which a bandpass filter selects frequencies in a higher Nyquist zone, which the ADC then aliases down, but these are uncommon.
How Oversampling Makes Anti-Aliasing Easier
In order to get good accuracy, delta-sigma converters need to run at a much higher frequency than the input signal. The first part of this series introduced an idea for an ADC built from a
delta-sigma modulator followed by a digital counter:
For this to work, the delta-sigma modulator has to <i>oversample</i>. In order to have the counter reflect the input signal accurately, there have to be many 1 and 0 bits in the modulator’s output
for the counter to count. In other words, each sample from the counter’s output has to reflect many samples in the modulator. This is called oversampling.
The nice thing about oversampling is that the Nyquist frequency goes up with the sample rate, even when oversampling! The classic example of this happens in CD audio systems. CDs carry audio
signals of up to 20 kHz, with a 44.1 ksps sample rate and a Nyquist frequency of 22.05 kHz. Furthermore, CD audio has a dynamic range of about 97 dB. In order to avoid aliasing signals back into
the 0 Hz – 20 kHz audio band, the antialiasing filter at the input of a CD audio ADC needs to have a rolloff frequency (-3 dB) at 20 kHz, and should be down to -97 dB by 24.1 kHz. This is an
impractical filter to design and build.
However, if an oversampled delta-sigma ADC is used, then the Nyquist frequency goes up to one-half the oversampled sample rate. A typical choice might be 64x oversampling, in which case the ADC will
sample at 2.8224 MHz. Then the Nyquist frequency is 1.4112 MHz. The anti-aliasing filter still needs to have its -3 dB rolloff at 20 kHz, but it does not need to be -97 dB down until 2.82 MHz.
That is a much easier filter to design. In fact, a 3-pole filter, easily and cheaply implemented with an op amp, is sufficient.
Moving Anti-Aliasing to the Digital Domain
To go from 2.8 Msps to 44.1 ksps requires another round of sampling, this time in the digital domain. Remember the counter? The process of reading its count, then resetting it for another round of
counting, is a form of sampling, and aliasing can result. The figure below shows an example. In this case, an 8 Hz sine wave is being sampled at 70 sps by an oversampling ADC. Then, that digital
signal is being downsampled, by taking every tenth sample, to 7 sps. The result is that the 8 Hz input is aliased to 1 Hz, just as if it was sampled at 7 sps in the first place.
Just as in the analog domain, there are times when the aliasing resulting from downsampling can be useful, but often it is not. To prevent it, we need a low-pass filter, but this time the filter can
be digital. In the CD-audio example, the filter needs to have the same rolloff characteristics as the challenging analog filter (-3 dB at 20 kHz, and -97 dB at 24.1 kHz). Doing that in a digital
filter, though, is much easier than in analog. Digital arithmetic can produce a filter of arbitrarily good performance, without the precision components or careful tuning adjustments that might be
required in analog. All it takes is throwing enough logic gates at the problem, and thanks to Moore’s Law, logic gates are cheap.
Since low-pass filters have an averaging effect, the filter will turn the bitstream of 1’s and 0’s into a series of multi-bit samples. The counter becomes unnecessary. Instead, it is enough to keep
one sample from the low-pass filter’s output every so often, discarding the rest. The ADC now looks like this. (A simple anti-aliasing filter before the input is needed, but not shown.)
The figure below shows the principle. A 1 Hz sine wave is oversampled at 70 Hz in the top graph, then 9 out of every 10 samples are discarded, leaving only the highlighted ones. Those samples are
plotted in the bottom graph. The result is identical to the sine wave samples in the first graph at the top of this article.
Technology similar to this, with some additional improvements to reduce the amount of math needed, makes it possible to put CD-quality ADCs in every desktop and laptop computer. Instead of an
expensive analog filter, cheap digital gates on an IC provide most of the filtering, reducing the cost of the ADC to only a dollar or two.
Wrapping Up
In this article, we have come full circle to find out how delta-sigma techniques can make analog design easier. I’ve shown how aliasing happens and explained the need for anti-aliasing filters.
Then we looked at the oversampling inherent to delta-sigma modulators, and how that permits a simpler analog antialiasing filter, at long as a digital low-pass filter is included after the
modulator. This is just one of the reasons why delta-sigma principles are very cool. Coming up in this series: Simulating a delta-sigma modulator and an introduction to noise shaping.
Bourdopoulos, George I., Aristodemos Pnevmatikakis, Vassilis Anastassopoulos, and Theodore Deliyannis. Delta-Sigma Modulators: Modeling, Design and Applications. London: Imperial College Press, 2003 | {"url":"http://skywired.net/blog/2011/07/how-delta-sigma-works-anti-aliasing-advantage/","timestamp":"2024-11-06T21:37:59Z","content_type":"text/html","content_length":"44890","record_id":"<urn:uuid:859eeb14-f820-4e77-a557-356f40894beb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00304.warc.gz"} |
Competency Based Questions Chapter 10 Mensuration - Rankers Study TutorialCompetency Based Questions Chapter 10 Mensuration
Hint: Give example(s) in order to define perimeter of closed figures.
Question.1. Jatin is finding the perimeter of a figure. Which of these could be the figure?
Question.2. In which of these situations will perimeter be calculated?
(a) Length of lace border needed to put around a rectangular table cover.
(b) Length of rope required to fence three sides of a rectangular backyard.
(c) Amount of water needed to fill a container.
(d) Amount of paint required to paint a wall.
Hint: Deduce and apply the formula to determine the perimeter of a rectangle.
Question.3. Aditi wants to put decorative tape around the borders of a rectangular cardboard which is 50 cm long and 45 cm wide. Which of these expressions represents the length of tape, in cm,
required to cover its borders?
(a) (50 × 45)
(b) (50 + 45)
(c) 2(50 × 45)
(d) 2(50 + 45)
Question.4. A student has to form distinct rectangles by using coloured ribbon and paste it on a sheet. If she uses 20 centimetres of ribbon for each rectangle, how many distinct rectangles she can
form of dimensions of positive integers?
(a) 1
(b) 5
(c) 6
(d) 10
Hint: Deduce and apply the formula to determine the perimeter of a square.
Question.5. Arjun wants to fence his square backyard of side length 11 m using rope. He makes 3 complete rounds using the rope to fence. What is the total length of rope used?
(a) 44 m
(b) 66 m
(c) 132 m
(d) 363 m
Question.6. The perimeter of a square is 2k cm. If the perimeter of square becomes \frac{1}{2}k cm, how will the side length of the square change?
(a) It will become 4 times
(b) It will become 8 times
(c) It will become one-fourth
(d) It will become one-eighth
Hint: Deduce and generalize the formula to determine the perimeter of a regular polygon.
Question.7. A wire of length 56 cm is made into the shape of a heptagon. What is the side length of the heptagon?
(a) 7 cm
(b) 8 cm
(c) 14 cm
(d) 49 cm
Question.8. The perimeter of a regular hexagon is 14 cm less than the perimeter of a regular octagon. If the side length of the hexagon is (2k+3) cm, what is the side length of the octagon?
(a) (2k+5) cm
(b) (2k-13) cm
(c) \frac{1}{2}(3k+1) cm
(d) \left(\frac{3}{2}k+4\right) cm
Hint: Give examples in order to defend that different shapes can have the same perimeter.
Question.9. Consider two shapes shown.x?
(a) 5
(b) 8
(c) 10
(d) 16
Question.10. Consider a figure below.
(a) A rectangle of length 12 cm and breadth 6 cm.
(b) An equilateral triangle of side length 24 cm.
(c) A regular pentagon of side length 9 cm.
(d) A square of side length 36 cm.
Hint: Count the squares in order to estimate the area of the given closed curve in the squares grid sheet.
Question.11. What is the area of the following figure?
(a) 13 square units
(b) 15 square units
(c) 16 square units
(d) 17 square units
Question.12. A farmer needs to buy seeds for a piece of agricultural land represented on a rectangular grid as shown below.
(a) 18.5
(b) 22.5
(c) 74
(d) 78
Hint: Deduce and apply the formula in order to determine the area of a rectangle.
Question.13. The length of a rectangle is twice its breadth. Given that the length of the rectangle is 8 cm, what is the area of the rectangle?
(a) 24 cm^{2}
(b) 32 cm^{2}
(c) 48 cm^{2}
(d) 128 cm^{2}
Question.14. The length and breadth of a rectangle are changed such that the area of the rectangle changes from 2k to k. If the length and breadth of the original rectangle are l and b respectively,
which of these could be the length and breadth of the new rectangle?
(a) \frac{l}{4} and 2b
(b) \frac{l}{2} and \frac{b}{2}
(c) \frac{l}{2} and 4b
(d) \frac{l}{4} and 4b
Hint: Deduce and apply the formula in order to determine the area of a square.
Question.15. Observe the figure below:
(a) 14 cm^{2}
(b) 28 cm^{2}
(c) 301 cm^{2}
(d) 949 cm^{2}
Question.16. If the side length of a square becomes one-third of the original side length, what is the ratio of the area of the original square to the area of square with changed side length?
(a) 1:3
(b) 1:9
(c) 3:1
(d) 9:1 | {"url":"https://rstudytutorial.com/cbe-questions-maths-class-6-chapter-10-mensuration/","timestamp":"2024-11-08T21:05:15Z","content_type":"text/html","content_length":"225655","record_id":"<urn:uuid:27485b1f-b7bf-4a75-b43a-19960a21153a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00814.warc.gz"} |
How to Save A Numpy Array As A Tensorflow Variable?
To save a numpy array as a tensorflow variable, you can use the tf.assign function. First, create a tensorflow variable using tf.Variable and then assign the numpy array to it using the tf.assign
function. This will allow you to save the numpy array as a tensorflow variable and use it in your tensorflow computations.
What is the importance of data integrity checks when saving a numpy array as a tensorflow variable?
Data integrity checks are crucial when saving a numpy array as a TensorFlow variable because they help ensure that the data being passed from one system to another is accurate, reliable, and
consistent. By performing data integrity checks, the integrity and validity of the data are verified, thus preventing any potential data corruption, loss, or inaccurate representation.
In the context of saving a numpy array as a TensorFlow variable, data integrity checks help ensure that the array is correctly converted and stored in the TensorFlow variable format without any loss
of information or corruption of data. This is especially important when dealing with large datasets or complex neural network models, where even a minor error in the data can lead to significant
issues in the model's performance.
Data integrity checks also help identify any potential errors or inconsistencies in the data before saving it as a TensorFlow variable, allowing for prompt remediation and ensuring the overall
quality of the data. This can help optimize the performance of the neural network model and prevent any potential issues during training or inference.
Overall, data integrity checks play a critical role in ensuring the accuracy, reliability, and consistency of data when saving a numpy array as a TensorFlow variable, ultimately leading to improved
performance and reliability of the neural network model.
How can I store a numpy array as a tensorflow variable in TensorFlow?
You can store a numpy array as a TensorFlow variable by using the tf.Variable function in TensorFlow. Here's an example of how you can do this:
1 import tensorflow as tf
2 import numpy as np
4 # Create a numpy array
5 array = np.array([[1, 2], [3, 4]])
7 # Convert the numpy array to a TensorFlow variable
8 tensor = tf.Variable(array)
10 # Initialize the TensorFlow variable
11 init = tf.global_variables_initializer()
13 # Create a TensorFlow session
14 with tf.Session() as sess:
15 # Run the initialization operation
16 sess.run(init)
18 # Print the TensorFlow variable
19 print(sess.run(tensor))
In this example, we first create a numpy array array. We then convert this numpy array to a TensorFlow variable tensor using the tf.Variable function. We initialize the variable using
tf.global_variables_initializer() and run it in a TensorFlow session. Finally, we print the value of the TensorFlow variable tensor.
How to serialize a numpy array before saving it as a tensorflow variable?
To serialize a numpy array before saving it as a TensorFlow variable, you can use the numpy.save function to save the array to a file in binary format. Here's an example of how you can do this:
1 import numpy as np
3 # Create a numpy array
4 arr = np.array([[1, 2, 3], [4, 5, 6]])
6 # Serialize the array
7 np.save('array.npy', arr)
9 # Load the serialized array
10 loaded_arr = np.load('array.npy')
12 # Now you can create a TensorFlow variable using the loaded array
13 import tensorflow as tf
15 tf_var = tf.Variable(loaded_arr)
In this example, we first create a numpy array arr and save it to a file named array.npy using np.save. We then load the serialized array using np.load and create a TensorFlow variable tf_var using
the loaded array. This variable can now be used in TensorFlow operations.
What steps are involved in deserializing a numpy array saved as a tensorflow variable?
To deserialize a numpy array saved as a tensorflow variable, you can follow these steps:
1. Load the saved tensorflow variable using tf.train.load_variable() function. This function takes the path to the saved checkpoint file and the name of the variable as input and return the value of
the variable as a numpy array.
2. Convert the numpy array into a tensorflow variable using tf.Variable() function. This function takes the numpy array as input and creates a new tensorflow variable with the same value.
Here's an example code that demonstrates the deserialization process:
1 import tensorflow as tf
2 import numpy as np
4 # Load the saved tensorflow variable
5 saved_var = tf.train.load_variable('./saved_model.ckpt', 'my_variable')
7 # Convert the numpy array into a tensorflow variable
8 tf_var = tf.Variable(saved_var)
10 # Now you can use tf_var as a tensorflow variable in your code
Make sure to replace './saved_model.ckpt' with the actual path to your saved checkpoint file and 'my_variable' with the name of the variable you want to deserialize.
What precautions should I take while saving a numpy array as a tensorflow variable?
When saving a numpy array as a TensorFlow variable, there are a few precautions you should take to ensure the data is saved correctly and can be properly loaded back in the future:
1. Check the data type: Make sure that the data type of the numpy array is compatible with TensorFlow. TensorFlow typically works with float32 or int32 data types, so you may need to convert your
numpy array to one of these types before saving it as a TensorFlow variable.
2. Normalize the data: If necessary, normalize the data in your numpy array before saving it as a TensorFlow variable. Normalizing the data can help improve the training process and prevent issues
with numerical stability.
3. Ensure the shape is compatible: Make sure that the shape of your numpy array is compatible with the shape expected by TensorFlow. Check the shape of the TensorFlow variable you are saving the
array as, and reshape your numpy array if necessary to match this shape.
4. Save the array using TensorFlow functions: To save a numpy array as a TensorFlow variable, use TensorFlow functions like tf.constant() or tf.Variable(). These functions will ensure that the data
is saved in the correct format and can be loaded back into a TensorFlow session properly.
5. Use the correct file format: When saving the TensorFlow variable to disk, use the appropriate file format such as .ckpt or .pb. This will help ensure that the data is saved correctly and can be
easily loaded back into a TensorFlow session in the future.
By following these precautions, you can save your numpy array as a TensorFlow variable in a way that ensures the data is preserved and can be properly loaded back into a TensorFlow session for future | {"url":"https://topminisite.com/blog/how-to-save-a-numpy-array-as-a-tensorflow-variable","timestamp":"2024-11-13T09:41:29Z","content_type":"text/html","content_length":"372646","record_id":"<urn:uuid:17910c4a-c73e-46a7-b7ff-f1d374684406>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00100.warc.gz"} |
“Stringing Along”
This large assignment aims to give you experience with computational design. You will use computer programming in Processing to write algorithms and procedures with turtle geometry and L-systems to
generate creative designs. Some of the work you create as part of this assignment will be fabricated (laser-cut) in the next assignment.
GitHub repository:
TASK 1: Processing functions
Use Processing functions to create a composite image with the simple geometries listed below in any colors (e.g., background, stroke, fill) of your choosing:
• Draw a point that is not at the origin.
• Use the above-generated point as a starting point to create a line.
• Draw a circle with a well-defined radius.
• Draw a triangle.
• Draw a rectangle (with unequal side lengths).
void setup() {
//setup code - gets called once
size (500, 500);
background (64);
void draw() {
// draw a green circle on the left
circle(200, 200, 100);
// draw a line
line(230, 110, 230, 400);
//draw a triangle behind the other shapes
fill(#ff7509, 148);
triangle(150, 300, 350, 200, 450, 300);
// draw a rectangle below without fill, just stroke
fill(0, 200);
stroke(5, 20);
rect(70, 280, 300, 50, 5);
//draw a point
point (230, 110);
TASK 2: Turtle Library
Use the Turtle library (in Processing) to create the following geometries listed below:
• Draw the shapes for capitalized letters “ I “ and “T”. Ensure that you don’t overwrite over paths in order to create these shapes. Hint: You should use the turtle’s penUp and penDown functions.
• Draw a triangle using turtle movement.
• Draw a regular pentagon using the movement of the turtle. Recall that for a regular pentagon, each of the internal angles measures 108°. In general for a regular polynomial with a number of sides
‘k’, the internal angle is given by [(k-2)/k]*180^°.
• Draw a circle with a well-defined radius using only turtle movement (do not use built-in functions in Processing).
See code on Github
TASK 3: L-system
• Define each of them by — the vocabulary (variables and constants), axiom, and production rules.
For each L-system, list the first three strings produced after the axiom (beyond iteration n=0).
• Use Processing, the Turtle Library, and LA1_LSystemBase (base code on GitHub) to program a representation of your system.
• In your implementation, be sure to use multiple variables and constants (“F“, “B“, “-“, “+“, and others, etc.) along with push/pop (e.g., “[“, “]“) to give your L-System a memory (see lecture
slides for more details).
See code on Github | {"url":"https://elazaro.uber.space/?p=902","timestamp":"2024-11-03T19:52:04Z","content_type":"text/html","content_length":"95300","record_id":"<urn:uuid:5746334f-3cf9-48db-bd97-e13ebe933fe9>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00628.warc.gz"} |
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Question Text बराबर है।
Updated On Nov 8, 2022
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 149
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Jarot Prasetyo and Anis Marjukah and Abdul Hadi (2022) SIMULASI MATEMATIS OPTIMALISASI KOMPOSISI PROPORSI PORTOFOLIO SAHAM MENGGUNAKAN METODA OPTIMALITY LAGRANGE MULTIPLIER. Widya Dharma Journal of
Business, 1 (1). pp. 35-50. ISSN 2829-3436
artikel_wijob SIMULASI MATEMATIS.pdf
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This study aims to determine the composition of the proportion of funds (W) that should be invested in a stock portfolio. The method of determining it is using a simulation called the Optimization of
the Optimization of the Share Portfolio Proportion Composition Using the Lagrange Multiplier Optimality Method. In this way it is expected that the proportion of funds provided by potential investors
will be optimal so that the portfolios that are formed will be efficient, that is, at a certain level of risk, the portfolio will provide the maximum expected return; or at a certain level of
expected return will result in minimal portfolio risk. Through this simulation, several equations of the proportion of funds for each portfolio forming share will be generated. If the value of the
expected return of each share is included in these equations, the proportion of funds generated will be optimal. Using the stock data included in the LQ45 index of 15 types of stocks, the simulation
produces an equation of the proportion for each type of stock at level E (Rp) of 0.014806, so a portfolio risk will be obtained of 6.50521E-17. This level of risk is much lower when compared to
portfolio risk before using the ideal proportion, which is 0.00028591 Keywords : stock portfolio, expected results, stock portfolio risk, Lagrange Multiplier Optimality
Actions (login required) | {"url":"http://repository.unwidha.com:880/3202/","timestamp":"2024-11-08T01:12:05Z","content_type":"application/xhtml+xml","content_length":"24523","record_id":"<urn:uuid:913f9d43-c8b5-414a-a357-3eb96bcc1376>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00419.warc.gz"} |
Entropy | Post-16 thermodynamics tutorials
Try this tutorial with accompanying videos to help teach students on post-16 courses about the concept of entropy and the second law of thermodynamics
This tutorial is designed to develop and consolidate students’ understanding of entropy as a fundamental concept in thermodynamics. Students build on their knowledge about the chance behaviour of
particles and energy to learn about:
• The second law of thermodynamics
• The role of energy in determining the entropy or randomness of a chemical system
• Entropy change and how to calculate it
The tutorial also features video demonstrations and animations to illustrate key ideas.
The interactive ‘simulations’ for this tutorial are currently unavailable. However, you may find it helpful to read any accompanying instructions, observations and conclusions relating to the
simulations below.
Introducing entropy
Scientists have a mathematical way of measuring randomness – it is called ‘entropy’ and is related to the number of arrangements of particles (such as molecules) that lead to a particular state. (By
a ‘state’, we mean an observable situation, such as a particular number of particles in each of two boxes.)
As the numbers of particles increases, the number of possible arrangements in any particular state increases astronomically so we need a scaling factor to produce numbers that are easy to work with.
Entropy, S, is defined by the equation:
S = k lnW
where W is the number of ways of arranging the particles that gives rise to a particular observed state of the system, and k is a constant called Boltzmann’s constant which has the value 1.38 x 10^
-23 J K^-1. In the expression above, k has the effect of scaling the vast number W to a smaller, more manageable number.
The natural logarithm, ln, also has the effect of scaling a vast number to a small one – the natural log of 10^-23 is 52.95, for example.
Just as logarithms to the base 10 are derived from 10^x, natural logarithms are derived from the exponent of the function e^x, where e has the value 2.718. There are certain mathematical advantages
to using this base number.
Don’t let students worry about ‘ln’, just get them to use the correct button on their calculators. Some examples of calculating the ln of large numbers might help students to see the scaling effect.
Entropies are measured in joules per kelvin per mole (J K^-1 mol^-1). Notice the difference between the units of entropy and those of ‘enthalpy’ (energy), kilojoules per mole (kJ mol^-1).
The key point to remember is that entropy is a figure that measures randomness and gases, where the particles could be anywhere, tend to have greater entropies than liquids, which tend to have
greater entropies than solids, where the particles are very regularly arranged. You can see this general trend from the animations of the three states below.
Some values of entropies
Substance Physical state at standard conditions Entropy, S J K^-1 mol^-1
Carbon (diamond) solid 2.4
Copper solid 33
Calcium oxide solid 40
Calcium carbonate solid 93
Ammonium chloride solid 95
Water (ice) solid 48
Water liquid 70
Water (steam) gas 189
Hydrogen chloride gas 187
Ammonia gas 192
Carbon dioxide gas 214
Observe the not all solids have smaller entropy values than all liquids nor do all liquids have smaller values than all gases.
There is, however, a general trend that:
S[solids] < S[liquids] < S[gasses]
The second law of thermodynamics
Since processes take place by chance alone, and this leads to increasing randomness, we can say that in any spontaneous process (one that takes place of its own accord and is not driven by outside
influences) entropy increases.
This is called the second law of thermodynamics, and is probably the most fundamental physical law. So processes that involve spreading out and increasing disorder are favoured over those where
things become more ordered – students may have noticed this in their bedrooms!
The word ‘feasible’ is also used to mean the same as ‘spontaneous’. The terms have nothing to do with the rate of a process. So a reaction may be feasible (spontaneous) but occur so slowly that in
practice it does not occur at all. For instance, the reaction of diamond (pure carbon) with oxygen to form carbon dioxide is feasible (spontaneous) but we do not have to worry about jewellery burning
in air at room temperature – ‘diamonds are forever’!
Illustrating the second law
Watch the two videos below. You would have no difficulty in deciding which one is being played in reverse – randomly arranged fragments do not spontaneously arrange themselves into an ordered
Video 1
Video 2
Entropy and solutions
So the idea of entropy increasing seems to explain why a crystal of salt (sodium chloride) will dissolve in a beaker of water of its own accord (spontaneously). The resulting solution in which sodium
and chloride ions are thoroughly mixed with water molecules is more random than a beaker of pure water and a crystal of solid sodium chloride in which the ions are in a highly ordered crystal
There is some local increase in order on forming the solution as (disordered) water molecules cluster round the Na^+ and Cl^– ions but this is outweighed by the large increase in disorder as the
sodium chloride lattice breaks up.
It also seems to explain our initial query, or why the reaction between magnesium and hydrochloric acid,
Mg[(s)] + 2HCl[(aq) ]→ H[2(g)] + MgCl[2(aq)]
occurs, but the reverse reaction,
H[2(g)] + MgCl[2(aq)]→ Mg[(s)] + 2HCl[(aq)]
does not. In the first case, the production of a gas from a solid clearly involves an increase in entropy while the reverse has a decrease.
Is the second law wrong?
It is not difficult to find examples of chemical reactions that appear to contradict the rule that entropy increases in spontaneous processes.
Take, for example, the demonstration illustrated in the video below, where the two gases, hydrogen chloride and ammonia, diffuse along a tube and produce a white ring of solid ammonium chloride:
HCl[(g)] + NH[3(g)]→ NH[4] Cl[(s)]
The two gases forming a solid clearly involve a decrease in entropy, yet the reaction occurs spontaneously.
In fact, we can calculate the numerical value of the entropy change from the figures in the table above (see Introducing entropy):
• Total entropy of starting materials 187 + 192 = 379 J K^-1 mol^-1
• Entropy of product = –95 J K^-1 mol^-1
• Entropy change = –284 J K^-1 mol^-1
As expected, a significant decrease. (Remember, we expect spontaneous reactions to have an increase in entropy.) Does this mean the second law is wrong?
The role of energy
Energy also has a role to play in the entropy or randomness of a chemical system, by which we mean a quantity of substance or substances (such as a reaction mixture).
Energy exists in ‘packets’ called quanta. You can have any whole number of quanta of energy but not 11/2 or 3.25 quanta.
Like the distribution of atoms in space, the distribution of quanta of energy between molecules is also random because, like molecules, energy quanta do not ‘know’ where they are supposed to be –
energy ‘doesn’t care’. We can simulate this distribution.
We have seen that the number of ways of arranging particles contributes to the entropy of a physical system. The distribution of energy quanta also contributes to the system’s entropy.
The distribution of energy simulation
Introduction and instructions
This is a simple example is to consider quanta of energy distributed between the vibrational energy levels of a set of diatomic molecules. Such energy levels are evenly spaced and can be represented
like the rungs of ladders. How many ways are there of distributing x quanta of energy betweeny molecules?
• Here you can vary the number of molecules and the number of quanta available to be distributed between them and the simulation will allow the energy to be exchanged between the molecules in all
possible ways at random.
• Start by setting the number of quanta and molecules to a low value. Use the step button to move through all the combinations (for example: there are four ways to distribute three quanta among two
Key observations
The more quanta of energy there are to be shared between a given number of molecules, the more ways there are of arranging them. Also, the more molecules there are, the more ways there are of
The tables below show the possible arrangements of four quanta between two molecules, five quanta between two molecules and three quanta between three molecules respectively. Try exploring these
using the simulator.
Two molecules, four
quanta (five
Molecule 1 Molecule 2
Two molecules, five
quanta (six
Molecule 1 Molecule 2
Three molecules, three quanta (ten
Molecule 1 Molecule 2 Molecule 3
The distribution of energy quanta also contributes to entropy because of the relationship S = k lnW. The more heat energy we put into anything, the more its entropy increases because there are more
quanta and thus more ways, W, to distribute them.
Most chemical reactions involve a change of heat energy (enthalpy), either given out from the reactants to the surroundings or taken in from the surroundings into the products. So we must also take
this into account when we are considering the entropy change of a chemical reaction. It is not just the chemical reaction that matters but the surroundings as well.
The system and its surroundings
Here is the solution to the puzzle about the ammonia–hydrogen chloride reaction. This reaction is strongly exothermic (gives out a lot of heat to the surroundings – in fact ΔH is –176 kJ mol^-1). The
key is the idea of the surroundings.
For each mole of ammonium chloride that is formed, 176 kJ of heat energy is transferred to the surroundings. As we have seen, this increases the entropy of the surroundings because of the increased
number of ways of arranging the quanta of energy. So, within the reaction itself (ie starting materials and products), entropy decreases but, because of the heat energy passed to the surroundings,
the entropy of the surroundings increases, and more than compensates for the decrease in entropy in the reaction.
In other words, this increase in the entropy of the surroundings is more than the decrease in entropy of the reaction and thus there is an overall increase in entropy.
We call the reaction itself ‘the system’ and everything else ‘the surroundings’. In principle, ‘the surroundings’ is literally the rest of the Universe, but in practice it is the area immediately
around the reaction vessel.
So, the second law of thermodynamics is not broken by the ammonia–hydrogen chloride reaction; the problem was that we had forgotten the surroundings. This is because, as chemists, we are used to
concentrating only on what happens inside our reaction vessel.
How does this affect our understanding of the second law?
We can build on these insights to make a better statement of the second law of thermodynamics: in a spontaneous change, the entropy of the Universe increases, ie the sum of the entropy of the system
and the entropy of the surroundings increases.
Total entropy change
As we have seen above, the entropy change of the ammonia–hydrogen chloride reaction (‘the system’) is –284 J K^-1 mol^-1. It is negative as we have calculated (and predicted from the reaction being
two gases going to a solid).
But how can we evaluate the entropy change caused by ‘dumping’ 176 kJ mol^-1 of heat energy into the surroundings? (Notice that this is 176 000 J mol^-1). It must be positive as more quanta of energy
lead to more possible arrangements and it must be more than 284 J K^-1 mol^-1.
The formal derivation is complex but leads to the expression ΔS of the surroundings:
ΔS[surroundings] = –ΔH/ T
This makes sense because:
1. The negative sign means that an exothermic reaction (ΔH is negative, heat given out) produces an increase in the entropy of the surroundings.
2. The more negative the value of ΔH, the more positive the entropy increase of the surroundings.
3. The same amount of energy dumped into the surroundings will make more difference at lower temperature – this rationalises the ‘division by T’.
For the ammonia–hydrogen chloride reaction at 298 K:
ΔS[surroundings] = –ΔH/ T = –(–176 000) / 298
ΔS[surroundings] = 591 J K^-1 mol^-1, more than enough to outweigh the value of ΔS[system] of –284 J K^-1 mol^-1
So the total entropy change (of the Universe, ie system + surroundings) brought about by the reaction is +307 J K^-1 mol^-1.
If we want to predict the direction of a chemical reaction, we must take account of the total entropy change of the system and the surroundings, and that includes the effect on entropy of any heat
change from the system to the surroundings (or, in the other direction, heat taken in from surroundings to system).
The equation for total entropy change
Total entropy change is the sum of the entropy changes of the system and its surroundings:
ΔS[total] = ΔS[system] + ΔS[surroundings]
If ΔS[total] for a reaction is positive, the reaction will be feasible, if negative it will not be feasible.
Additional information
This resource was originally published by the Royal Society of Chemistry in 2015 as part of the Quantum Casino website, with support from Reckitt Benckiser.
Post-16 thermodynamics tutorials
• 1
• 2
• 3
Currently reading
• 4 | {"url":"https://edu.rsc.org/resources/entropy-post-16-thermodynamics-tutorials/4012924.article?section=tutorial&article=5","timestamp":"2024-11-02T12:32:23Z","content_type":"text/html","content_length":"279025","record_id":"<urn:uuid:7a88244f-986b-42e4-bedb-3b711559d79e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00363.warc.gz"} |
Deep Venous Thrombosis (DVT) – TheNNT
Positive Findings (Patient Has This)
Finding (Sign/Symptom/Lab/Study) Number Needed to Diagnose(Positive Likelihood Ratio)
High Sensitivity D-Dimer with Low Pre-Test Probability 2.4×
High Sensitivity D-Dimer with Mod Pre-Test Probability 1.7×
High Sensitivity D-Dimer with High Pre-Test Probability 1.5×
Negative Findings (Patient Doesn't Have This)
Finding (Sign/Symptom/Lab/Study) Number Needed to Diagnose(Negative Likelihood Ratio)
High Sensitivity D-Dimer with Mod Pre-Test Probability 0.05×
High Sensitivity D-Dimer with High Pre-Test Probability 0.07×
High Sensitivity D-Dimer with Low Pre-Test Probability 0.10× | {"url":"https://thennt.com/lr/deep-venous-thrombosis-dvt/","timestamp":"2024-11-05T17:01:54Z","content_type":"text/html","content_length":"246280","record_id":"<urn:uuid:1d582561-9d59-48b4-8f1f-9cd66c754567>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00108.warc.gz"} |
The relative rank $\rank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $\langle A\cup B\rangle=S$. It follows from a result of Sierpiński that, if $X$ is
infinite, the relative rank of a subset of the full transformation semigroup $\mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $\mathcal{B}_{X}$ of
binary relations on $X$.
A subset $S$ of $\mathcal{T}_{\mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $\mathcal{T}_{\mathbb{N}}$ with the property that for each $\sigma$ in $S$ there exists $\
mu$ in $U$ such that $i\sigma\le i\mu$ for all $i$ in $\mathbb{N}$. It is shown that every dominated subset of $\mathcal{T}_{\mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid
of all contractions in $\mathcal{T}_{\mathbb{N}}$ (mappings $\alpha$ with the property that $|i\alpha-j\alpha|\le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.
It is shown (among other results) that $\rank(\mathcal{B}_{X}:\mathcal{T}_{X})=1$ and that $\rank(\mathcal{B}_{X}:\mathcal{I}_{X})=1$ (where $\mathcal{I}_{X}$ is the symmetric inverse semigroup on
$X$). By contrast, if $\mathcal{S}_{X}$ is the symmetric group, $\rank(\mathcal{B}_{X}:\mathcal{S}_{X})=2$.
AMS 2000 Mathematics subject classification: Primary 20M20 | {"url":"https://core-cms.prod.aop.cambridge.org/core/search?filters%5BauthorTerms%5D=N.%20Ruskuc&eventCode=SE-AU","timestamp":"2024-11-12T00:08:37Z","content_type":"text/html","content_length":"1003861","record_id":"<urn:uuid:d3fde79e-d877-4950-81ad-99ad3365cf60>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00161.warc.gz"} |
EE 569: Homework #1 solved
Problem 1: Image Demosaicing and Histogram Manipulation (50%)
(a) Bilinear Demosaicing (10%)
To capture color images, digital camera sensors are usually arranged in form of a color filter array
(CFA), called the Bayer array, as shown in Figure 1. Since each sensor at a pixel location only captures
one of the three primary colors (R, G, B), the other two colors have to be re-constructed based on their
neighbor pixel values to obtain the full color. Demosaicing is the process of translating this Bayer array
of primary colors into a color image that contains the R, G, B values at each pixel.
Figure 1: (a) Single CCD sensor covered by a CFA and (b) Bayer pattern [1].
Implement the simplest demosaicing method based on bilinear interpolation. Exemplary demosaicing
results are given in Figure 2. With this method, the missing color value at each pixel is approximated by
bilinear interpolation using the average of its two or four adjacent pixels of the same color. To give an
example, the missing blue and green values at pixel R3,4 are estimated as:
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 2 of 10
As for pixel G3,3, the blue and red values are calculated as:
(1) Apply the bilinear demosaicing to the cat image in Figure 3 and show your results.
(2) Do you observe any artifacts? If yes, explain the cause of the artifacts and provide your ideas to
improve the demosaicing performance.
Figure 2: (a) The original lighthouse image and (b) the demosaiced lighthouse image
by bilinear interpolation.
Figure 3: The cat image with the CFA sensor input.
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 3 of 10
(b) Malvar-He-Cutler (MHC) Demosaicing (20%)
Malvar et al. [1] proposed an improved linear interpolation demosaicing algorithm. It yields a higher
quality demosaicing result by adding a 2
-order cross-channel correction term to the basic bilinear
demosaicing result. Both the bilinear and the MHC demosaicing results of the Fruit_Shop image are
shown in Figure 4.
Figure 4: Demosacing results of Fruit_Shop image: the CFA input (left), the bilinear demosaicing
result (middle) and the MHC demosaicing result (right)
The MHC algorithm is stated below.
To estimate a green component at a red pixel location, we have
(i, j) = G
ˆ bl(i, j)+aDR (i, j)
where the 1st term at the right-hand-side (RHS) is the bilinear interpolation result given in (1) and the
nd term is a correction term. For the 2
nd term, alpha is a weight factor, and is the discrete 5-point
Laplacian of the red channel:
To estimate a red component at a green pixel location, we have
(i, j) = R
ˆ bl(i, j)+ bDG (i, j)
where ΔG is a discrete 9-point Laplacian of the green channel.
To estimate a red component at a blue pixel location,
where ΔB is a discrete 9-point Laplacian of the blue channel. The weights control how much
correction is applied, and their default values are:
a =
,b =
,g =
DR (i, j) = R(i, j) –
(R(i – 2, j) + R(i + 2, j) + R(i, j – 2) + R(i, j + 2))
(i, j) = R
(i, j)+gDB (i, j)
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 4 of 10
The above formulas can be generalized to missing color components at each sensor location.
Consequently, the MHC demosaicing can be implemented by convolution with a set of linear filters.
There are eight different filters for interpolating the different color components at different locations as
illustrated in Figure 5 [1].
Figure 5: Filter coefficients
(1) Implement the MHC linear demosaicing algorithm and apply it to the Cat image in Figure 3.
Show your results.
(2) Compare the MHC and the bilinear demosaicing results and explain the performance differences
between these two algorithms in your own words.
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 5 of 10
(c) Histogram Manipulation (20%)
Implement two histogram equalization techniques:
Method A: the transfer-function-based histogram equalization method
Method B: the cumulative-probability-based histogram equalization method
Enhance the contrast of images rose_dark and rose_bright in Figure 6 using both methods. Describe the
procedure and show final results with the following steps.
(1) Plot the histograms of all images (2 input and 4 output images). The figure should have the
intensity value as the x-axis and the number of pixels as the y-axis.
(2) Apply Method A to rose_dark and rose_bright and show the enhanced image. Plot the transfer
function for each testing image.
(3) Apply Method B to rose_dark and rose_bright and show the enhanced image. Plot the
cumulative histogram for each testing image.
(4) Discuss your observations on these two enhancement results. Do you have any idea to improve
the current result?
(5) Apply your implemented Method A and Method B to rose_mix in Figure 4 and show the result.
Can you get similar result as in previous part? If not, how to modify your implementation?
Please justify your answer with discussion.
Note that MATLAB users CANNOT use functions from the Image Processing Toolbox except
displaying function like imshow().
(a) rose_dark (b) rose_bright (c) rose_mix
Figure 6: Three rose images with different gray-scale histograms.
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 6 of 10
Problem 2: Image Denoising (50%)
(a) Gray-level image (20%)
Remove noise in the image in Figure 7(b), compare it with the original image in Figure 7(a), and answer
the following questions:
(a) (b)
Figure 7: (a) the original pepper image and (b) the pepper image with added noise.
(1) What is the type of embedded noise in Figure 7(b)?
(2) Apply a linear filter of size N by N to the noisy image. Compare the performance of two choices
of the filter parameters – the uniform weight function and the Gaussian weight function and
study the effect of the window size (i.e. parameter N). Plot the peak-to-signal-noise (PSNR)
value as a function of N for both weighting functions.
Note: The PSNR value between images I and Y can be calculated as:
PSNR(dB) =
MSE =
and where I is the noise-free image of size N M, Y is the filtered image of size N M, Max
for 8-bit image is 255.
(3) In most low-pass linear filters, we often see degradation of edges. However, using some nonlinear filters, we can preserve the edges. Bilateral filters are one such kind of filters. A discrete
bilateral filter is given by:
where is the neighboring pixel location within the window centered around , I is the
image with noise, Y is the filtered image. and are the spread parameters. Implement the
bilateral denoising filter and apply it to the noisy image.
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 7 of 10
(4) The non-local mean filter utilizes the pixel value from a larger region rather the mean of a local
window centered around the target pixel. A discrete non-local mean filter with Gaussian
weighting function is as follows:
where I, Y are the noisy and filtered images respectively, is the window centered around
location (x, y), and is the filtering parameter,
and denote the window size of
your choice.
The Gaussian weighted Euclidian distance between window and is defined as:
where denotes the local neighborhood centered at the origin, denotes the relative
position in the neighborhood window. a > 0 is the standard deviation of the Gaussian kernel.
Implement the non-local mean filter and apply to the noisy image.
(a) (b)
Figure 8: (a) the original rose image (b) the rose image with mixed noise.
(b) Color image (20%)
Figure 8 (b) is a noisy color image, where each channel is corrupted by both impulse and uniform
noises. Try your best to remove noise and compare the result with the original image as shown in Figure
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 8 of 10
8(a). Discuss your denoising strategy and answer the following questions:
(1) Should you perform filtering on individual channels separately for both noise types?
(2) What filters would you like use to remove mixed noise?
(3) Can you cascade these filters in any order? Justify your answer.
(4) It could be difficult to remove uniform noise satisfactorily. Filters may blur the object’s edge
when smoothing noise. A successful design of a uniform noise filter depends on its ability to
distinguish the pattern between the noise and the edges. Please suggest an alternative to low pass
filter with Gaussian weight function and discuss why such solution can potentially work better.
(You are welcome to implement the filter, but it is not required)
(c) Shot noise (10%)
Electronic camera image sensor, especially the CMOS sensor typically has noise in the dark parts of the
captured image. Such noise is called shot noise, which is caused by statistical quantum (the photon)
(a) (b)
Figure 9: (a) the original pepper image and (b) the pepper image with shot noise.
Figure 9(b) shows a typical image with shot noise. Unlike uniform noise, shot noise is Poisson
distributed. A common solution is firstly applying the Anscombe root transformation on each pixel z:
to the input image, resulting an image with additive Gaussian noise of unit variance. Then, a
conventional denoising filter for Gaussian noise is used. The denoised image is finally inverse
transformed. The detailed denoising steps can be found in [3].
(1) Implement the denoising method in [3] and apply it to Figure 9(b), where you can remove the
additive Gaussian noise using two methods.
Use a Gaussian low pass filter
Use the MATLAB code for block-matching and 3-D (BM3D) transform from [4].
(2) Show the final noise-removed pepper images and compare the PSNR values and visual quality of
the resulting images.
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 9 of 10
EE 569 Digital Image Processing: Homework #1
Professor C.-C. Jay Kuo Page 10 of 10
Problem 1: Image Demosaicing and Histogram Manipulation
cat_ori.raw 390x300x3 24-bit color(RGB)
cat.raw 390×300 8-bit gray
rose_ori.raw 400×400 8-bit gray
rose_dark.raw 400×400 8-bit gray
rose_bright.raw 400×400 8-bit gray
rose_mix.raw 400×400 8-bit gray
Problem 2: Noise Removal
pepper.raw 256×256 8-bit gray
pepper_noise.raw 256×256 8-bit gray
rose_color.raw 256x256x3 8-bit color(RGB)
rose_color_noise.raw 256x256x3 8-bit color(RGB)
pepper_dark.raw 256×256 8-bit gray
pepper_dark_noise.raw 256×256 8-bit gray
Reference Images
All images in this homework are from the USC-SIPI image database [5] or Google images [6].
[1] M. E. Celebi et al. (eds.), Color Image and Video Enhancement
[2] Malvar, Henrique S., Li-wei He, and Ross Cutler. “High-quality linear interpolation for demosaicing
of Bayer-patterned color images.” Acoustics, Speech, and Signal Processing, 2004. Proceedings.
(ICASSP’04). IEEE International Conference on. Vol. 3. IEEE, 2004.
[3] Makitalo, Markku, and Alessandro Foi. “Optimal inversion of the Anscombe transformation in lowcount Poisson image denoising.” IEEE transactions on Image Processing 20.1 (2011): 99-109.
[4] [Online]. Available: http://www.cs.tut.fi/~foi/GCF-BM3D/
[5] [Online] http://sipi.usc.edu/database/
[6] [Online] http://images.google.com/ | {"url":"https://codeshive.com/questions-and-answers/ee-569-homework-1-solved-2/","timestamp":"2024-11-04T02:58:39Z","content_type":"text/html","content_length":"119190","record_id":"<urn:uuid:08a10dea-c941-4198-9cd1-62adb945886c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00548.warc.gz"} |
The Mis-Education of Mathematics Teachers?
In Vol. 58, No. 3 of the Notices of the American Mathematical Society, a very disturbing article, The Mis-Education of Mathematics Teachers, by H. Wu, appeared. WU began by posing the rhetorical
question: "If we want to produce good French teachers in schools, should we require them to learn Latin in college, but not French?" Good point! Because it seems in training American math teachers,
this is equivalent to what is being done!
Wu gives the example of teaching fractions, say in grades 5-7. And say one wishes to add:
1/2 + 3/4
Easy, no?
Not according to Wu. And the reason is that the maths educators seem to be required to take courses that focus on the abstractions as opposed to the mechanics and operations.
Thus, he notes that Math professors may wonder why teachers of math must be provided with a knowledge of fractions relevant to the classroom. They may ask:
'What's so hard about equivalence classes of ordered pairs of integers?'
In this approach, to remind readers who may be unaware, we let Z be the integers and S the subset of ordered pairs of integers: Z X Z, consisting of all the elements (x.y) so that y not equal zero.
One then introduces an equivalence relation (~) in S by defining:
(x,y) ~ (z, w) if xw = yz
Then one denotes the equivalence class of (x,y) in S by (x/y) so that y not = 0.
We call the set of all such (x/y) the rational numbers, Q, because they are designated by a RATIO.
One then identifies Z with the set of all elements of the fom: (x/ 1) and one has:
Z ( Q (Z a subset of Q)
Finally, we convert Q into a ring by defining addition and multiplication in Q by:
(x/y) + (z/w) = (xw + zy)/ yw
(x/y) * (z/w) = xz/ yw
Is this needed to teach fractional addition (or multiplication) to kids in grades 5-7? HELL NO!
But, as Wu notes, this is normally how we teach our math majors (in 2-3 lectures) to do it. Can you do this with middle school students? Well, if you try either get set to have all the ipods pulled
out with you tuned out, or be looked at as some kind of freak.
As Wu points out, it's "totally consistent with the fundamental principles of mathematics" but not much use in pedagogy!
Why? A number of reasons:
1) It requires an understanding of the partition of S into equivalence classes
2) It requires the ability to consider each such class as one element.
3) It requires understanding yet another level of sophistication in terms of the identification of Z with {x/1: x in Z}.
Wu is correct when he notes that schools were in existence (and teaching fractions) long before their formal representation as equivalence classes of ordered pairs of integers, but this doesn't mean
the teaching preceding that formal presentation is any less valid. It merely means that the teaching was devoid of the abstraction and proofs.
In this sense, I disagree with Wu that the informal teaching of fractions is not teaching mathematics but a pretense of such. No, it is teaching math, but not predicated on a formal abstract theory.
In the same way, I can teach physics students (say at A-level) the principles of atomic behavior and how emission or absorption spectra are formed (including using experiments) without introducing
all the quantum mechanics abstractions, to the effect atomic states really entail "probability waves" and one has Hilbert spaces to define the behavior of the wave functions.
Thus, cutoff points for pedagogical abstraction don't mean one is operating under "pretense" but more accurately, a lower conceptual domain. (This would be in accord for Jean Piaget's notable levels
whereby a student may grasp a thing at one, but not another).
But to call the rudimentary mechanical teaching "pretense" is to not understand how teaching differs from mathematical research in the first place.
Wu is correct that "mathematics depends on precise and literal definitions"- but again, one must not let oneself become a hostage to technical formalisms and procedures or abstract pedantry. Physics
also depends on precise and literal definitions, but I wouldn't subject an A-level physics student to the definition of a quantum superposition or even the Heisenberg Uncertainty Principle (or
Principle of Complementarity) using Poisson brackets, before letting him examine radioactivity - say from a decaying alpha source near a Geiger-Müller counter.
It is here I believe that common sense in teaching must prevail, and yes, often details must be surrendered.
The trick, whether in math or physics, is knowing where the details end and the principles begin, and also if putting the principles in a very rudimentary form constitutes "pretense". | {"url":"https://brane-space.blogspot.com/2011/03/mis-education-of-mathematics-teachers.html","timestamp":"2024-11-04T05:45:26Z","content_type":"text/html","content_length":"119781","record_id":"<urn:uuid:dd4b2d9b-c9f2-4011-980d-46a2917b164a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00004.warc.gz"} |
Quantum algorithms and the finite element method
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a
large system of linear equations. Here we investigate the extent to which the finite element method can be accelerated using an efficient quantum algorithm for solving linear equations. We consider
the representative general question of approximately computing a linear functional of the solution to a boundary value problem, and compare the quantum algorithm's theoretical performance with that
of a standard classical algorithm -- the conjugate gradient method. Prior work had claimed that the quantum algorithm could be exponentially faster, but did not determine the overall classical and
quantum runtimes required to achieve a predetermined solution accuracy. Taking this into account, we find that the quantum algorithm can achieve a polynomial speedup, the extent of which grows with
the dimension of the partial differential equation. In addition, we give evidence that no improvement of the quantum algorithm could lead to a super-polynomial speedup when the dimension is fixed and
the solution satisfies certain smoothness properties.
Dive into the research topics of 'Quantum algorithms and the finite element method'. Together they form a unique fingerprint. | {"url":"https://research-information.bris.ac.uk/en/publications/quantum-algorithms-and-the-finite-element-method","timestamp":"2024-11-08T23:53:49Z","content_type":"text/html","content_length":"72934","record_id":"<urn:uuid:d04bf50c-3c3e-4fea-893e-c2707d74ca7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00274.warc.gz"} |
nForum - Discussion Feed (model theory and physics)trent comments on "model theory and physics" (51415)Urs comments on "model theory and physics" (51409)
I hardly know anything about model theory, but just wanted to point out that terrence tao has an interesting comment somewhere on his blog where (in different words) he speculates whether (a la
computational trinitarianism) there is a “mathematical trinitirian” type relationship between category theory, homotopy theory or type theory (can’t remember which), and model theory. | {"url":"https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=6405&FeedTitle=Discussion+Feed+%28model+theory+and+physics%29","timestamp":"2024-11-02T18:15:41Z","content_type":"application/atom+xml","content_length":"4295","record_id":"<urn:uuid:dbf49660-2b1b-4ed7-974e-afb2da1030f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00473.warc.gz"} |
Mechanical Engineering
Mechanical Engineering - Thermodynamics - Discussion
Discussion Forum : Thermodynamics - Section 3 (Q.No. 4)
The change of entropy, when heat is removed from the gas, is negative.
14 comments
Sunny said: 4 months ago
As temperature increases entropy increases temp decreases entropy decreases.
Sagar singh said: 2 years ago
Entropy is always increasing when the process is irreversible.
In some cases, if heat emitting is more than heat absorbing then the entropy is negative.
M S NAYAK said: 7 years ago
Entropy is never negative for irreversible process even heat is removed so,
Correct answer is B.
Aniket pisal said: 7 years ago
Entropy =change in heat /temperature.
Δs=delta Q/T.
As heat is removed from gas change in heat is negative hence,
Δs= -deltaQ/T.
Hence deltaQ/T= -Δs.
ie. Entropy is negative.
Nagendra singh said: 8 years ago
Entropy of system will decrease by removal of heat & increase by adding of heat. But entropy of universe is always increased.
Harsh said: 8 years ago
For reversible process entropy is + ve, - ve, and 0.
For irreversible process always +ve.
Shekhar Ghotekar said: 8 years ago
For a process, entropy can have positive, negative or zero value?
Diwakar Chaurasia said: 9 years ago
Yes @Ototo erick you are right.
The change in entropy can be negative it is possible somehow but the entropy can't be negative anyhow.
Ototo Erick said: 9 years ago
Change in entropy must be negative if temperature is remove and vice versa.
Krishna said: 9 years ago
Change in entropy can be negative but entropy can't be.
Quick links
Quantitative Aptitude
Verbal (English)
Placement Papers | {"url":"https://www.indiabix.com/mechanical-engineering/thermodynamics/discussion-1467-1","timestamp":"2024-11-13T14:54:01Z","content_type":"text/html","content_length":"45947","record_id":"<urn:uuid:a36d17c5-494a-45e5-acc4-22b696a93ed4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00380.warc.gz"} |
M1_P5. Cattle are sent to a feedlot to be grain-fed before being
processed into beef. The...
LPP formulation:
Decision Variable:
For the given situation, decision is regarding how much gram of feeds X, Y, and Z should be blended such that requirement are satisfied and total cost of mix is minimized.
X = number of lbs of feed X to be mixed
Y = number of lbs of feed Y to be mixed
Z = number of lbs of feed Z to be mixed
Objective function
The objective is to provide feed to cattles by blending three feeds such that total cost is minimized. The cost function is given as follows:
Cost = cost per lbs x lbs of feed i
The objective function is given as follows:
Minimize Z = ($5)X + ($4.5)Y + ($3.5)Z
Subject To:
1 Minimum requirement of nutrient A Total lbs of A of blending >= 6 1/16(4X + 2Y + 5Z >= 6
2 Minimum requirement of B Total lbs of B of blending >= 5 lbs 1/16(2X + 3Y + 2Z) >= 5
3 Minimum requirement of C Total lbs of C of blending >= 2 lbs 1/16(3X + 0Y + 2Z >= 2
4 Minimum requirement of D Total lbs of D of blending >= 10 1/16(6X + 8Y + 5Z >= 10
5 Availability of feed Z Availability <= 5 Z <= 5
6 Non-negative constraint X, Y, Z>= 0
Excel Model:
Optimal Solution:
X = 9.13 lbs
Y = 17.25 lbs
Z = 5.00 lbs
Total cost = $140.75 per day | {"url":"https://wizedu.com/questions/1394467/m1_p5-cattle-are-sent-to-a-feedlot-to-be-grain","timestamp":"2024-11-06T12:01:50Z","content_type":"text/html","content_length":"47867","record_id":"<urn:uuid:1ca6d300-0023-4295-b840-18fc0911fe09>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00501.warc.gz"} |
HDU 1690 multi-source shortest path Bus System
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Engineering Acoustics/Print version - Wikibooks, open books for an open world
Note: current version of this book can be found at http://en.wikibooks.org/wiki/Engineering_Acoustics
Remember to click "refresh" to view this version.
Part 1: Lumped Acoustical Systems
Simple Oscillation
The Position Equation
This section shows how to form the equation describing the position of a mass on a spring.
For a simple oscillator consisting of a mass m attached to one end of a spring with a spring constant s, the restoring force, f, can be expressed by the equation
${\displaystyle f=-sx\,}$
where x is the displacement of the mass from its rest position. Substituting the expression for f into the linear momentum equation,
${\displaystyle f=ma=m{d^{2}x \over dt^{2}}\,}$
where a is the acceleration of the mass, we can get
${\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-sx}$
${\displaystyle {\frac {d^{2}x}{dt^{2}}}+{\frac {s}{m}}x=0}$
Note that the frequency of oscillation ${\displaystyle \omega _{0}}$ is given by
${\displaystyle \omega _{0}^{2}={s \over m}\,}$
To solve the equation, we can assume
${\displaystyle x(t)=Ae^{\lambda t}\,}$
The force equation then becomes
${\displaystyle (\lambda ^{2}+\omega _{0}^{2})Ae^{\lambda t}=0,}$
Giving the equation
${\displaystyle \lambda ^{2}+\omega _{0}^{2}=0,}$
Solving for ${\displaystyle \lambda }$
${\displaystyle \lambda =\pm j\omega _{0}\,}$
This gives the equation of x to be
${\displaystyle x=C_{1}e^{j\omega _{0}t}+C_{2}e^{-j\omega _{0}t}\,}$
Note that
${\displaystyle j=(-1)^{1/2}\,}$
and that C[1] and C[2] are constants given by the initial conditions of the system
If the position of the mass at t = 0 is denoted as x[0], then
${\displaystyle C_{1}+C_{2}=x_{0}\,}$
and if the velocity of the mass at t = 0 is denoted as u[0], then
${\displaystyle -j(u_{0}/\omega _{0})=C_{1}-C_{2}\,}$
Solving the two boundary condition equations gives
${\displaystyle C_{1}={\frac {1}{2}}(x_{0}-j(u_{0}/\omega _{0}))}$
${\displaystyle C_{2}={\frac {1}{2}}(x_{0}+j(u_{0}/\omega _{0}))}$
The position is then given by
${\displaystyle x(t)=x_{0}cos(\omega _{0}t)+(u_{0}/\omega _{0})sin(\omega _{0}t)\,}$
This equation can also be found by assuming that x is of the form
${\displaystyle x(t)=A_{1}cos(\omega _{0}t)+A_{2}sin(\omega _{0}t)\,}$
And by applying the same initial conditions,
${\displaystyle A_{1}=x_{0}\,}$
${\displaystyle A_{2}={\frac {u_{0}}{\omega _{0}}}\,}$
This gives rise to the same position equation
${\displaystyle x(t)=x_{0}cos(\omega _{0}t)+(u_{0}/\omega _{0})sin(\omega _{0}t)\,}$
Alternate Position Equation Forms
If A[1] and A[2] are of the form
${\displaystyle A_{1}=Acos(\phi )\,}$
${\displaystyle A_{2}=Asin(\phi )\,}$
Then the position equation can be written
${\displaystyle x(t)=Acos(\omega _{0}t-\phi )\,}$
By applying the initial conditions (x(0)=x[0], u(0)=u[0]) it is found that
${\displaystyle x_{0}=Acos(\phi )\,}$
${\displaystyle {\frac {u_{0}}{\omega _{0}}}=Asin(\phi )\,}$
If these two equations are squared and summed, then it is found that
${\displaystyle A={\sqrt {x_{0}^{2}+({\frac {u_{0}}{\omega _{0}}})^{2}}}\,}$
And if the difference of the same two equations is found, the result is that
${\displaystyle \phi =tan^{-1}({\frac {u_{0}}{x_{0}\omega _{0}}})\,}$
The position equation can also be written as the Real part of the imaginary position equation
${\displaystyle \mathbf {Re} [x(t)]=x(t)=Acos(\omega _{0}t-\phi )\,}$
Due to euler's rule (e^jφ = cosφ + jsinφ), x(t) is of the form
${\displaystyle x(t)=Ae^{j(\omega _{0}t-\phi )}\,}$
Example 1.1
GIVEN: Two springs of stiffness, ${\displaystyle s}$, and two bodies of mass, ${\displaystyle M}$
FIND: The natural frequencies of the systems sketched below
${\displaystyle s_{TOTAL}=s+s{\text{ (springs are in parallel)}}}$
${\displaystyle \omega _{0}={\sqrt {\frac {s_{TOTAL}}{m_{TOTAL}}}}={\sqrt {\frac {2s}{M}}}}$
${\displaystyle \mathbf {f_{0}} ={\frac {\omega _{0}}{2\pi }}=\mathbf {{\frac {1}{2\pi }}{\sqrt {\frac {2s}{M}}}} }$
${\displaystyle \omega _{0}={\sqrt {\frac {s_{TOTAL}}{m_{TOTAL}}}}={\sqrt {\frac {s}{2M}}}}$
${\displaystyle \mathbf {f_{0}} ={\frac {\omega _{0}}{2\pi }}=\mathbf {{\frac {1}{2\pi }}{\sqrt {\frac {s}{2M}}}} }$
${\displaystyle \mathbf {1.} {\text{ }}s(x_{1}-x_{2})=sx_{2}}$
${\displaystyle \mathbf {2.} {\text{ }}-s(x_{1}-x_{2})=m{\frac {d^{2}x}{dt^{2}}}}$
${\displaystyle {\frac {d^{2}x_{1}}{dt^{2}}}+{\frac {s}{2m}}x_{1}=0}$
${\displaystyle \omega _{0}={\sqrt {\frac {s}{2m}}}}$
${\displaystyle \mathbf {f_{0}={\frac {1}{2\pi }}{\sqrt {\frac {s}{2m}}}} }$
${\displaystyle \omega _{0}={\sqrt {\frac {2s}{m}}}}$
${\displaystyle \mathbf {f_{0}={\frac {1}{2\pi }}{\sqrt {\frac {2s}{m}}}} }$
Forced Oscillations(Simple Spring-Mass System)
Recap of Section 1.3
In the previous section, we discussed how adding a damping component (e. g. a dashpot) to an unforced, simple spring-mass system would affect the response of the system. In particular, we learned
that adding the dashpot to the system changed the natural frequency of the system from to a new damped natural frequency , and how this change made the response of the system change from a constant
sinusoidal response to an exponentially-decaying sinusoid in which the system either had an under-damped, over-damped, or critically-damped response.
In this section, we will digress a bit by going back to the simple (undamped) oscillator system of the previous section, but this time, a constant force will be applied to this system, and we will
investigate this system’s performance at low and high frequencies as well as at resonance. In particular, this section will start by introducing the characteristics of the spring and mass elements of
a spring-mass system, introduce electrical analogs for both the spring and mass elements, learn how these elements combine to form the mechanical impedance system, and reveal how the impedance can
describe the mechanical system’s overall response characteristics. Next, power dissipation of the forced, simple spring-mass system will be discussed in order to corroborate our use of electrical
circuit analogs for the forced, simple spring-mass system. Finally, the characteristic responses of this system will be discussed, and a parameter called the amplification ratio (AR) will be
introduced that will help in plotting the resonance of the forced, simple spring-mass system.
Forced Spring Element
Taking note of Figs. 1, we see that the equation of motion for a spring that has some constant, external force being exerted on it is...
${\displaystyle {\hat {F}}=s_{M}\Delta {\hat {x}}\qquad (1.4.1)\,}$
where ${\displaystyle s_{M}\,}$ is the mechanical stiffness of the spring.
Note that in Fig. 1(c), force ${\displaystyle {\hat {F}}}$ flows constantly (i.e. without decreasing) throughout a spring, but the velocity ${\displaystyle {\hat {u}}}$ of the spring decrease from $
{\displaystyle {\hat {u_{1}}}}$ to ${\displaystyle {\hat {u_{2}}}}$ as the force flows through the spring. This concept is important to know because it will be used in subsequent sections.
In practice, the stiffness of the spring ${\displaystyle s_{M}\,}$, also called the spring constant, is usually expressed as ${\displaystyle C_{M}={\frac {1}{s_{M}}}\,}$ , or the mechanical
compliance of the spring. Therefore, the spring is very stiff if ${\displaystyle s_{M}\,}$ is large ${\displaystyle \Rightarrow \;C_{M}}$ is small. Similarly, the spring is very loose or “bouncy” if
${\displaystyle s_{M}\,}$ is small ${\displaystyle \Rightarrow \;C_{M}}$ is large. Noting that force and velocity are analogous to voltage and current, respectively, in electrical systems, it turns
out that the characteristics of a spring are analogous to the characteristics of a capacitor in relation to, and, so we can model the “reactiveness” of a spring similar to the reactance of a
capacitor if we let ${\displaystyle C=C_{M}\,}$ as shown in Fig. 2 below.
${\displaystyle Reactance\ of\ Capacitor:\ X_{C}={\frac {1}{j\omega C}}\qquad (1.4.2a)}$
${\displaystyle Reactance\ of\ Spring:\ X_{MS}={\frac {1}{j\omega C_{M}}}\qquad (1.4.2b)}$
Forced Mass Element
Taking note of Fig. 3, the equation for a mass that has constant, external force being exerted on it is...
${\displaystyle {\hat {F}}=M_{M}{\hat {a}}=M_{M}{\hat {\dot {u}}}=M_{M}{\hat {\ddot {x}}}\qquad (1.4.3)}$
If the mass ${\displaystyle M_{M}\,}$ can vary its value and is oscillating in a mechanical system at max amplitude ${\displaystyle A_{M}\,}$ such that the input the system receives is constant at
frequency ${\displaystyle \omega \,}$, as ${\displaystyle M_{M}\,}$ increases, the harder it will be for the system to move the mass at ${\displaystyle \omega \,}$ at ${\displaystyle A_{M}\,}$ until,
eventually, the mass doesn’t oscillate at all . Another equivalently way to look at it is to let ${\displaystyle \omega \,}$ vary and hold ${\displaystyle M_{M}\,}$ constant. Similarly, as ${\
displaystyle \omega \,}$ increases, the harder it will be to get ${\displaystyle M_{M}\,}$ to oscillate at ${\displaystyle \omega \,}$ and keep the same amplitude ${\displaystyle A_{M}\,}$ until,
eventually, the mass doesn’t oscillate at all. Therefore, as ${\displaystyle \omega \,}$ increases, the “reactiveness” of mass ${\displaystyle M_{M}\,}$ decreases (i.e. ${\displaystyle M_{M}\,}$
starts to move less and less). Recalling the analogous relationship of force/voltage and velocity/current, it turns out that the characteristics of a mass are analogous to an inductor. Therefore, we
can model the “reactiveness” of a mass similar to the reactance of an inductor if we let ${\displaystyle L=M_{M}\,}$ as shown in Fig. 4.
${\displaystyle Reactance\ of\ Inductor:\ X_{L}=j\omega L\qquad (1.4.4a)}$
${\displaystyle Reactance\ of\ Mass:\ X_{MM}=j\omega L_{M}\qquad (1.4.4b)}$
Mechanical Impedance of Spring-Mass System
As mentioned twice before, force is analogous to voltage and velocity is analogous to current. Because of these relationships, this implies that the mechanical impedance for the forced, simple
spring-mass system can be expressed as follows:
${\displaystyle {\hat {Z_{M}}}={\frac {\hat {F}}{\hat {u}}}\qquad (1.4.5)\,}$
In general, an undamped, spring-mass system can either be “spring-like” or “mass-like”. “Spring-like” systems can be characterized as being “bouncy” and they tend to grossly overshoot their target
operating level(s) when an input is introduced to the system. These type of systems relatively take a long time to reach steady-state status. Conversely, “mass-like” can be characterized as being
“lethargic” and they tend to not reach their desired operating level(s) for a given input to the system...even at steady-state! In terms of complex force and velocity, we say that “ force LEADS
velocity” in mass-like systems and “velocity LEADS force” in spring-like systems (or equivalently “ force LAGS velocity” in mass-like systems and “velocity LAGS force” in spring-like systems). Figs.
5 shows this relationship graphically.
Power Transfer of a Simple Spring-Mass System
From electrical circuit theory, the average complex power ${\displaystyle P_{E}\,}$ dissipated in a system is expressed as ...
${\displaystyle P_{E}={\frac {1}{2}}\mathbf {Re} \left\{{\hat {V}}{\hat {I^{*}}}\right\}\qquad (1.4.6)\,}$
where ${\displaystyle {\hat {V}}\,}$ and ${\displaystyle {\hat {I^{*}}}\;}$ represent the (time-invariant) complex voltage and complex conjugate current, respectively. Analogously, we can express the
net power dissipation of the mechanical system ${\displaystyle {\hat {P}}_{E}\,}$ in general along with the power dissipation of a spring-like system ${\displaystyle {\hat {P}}_{MS}\,}$ or mass-like
system ${\displaystyle {\hat {P}}_{MM}\,}$ as...
${\displaystyle {\hat {P}}_{E}={\frac {1}{2}}\mathbf {Re} \left\{{\hat {F}}{\hat {u^{*}}}\right\}\qquad \qquad \qquad (1.4.7a)\,}$
${\displaystyle {\hat {P}}_{MS}={\frac {1}{2}}\mathbf {Re} \left\{{\hat {F}}\left({\frac {j{\hat {F}}\omega }{s_{M}}}\right)^{*}\right\}\qquad (1.4.7b)}$
${\displaystyle {\hat {P}}_{MM}={\frac {1}{2}}\mathbf {Re} \left\{{\hat {F}}\left({\frac {\hat {F}}{j\omega M_{M}}}\right)^{*}\right\}\qquad (1.4.7c)}$
In equations 1.4.7, we see that the product of complex force and velocity are purely imaginary. Since reactive elements, or commonly called, lossless elements, cannot dissipate energy, this implies
that the net power dissipation of the system is zero. This means that in our simple spring-mass system, power can only be (fully) transferred back and forth between the spring and the mass. But this
is precisely what a simple spring-mass system does. Therefore, by evaluating the power dissipation, this corroborates the notion of using electrical circuit elements to model mechanical elements in
our spring-mass system.
Responses For Forced, Simple Spring-Mass System
Fig. 6 below illustrates a simple spring-mass system with a force exerted on the mass.
This system has response characteristics similar to that of the undamped oscillator system, with the only difference being that at steady-state, the system oscillates at the constant force magnitude
and frequency versus exponentially decaying to zero in the unforced case. Recalling equations 1.4.2b and 1.4.4b, letting be the natural (resonant) frequency of the spring-mass system, and letting ${\
displaystyle \omega _{n}\,}$ be frequency of the input received by the system, the characteristic responses of the forced spring-mass systems are presented graphically in Figs. 7 below.
${\displaystyle \mathbf {Figs.7} \,}$
Amplification Ratio
The amplification ratio is a useful parameter that allows us to plot the frequency of the spring-mass system with the purports of revealing the resonant freq of the system solely based on the force
experienced by each, the spring and mass elements of the system. In particular, AR is the magnitude of the ratio of the complex force experienced by the spring and the complex force experienced by
the mass, i.e.
${\displaystyle \mathbf {AR} =\left|{\frac {s_{M}{\hat {x}}}{M_{M}{\hat {a}}}}\right|=\left|{\frac {s_{M}{\hat {x}}}{M_{M}{\hat {\dot {u}}}}}\right|=\left|{\frac {s_{M}{\hat {x}}}{M_{M}{\hat {\ddot
{x}}}}}\right|\qquad (1.4.8)\,}$
If we let ${\displaystyle \zeta ={\frac {\omega }{\omega _{n}}}}$, be the frequency ratio, it turns out that AR can also be expressed as...
${\displaystyle \mathbf {AR} ={\frac {1}{1-\zeta ^{2}}}\qquad (1.4.9)\,}$
AR will be at its maximum when ${\displaystyle \left|X_{MS}\right|=\left|X_{MM}\right|\,}$. This happens precisely when ${\displaystyle \zeta ^{2}=1\,}$ . An example of an AR plot is shown below in
Fig 8.
Mechanical Resistance
Mechanical Resistance
For most systems, a simple oscillator is not a very accurate model. While a simple oscillator involves a continuous transfer of energy between kinetic and potential form, with the sum of the two
remaining constant, real systems involve a loss, or dissipation, of some of this energy, which is never recovered into kinetic nor potential energy. The mechanisms that cause this dissipation are
varied and depend on many factors. Some of these mechanisms include drag on bodies moving through the air, thermal losses, and friction, but there are many others. Often, these mechanisms are either
difficult or impossible to model, and most are non-linear. However, a simple, linear model that attempts to account for all of these losses in a system has been developed.
The most common way of representing mechanical resistance in a damped system is through the use of a dashpot. A dashpot acts like a shock absorber in a car. It produces resistance to the system's
motion that is proportional to the system's velocity. The faster the motion of the system, the more mechanical resistance is produced.
As seen in the graph above, a linear relationship is assumed between the force of the dashpot and the velocity at which it is moving. The constant that relates these two quantities is ${\displaystyle
R_{M}}$, the mechanical resistance of the dashpot. This relationship, known as the viscous damping law, can be written as:
${\displaystyle F=R\cdot u}$
Also note that the force produced by the dashpot is always in phase with the velocity.
The power dissipated by the dashpot can be derived by looking at the work done as the dashpot resists the motion of the system:
${\displaystyle P_{D}={\frac {1}{2}}\Re \left[{\hat {F}}\cdot {\hat {u^{*}}}\right]={\frac {|{\hat {F}}|^{2}}{2R_{M}}}}$
Modeling the Damped Oscillator
In order to incorporate the mechanical resistance (or damping) into the forced oscillator model, a dashpot is placed next to the spring. It is connected to the mass (${\displaystyle M_{M}}$) on one
end and attached to the ground on the other end. A new equation describing the forces must be developed:
${\displaystyle F-S_{M}x-R_{M}u=M_{M}a\rightarrow F=S_{M}x+R_{M}{\dot {x}}+M_{M}{\ddot {x}}}$
It's phasor form is given by the following:
${\displaystyle {\hat {F}}e^{j\omega t}={\hat {x}}e^{j\omega t}\left[S_{M}+j\omega R_{M}+\left(-\omega ^{2}\right)M_{M}\right]}$
Mechanical Impedance for Damped Oscillator
Previously, the impedance for a simple oscillator was defined as ${\displaystyle \mathbf {\frac {F}{u}} }$. Using the above equations, the impedance of a damped oscillator can be calculated:
${\displaystyle {\hat {Z_{M}}}={\frac {\hat {F}}{\hat {u}}}=R_{M}+j\left(\omega M_{M}-{\frac {S_{M}}{\omega }}\right)=|{\hat {Z_{M}}}|e^{j\Phi _{Z}}}$
For very low frequencies, the spring term dominates because of the ${\displaystyle {\frac {1}{\omega }}}$ relationship. Thus, the phase of the impedance approaches ${\displaystyle {\frac {-\pi }{2}}}
$ for very low frequencies. This phase causes the velocity to "lag" the force for low frequencies. As the frequency increases, the phase difference increases toward zero. At resonance, the imaginary
part of the impedance vanishes, and the phase is zero. The impedance is purely resistive at this point. For very high frequencies, the mass term dominates. Thus, the phase of the impedance approaches
${\displaystyle {\frac {\pi }{2}}}$ and the velocity "leads" the force for high frequencies.
Based on the previous equations for dissipated power, we can see that the real part of the impedance is indeed ${\displaystyle R_{M}}$. The real part of the impedance can also be defined as the
cosine of the phase times its magnitude. Thus, the following equations for the power can be obtained.
${\displaystyle W_{R}={\frac {1}{2}}\Re \left[{\hat {F}}{\hat {u^{*}}}\right]={\frac {1}{2}}R_{M}|{\hat {u}}|^{2}={\frac {1}{2}}{\frac {|{\hat {F}}|^{2}}{|{\hat {Z_{M}}}|^{2}}}R_{M}={\frac {1}{2}}{\
frac {|{\hat {F}}|^{2}}{|{\hat {Z_{M}}}|}}cos(\Phi _{Z})}$
Characterizing Damped Mechanical Systems
Characterizing Damped Mechanical Systems
Characterizing the response of Damped Mechanical Oscillating system can be easily quantified using two parameters. The system parameters are the resonance frequency (${\displaystyle '''wresonance'''}
$ and the damping of the system ${\displaystyle '''Q(qualityfactor)orB(TemporalAbsorption''')}$. In practice, finding these parameters would allow for quantification of unknown systems and allow you
to derive other parameters within the system.
Using the mechanical impedance in the following equation, notice that the imaginary part will equal zero at resonance.
(${\displaystyle Z_{m}=F/u=R_{m}+j(w*M_{m}-s/w)}$)
Resonance case:(${\displaystyle w*M_{m}=s/w}$)
Calculating the Mechanical Resistance
The decay time of the system is related to 1 / B where B is the Temporal Absorption. B is related to the mechancial resistance and to the mass of the system by the following equation.
${\displaystyle B=Rm/2*Mm}$
The mechanical resistance can be derived from the equation by knowing the mass and the temporal absorption.
Critical Damping
The system is said to be critically damped when:
${\displaystyle Rc=2*M*sqrt(s/Mm)=2*sqrt(s*Mm)=2*Mm*wn}$
A critically damped system is one in which an entire cycle is never completed. The absorption coefficient in this type of system equals the natural frequency. The system will begin to oscillate,
however the amplitude will decay exponentially to zero within the first oscillation.
Damping Ratio
${\displaystyle DampingRatio=Rm/Rc}$
The damping ratio is a comparison of the mechanical resistance of a system to the resistance value required for critical damping. Rc is the value of Rm for which the absorption coefficient equals the
natural frequency (critical damping). A damping ratio equal to 1 therefore is critically damped, because the mechanical resistance value Rm is equal to the value required for critical damping Rc. A
damping ratio greater than 1 will be overdamped, and a ratio less than 1 will be underdamped.
Quality Factor
The Quality Factor (Q) is way to quickly characterize the shape of the peak in the response. It gives a quantitative representation of power dissipation in an oscillation.
${\displaystyle Q=wresonance/(wu-wl)}$
Wu and Wl are called the half power points. When looking at the response of a system, the two places on either side of the peak where the point equals half the power of the peak power defines Wu and
Wl. The distance in between the two is called the half-power bandwidth. So, the resonant frequency divided by the half-power bandwidth gives you the quality factor. Mathematically, it takes Q/pi
oscillations for the vibration to decay to a factor of 1/e of its original amplitude.
Electro-Mechanical Analogies
Why Circuit Analogs?
Acoustic devices are often combinations of mechanical and electrical elements. A common example of this would be a loudspeaker connected to a power source. It is useful in engineering applications to
model the entire system with one method. This is the reason for using a circuit analogy in a vibrating mechanical system. The same analytic method can be applied to Electro-Acoustic Analogies.
How Electro-Mechanical Analogies Work
An electrical circuit is described in terms of its potential (voltage) and flux (current). To construct a circuit analog of a mechanical system we define flux and potential for the system. This leads
to two separate analog systems. The Impedance Analog denotes the force acting on an element as the potential and the velocity of the element as the flux. The Mobility Analog equates flux with the
force and velocity with potential.
Mechanical Electrical Equivalent
Impedance Analog
Potential: Force Voltage
Flux: Velocity Current
Mobility Analog
Potential: Velocity Voltage
Flux: Force Current
For many, the mobility analog is considered easier for a mechanical system. It is more intuitive for force to flow as a current and for objects oscillating the same frequency to be wired in parallel.
However, either method will yield equivalent results and can also be translated using the dual (dot) method.
The Basic Elements of an Oscillating Mechanical System
The Mechanical Spring:
The ideal spring is considered to be operating within its elastic limit, so the behavior can be modeled with Hooke's Law. It is also assumed to be massless and have no damping effects.
${\displaystyle F=-cx,\ }$
The Mechanical Mass
In a vibrating system, a mass element opposes acceleration. From Newton's Second Law:
${\displaystyle F=mx^{\prime \prime }=ma=m{\frac {du}{dt}}}$
${\displaystyle F=K\int \,udt}$
The Mechanical Resistance
The dashpot is an ideal viscous damper which opposes velocity.
${\displaystyle F=Ru\displaystyle }$
Ideal Generators
The two ideal generators which can drive any system are an ideal velocity and ideal force generator. The ideal velocity generator can be denoted by a drawing of a crank or simply by declaring ${\
displaystyle u(t)=f(t)}$, and the ideal force generator can be drawn with an arrow or by declaring ${\displaystyle F(t)=f(t)}$
Simple Damped Mechanical Oscillators
In the following sections we will consider this simple mechanical system as a mobility and impedance analog. It can be driven either by an ideal force or an ideal velocity generator, and we will
consider simple harmonic motion. The m in the subscript denotes a mechanical system, which is currently redundant, but can be useful when combining mechanical and acoustic systems.
The Impedance Analog
The Mechanical Spring
In a spring, force is related to the displacement from equilibrium. By Hooke's Law,
${\displaystyle F(t)=c_{m}\Delta x=c_{m}\int _{0}^{t}u(\tau )d\tau }$
The equivalent behaviour in a circuit is a capacitor:
${\displaystyle V(t)={\frac {1}{C}}\int _{0}^{t}\,i(\tau )d\tau }$
The Mechanical Mass
The force on a mass is related to the acceleration (change in velocity). The behaviour, by Newton's Second Law, is:
${\displaystyle F(t)=m_{m}a=m_{m}{\frac {d}{dt}}u(t)}$
The equivalent behaviour in a circuit is an inductor:
${\displaystyle V(t)=L{\frac {d}{dt}}i(t)}$
The Mechanical Resistance
For a viscous damper, the force is directly related to the velocity
${\displaystyle F=R_{m}u\displaystyle }$
The equivalent is a simple resistor of value ${\displaystyle R_{m}\displaystyle }$
${\displaystyle V=Ri\displaystyle }$
Thus the simple mechanical oscillator in the previous section becomes a series RCL Circuit:
The current through all three elements is equal (they are at the same velocity) and that the sum of the potential drops across each element will equal the potential at the generator (the driving
force). The ideal voltage generator depicted here would be equivalent to an ideal force generator.
IMPORTANT NOTE: The velocity measured for the spring and dashpot is the relative velocity ( velocity of one end minus the velocity of the other end). The velocity of the mass, however, is the
absolute velocity.
Element Impedance
Spring Capacitor ${\displaystyle Z_{c}={\frac {V_{c}}{I_{c}}}={\frac {c_{m}}{j\omega }}}$
Mass Inductor ${\displaystyle Z_{m}={\frac {V_{m}}{I_{m}}}=j\omega m_{m}}$
Dashpot Resistor ${\displaystyle Z_{d}={\frac {V_{m}}{I_{m}}}=R_{m}}$
The Mobility Analog
Like the Impedance Analog above, the equivalent elements can be found by comparing their fundamental equations with the equations of circuit elements. However, since circuit equations usually define
voltage in terms of current, in this case the analogy would be an expression of velocity in terms of force, which is the opposite of convention. However, this can be solved with simple algebraic
The Mechanical Spring
${\displaystyle F(t)=c_{m}\int u(t)dt}$
The equivalent behavior for this circuit is the behavior of an inductor.
${\displaystyle \int Vdt=\int L{\frac {d}{dt}}i(t)dt}$
${\displaystyle i={\frac {1}{L}}\int \,Vdt}$
The Mechanical Mass
${\displaystyle F=m_{m}a=m_{m}{\frac {d}{dt}}u(t)}$
Similar to the spring element, if we take the general equation for a capacitor and differentiate,
${\displaystyle {\frac {d}{dt}}V(t)={\frac {d}{dt}}{\frac {1}{C}}\int \,i(t)dt}$
${\displaystyle i(t)=C{\frac {d}{dt}}V(t)}$
The Mechanical Resistance
Since the relation between force and velocity is proportionate, the only difference is that the mechanical resistance becomes inverted:
${\displaystyle F={\frac {1}{r_{m}}}u=R_{m}u}$
${\displaystyle i={\frac {1}{R}}V}$
The simple mechanical oscillator drawn above would become a parallel RLC Circuit. The potential across each element is the same because they are each operating at the same velocity. This is often the
more intuitive of the two analogy methods to use, because you can visualize force "flowing" like a flux through your system. The ideal voltage generator in this drawing would correspond to an ideal
velocity generator.
IMPORTANT NOTE: Since the measure of the velocity of a mass is absolute, a capacitor in this analogy must always have one terminal grounded. A capacitor with both terminals at a potential other than
ground may be realized physically as an inverter, which completes all elements of this analogy.
Element Impedance
Spring Inductor ${\displaystyle Z_{c}={\frac {V_{m}}{I_{m}}}={\frac {j\omega }{c_{m}}}}$
Mass Capacitor ${\displaystyle Z_{m}={\frac {V_{c}}{I_{c}}}={\frac {1}{j\omega m_{m}}}}$
Dashpot Resistor ${\displaystyle Z_{d}={\frac {V_{m}}{I_{m}}}=r_{m}={\frac {1}{R_{m}}}}$
Methods for checking Electro-Mechanical Analogies
After drawing the electro-mechanical analogy of a mechanical system, it is always safe to check the circuit. There are two methods to accomplish this:
Review of Circuit Solving Methods
Kirchkoff's Voltage law
"The sum of the potential drops around a loop must equal zero."
${\displaystyle v_{1}+v_{2}+v_{3}+v_{4}=0\displaystyle }$
Kirchkoff's Current Law
"The Sum of the currents at a node (junction of more than two elements) must be zero"
${\displaystyle -i_{1}+i_{2}+i_{3}-i_{4}=0\displaystyle }$
Hints for solving circuits:
Remember that certain elements can be combined to simplify the circuit (the combination of like elements in series and parallel)
If solving a circuit that involves steady-state sources, use impedances. Any circuit can eventually be combined into a single impedance using the following identities:
Impedances in series: ${\displaystyle Z_{\mathrm {eq} }=Z_{1}+Z_{2}+\,\cdots \,+Z_{n}.}$
Impedances in parallel: ${\displaystyle {\frac {1}{Z_{\mathrm {eq} }}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\,\cdots \,+{\frac {1}{Z_{n}}}.}$
Dot Method: (Valid only for planar network)
This method helps obtain the dual analog (one analog is the dual of the other). The steps for the dot product are as follows:
1) Place one dot within each loop and one outside all the loops.
2) Connect the dots. Make sure that there is only one line through each element and that no lines cross more than one element.
3) Draw in each line that crosses an element its dual element, including the source.
4) The circuit obtained should have an equivalent behavior as the dual analog of the original electro-mechanical circuit.
The parallel RLC Circuit above is equivalent to a series RLC driven by an ideal current source
Low-Frequency Limits
This method looks at the behavior of the system for very large or very small values of the parameters and compares them with the expected behavior of the mechanical system. For instance, you can
compare the mobility circuit behavior of a near-infinite inductance with the mechanical system behavior of a near-infinite stiffness spring.
Very High Value Very Low Value
Capacitor Short Circuit Open Circuit
Inductor Open Circuit Closed Circuit
Resistor Open Circuit Short Circuit
Additional Resources for solving linear circuits
Thomas & Rosa, "The Analysis and Design of Linear Circuits", Wiley, 2001
Hayt, Kemmerly & Durbin, "Engineering Circuit Analysis", 6th ed., McGraw Hill, 2002
Examples of Electro-Mechanical Analogies
Example of Electro-Mechanical Analogies
Note: The crank indicates an ideal velocity generator, with an amplitude of ${\displaystyle u_{0}}$ rotating at ${\displaystyle \omega }$ rad/s.
Impedance Analog Solution
Mobility Analog Solution
Primary variables of interest
Basic Assumptions
Consider a piston moving in a tube. The piston starts moving at time t=0 with a velocity u=${\displaystyle u_{p}}$. The piston fits inside the tube smoothly without any friction or gap. The motion of
the piston creates a planar sound wave or acoustic disturbance traveling down the tube at a constant speed c>>${\displaystyle u_{p}}$. In a case where the tube is very small, one can neglect the time
it takes for acoustic disturbance to travel from the piston to the end of the tube. Hence, one can assume that the acoustic disturbance is uniform throughout the tube domain.
1. Although sound can exist in solids or fluid, we will first consider the medium to be a fluid at rest. The ambient, undisturbed state of the fluid will be designated using subscript zero. Recall
that a fluid is a substance that deforms continuously under the application of any shear (tangential) stress.
2. Disturbance is a compressional one (as opposed to transverse).
3. Fluid is a continuum: infinitely divisible substance. Each fluid property assumed to have definite value at each point.
4. The disturbance created by the motion of the piston travels at a constant speed. It is a function of the properties of the ambient fluid. Since the properties are assumed to be uniform (the same
at every location in the tube) then the speed of the disturbance has to be constant. The speed of the disturbance is the speed of sound, denoted by letter ${\displaystyle c_{0}}$ with subscript zero
to denote ambient property.
5. The piston is perfectly flat, and there is no leakage flow between the piston and the tube inner wall. Both the piston and the tube walls are perfectly rigid. Tube is infinitely long, and has a
constant area of cross section, A.
6. The disturbance is uniform. All deviations in fluid properties are the same across the tube for any location x. Therefore the instantaneous fluid properties are only a function of the Cartesian
coordinate x (see sketch). Deviations from the ambient will be denoted by primed variables.
Variables of interest
Pressure (force / unit area)
Pressure is defined as the normal force per unit area acting on any control surface within the fluid.
${\displaystyle p={\frac {{\tilde {F}}.{\tilde {n}}}{dS}}}$
For the present case,inside a tube filled with a working fluid, pressure is the ratio of the surface force acting onto the fluid in the control region and the tube area. The pressure is decomposed
into two components - a constant equilibrium component, ${\displaystyle p_{0}}$, superimposed with a varying disturbance ${\displaystyle p^{'}(x)}$. The deviation ${\displaystyle p^{'}}$is also
called the acoustic pressure. Note that ${\displaystyle p^{'}}$ can be positive or negative. Unit: ${\displaystyle kg/ms^{2}}$. Acoustical pressure can be measured using a microphone.
Density is mass of fluid per unit volume. The density, ρ, is also decomposed into the sum of ambient value (usually around ρ0= 1.15 kg/m3) and a disturbance ρ’(x). The disturbance can be positive or
negative, as for the pressure. Unit: ${\displaystyle kg/m^{3}}$
Acoustic volume velocity
Rate of change of fluid particles position as a function of time. Its the well known fluid mechanics term, flow rate.
${\displaystyle U=\int _{s}{\tilde {u}}.{\tilde {n}}\,dS}$
In most cases, the velocity is assumed constant over the entire cross section (plug flow), which gives acoustic volume velocity as a product of fluid velocity ${\displaystyle {\tilde {u}}}$ and cross
section S.
${\displaystyle U={\tilde {u}}.S}$
Electro-acoustic analogies
Electro-acoustical Analogies
Acoustical Mass
Consider a rigid tube-piston system as following figure.
Piston is moving back and forth sinusoidally with frequency of f. Assuming ${\displaystyle f<<{\frac {c}{l\ or\ {\sqrt {S}}}}}$ (where c is sound velocity ${\displaystyle c={\sqrt {\gamma RT_{0}}}}$
), volume of fluid in tube is,
${\displaystyle \Pi _{v}=S\ l}$
Then mass (mechanical mass) of fluid in tube is given as,
${\displaystyle M_{M}=\Pi _{v}\rho _{0}=\rho _{0}S\ l}$
For sinusoidal motion of piston, fluid move as rigid body at same velocity as piston. Namely, every point in tube moves with the same velocity.
Applying the Newton's second law to the following free body diagram,
${\displaystyle SP'=(\rho _{0}Sl){\frac {du}{dt}}}$
${\displaystyle {\hat {P}}=\rho _{0}l(j\omega ){\hat {u}}=j\omega ({\frac {\rho _{0}l}{S}}){\hat {U}}}$
Where, plug flow assumption is used.
"Plug flow" assumption:
Frequently in acoustics, the velocity distribution along the normal surface of
fluid flow is assumed uniform. Under this assumption, the acoustic volume velocity U is
simply product of velocity and entire surface. ${\displaystyle U=Su}$
Acoustical Impedance
Recalling mechanical impedance,
${\displaystyle {\hat {Z}}_{M}={\frac {\hat {F}}{\hat {u}}}=j\omega (\rho _{0}Sl)}$
acoustical impedance (often termed an acoustic ohm) is defined as,
${\displaystyle {\hat {Z}}_{A}={\frac {\hat {P}}{\hat {U}}}={\frac {Z_{M}}{S^{2}}}=j\omega ({\frac {\rho _{0}l}{S}})\quad \left[{\frac {Ns}{m^{5}}}\right]}$
where, acoustical mass is defined.
${\displaystyle M_{A}={\frac {\rho _{0}l}{S}}}$
Acoustical Mobility
Acoustical mobility is defined as,
${\displaystyle {\hat {\xi }}_{A}={\frac {1}{{\hat {Z}}_{A}}}={\frac {\hat {U}}{\hat {P}}}}$
Impedance Analog vs. Mobility Analog
Acoustical Resistance
Acoustical resistance models loss due to viscous effects (friction) and flow resistance (represented by a screen).
File:Ra analogs.png r[A] is the reciprocal of R[A] and is referred to as responsiveness.
Acoustical Generators
The acoustical generator components are pressure, P and volume velocity, U, which are analogus to force, F and velocity, u of electro-mechanical analogy respectively. Namely, for impedance analog,
pressure is analogous to voltage and volume velocity is analogus to current, and vice versa for mobility analog. These are arranged in the following table.
Impedance and Mobility analogs for acoustical generators of constant pressure and constant volume velocity are as follows:
File:Acoustic gen.png
Acoustical Compliance
Consider a piston in an enclosure.
File:Enclosed Piston.png
When the piston moves, it displaces the fluid inside the enclosure. Acoustic compliance is the measurement of how "easy" it is to displace the fluid.
Here the volume of the enclosure should be assumed to be small enough that the fluid pressure remains uniform.
Assume no heat exchange 1.adiabatic 2.gas compressed uniformly , p prime in cavity everywhere the same.
from thermo equitation File:Equ1.jpg it is easy to get the relation between disturbing pressure and displacement of the piston File:Equ3.gif where U is volume rate, P is pressure according to the
definition of the impendance and mobility, we can getFile:Equ4.gif
Mobility Analog VS Impedance Analog
Examples of Electro-Acoustical Analogies
Example 1: Helmholtz Resonator
Assumptions - (1) Completely sealed cavity with no leaks. (2) Cavity acts like a rigid body inducing no vibrations.
- Impedance Analog -
Example 2: Combination of Side-Branch Cavities
- Impedance Analog -
Transducers - Loudspeaker
Acoustic transducer
The purpose of an acoustic transducer is to convert electrical energy into acoustic energy. Many variations of acoustic transducers exist, such as electrostatic, balanced armature and moving-coil
loudspeakers. This article focuses on moving-coil loudspeakers since they are the most commonly used type of acoustic transducer. First, the physical construction and principle of a typical moving
coil transducer are discussed briefly. Second, electro-mechano-acoustical modeling of each element composing the loudspeaker is presented in a tutorial way to reinforce and supplement the theory on
electro-mechanical analogies and electro-acoustic analogies previously seen in other sections. Third, the equivalent circuit is analyzed to introduce the theory behind Thiele-Small parameters, which
are very useful when designing loudspeaker enclosures. A method to experimentally determine Thiele-Small parameters is also included.
Moving-coil loudspeaker construction and principle
The classic moving-coil loudspeaker driver can be divided into three key components:
1) The magnet motor drive system, comprising the permanent magnet, the center pole and the voice coil acting together to produce a mechanical force on the diaphragm from an electrical current.
2) The loudspeaker cone system, comprising the diaphragm and dust cap, permitting mechanical force to be translated into acoustic pressure;
3) The loudspeaker suspension, comprising the spider and surround, preventing the diaphragm from breaking due to over excursion, allowing only translational movement and tending to bring the
diaphragm back to its rest position.
The following illustration shows a cut-away view of a typical moving coil-permanent magnet loudspeaker. A coil is mechanically coupled to a diaphragm, also called cone, and rests in a fixed magnetic
field produced by a magnet. When an electrical current flows through the coil, a corresponding magnetic field is emitted, interacting with the fixed field of the magnet and thus applying a force to
the coil, pushing it away or towards the magnet. Since the cone is mechanically coupled to the coil, it will push or pull the air it is facing, causing pressure changes and emitting a sound wave.
Figure 1: A cross-sectional view of a typical moving-coil loudspeaker
An equivalent circuit can be obtained to model the loudspeaker as a lumped system. This circuit can be used to drive the design of a complete loudspeaker system, including an enclosure and sometimes
even an amplifier that is matched to the properties of the driver. The following section shows how such an equivalent circuit can be obtained.
Electro-mechano-acoustical equivalent circuit
Electro-mechanico-acoustical systems such as loudspeakers can be modeled as equivalent electrical circuits as long as each element moves as a whole. This is usually the case at low frequencies or at
frequencies where the dimensions of the system are small compared to the wavelength of interest. To obtain a complete model of the loudspeaker, the interactions and properties of electrical,
mechanical, and acoustical subsystems composing the loudspeaker driver must each be modeled. The following sections detail how the circuit may be obtained starting with the amplifier and ending with
the acoustical load presented by air. A similar development can be found in [1] or [2].
Electrical subsystem
The electrical part of the system is composed of a driving amplifier and a voice coil. Most amplifiers can be approximated as a perfect voltage source in series with the amplifier output impedance.
The voice coil exhibits an inductance and a resistance that may be directly modeled as a circuit.
Figure 2: The amplifier and loudspeaker electrical elements modeled as a circuit
Electrical to mechanical subsystem
When the loudspeaker is fed an electrical signal, the voice coil and magnet convert current to force. Similarly, voltage is related to the velocity. This relationship between the electrical side and
the mechanical side can be modeled by a transformer.
${\displaystyle {\tilde {f_{c}}}=Bl{\tilde {i}}}$; ${\displaystyle {\tilde {u_{c}}}={\dfrac {\tilde {e}}{Bl}}}$
Figure 3: A transformer modeling transduction from the electrical impedance to mechanical mobility analogy
Mechanical subsystem
In a first approximation, a moving coil loudspeaker may be thought of as a mass-spring system where the diaphragm and the voice coil constitute the mass and the spider and surround constitute the
spring element. Losses in the suspension can be modeled as a resistor.
Figure 4: Mass spring system and associated circuit analogies of the impedance and mobility type.
The equation of motion gives us :
${\displaystyle {\tilde {f_{c}}}=R_{m}{\tilde {u_{c}}}+{\dfrac {\tilde {u_{c}}}{j\omega C_{MS}}}+j\omega M_{MD}{\tilde {u_{c}}}}$
${\displaystyle {\dfrac {\tilde {f_{c}}}{\tilde {u_{c}}}}=R_{m}+{\dfrac {1}{j\omega C_{MS}}}+j\omega M_{MD}}$
Which yields the mechanical impedance type analogy in the form of a series RLC circuit. A parallel RLC circuit may also be obtained to get the mobility analog following mathematical manipulation:
${\displaystyle {\dfrac {\tilde {u_{c}}}{\tilde {f_{c}}}}={\dfrac {1}{R_{m}+{\dfrac {1}{j\omega C_{MS}}}+j\omega M_{MD}}}}$
${\displaystyle {\dfrac {\tilde {u_{c}}}{\tilde {f_{c}}}}={\dfrac {1}{{\dfrac {1}{G_{m}}}+{\dfrac {1}{j\omega C_{MS}}}+{\dfrac {1}{\dfrac {1}{j\omega M_{MD}}}}}}}$
Which expresses the mechanical mobility type analogy in the form of a parallel RLC circuit where the denominator elements are respectively a parallel conductance, inductance, and compliance.
Mechanical to acoustical subsystem
A loudspeaker’s diaphragm may be thought of as a piston that pushes and pulls on the air facing it, converting mechanical force and velocity into acoustic pressure and volume velocity. The equations
are as follow:
${\displaystyle {\tilde {P_{d}}}={\dfrac {\tilde {f_{c}}}{\tilde {S_{D}}}}}$; ${\displaystyle {\tilde {U_{c}}}={\tilde {u_{c}}}{S_{D}}}$
These equations can be modeled by a transformer.
Figure 5: A transformer modeling the transduction from mechanical mobility to acoustical mobility analogy performed by a loudspeaker's diaphragm
Acoustical subsystem
The impedance presented by the air load on the loudspeaker's diaphragm is both resistive due to sound radiation and reactive due to the air mass that is being pushed radially but does not contribute
to sound radiation to the far field. The air load on the diaphragm can be modeled as an impedance or an admittance. Specific values and approximations can be found in [1], [2] or [3]. Note that the
air load depends on the mounting conditions of the loudspeaker. If the loudspeaker is mounted in a baffle, the air load will be the same on each side of the diaphragm. Then, if the air load on one
side is ${\displaystyle Y_{AR}}$ in the admittance analogy, then the total air load is ${\displaystyle Y_{AR}/2}$ as both loads are in parallel.
Complete electro-mechano-acoustical equivalent circuit
Using electrical impedance, mechanical mobility and acoustical admittance yield the following equivalent circuit, modeling the entire loudspeaker drive unit.
Figure 6: A complete electro-mechano-acoustical equivalent circuit of a loudspeaker drive unit
This circuit can be reduced by substituting the transformers and connected loads by an equivalent loading that would present the same impedance as the loaded transformer. An example of this is shown
on figure 7, where acoustical and electrical loads and sources have been "brought over" to the mechanical side.
Figure 7: Mechanical equivalent circuit modeling of a loudspeaker drive unit
The advantage of doing such manipulations is that we can then directly relate electrical measurements with elements in the circuit. This will later allow us to obtain values for the different
components of the model and match this model to real loudspeaker drivers. We can further simplify this circuit by using Norton's theorem and converting the series electrical components and voltage
source into an equivalent current source and parallel electrical components. Then, using a technique called the Dot method, presented in section Solution Methods: Electro-Mechanical Analogies, we can
obtain a single loop series circuit which is the dual of the parallel circuit previously obtained with Norton's theorem. If we are mainly interested in the low frequency behavior of the loudspeaker,
as should be the case when using lumped element modeling, we can neglect the effect of the voice coil inductance, which has an effect only at high frequencies. Furthermore, the air load impedance at
low frequencies is mass-like and can be modeled by a simple inductance ${\displaystyle M_{M1}}$. This results in a simplified low frequency model equivalent circuit, shown of figure 8, which is
easier to manipulate than the circuit of figure 7. Note that the analogy used for this circuit is of the impedance type.
Figure 8: Low frequency approximation mechanical equivalent circuit of a loudspeaker drive unit
Where ${\displaystyle M_{M1}=2.67a^{3}\rho }$ if ${\displaystyle a}$ is the radius of the loudspeaker and ${\displaystyle \rho }$, the density of air. Mass elements, in this case the mass of the
diaphragm and voice coil ${\displaystyle M_{MS}}$ and the air mass loading the diaphragm ${\displaystyle 2M_{M1}}$ can be regrouped in a single element:
${\displaystyle M_{MS}=M_{MD}+2M_{M1}}$
Thiele-Small Parameters
The complete low frequency behavior of a loudspeaker drive unit can be modeled with just six parameters, called Thiele-Small parameters. Most of these parameters result from algebraic manipulation of
the equations of the circuit of figure 8. Loudspeaker driver manufacturers seldom provide electro-mechano-acoustical parameters directly and rather provide Thiele-Small parameters in datasheets, but
conversion from one to the other is quite simple. The Thiele-Small parameters are as follow:
1. ${\displaystyle R_{e}}$, the voice coil DC resistance;
2. ${\displaystyle Q_{ES}}$, the electrical Q factor;
3. ${\displaystyle Q_{MS}}$, the mechanical Q factor;
4. ${\displaystyle f_{s}}$, the loudspeaker resonance frequency;
5. ${\displaystyle S_{D}}$, the effective surface area of the diaphragm;
6. ${\displaystyle V_{AS}}$, the equivalent suspension volume: the volume of air that has the same acoustic compliance as the suspension of the loudspeaker driver.
These parameters can be related directly from the low frequency approximation circuit of figure 8, with ${\displaystyle R_{e}}$ and ${\displaystyle S_{D}}$ being explicit.
${\displaystyle Q_{MS}={\dfrac {1}{R_{MS}}}{\sqrt {\dfrac {M_{MS}}{C_{MS}}}}}$; ${\displaystyle Q_{ES}={\dfrac {R_{g}+R_{e}}{(Bl)^{2}}}{\sqrt {\dfrac {M_{MS}}{C_{MS}}}}}$ ; ${\displaystyle f_{s}={\
dfrac {1}{2\pi {\sqrt {M_{MS}C_{MS}}}}}}$; ${\displaystyle V_{AS}=C_{MS}S_{D}^{2}\rho c^{2}}$
Where ${\displaystyle \rho c^{2}}$ is the Bulk modulus of air. It follows that, if given Thiele-Small parameters, one can extract the values of each component of the circuit of figure 8 using the
following equations :
${\displaystyle C_{MS}={\dfrac {V_{AS}}{S_{D}^{2}\rho c^{2}}}}$; ${\displaystyle M_{MS}={\dfrac {1}{(2\pi f_{s})^{2}C_{MS}}}}$; ${\displaystyle R_{MS}={\dfrac {1}{Q_{MS}}}{\sqrt {\dfrac {M_{MS}}{C_
{MS}}}}}$; ${\displaystyle Bl={\sqrt {\dfrac {R_{e}}{2\pi f_{s}Q_{ES}C_{MS}}}}}$; ${\displaystyle M_{MD}=M_{MS}-2M_{M1}}$;
Many methods can be used to measure Thiele-Small parameters of drivers. Measurement of Thiele-Small parameters is sometimes necessary if a manufacturer does not provide them. Also, the actual
Thiele-Small parameters of a given loudspeaker can differ from nominal values significantly. The method described in this section comes from [2]. Note that for this method, the loudspeaker is
considered to be mounted in an infinite baffle. In practice, a baffle with a diameter of four times that of the loudspeaker is sufficient. Measurements without a baffle are also possible: the air
mass loading will simply be halved and can be easily accounted for. The setup for this method includes an FFT analyzer or a mean to obtain an impedance curve. A signal generator of variable frequency
and an AC meter can also be used.
Figure 9: Simple experimental setup to measure the impedance of a loudspeaker drive unit
${\displaystyle Z_{spk}=R{\dfrac {V_{spk}}{V_{s}\left(1-{\dfrac {V_{spk}}{V_{s}}}\right)}}}$
Figure 10: A typical loudspeaker drive unit impedance curve
Once the impedance curve of the loudspeaker is measured, ${\displaystyle R_{e}}$ and ${\displaystyle f_{s}}$ can be directly identified by looking at the low frequency asymptote of the impedance
value and the center frequency of the resonance peak. If the frequencies where ${\displaystyle Z_{spk}={\sqrt {R_{e}R_{c}}}}$ are identified as ${\displaystyle f_{l}}$ and ${\displaystyle f_{h}}$, Q
factors can be calculated.
${\displaystyle Q_{MS}={\dfrac {f_{s}}{f_{h}-f_{l}}}{\sqrt {\dfrac {R_{c}}{R_{e}}}}}$
${\displaystyle Q_{ES}={\dfrac {Q_{MS}}{{\dfrac {R_{c}}{R_{e}}}-1}}}$
${\displaystyle S_{D}}$ can simply be approximated by ${\displaystyle \pi a^{2}}$, where ${\displaystyle a}$ is the radius of the loudspeaker driver. The last remaining Thiele-Small parameter, ${\
displaystyle V_{AS}}$ is slightly trickier to measure. The idea is to either increase mass or reduce compliance of the loudspeaker drive unit and note the shift in resonance frequency. If a known
mass ${\displaystyle M_{x}}$ is added to the loudspeaker diaphragm, the new resonance frequency will be:
${\displaystyle f_{s}^{'}={\dfrac {1}{2\pi {\sqrt {(M_{MS}+M_{x})C_{MS}}}}}}$
And the equivalent suspension volume may be obtained with:
${\displaystyle V_{AS}=\left(1-{\dfrac {f_{s}^{'2}}{f_{s}^{2}}}\right){\dfrac {S_{D}^{2}\rho c^{2}}{(2\pi f_{s}^{'})^{2}M_{x}}}}$
Hence, all Thiele-Small parameters modeling the low frequency behavior of the loudspeaker drive unit can be obtained from a fairly simple setup. These parameters are of tremendous help in loudspeaker
enclosure design.
Numerical example
This section presents a numerical example of obtaining Thiele-Small parameters from impedance curves. The impedance curves presented in this section have been obtained from simulations using nominal
Thiele-Small parameters of a real woofer loudspeaker. Firsy, these Thiele-Small parameters have been transformed into an electro-mechano-acoustical circuit using the equation presented before.
Second, the circuit was treated as a black box and the method to extract Thiele-Small parameters was used. The purpose of this simulation is to present the method, step by step, using realistic
values so that the reader can get more familiar with the process, the magnitude of the values and with what to expect when performing such measurements.
For this simulation, a loudspeaker of radius ${\displaystyle a=6.55cm}$ is mounted on a baffle sufficiently large to act as an infinite baffle. Its impedance is obtained and plotted in figure 11,
where important cursors have already been placed.
Figure 11: Simulated measurement of an impedance curve for a woofer loudspeaker
The low frequency asymptote is immediately identified as ${\displaystyle Re=6.6\Omega }$. The resonance is clear and centered at ${\displaystyle fs=33Hz}$. The value of the impedance at this
frequency is about ${\displaystyle 66\Omega }$. This yields ${\displaystyle {\sqrt {R_{e}R_{c}}}=20.8\Omega }$, which occurs at ${\displaystyle f_{l}=19.5Hz}$ and ${\displaystyle f_{h}=52.5Hz}$. With
this information, we can compute some of the Thiele-Small parameters.
${\displaystyle Q_{MS}={\dfrac {f_{s}}{f_{h}-f_{l}}}{\sqrt {\dfrac {R_{c}}{R_{e}}}}={\dfrac {33}{52.5-19.5}}*{\sqrt {\dfrac {66}{6.6}}}=3.1}$
${\displaystyle Q_{ES}={\dfrac {Q_{MS}}{{\dfrac {R_{c}}{R_{e}}}-1}}={\dfrac {3.1}{{\dfrac {66}{6.6}}-1}}=0.35}$
As a next step, a mass of ${\displaystyle M_{x}=10g}$ is fixed to the loudspeaker diaphragm. This shifts the resonance frequency and yields a new impedance curve, as shown on figure 12.
Figure 12: Simulated measurement of an impedance curve for a woofer loudspeaker
${\displaystyle S_{D}=\pi a^{2}=0.0135m^{2}}$
${\displaystyle V_{AS}=\left(1-{\dfrac {27.5^{2}}{33^{2}}}\right){\dfrac {0.0135^{2}*1.18*344^{2}}{(2\pi 27.5)^{2}*0.01}}=0.0272m^{3}}$
Once all six Thiele-Small parameters have been obtained, it is possible to calculate values for the electro-mechano-acoustical circuit modeling elements of figure 6 or 7. From then, the design of an
enclosure can start. This is discussed in application sections Sealed box subwoofer design and Bass reflex enclosure design.
[1] Kleiner, Mendel. Electroacoustics. CRC Press, 2013.
[2] Beranek, Leo L., and Tim Mellow. Acoustics: sound fields and transducers. Academic Press, 2012.
[3] Kinsler, Lawrence E., et al. Fundamentals of Acoustics, 4th Edition. Wiley-VCH, 1999.
[4] Small, Richard H. "Direct radiator loudspeaker system analysis." Journal of the Audio Engineering Society 20.5 (1972): 383-395.
Moving Resonators
Moving Resonators
Consider the situation shown in the figure below. We have a typical Helmholtz resonator driven by a massless piston which generates a sinusoidal pressure ${\displaystyle P_{G}}$, however the cavity
is not fixed in this case. Rather, it is supported above the ground by a spring with compliance ${\displaystyle C_{M}}$. Assume the cavity has a mass ${\displaystyle M_{M}}$.
Recall the Helmholtz resonator (see Module #9). The difference in this case is that the pressure in the cavity exerts a force on the bottom of the cavity, which is now not fixed as in the original
Helmholtz resonator. This pressure causes a force that acts upon the cavity bottom. If the surface area of the cavity bottom is ${\displaystyle S_{C}}$, then Newton's Laws applied to the cavity
bottom give
${\displaystyle \sum {F}=p_{C}S_{C}-{\frac {x}{C_{M}}}=M_{M}{\ddot {x}}\Rightarrow p_{C}S_{C}=\left[{\frac {1}{j\omega C_{M}}}+j\omega M_{M}\right]u}$
In order to develop the equivalent circuit, we observe that we simply need to use the pressure (potential across ${\displaystyle C_{A}}$) in the cavity to generate a force in the mechanical circuit.
The above equation shows that the mass of the cavity and the spring compliance should be placed in series in the mechanical circuit. In order to convert the pressure to a force, the transformer is
used with a ratio of ${\displaystyle 1:S_{C}}$.
A practical example of a moving resonator is a marimba. A marimba is a similar to a xylophone but has larger resonators that produce deeper and richer tones. The resonators (seen in the picture as
long, hollow pipes) are mounted under an array of wooden bars which are struck to create tones. Since these resonators are not fixed, but are connected to the ground through a stiffness (the stand),
it can be modeled as a moving resonator. Marimbas are not tunable instruments like flutes or even pianos. It would be interesting to see how the tone of the marimba changes as a result of changing
the stiffness of the mount.
For more information about the acoustics of marimbas see http://www.mostlymarimba.com/techno1.html
Part 2: One-Dimensional Wave Motion
Transverse vibrations of strings
This section deals with the wave nature of vibrations constrained to one dimension. Examples of this type of wave motion are found in objects such a pipes and tubes with a small diameter (no
transverse motion of fluid) or in a string stretched on a musical instrument.
Stretched strings can be used to produce sound (e.g. music instruments like guitars). The stretched string constitutes a mechanical system that will be studied in this chapter. Later, the
characteristics of this system will be used to help to understand by analogies acoustical systems.
What is a wave equation?
There are various types of waves (i.e. electromagnetic, mechanical, etc.) that act all around us. It is important to use wave equations to describe the time-space behavior of the variables of
interest in such waves. Wave equations solve the fundamental equations of motion in a way that eliminates all variables but one. Waves can propagate longitudinal or parallel to the propagation
direction or perpendicular (transverse) to the direction of propagation. To visualize the motion of such waves click here (Acoustics animations provided by Dr. Dan Russell,Kettering University)
One dimensional Case
Assumptions :
- the string is uniform in size and density
- stiffness of string is negligible for small deformations
- effects of gravity neglected
- no dissipative forces like frictions
- string deforms in a plane
- motion of the string can be described by using one single spatial coordinate
Spatial representation of the string in vibration:
The following is the free-body diagram of a string in motion in a spatial coordinate system:
From the diagram above, it can be observed that the tensions in each side of the string will be the same as follows:
Using Taylor series to expand we obtain:
Characterization of the mechanical system
A one dimensional wave can be described by the following equation (called the wave equation):
${\displaystyle \left({\frac {\partial ^{2}y}{\partial x^{2}}}\right)=\left({\frac {1}{c^{2}}}\right)\left({\frac {\partial ^{2}y}{\partial t^{2}}}\right)}$
${\displaystyle y(x,t)=f(\xi )+g(\eta )\,}$ is a solution,
With ${\displaystyle \xi =ct-x\,}$ and ${\displaystyle \eta =ct+x\,}$
This is the D'Alambert solution, for more information see: [1]
Another way to solve this equation is the Method of separation of variables. This is useful for modal analysis. This assumes the solution is of the form:
${\displaystyle y(x,t)=f(x)g(t)\ }$
The result is the same as above, but in a form that is more convenient for modal analysis.
For more information on this approach see: Eric W. Weisstein et al. "Separation of Variables." From MathWorld—A Wolfram Web Resource. [2]
Please see Wave Properties for information on variable c, along with other important properties.
For more information on wave equations see: Eric W. Weisstein. "Wave Equation." From MathWorld—A Wolfram Web Resource. [3]
Example with the function ${\displaystyle f(\xi )}$ :
Example: Java String simulation
This show a simple simulation of a plucked string with fixed ends.
Time-Domain Solutions
d'Alembert Solutions
In 1747, Jean Le Rond d'Alembert published a solution to the one-dimensional wave equation.
The general solution, now known as the d'Alembert method, can be found by introducing two new variables:
${\displaystyle \xi =ct-x\,}$ and ${\displaystyle \eta =ct+x\,}$
and then applying the chain rule to the general form of the wave equation.
From this, the solution can be written in the form:
${\displaystyle y(\xi ,\eta )=f(\xi )+g(\eta )\,=f(x+ct)+g(x-ct)}$
where f and g are arbitrary functions, that represent two waves traveling in opposing directions.
A more detailed look into the proof of the d'Alembert solution can be found here.
Example of Time Domain Solution
If f(ct-x) is plotted vs. x for two instants in time, the two waves are the same shape but the second displaced by a distance of c(t2-t1) to the right.
The two arbitrary functions could be determined from initial conditions or boundary values.
Boundary Conditions and Forced Vibrations
Boundary Conditions
The functions representing the solutions to the wave equation previously discussed,
i.e. ${\displaystyle y(x,t)=f(\xi )+g(\eta )\,}$ with ${\displaystyle \xi =ct-x\,}$ and ${\displaystyle \eta =ct+x\,}$
are dependent upon the boundary and initial conditions. If it is assumed that the wave is propagating through a string, the initial conditions are related to the specific disturbance in the string at
t=0. These specific disturbances are determined by location and type of contact and can be anything from simple oscillations to violent impulses. The effects of boundary conditions are less subtle.
The most simple boundary conditions are the Fixed Support and Free End. In practice, the Free End boundary condition is rarely encountered since it is assumed there are no transverse forces holding
the string (e.g. the string is simply floating).
For a Fixed Support
The overall displacement of the waves travelling in the string, at the support, must be zero. Denoting x=0 at the support, This requires:
${\displaystyle y(0,t)=f(ct-0)+g(ct+0)=0\,}$
Therefore, the total transverse displacement at x=0 is zero.
The sequence of wave reflection for incident, reflected and combined waves are illustrated below. Please note that the wave is traveling to the left (negative x direction) at the beginning. The
reflected wave is ,of course, traveling to the right (positive x direction).
For a Free Support
Unlike the Fixed Support boundary condition, the transverse displacement at the support does not need to be zero, but must require the sum of transverse forces to cancel. If it is assumed that the
angle of displacement is small,
${\displaystyle \sin(\theta )\approx \theta =\left({\frac {\partial y}{\partial x}}\right)\,}$
and so,
${\displaystyle \sum F_{y}=T\sin(\theta )\approx T\left({\frac {\partial y}{\partial x}}\right)=0\,}$
But of course, the tension in the string, or T, will not be zero and this requires the slope at x=0 to be zero:
i.e. ${\displaystyle \left({\frac {\partial y}{\partial x}}\right)=0\,}$
Again for free boundary, the sequence of wave reflection for incident, reflected and combined waves are illustrated below:
Other Boundary Conditions
There are many other types of boundary conditions that do not fall into our simplified categories. As one would expect though, it isn't difficult to relate the characteristics of numerous "complex"
systems to the basic boundary conditions. Typical or realistic boundary conditions include mass-loaded, resistance-loaded, damping-loaded, and impedance-loaded strings. For further information, see
Kinsler, Fundamentals of Acoustics, pp 54–58.
Here is a website with nice movies of wave reflection at different BC's: Wave Reflection
Wave Properties
To begin with, a few definitions of useful variables will be discussed. These include; the wave number, phase speed, and wavelength characteristics of wave travelling through a string.
The speed that a wave propagates through a string is given in terms of the phase speed, typically in m/s, given by:
${\displaystyle c={\sqrt {T/\rho _{L}}}\,}$ where ${\displaystyle \rho _{L}\,}$ is the density per unit length of the string.
The wave number is used to reduce the transverse displacement equation to a simpler form and for simple harmonic motion, is multiplied by the lateral position. It is given by:
${\displaystyle k=\left({\frac {\omega }{c}}\right)\,}$ where ${\displaystyle \omega =2\pi f\,}$
Lastly, the wavelength is defined as:
${\displaystyle \lambda =\left({\frac {2\pi }{k}}\right)=\left({\frac {c}{f}}\right)\,}$
and is defined as the distance between two points, usually peaks, of a periodic waveform.
These "wave properties" are of practical importance when calculating the solution of the wave equation for a number of different cases. As will be seen later, the wave number is used extensively to
describe wave phenomenon graphically and quantitatively.
For further information: Wave Properties
Forced Vibrations
1.forced vibrations of infinite string suppose there is a string very long , at x=0 there is a force exerted on it.
use the boundary condition at x=0,
neglect the reflected wave
it is easy to get the wave form
where w is the angular velocity, k is the wave number.
according to the impedance definition
it represents the characteristic impedance of the string. obviously, it is purely resistive, which is like the resistance in the mechanical system.
The dissipated power
Note: along the string, all the variables propagate at same speed.
link title a useful link to show the time-space property of the wave.
Some interesting animations of the wave at different boundary conditions.
1.hard boundary (which is like a fixed end)
2.soft boundary (which is like a free end)
3.from low density to high density string
4.from high density to low density string
Part 3: Applications
Room Acoustics and Concert Halls
Room Acoustics and Concert Halls
From performing on many different rooms and stages all over the United States, I thought it would be nice to have a better understanding and source about the room acoustics. This Wikibook page is
intended to help the user with basic technical questions/answers about room acoustics. Main topics that will be covered are: what really makes a room sound good or bad, alive or dead. This will lead
into absorption and transmission coefficients, decay of sound in the room, and reverberation. Different use of materials in rooms will be mentioned also. There is no intention of taking work from
another. This page is a switchboard source to help the user find information about room acoustics.
Sound Fields
Two types of sound fields are involved in room acoustics: Direct Sound and Reverberant Sound.
Direct Sound
The component of the sound field in a room that involves only a direct path between the source and the receiver, before any reflections off walls and other surfaces.
Reverberant Sound
The component of the sound field in a room that involves the direct path and the path after it reflects off of walls or any other surfaces. How the waves deflect off of the mediums all depends on the
absorption and transmission coefficients.
Good example pictures are shown at Crutchfield Advisor, a Physics Site from MTSU, and Voiceteacher.com
Room Coefficients
In a perfect world, if there is a sound shot right at a wall, the sound should come right back. But because sounds hit different materials types of walls, the sound does not have perfect reflection.
From 1, these are explained as follows:
Absorption & Transmission Coefficients
The best way to explain how sound reacts to different mediums is through acoustical energy. When sound impacts on a wall, acoustical energy will be reflected, absorbed, or transmitted through the
Absorption Coefficient: NB: this chemical structure is unrelated to the acoustics being discussed.
If all of the acoustic energy hits the wall and none is reflected, the alpha would equal 1. The energy had zero reflection and was absorbed or transmitted. This would be an example of a dead or soft
wall because it takes in everything and doesn't reflect anything back. Rooms that are like this are called Anechoic Rooms which looks like this from Axiomaudio.
If all of the acoustic energy hits the wall and all reflects back, the alpha would equal 0. This would be an example of a live or hard wall because the sound bounces right back and does not go
through the wall. Rooms that are like this are called Reverberant Rooms like this McIntosh room. Look how the walls have nothing attached to them. More room for the sound waves to bounce off the
Room Averaged Sound Absorption Coefficient
Not all rooms have the same walls on all sides. The room averaged sound absorption coefficient can be used to have different types of materials and areas of walls averaged together.
Absorption Coefficients for Specific Materials
Basic sound absorption Coefficients are shown here at Acoustical Surfaces.
Brick, unglazed, painted alpha ~ .01 - .03 -> Sound reflects back
An open door alpha equals 1 -> Sound goes through
Units are in Sabins.
Sound Decay and Reverberation Time
In a large reverberant room, a sound can still propagate after the sound source has been turned off. This time when the sound intensity level has decay 60 dB is called the reverberation time of the
Great Halls in the World
Pick Staiger at Northwestern U
[1] Lord, Gatley, Evensen; Noise Control for Engineers, Krieger Publishing, 435 pgs
Back to Engineering Acoustics
Created by Kevin Baldwin
Bass Reflex Enclosure Design
Bass-reflex enclosures improve the low-frequency response of loudspeaker systems. Bass-reflex enclosures are also called "vented-box design" or "ported-cabinet design". A bass-reflex enclosure
includes a vent or port between the cabinet and the ambient environment. This type of design, as one may observe by looking at contemporary loudspeaker products, is still widely used today. Although
the construction of bass-reflex enclosures is fairly simple, their design is not simple, and requires proper tuning. This reference focuses on the technical details of bass-reflex design. General
loudspeaker information can be found here.
Effects of the Port on the Enclosure Response
Before discussing the bass-reflex enclosure, it is important to be familiar with the simpler sealed enclosure system performance. As the name suggests, the sealed enclosure system attaches the
loudspeaker to a sealed enclosure (except for a small air leak included to equalize the ambient pressure inside). Ideally, the enclosure would act as an acoustical compliance element, as the air
inside the enclosure is compressed and rarified. Often, however, an acoustic material is added inside the box to reduce standing waves, dissipate heat, and other reasons. This adds a resistive
element to the acoustical lumped-element model. A non-ideal model of the effect of the enclosure actually adds an acoustical mass element to complete a series lumped-element circuit given in Figure
1. For more on sealed enclosure design, see the Sealed Box Subwoofer Design page.
Figure 1. Sealed enclosure acoustic circuit.
In the case of a bass-reflex enclosure, a port is added to the construction. Typically, the port is cylindrical and is flanged on the end pointing outside the enclosure. In a bass-reflex enclosure,
the amount of acoustic material used is usually much less than in the sealed enclosure case, often none at all. This allows air to flow freely through the port. Instead, the larger losses come from
the air leakage in the enclosure. With this setup, a lumped-element acoustical circuit has the following form.
In this figure, ${\displaystyle Z_{RAD}}$ represents the radiation impedance of the outside environment on the loudspeaker diaphragm. The loading on the rear of the diaphragm has changed when
compared to the sealed enclosure case. If one visualizes the movement of air within the enclosure, some of the air is compressed and rarified by the compliance of the enclosure, some leaks out of the
enclosure, and some flows out of the port. This explains the parallel combination of ${\displaystyle M_{AP}}$, ${\displaystyle C_{AB}}$, and ${\displaystyle R_{AL}}$. A truly realistic model would
incorporate a radiation impedance of the port in series with ${\displaystyle M_{AP}}$, but for now it is ignored. Finally, ${\displaystyle M_{AB}}$, the acoustical mass of the enclosure, is included
as discussed in the sealed enclosure case. The formulas which calculate the enclosure parameters are listed in Appendix B.
It is important to note the parallel combination of ${\displaystyle M_{AP}}$ and ${\displaystyle C_{AB}}$. This forms a Helmholtz resonator (click here for more information). Physically, the port
functions as the “neck” of the resonator and the enclosure functions as the “cavity.” In this case, the resonator is driven from the piston directly on the cavity instead of the typical Helmholtz
case where it is driven at the “neck.” However, the same resonant behavior still occurs at the enclosure resonance frequency, ${\displaystyle f_{B}}$. At this frequency, the impedance seen by the
loudspeaker diaphragm is large (see Figure 3 below). Thus, the load on the loudspeaker reduces the velocity flowing through its mechanical parameters, causing an anti-resonance condition where the
displacement of the diaphragm is a minimum. Instead, the majority of the volume velocity is actually emitted by the port itself instead of the loudspeaker. When this impedance is reflected to the
electrical circuit, it is proportional to ${\displaystyle 1/Z}$, thus a minimum in the impedance seen by the voice coil is small. Figure 3 shows a plot of the impedance seen at the terminals of the
loudspeaker. In this example, ${\displaystyle f_{B}}$ was found to be about 40 Hz, which corresponds to the null in the voice-coil impedance.
Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.
Quantitative Analysis of Port on Enclosure
The performance of the loudspeaker is first measured by its velocity response, which can be found directly from the equivalent circuit of the system. As the goal of most loudspeaker designs is to
improve the bass response (leaving high-frequency production to a tweeter), low frequency approximations will be made as much as possible to simplify the analysis. First, the inductance of the voice
coil, ${\displaystyle {\it {L_{E}}}}$, can be ignored as long as ${\displaystyle \omega \ll R_{E}/L_{E}}$. In a typical loudspeaker, ${\displaystyle {\it {L_{E}}}}$ is of the order of 1 mH, while ${\
displaystyle {\it {R_{E}}}}$ is typically 8${\displaystyle \Omega }$, thus an upper frequency limit is approximately 1 kHz for this approximation, which is certainly high enough for the frequency
range of interest.
Another approximation involves the radiation impedance, ${\displaystyle {\it {Z_{RAD}}}}$. It can be shown [1] that this value is given by the following equation (in acoustical ohms):
${\displaystyle Z_{RAD}={\frac {\rho _{0}c}{\pi a^{2}}}\left[\left(1-{\frac {J_{1}(2ka)}{ka}}\right)+j{\frac {H_{1}(2ka)}{ka}}\right]}$
Where ${\displaystyle J_{1}(x)}$ and ${\displaystyle H_{1}(x)}$ are types of Bessel functions. For small values of ka,
${\displaystyle J_{1}(2ka)\approx ka} and ${\displaystyle H_{1}(2ka)\approx {\frac {8(ka)^{2}}{3\pi ${\displaystyle \Rightarrow Z_{RAD}\approx j{\frac {8\rho _{0}\omega }{3\pi ^{2}a}}=jM_
$ }}}$ {A1}}$
Hence, the low-frequency impedance on the loudspeaker is represented with an acoustic mass ${\displaystyle M_{A1}}$ [1]. For a simple analysis, ${\displaystyle R_{E}}$, ${\displaystyle M_{MD}}$, ${\
displaystyle C_{MS}}$, and ${\displaystyle R_{MS}}$ (the transducer parameters, or Thiele-Small parameters) are converted to their acoustical equivalents. All conversions for all parameters are given
in Appendix A. Then, the series masses, ${\displaystyle M_{AD}}$, ${\displaystyle M_{A1}}$, and ${\displaystyle M_{AB}}$, are lumped together to create ${\displaystyle M_{AC}}$. This new circuit is
shown below.
Unlike sealed enclosure analysis, there are multiple sources of volume velocity that radiate to the outside environment. Hence, the diaphragm volume velocity, ${\displaystyle U_{D}}$, is not analyzed
but rather ${\displaystyle U_{0}=U_{D}+U_{P}+U_{L}}$. This essentially draws a “bubble” around the enclosure and treats the system as a source with volume velocity ${\displaystyle U_{0}}$. This
“lumped” approach will only be valid for low frequencies, but previous approximations have already limited the analysis to such frequencies anyway. It can be seen from the circuit that the volume
velocity flowing into the enclosure, ${\displaystyle U_{B}=-U_{0}}$, compresses the air inside the enclosure. Thus, the circuit model of Figure 3 is valid and the relationship relating input voltage,
${\displaystyle V_{IN}}$ to ${\displaystyle U_{0}}$ may be computed.
In order to make the equations easier to understand, several parameters are combined to form other parameter names. First, ${\displaystyle \omega _{B}}$ and ${\displaystyle \omega _{S}}$, the
enclosure and loudspeaker resonance frequencies, respectively, are:
${\displaystyle \omega _{B}={\frac {1}{\sqrt {M_{AP}C_{AB}}}}}$ ${\displaystyle \omega _{S}={\frac {1}{\sqrt {M_{AC}C_{AS}}}}}$
Based on the nature of the derivation, it is convenient to define the parameters ${\displaystyle \omega _{0}}$ and h, the Helmholtz tuning ratio:
${\displaystyle \omega _{0}={\sqrt {\omega _{B}\omega _{S}}}}$ ${\displaystyle h={\frac {\omega _{B}}{\omega _{S}}}}$
A parameter known as the compliance ratio or volume ratio, ${\displaystyle \alpha }$, is given by:
${\displaystyle \alpha ={\frac {C_{AS}}{C_{AB}}}={\frac {V_{AS}}{V_{AB}}}}$
Other parameters are combined to form what are known as quality factors:
${\displaystyle Q_{L}=R_{AL}{\sqrt {\frac {C_{AB}}{M_{AP}}}}}$ ${\displaystyle Q_{TS}={\frac {1}{R_{AE}+R_{AS}}}{\sqrt {\frac {M_{AC}}{C_{AS}}}}}$
This notation allows for a simpler expression for the resulting transfer function [1]:
${\displaystyle {\frac {U_{0}}{V_{IN}}}=G(s)={\frac {(s^{3}/\omega _{0}^{4})}{(s/\omega _{0})^{4}+a_{3}(s/\omega _{0})^{3}+a_{2}(s/\omega _{0})^{2}+a_{1}(s/\omega _{0})+1}}}$
${\displaystyle a_{1}={\frac {1}{Q_{L}{\sqrt {h}}}}+{\frac {\sqrt ${\displaystyle a_{2}={\frac {\alpha +1}{h}}+h+{\frac {1}{Q_ ${\displaystyle a_{3}={\frac {1}{Q_{TS}{\sqrt {h}}}}+{\frac {\sqrt
{h}}{Q_{TS}}}}$ {L}Q_{TS}}}}$ {h}}{Q_{L}}}}$
Development of Low-Frequency Pressure Response
It can be shown [2] that for ${\displaystyle ka<1/2}$, a loudspeaker behaves as a spherical source. Here, a represents the radius of the loudspeaker. For a 15” diameter loudspeaker in air, this low
frequency limit is about 150 Hz. For smaller loudspeakers, this limit increases. This limit dominates the limit which ignores ${\displaystyle L_{E}}$, and is consistent with the limit that models ${\
displaystyle Z_{RAD}}$ by ${\displaystyle M_{A1}}$.
Within this limit, the loudspeaker emits a volume velocity ${\displaystyle U_{0}}$, as determined in the previous section. For a simple spherical source with volume velocity ${\displaystyle U_{0}}$,
the far-field pressure is given by [1]:
${\displaystyle p(r)\simeq j\omega \rho _{0}U_{0}{\frac {e^{-jkr}}{4\pi r}}}$
It is possible to simply let ${\displaystyle r=1}$ for this analysis without loss of generality because distance is only a function of the surroundings, not the loudspeaker. Also, because the
transfer function magnitude is of primary interest, the exponential term, which has a unity magnitude, is omitted. Hence, the pressure response of the system is given by [1]:
${\displaystyle {\frac {p}{V_{IN}}}={\frac {\rho _{0}s}{4\pi }}{\frac {U_{0}}{V_{IN}}}={\frac {\rho _{0}Bl}{4\pi S_{D}R_{E}M_{A}S}}H(s)}$
Where ${\displaystyle H(s)=sG(s)}$. In the following sections, design methods will focus on ${\displaystyle |H(s)|^{2}}$ rather than ${\displaystyle H(s)}$, which is given by:
${\displaystyle |H(s)|^{2}={\frac {\Omega ^{8}}{\Omega ^{8}+\left(a_{3}^{2}-2a_{2}\right)\Omega ^{6}+\left(a_{2}^{2}+2-2a_{1}a_{3}\right)\Omega ^{4}+\ ${\displaystyle \Omega ={\frac {\omega }{\
left(a_{1}^{2}-2a_{2}\right)\Omega ^{2}+1}}}$ omega _{0}}}}$
This also implicitly ignores the constants in front of ${\displaystyle |H(s)|}$ since they simply scale the response and do not affect the shape of the frequency response curve.
A popular way to determine the ideal parameters has been through the use of alignments. The concept of alignments is based upon well investigated electrical filter theory. Filter development is a
method of selecting the poles (and possibly zeros) of a transfer function to meet a particular design criterion. The criteria are the desired properties of a magnitude-squared transfer function,
which in this case is ${\displaystyle |H(s)|^{2}}$. From any of the design criteria, the poles (and possibly zeros) of ${\displaystyle |H(s)|^{2}}$ are found, which can then be used to calculate the
numerator and denominator. This is the “optimal” transfer function, which has coefficients that are matched to the parameters of ${\displaystyle |H(s)|^{2}}$ to compute the appropriate values that
will yield a design that meets the criteria.
There are many different types of filter designs, each which have trade-offs associated with them. However, this design approach is limited because of the structure of ${\displaystyle |H(s)|^{2}}$.
In particular, it has the structure of a fourth-order high-pass filter with all zeros at s = 0. Therefore, only those filter design methods which produce a low-pass filter with only poles will be
acceptable methods to use. From the traditional set of algorithms, only Butterworth and Chebyshev low-pass filters have only poles. In addition, another type of filter called a quasi-Butterworth
filter can also be used, which has similar properties to a Butterworth filter. These three algorithms are fairly simple, thus they are the most popular. When these low-pass filters are converted to
high-pass filters, the ${\displaystyle s\rightarrow 1/s}$ transformation produces ${\displaystyle s^{8}}$ in the numerator.
More details regarding filter theory and these relationships can be found in numerous resources, including [5].
Butterworth Alignment
The Butterworth algorithm is designed to have a maximally flat pass band. Since the slope of a function corresponds to its derivatives, a flat function will have derivatives equal to zero. Since as
flat of a pass band as possible is optimal, the ideal function will have as many derivatives equal to zero as possible at s = 0. Of course, if all derivatives were equal to zero, then the function
would be a constant, which performs no filtering.
Often, it is better to examine what is called the loss function. Loss is the reciprocal of gain, thus
${\displaystyle |{\hat {H}}(s)|^{2}={\frac {1}{|H(s)|^{2}}}}$
The loss function can be used to achieve the desired properties, then the desired gain function is recovered from the loss function.
Now, applying the desired Butterworth property of maximal pass-band flatness, the loss function is simply a polynomial with derivatives equal to zero at s = 0. At the same time, the original
polynomial must be of degree eight (yielding a fourth-order function). However, derivatives one through seven can be equal to zero if [3]
${\displaystyle |{\hat {H}}(\Omega )|^{2}=1+\Omega ^{8}\Rightarrow |H(\Omega )|^{2}={\frac {1}{1+\Omega ^{8}}}}$
With the high-pass transformation ${\displaystyle \Omega \rightarrow 1/\Omega }$,
${\displaystyle |H(\Omega )|^{2}={\frac {\Omega ^{8}}{\Omega ^{8}+1}}}$
It is convenient to define ${\displaystyle \Omega =\omega /\omega _{3dB}}$, since ${\displaystyle \Omega =1\Rightarrow |H(s)|^{2}=0.5}$ or -3 dB. This definition allows the matching of coefficients
for the ${\displaystyle |H(s)|^{2}}$ describing the loudspeaker response when ${\displaystyle \omega _{3dB}=\omega _{0}}$. From this matching, the following design equations are obtained [1]:
${\displaystyle a_{1}=a_{3}={\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle a_{2}=2+{\sqrt {2}}}$
Quasi-Butterworth Alignment
The quasi-Butterworth alignments do not have as well-defined of an algorithm when compared to the Butterworth alignment. The name “quasi-Butterworth” comes from the fact that the transfer functions
for these responses appear similar to the Butterworth ones, with (in general) the addition of terms in the denominator. This will be illustrated below. While there are many types of quasi-Butterworth
alignments, the simplest and most popular is the 3rd order alignment (QB3). The comparison of the QB3 magnitude-squared response against the 4th order Butterworth is shown below.
${\displaystyle \left|H_{QB3}(\omega )\right|^{2}={\frac {(\omega /\omega _{3dB})^{8}}{(\omega /\omega _{3dB}) ${\displaystyle \left|H_{B4}(\omega )\right|^{2}={\frac {(\omega /\omega _{3dB})^{8}}
^{8}+B^{2}(\omega /\omega _{3dB})^{2}+1}}}$ {(\omega /\omega _{3dB})^{8}+1}}}$
Notice that the case ${\displaystyle B=0}$ is the Butterworth alignment. The reason that this QB alignment is called 3rd order is due to the fact that as B increases, the slope approaches 3 dec/dec
instead of 4 dec/dec, as in 4th order Butterworth. This phenomenon can be seen in Figure 5.
Figure 5: 3rd-Order Quasi-Butterworth Response for ${\displaystyle 0.1\leq B\leq 3}$
Equating the system response ${\displaystyle |H(s)|^{2}}$ with ${\displaystyle |H_{QB3}(s)|^{2}}$, the equations guiding the design can be found [1]:
${\displaystyle B^{2}=a_{1}^{2}-2a_{2}}$ ${\displaystyle a_{2}^{2}+2=2a_{1}a_{3}}$ ${\displaystyle a_{3}={\sqrt {2a_{2}}}}$ ${\displaystyle a_{2}>2+{\sqrt {2}}}$
Chebyshev Alignment
The Chebyshev algorithm is an alternative to the Butterworth algorithm. For the Chebyshev response, the maximally-flat passband restriction is abandoned. Now, a ripple, or fluctuation is allowed in
the pass band. This allows a steeper transition or roll-off to occur. In this type of application, the low-frequency response of the loudspeaker can be extended beyond what can be achieved by
Butterworth-type filters. An example plot of a Chebyshev high-pass response with 0.5 dB of ripple against a Butterworth high-pass response for the same ${\displaystyle \omega _{3dB}}$ is shown below.
The Chebyshev response is defined by [4]:
${\displaystyle |{\hat {H}}(j\Omega )|^{2}=1+\epsilon ^{2}C_{n}^{2}(\Omega )}$
${\displaystyle C_{n}(\Omega )}$ is called the Chebyshev polynomial and is defined by [4]:
${\displaystyle C_{n}(\Omega )={\big \lbrace }}$ ${\displaystyle {\rm {{cos}[{\it {{n}{\rm {{cos}^{-1}(\Omega )]}}}}}}}$ ${\displaystyle |\Omega |<1}$
${\displaystyle {\rm {{cosh}[{\it {{n}{\rm {{cosh}^{-1}(\Omega )]}}}}}}}$ ${\displaystyle |\Omega |>1}$
Fortunately, Chebyshev polynomials satisfy a simple recursion formula [4]:
${\displaystyle C_{0}(x)=1}$ ${\displaystyle C_{1}(x)=x}$ ${\displaystyle C_{n}(x)=2xC_{n-1}-C_{n-2}}$
For more information on Chebyshev polynomials, see the Wolfram Mathworld: Chebyshev Polynomials page.
When applying the high-pass transformation to the 4th order form of ${\displaystyle |{\hat {H}}(j\Omega )|^{2}}$, the desired response has the form [1]:
${\displaystyle |H(j\Omega )|^{2}={\frac {1+\epsilon ^{2}}{1+\epsilon ^{2}C_{4}^{2}(1/\Omega )}}}$
The parameter ${\displaystyle \epsilon }$ determines the ripple. In particular, the magnitude of the ripple is ${\displaystyle 10{\rm {{log}[1+\epsilon ^{2}]}}}$ dB and can be chosen by the designer,
similar to B in the quasi-Butterworth case. Using the recursion formula for ${\displaystyle C_{n}(x)}$,
${\displaystyle C_{4}\left({\frac {1}{\Omega }}\right)=8\left({\frac {1}{\Omega }}\right)^{4}-8\left({\frac {1}{\Omega }}\right)^{2}+1}$
Applying this equation to ${\displaystyle |H(j\Omega )|^{2}}$ [1],
${\displaystyle \Rightarrow |H(\Omega )|^{2}={\frac {{\frac {1+\epsilon ^{2}}{64\epsilon ^{2}}}\Omega ^{8}}{{\frac {1+\epsilon ^{2}}{64\epsilon ^{2}}}\Omega ^{8}+{\frac {1}{4}}\Omega ^{6}+{\frac {5}
{4}}\Omega ^{4}-2\Omega ^{2}+1}}}$
${\displaystyle \Omega ={\frac {\omega }{\omega _{n}}}}$ ${\displaystyle \omega _{n}={\frac {\omega _{3dB}}{2}}{\sqrt {2+{\sqrt {2+2{\sqrt {2+{\frac {1}{\epsilon ^{2}}}}}}}}}}$
Thus, the design equations become [1]:
${\displaystyle \omega _{0}=\omega _{n}{\sqrt[{8}]{\frac ${\displaystyle k={\rm {{tanh}\left[{\frac {1}{4}}{\rm {{sinh}^{-1}\left ${\displaystyle D={\frac {k^{4}+6k^{2}+1}{8}}}$
{64\epsilon ^{2}}{1+\epsilon ^{2}}}}}$ ({\frac {1}{\epsilon }}\right)}}\right]}}}$
${\displaystyle a_{1}={\frac {k{\sqrt {4+2{\sqrt {2}}}}}{\ ${\displaystyle a_{2}={\frac {1+k^{2}(1+{\sqrt {2}})}{\sqrt {D}}}}$ ${\displaystyle a_{3}={\frac {a_{1}}{\sqrt {D}}}\left[1-{\
sqrt[{4}]{D}}},}$ frac {1-k^{2}}{2{\sqrt {2}}}}\right]}$
Choosing the Correct Alignment
With all the equations that have already been presented, the question naturally arises, “Which one should I choose?” Notice that the coefficients ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, and
${\displaystyle a_{3}}$ are not simply related to the parameters of the system response. Certain combinations of parameters may indeed invalidate one or more of the alignments because they cannot
realize the necessary coefficients. With this in mind, general guidelines have been developed to guide the selection of the appropriate alignment. This is very useful if one is designing an enclosure
to suit a particular transducer that cannot be changed.
The general guideline for the Butterworth alignment focuses on ${\displaystyle Q_{L}}$ and ${\displaystyle Q_{TS}}$. Since the three coefficients ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, and
${\displaystyle a_{3}}$ are a function of ${\displaystyle Q_{L}}$, ${\displaystyle Q_{TS}}$, h, and ${\displaystyle \alpha }$, fixing one of these parameters yields three equations that uniquely
determine the other three. In the case where a particular transducer is already given, ${\displaystyle Q_{TS}}$ is essentially fixed. If the desired parameters of the enclosure are already known,
then ${\displaystyle Q_{L}}$ is a better starting point.
In the case that the rigid requirements of the Butterworth alignment cannot be satisfied, the quasi-Butterworth alignment is often applied when ${\displaystyle Q_{TS}}$ is not large enough.. The
addition of another parameter, B, allows more flexibility in the design.
For ${\displaystyle Q_{TS}}$ values that are too large for the Butterworth alignment, the Chebyshev alignment is typically chosen. However, the steep transition of the Chebyshev alignment may also be
utilized to attempt to extend the bass response of the loudspeaker in the case where the transducer properties can be changed.
In addition to these three popular alignments, research continues in the area of developing new algorithms that can manipulate the low-frequency response of the bass-reflex enclosure. For example, a
5th order quasi-Butterworth alignment has been developed [6]; its advantages include improved low frequency extension, and much reduced driver excursion at low frequencies and typically bi-amping or
tri-amping, while its disadvatages include somewhat difficult mathematics and electronic complication (electronic crossovers are typically required). Another example [7] applies root-locus techniques
to achieve results. In the modern age of high-powered computing, other researchers have focused their efforts in creating computerized optimization algorithms that can be modified to achieve a
flatter response with sharp roll-off or introduce quasi-ripples which provide a boost in sub-bass frequencies [8].
[1] Leach, W. Marshall, Jr. Introduction to Electroacoustics and Audio Amplifier Design. 2nd ed. Kendall/Hunt, Dubuque, IA. 2001.
[2] Beranek, L. L. Acoustics. 2nd ed. Acoustical Society of America, Woodbridge, NY. 1993.
[3] DeCarlo, Raymond A. “The Butterworth Approximation.” Notes from ECE 445. Purdue University. 2004.
[4] DeCarlo, Raymond A. “The Chebyshev Approximation.” Notes from ECE 445. Purdue University. 2004.
[5] VanValkenburg, M. E. Analog Filter Design. Holt, Rinehart and Winston, Inc. Chicago, IL. 1982.
[6] Kreutz, Joseph and Panzer, Joerg. "Derivation of the Quasi-Butterworth 5 Alignments." Journal of the Audio Engineering Society. Vol. 42, No. 5, May 1994.
[7] Rutt, Thomas E. "Root-Locus Technique for Vented-Box Loudspeaker Design." Journal of the Audio Engineering Society. Vol. 33, No. 9, September 1985.
[8] Simeonov, Lubomir B. and Shopova-Simeonova, Elena. "Passive-Radiator Loudspeaker System Design Software Including Optimization Algorithm." Journal of the Audio Engineering Society. Vol. 47, No.
4, April 1999.
Appendix A: Equivalent Circuit Parameters
┃ Name │ Electrical Equivalent │ Mechanical Equivalent │ Acoustical Equivalent ┃
┃ Voice-Coil Resistance │ ${\displaystyle R_{E}}$ │ ${\displaystyle R_{ME}={\frac {(Bl)^{2}}{R_ │${\displaystyle R_{AE}={\frac {(Bl)^{2}}{R_{E}S_{D}^┃
┃ │ │ {E}}}}$ │ {2}}}}$ ┃
┃ Driver (Speaker) Mass │ See ${\displaystyle C_{MEC}}$ │ ${\displaystyle M_{MD}}$ │${\displaystyle M_{AD}={\frac {M_{MD}}{S_{D}^{2}}}}$┃
┃ Driver (Speaker) Suspension │ ${\displaystyle L_{CES}=(Bl)^{2}C_{MS}}$ │ ${\displaystyle C_{MS}}$ │ ${\displaystyle C_{AS}=S_{D}^{2}C_{MS}}$ ┃
┃ Compliance │ │ │ ┃
┃ Driver (Speaker) Suspension │ ${\displaystyle R_{ES}={\frac {(Bl)^{2}}{R_{MS}}}}$ │ ${\displaystyle R_{MS}}$ │${\displaystyle R_{AS}={\frac {R_{MS}}{S_{D}^{2}}}}$┃
┃ Resistance │ │ │ ┃
┃ Enclosure Compliance │${\displaystyle L_{CEB}={\frac {(Bl)^{2}C_{AB}}{S_{D}^{2}}}}$│ ${\displaystyle C_{MB}={\frac {C_{AB}}{S_{D}^ │ ${\displaystyle C_{AB}}$ ┃
┃ │ │ {2}}}}$ │ ┃
┃ Enclosure Air-Leak Losses │${\displaystyle R_{EL}={\frac {(Bl)^{2}}{S_{D}^{2}R_{AL}}}}$ │ ${\displaystyle R_{ML}=S_{D}^{2}R_{AL}}$ │ ${\displaystyle R_{AL}}$ ┃
┃ Acoustic Mass of Port │${\displaystyle C_{MEP}={\frac {S_{D}^{2}M_{AP}}{(Bl)^{2}}}}$│ ${\displaystyle M_{MP}=S_{D}^{2}M_{AP}}$ │ ${\displaystyle M_{AP}}$ ┃
┃ Enclosure Mass Load │ See ${\displaystyle C_{MEC}}$ │ See ${\displaystyle M_{MC}}$ │ ${\displaystyle M_{AB}}$ ┃
┃Low-Frequency Radiation Mass Load│ See ${\displaystyle C_{MEC}}$ │ See ${\displaystyle M_{MC}}$ │ ${\displaystyle M_{A1}}$ ┃
┃ │${\displaystyle C_{MEC}={\frac {S_{D}^{2}M_{AC}}{(Bl)^{2}}}}$│${\displaystyle M_{MC}=S_{D}^{2}(M_{AB}+M_{A1}) │ ${\displaystyle M_{AC}=M_{AD}+M_{AB}+M_{A1}}$ ┃
┃ Combination Mass Load │ ${\displaystyle ={\frac {S_{D}^{2}(M_{AB}+M_{A1})+M_{MD}} │ +M_{MD}}$ │${\displaystyle ={\frac {M_{MD}}{S_{D}^{2}}}+M_{AB} ┃
┃ │ {(Bl)^{2}}}}$ │ │ +M_{A1}}$ ┃
Appendix B: Enclosure Parameter Formulas
Based on these dimensions [1],
${\displaystyle C_{AB}={\frac {V_{AB}}{\rho _{0}c_{0}^{2}}}}$ ${\displaystyle M_{AB}={\frac {B\rho _{eff}}{\pi a}}}$
${\displaystyle B={\frac {d}{3}}\left({\frac {S_{D}}{S_{B}}}\right)^{2}{\sqrt {\frac {\pi }{S_{D}}}}+{\ ${\displaystyle \rho _{0}\leq \rho _{eff}\leq \rho _{0}\left(1-{\frac {V_{fill}}{V_{B}}}\
frac {8}{3\pi }}\left[1-{\frac {S_{D}}{S_{B}}}\right]}$ right)+\rho _{fill}{\frac {V_{fill}}{V_{B}}}}$
${\displaystyle V_{AB}=V_{B}\left[1-{\frac {V_{fill}}{V_{B}}}\right]\left[1+{\frac {\gamma -1}{1+\gamma \left({\frac {V_{B}}{V_{fill}}}-1\right){\frac {\rho _{0}c_{air}}{\rho _{fill}c_{fill}}}}}\
${\displaystyle V_{B}=hwd}$ (inside enclosure gross volume) ${\displaystyle S_{B}=wh}$ (baffle area of the side the speaker is mounted on)
${\displaystyle c_{air}=}$specific heat of air at constant isovolumetric process (about ${\displaystyle ${\displaystyle c_{fill}=}$specific heat of filling at constant volume (${\displaystyle V_
0.718{\frac {\rm {kJ}}{\rm {kg.K}}}}$ at 300 K) {filling}}$)
${\displaystyle \rho _{0}=}$mean density of air (about ${\displaystyle 1.3{\frac {\rm {kg}}{\rm {m^{3}}}}} ${\displaystyle \rho _{fill}=}$ density of filling
$ at 300 K)
${\displaystyle \gamma =}$ ratio of specific heats (Isobaric/Isovolumetric processes) for air (about 1.4 ${\displaystyle c_{0}=}$ speed of sound in air (about 344 m/s)
at 300 K)
${\displaystyle \rho _{eff}}$ = effective density of enclosure. If little or no filling (acceptable assumption in a bass-reflex system but not for sealed enclosures), ${\displaystyle \rho _{eff}\
approx \rho _{0}}$
New Acoustic Filter For Ultrasonics Media
Acoustic filters are used in many devices such as mufflers, noise control materials (absorptive and reactive), and loudspeaker systems to name a few. Although the waves in simple (single-medium)
acoustic filters usually travel in gases such as air and carbon-monoxide (in the case of automobile mufflers) or in materials such as fiberglass, polyvinylidene fluoride (PVDF) film, or polyethylene
(Saran Wrap), there are also filters that couple two or three distinct media together to achieve a desired acoustic response. General information about basic acoustic filter design can be perused at
the following wikibook page Acoustic Filter Design & Implementation. The focus of this article will be on acoustic filters that use multilayer air/polymer film-coupled media as its acoustic medium
for sound waves to propagate through; concluding with an example of how these filters can be used to detect and extrapolate audio frequency information in high-frequency "carrier" waves that carry an
audio signal. However, before getting into these specific type of acoustic filters, we need to briefly discuss how sound waves interact with the medium(media) in which it travels and how these
factors can play a role when designing acoustic filters.
Changes in Media Properties Due to Sound Wave Characteristics
As with any system being designed, the filter response characteristics of an acoustic filter are tailored based on the frequency spectrum of the input signal and the desired output. The input signal
may be infrasonic (frequencies below human hearing), sonic (frequencies within human hearing range), or ultrasonic (frequencies above human hearing range). In addition to the frequency content of the
input signal, the density, and, thus, the characteristic impedance of the medium (media) being used in the acoustic filter must also be taken into account. In general, the characteristic impedance
${\displaystyle Z_{0}\,}$
for a particular medium is expressed as...
${\displaystyle Z_{0}=\pm \rho _{0}c\,}$ ${\displaystyle (Pa\cdot s/m)}$
${\displaystyle \pm \rho _{0}\,}$ = (equilibrium) density of medium ${\displaystyle (kg/m^{3})\,}$
${\displaystyle c\,}$ = speed of sound in medium ${\displaystyle (m/s)\,}$
The characteristic impedance is important because this value simultaneously gives an idea of how fast or slow particles will travel as well as how much mass is "weighting down" the particles in the
medium (per unit area or volume) when they are excited by a sound source. The speed in which sound travels in the medium needs to be taken into consideration because this factor can ultimately affect
the time response of the filter (i.e. the output of the filter may not radiate or attentuate sound fast or slow enough if not designed properly). The intensity
${\displaystyle I_{A}\,}$
of a sound wave is expressed as...
${\displaystyle I_{A}={\frac {1}{T}}\int _{0}^{T}pu\quad dt=\pm {\frac {P^{2}}{2\rho _{0}c}}\,}$ ${\displaystyle (W/m^{2})\,}$.
${\displaystyle I_{A}\,}$ is interpreted as the (time-averaged) rate of energy transmission of a sound wave through a unit area normal to the direction of propagation, and this parameter is also an
important factor in acoustic filter design because the characteristic properties of the given medium can change relative to intensity of the sound wave traveling through it. In other words, the
reaction of the particles (atoms or molecules) that make up the medium will respond differently when the intensity of the sound wave is very high or very small relative to the size of the control
area (i.e. dimensions of the filter, in this case). Other properties such as the elasticity and mean propagation velocity (of a sound wave) can change in the acoustic medium as well, but focusing on
frequency, impedance, and/or intensity in the design process usually takes care of these other parameters because most of them will inevitably be dependent on the aforementioned properties of the
Why Coupled Acoustic Media in Acoustic Filters?
In acoustic transducers, media coupling is employed in acoustic transducers to either increase or decrease the impedance of the transducer, and, thus, control the intensity and speed of the signal
acting on the transducer while converting the incident wave, or initial excitation sound wave, from one form of energy to another (e.g. converting acoustic energy to electrical energy). Specifically,
the impedance of the transducer is augmented by inserting a solid structure (not necessarily rigid) between the transducer and the initial propagation medium (e.g. air). The reflective properties of
the inserted medium is exploited to either increase or decrease the intensity and propagation speed of the incident sound wave. It is the ability to alter, and to some extent, control, the impedance
of a propagation medium by (periodically) inserting (a) solid structure(s) such as thin, flexible films in the original medium (air) and its ability to concomitantly alter the frequency response of
the original medium that makes use of multilayer media in acoustic filters attractive. The reflection factor and transmission factor ${\displaystyle {\hat {R}}\,}$ and ${\displaystyle {\hat {T}}\,}$,
respectively, between two media, expressed as...
${\displaystyle {\hat {R}}={\frac {pressure\ of\ reflected\ portion\ of\ incident\ wave}{pressure\ of\ incident\ wave}}={\frac {\rho c-Z_{in}}{\rho c+Z_{in}}}\,}$
${\displaystyle {\hat {T}}={\frac {pressure\ of\ transmitted\ portion\ of\ incident\ wave}{pressure\ of\ incident\ wave}}=1+{\hat {R}}\,}$,
are the tangible values that tell how much of the incident wave is being reflected from and transmitted through the junction where the media meet. Note that ${\displaystyle Z_{in}\,}$ is the (total)
input impedance seen by the incident sound wave upon just entering an air-solid acoustic media layer. In the case of multiple air-columns as shown in Fig. 2, ${\displaystyle Z_{in}\,}$ is the
aggregate impedance of each air-column layer seen by the incident wave at the input. Below in Fig. 1, a simple illustration explains what happens when an incident sound wave propagating in medium (1)
and comes in contact with medium (2) at the junction of the both media (x=0), where the sound waves are represented by vectors.
As mentioned above, an example of three such successive air-solid acoustic media layers is shown in Fig. 2 and the electroacoustic equivalent circuit for Fig. 2 is shown in Fig. 3 where ${\
displaystyle L=\rho _{s}h_{s}\,}$ = (density of solid material)(thickness of solid material) = unit-area (or volume) mass, ${\displaystyle Z=\rho c=\,}$ characteristic acoustic impedance of medium,
and ${\displaystyle \beta =k=\omega /c=\,}$ wavenumber. Note that in the case of a multilayer, coupled acoustic medium in an acoustic filter, the impedance of each air-solid section is calculated by
using the following general purpose impedance ratio equation (also referred to as transfer matrices)...
${\displaystyle {\frac {Z_{a}}{Z_{0}}}={\frac {\left({\frac {Z_{b}}{Z_{0}}}\right)+j\ \tan(kd)}{1+j\ \left({\frac {Z_{b}}{Z_{0}}}\right)\tan(kd)}}\,}$
where ${\displaystyle Z_{b}\,}$ is the (known) impedance at the edge of the solid of an air-solid layer (on the right) and ${\displaystyle Z_{a}\,}$ is the (unknown) impedance at the edge of the air
column of an air-solid layer.
Effects of High-Intensity, Ultrasonic Waves in Acoustic Media in Audio Frequency Spectrum
When an ultrasonic wave is used as a carrier to transmit audio frequencies, three audio effects are associated with extrapolating the audio frequency information from the carrier wave: (a) beating
effects, (b) parametric array effects, and (c) radiation pressure.
Beating occurs when two ultrasonic waves with distinct frequencies ${\displaystyle f_{1}\,}$ and ${\displaystyle f_{2}\,}$ propagate in the same direction, resulting in amplitude variations which
consequently make the audio signal information go in and out of phase, or “beat”, at a frequency of ${\displaystyle f_{1}-f_{2}\,}$.
Parametric array effects occur when the intensity of an ultrasonic wave is so high in a particular medium that the high displacements of particles (atoms) per wave cycle changes properties of that
medium so that it influences parameters like elasticity, density, propagation velocity, etc. in a non-linear fashion. The results of parametric array effects on modulated, high-intensity, ultrasonic
waves in a particular medium (or coupled media) is the generation and propagation of audio frequency waves (not necessarily present in the original audio information) that are generated in a manner
similar to the nonlinear process of amplitude demodulation commonly inherent in diode circuits (when diodes are forward biased).
Another audio effect that arises from high-intensity ultrasonic beams of sound is a static (DC) pressure called radiation pressure. Radiation pressure is similar to parametric array effects in that
amplitude variations in the signal give rise to audible frequencies via amplitude demodulation. However, unlike parametric array effects, radiation pressure fluctuations that generate audible signals
from amplitude demodulation can occur due to any low-frequency modulation and not just from pressure fluctuations occurring at the modulation frequency ${\displaystyle \omega _{M}\,}$ or beating
frequency ${\displaystyle f_{1}-f_{2}\,}$.
An Application of Coupled Media in Acoustic Filters
Figs. 1 - 3 were all from a research paper entitled New Type of Acoustics Filter Using Periodic Polymer Layers for Measuring Audio Signal Components Excited by Amplitude-Modulated High_Intensity
Ultrasonic Waves submitted to the Audio Engineering Society (AES) by Minoru Todo, Primary Innovator at Measurement Specialties, Inc., in the October 2005 edition of the AES Journal. Figs. 4 and 5
below, also from this paper, are illustrations of test setups referred to in this paper. Specifically, Fig. 4 is a test setup used to measure the transmission (of an incident ultrasonic sound wave)
through the acoustic filter described by Figs. 1 and 2. Fig. 5 is a block diagram of the test setup used for measuring radiation pressure, one of the audio effects mentioned in the previous section.
It turns out that out of all of the audio effects mentioned in the previous section that are caused by high-intensity ultrasonic waves propagating in a medium, sound waves produced from radiated
pressure are the hardest to detect when microphones and preamplifiers are used in the detection/receiver system. Although nonlinear noise artifacts occur due to overloading of the preamplifier
present in the detection/receiver system, the bulk of the nonlinear noise comes from the inherent nonlinear noise properties of microphones. This is true because all microphones, even specialized
measurement microphones designed for audio spectrum measurements that have sensitivity well beyond the threshold of hearing, have nonlinearities artifacts that (periodically) increase in magnitude
with respect to increase at ultrasonic frequencies. These nonlinearities essentially mask the radiation pressure generated because the magnitude of these nonlinearities are orders of magnitude
greater than the radiation pressure. The acoustic (low-pass) filter referred to in this paper was designed in order to filter out the "detrimental" ultrasonic wave that was inducing high nonlinear
noise artifacts in the measurement microphones. The high-intensity, ultrasonic wave was producing radiation pressure (which is audible) within the initial acoustic medium (i.e. air). By filtering out
the ultrasonic wave, the measurement microphone would only detect the audible radiation pressure that the ultrasonic wave was producing in air. Acoustic filters like these could possibly be used to
detect/receive any high-intensity, ultrasonic signal that may carry audio information which may need to be extrapolated with an acceptable level of fidelity.
[1] Minoru Todo, "New Type of Acoustic Filter Using Periodic Polymer Layers for Measuring Audio Signal Components Excited by Amplitude-Modulated High-Intensity Ultrasonic Waves," Journal of Audio
Engineering Society, Vol. 53, pp. 930–41 (2005 October)
[2] Fundamentals of Acoustics; Kinsler et al., John Wiley & Sons, 2000
[3] ME 513 Course Notes, Dr. Luc Mongeau, Purdue University
[4] http://www.ieee-uffc.org/archive/uffc/trans/Toc/abs/02/t0270972.htm
Noise in Hydraulic Systems
Noise in Hydraulic Systems
Hydraulic systems are the most preferred source of power transmission in most of the industrial and mobile equipments due to their advantages in power density, compactness, flexibility, fast response
and efficiency. The field hydraulics and pneumatics is also known as 'Fluid Power Technology'. Fluid power systems have a wide range of applications which include industrial, off-road vehicles,
automotive system and aircrafts. In spite of these advantages, there are also some disadvantages. One of the main drawbacks with the hydraulic fluid power systems is the vibration and noise generated
by them. The health and safety issues relating to noise, vibration and harshness (NVH) have been recognized for many years and legislation is now placing clear demands on manufacturers to reduce
noise levels [1]. Hence, a lot of attention has been paid on reducing the noise of hydraulic fluid power systems both from the industrial and academic researchers. A good understanding of the noise
generation, transmission and propagation is very important in order to improve the NVH performance of hydraulic fluid power systems.
Sound in fluids
The speed of sound in fluids can be determined using the following relation.
${\displaystyle c={\sqrt {\frac {K}{\rho }}}}$ where K - fluid bulk modulus, ${\displaystyle \rho }$- fluid density, c - velocity of sound
Typical value of bulk modulus range from 2e9 to 2.5e9 N/m2. For a particular oil, with a density of 889 kg/m3,
speed of sound ${\displaystyle c={\sqrt {\frac {2e9}{889}}}=1499.9m/s}$
Source of Noise
The main source of noise in hydraulic systems is the pump which supplies the flow. Most of the pumps used are positive displacement pumps. Of the positive displacement pumps, axial piston swash plate
type is mostly preferred due to their controllability and efficiency.
The noise generation in an axial piston pump can be classified under two categories:
(i) fluidborne noise and
(ii) Structureborne noise
Fluidborne Noise (FBN)
Among the positive displacement pumps, highest levels of FBN are generated by axial piston pumps and lowest levels by screw pumps and in between these lie the external gear pump and vane pump [1].
The discussion in this page is mainly focused on axial piston swash plate type pumps. An axial piston pump has a fixed number of displacement chambers arranged in a circular pattern separated from
each other by an angular pitch equal to ${\displaystyle \phi ={\frac {360}{n}}}$ where n is the number of displacement chambers. As each chamber discharges a specific volume of fluid, the discharge
at the pump outlet is sum of all the discharge from the individual chambers. The discontinuity in flow between adjacent chambers results in a kinematic flow ripple. The amplitude of the kinematic
ripple can be theoretically determined given the size of the pump and the number of displament chambers. The kinematic ripple is the main cause of the fluidborne noise. The kinematic ripples is a
theoretical value. The actual flow ripple at the pump outlet is much larger than the theoretical value because the kinematic ripple is combined with a compressibility component which is due to the
fluid compressibility. These ripples (also referred as flow pulsations) generated at the pump are transmitted through the pipe or flexible hose connected to the pump and travel to all parts of the
hydraulic circuit.
The pump is considered an ideal flow source. The pressure in the system will be decided by resistance to the flow or otherwise known as system load. The flow pulsations result in pressure pulsations.
The pressure pulsations are superimposed on the mean system pressure. Both the flow and pressure pulsations easily travel to all part of the circuit and affect the performance of the components like
control valve and actuators in the system and make the component vibrate, sometimes even resonate. This vibration of system components adds to the noise generated by the flow pulsations. The
transmission of FBN in the circuit is discussed under transmission below.
A typical axial piston pump with 9 pistons running at 1000 rpm can produce a sound pressure level of more than 70 dBs.
Structureborne Noise (SBN)
In swash plate type pumps, the main sources of the structureborne noise are the fluctuating forces and moments of the swas plate. These fluctuating forces arise as a result of the varying pressure
inside the displacement chamber. As the displacing elements move from suction stroke to discharge stroke, the pressure varies accordingly from few bars to few hundred bars. This pressure changes are
reflected on the displacement elements (in this case, pistons) as forces and these force are exerted on the swash plate causing the swash plate to vibrate. This vibration of the swash plate is the
main cause of structureborne noise. There are other components in the system which also vibrate and lead to structureborne noise, but the swash is the major contributor.
Fig. 1 shows an exploded view of axial piston pump. Also the flow pulsations and the oscillating forces on the swash plate, which cause FBN and SBN respectively are shown for one revolution of the
The transmission of FBN is a complex phenomenon. Over the past few decades, considerable amount of research had gone into mathematical modeling of pressure and flow transient in the circuit. This
involves the solution of wave equations, with piping treated as a distributed parameter system known as a transmission line [1] & [3].
Lets consider a simple pump-pipe-loading valve circuit as shown in Fig. 2. The pressure and flow ripple at ay location in the pipe can be described by the relations:
${\displaystyle {\frac {}{}}P=Ae^{-kx}+Be^{-kx}}$ .........(1)
${\displaystyle Q={\frac {1}{Z_{0}}}(Ae^{-kx}-Be^{-kx})}$.....(2)
where ${\displaystyle {\frac {}{}}A}$ and ${\displaystyle {\frac {}{}}B}$ are frequency dependent complex coefficients which are directly proportional to pump (source) flow ripple, but also functions
of the source impedance ${\displaystyle {\frac {}{}}Z_{s}}$, characteristic impedance of the pipe ${\displaystyle {\frac {}{}}Z_{0}}$ and the termination impedance ${\displaystyle {\frac {}{}}Z_{t}}$
. These impedances ,usually vary as the system operating pressure and flow rate changes, can be determined experimentally.
Fig.2 Schematic of a pump connected to a hydraulic line
For complex systems with several system components, the pressure and flow ripples are estimated using the transformation matrix approach. For this, the system components can be treated as lumped
impedances (a throttle valve or accumulator), or distributed impedances (flexible hose or silencer). Various software packages are available today to predict the pressure pulsations.
The transmission of SBN follows the classic source-path-noise model. The vibrations of the swash plate, the main cause of SBN, is transferred to the pump casing which encloses all the rotating group
in the pump including displacement chambers (also known as cylinder block), pistons and the swash plate. The pump case, apart from vibrating itself, transfers the vibration down to the mount on which
the pump is mounted. The mount then passes the vibrations down to the main mounted structure or the vehicle. Thus the SBN is transferred from the swash plate to the main strucuture or vehicle via
pumpcasing and mount.
Some of the machine structures, along the path of transmission, are good at transmitting this vribational energy and they even resonate and reinforce it. By converting only a fraction of 1% of the
pump structureborne noise into sound, a member in the transmission path could radiate more ABN than the pump itself [4].
Airborne noise (ABN)
Both FBN and SBN , impart high fatigue loads on the system components and make them vibrate. All of these vibrations are radiated as airborne noise and can be heard by a human operator. Also, the
flow and pressure pulsations make the system components such as a control valve to resonate. This vibration of the particular component again radiates airborne noise.
Noise reduction
The reduction of the noise radiated from the hydraulic system can be approached in two ways.
(i) Reduction at Source - which is the reduction of noise at the pump. A large amount of open literature are availabale on the reduction techniques with some techniques focusing on reducing FBN at
source and others focusing on SBN. Reduction in FBN and SBN at the source has a large influence on the ABN that is radiated. Even though, a lot of progress had been made in reducing the FBN and SBN
separately, the problem of noise in hydraulic systems is not fully solved and lot need to be done. The reason is that the FBN and SBN are interlated, in a sense that, if one tried to reduce the FBN
at the pump, it tends to affect the SBN characteristics. Currently, one of the main researches in noise reduction in pumps, is a systematic approach in understanding the coupling between FBN and SBN
and targeting them simultaneously instead of treating them as two separte sources. Such an unified approach, demands not only well trained researchers but also sophisticated computer based
mathematical model of the pump which can accurately output the necessary results for optimization of pump design. The amplitude of fluid pulsations can be reduced, at the source, with the use of an
hydraulic attenuator(5).
(ii) Reduction at Component level - which focuses on the reduction of noise from individual component like hose, control valve, pump mounts and fixtures. This can be accomplished by a suitable design
modification of the component so that it radiates least amount of noise. Optimization using computer based models can be one of the ways.
Hydraulic System noise
Fig.3 Domain of hydraulic system noise generation and transmission (Figure recreated from [1])
1. Designing Quieter Hydraulic Systems - Some Recent Developements and Contributions, Kevin Edge, 1999, Fluid Power: Forth JHPS International Symposium.
2. Fundamentals of Acoustics, L.E. Kinsler, A.R. Frey, A.B.Coppens, J.V. Sanders. Fourth Edition. John Wiley & Sons Inc.
3. Reduction of Axial Piston Pump Pressure Ripple, A.M. Harrison. PhD thesis, University of Bath. 1997
4. Noise Control of Hydraulic Machinery, Stan Skaistis, 1988. MARCEL DEKKER , INC.
5. Hydraulic Power System Analysis, A. Akers, M. Gassman, & R. Smith, Taylor & Francis, New York, 2006, ISBN 0-8247-9956-9
6. Experimental studies of the vibro-acoustic characteristics of an axial piston pump under run-up and steady-state operating conditions, Shaogan Ye et al., 2018, Measurement, 133.
7. Sound quality evaluation and prediction for the emitted noise of axial piston pumps, Junhui Zhang, Shiqi Xia, Shaogan Ye et al., 2018, Applied Acoustics 145:27-40.
Basic Acoustics of the Marimba
Marimba Band "La Gloria Antigueña", Antigua Guatemala, 1979
Like a xylophone, a marimba has octaves of wooden bars that are struck with mallets to produce tones. Unlike the harsh sound of a xylophone, a marimba produces a deep, rich tone. Marimbas are not
uncommon and are played in most high school bands.
Now, while all the trumpet and flute and clarinet players are busy tuning up their instruments, the marimba player is back in the percussion section with her feet up just relaxing. This is a bit
surprising, however, since the marimba is a melodic instrument that needs to be in tune to sound good. So what gives? Why is the marimba never tuned? How would you even go about tuning a marimba? To
answer these questions, the acoustics behind (or within) a marimba must be understood.
Components of Sound
What gives the marimba its unique sound? It can be boiled down to two components: the bars and the resonators. Typically, the bars are made of rosewood (or some synthetic version of wood). They are
cut to size depending on what note is desired, then the tuning is refined by shaving wood from the underside of the bar.
Example: Rosewood bar, middle C, 1 cm thick
The equation that relates the length of the bar with the desired frequency comes from the theory of modeling a bar that is free at both ends. This theory yields the following equation:
${\displaystyle Length={\sqrt {\frac {3.011^{2}\cdot \pi \cdot t\cdot v}{8\cdot {\sqrt {12}}\cdot f}}}}$
where t is the thickness of the bar, v is the speed of sound in the bar, and f is the frequency of the note. For rosewood, v = 5217 m/s. For middle C, f=262 Hz. Therefore, to make a middle C key for
a rosewood marimba, cut the bar to be:
${\displaystyle Length={\sqrt {\frac {3.011^{2}\cdot \pi \cdot .01\cdot 5217}{8\cdot {\sqrt {12}}\cdot 262}}}=.45m=45cm}$
The resonators are made from metal (usually aluminum) and their lengths also differ depending on the desired note. It is important to know that each resonator is open at the top but closed by a
stopper at the bottom end.
Example: Aluminum resonator, middle C
The equation that relates the length of the resonator with the desired frequency comes from modeling the resonator as a pipe that is driven at one end and closed at the other end. A "driven" pipe is
one that has a source of excitation (in this case, the vibrating key) at one end. This model yields the following:
${\displaystyle Length={\frac {c}{4\cdot f}}}$
where c is the speed of sound in air and f is the frequency of the note. For air, c = 343 m/s. For middle C, f = 262 Hz. Therefore, to make a resonator for the middle C key, the resonator length
should be:
${\displaystyle Length={\frac {343}{4\cdot 262}}=.327m=32.7cm}$
Resonator Shape
The shape of the resonator is an important factor in determining the quality of sound that can be produced. The ideal shape is a sphere. This is modeled by the Helmholtz resonator. (For more see
Helmholtz Resonator page) However, mounting big, round, beach ball-like resonators under the keys is typically impractical. The worst choices for resonators are square or oval tubes. These shapes
amplify the non-harmonic pitches sometimes referred to as “junk pitches”. The round tube is typically chosen because it does the best job (aside from the sphere) at amplifying the desired harmonic
and not much else.
As mentioned in the second example above, the resonator on a marimba can be modeled by a closed pipe. This model can be used to predict what type of sound (full and rich vs dull) the marimba will
produce. Each pipe is a "quarter wave resonator" that amplifies the sound waves produced by of the bar. This means that in order to produce a full, rich sound, the length of the resonator must
exactly match one-quarter of the wavelength. If the length is off, the marimba will produce a dull or off-key sound for that note.
Why would the marimba need tuning?
In the theoretical world where it is always 72 degrees with low humidity, a marimba would not need tuning. But, since weather can be a factor (especially for the marching band) marimbas do not always
perform the same way. Hot and cold weather can wreak havoc on all kinds of percussion instruments, and the marimba is no exception. On hot days, the marimba tends to be sharp and for cold days it
tends to be flat. This is the exact opposite of what happens to string instruments. Why? The tone of a string instrument depends mainly on the tension in the string, which decreases as the string
expands with heat. The decrease in tension leads to a flat note. Marimbas on the other hand produce sound by moving air through the resonators. The speed at which this air is moved is the speed of
sound, which varies proportionately with temperature! So, as the temperature increases, so does the speed of sound. From the equation given in example 2 from above, you can see that an increase in
the speed of sound (c) means a longer pipe is needed to resonate the same note. If the length of the resonator is not increased, the note will sound sharp. Now, the heat can also cause the wooden
bars to expand, but the effect of this expansion is insignificant compared to the effect of the change in the speed of sound.
Tuning Myths
It is a common myth among percussionists that the marimba can be tuned by simply moving the resonators up or down (while the bars remain in the same position.) The thought behind this is that by
moving the resonators down, for example, you are in effect lengthening them. While this may sound like sound reasoning, it actually does not hold true in practice. Judging by how the marimba is
constructed (cutting bars and resonators to specific lengths), it seems that there are really two options to consider when looking to tune a marimba: shave some wood off the underside of the bars, or
change the length of the resonator. For obvious reasons, shaving wood off the keys every time the weather changes is not a practical solution. Therefore, the only option left is to change the length
of the resonator. As mentioned above, each resonator is plugged by a stopper at the bottom end. So, by simply shoving the stopper farther up the pipe, you can shorten the resonator and sharpen the
note. Conversely, pushing the stopper down the pipe can flatten the note. Most marimbas do not come with tunable resonators, so this process can be a little challenging. (Broomsticks and hammers are
common tools of the trade.)
Example: Middle C Resonator lengthened by 1 cm
For ideal conditions, the length of the middle C (262 Hz) resonator should be 32.7 cm as shown in example 2. Therefore, the change in frequency for this resonator due to a change in length is given
${\displaystyle \Delta Frequency=262Hz-{\frac {c}{4\cdot (.327+\Delta L)}}}$
If the length is increased by 1 cm, the change in frequency will be:
${\displaystyle \Delta Frequency={\frac {343}{4\cdot (.327+.01)}}-262Hz=7.5Hz}$
The acoustics behind the tuning a marimba go back to the design that each resonator is to be ¼ of the total wavelength of the desired note. When marimbas get out of tune, this length is no longer
exactly equal to ¼ the wavelength due to the lengthening or shortening of the resonator as described above. Because the length has changed, resonance is no longer achieved, and the tone can become
muffled or off-key.
Some marimba builders are now changing their designs to include tunable resonators. There are in fact several marimba companies that have had tunable resonators for decades. However, only a few offer
full range tuning. Since any leak in the end-seal will cause major loss of volume and richness of the tone, this is proving to be a very difficult task. At least now, though, armed with the acoustic
background of their instruments, percussionists everywhere will now have something to do when the conductor says, “tune up!”
Links and References
How an Acoustic Guitar works
There are three main parts of the guitar that contribute to sound production.
First of all, there are strings. Any string that is under tension will vibrate at a certain frequency. The tension and gauge in the string determine the frequency at which it vibrates. The guitar
controls the length and tension of six differently weighted strings to cover a very wide range of frequencies.
Second of all, there is the body of the guitar. The guitar body is connected directly to one end of each of the strings. The body receives the vibrations of the strings and transmits them to the air
around the body. It is the body’s large surface area that allows it to “push” a lot more air than a string.
Finally, there is the air inside the body. This is very important for the lower frequencies of the guitar. The air mass just inside the soundhole oscillates, compressing and decompressing the
compliant air inside the body. In practice, this concept is called a Helmholtz resonator. Without this, it would difficult to produce the wonderful timbre of the guitar.
The Strings
The strings of the guitar vary in linear density, length, and tension. This gives the guitar a wide range of attainable frequencies. The larger the linear density is, the slower the string vibrates.
The same goes for the length; the longer the string is the slower it vibrates. This causes a low frequency. Inversely, if the strings are less dense and/or shorter they create a higher frequency. The
lowest resonance frequencies of each string can be calculated by
${\displaystyle f_{1}={\frac {1}{2L}}{\sqrt {\frac {T}{\rho _{1}}}}}$ where ${\displaystyle T}$= string tension, ${\displaystyle \rho _{1}}$=linear density, ${\displaystyle L}$ = string length
The string length, L, in the equation is what changes when a player presses on a string at a certain fret. This will shorten the string which in turn increases the frequency it produces when plucked.
The spacing of these frets is important. The length from the nut to bridge determines how much space goes between each fret. If the length is 25 inches, then the position of the first fret should be
located (25/17.817) inches from the nut. Then the second fret should be located (25-(25/17.817))/17.817 inches from the first fret. This results in the equation
When a string is plucked, a disturbance is formed and travels in both directions away from point where the string was plucked. These "waves" travel at a speed that is related to the tension and
linear density and can be calculated by
The waves travel until they reach the boundaries on each end where they are reflected back. The link below displays how the waves propagate in a string.
Plucked String @ www.phys.unsw.edu
The strings themselves do not produce very much sound because they are so thin. This is why they are connected to the top plate of the guitar body. They need to transfer the frequencies they are
producing to a large surface area which can create more intense pressure disturbances.
The Body
The body of the guitar transfers the vibrations of the bridge to the air that surrounds it. The top plate contributes to most of the pressure disturbances, because the player dampens the back plate
and the sides are relatively stiff. This is why it is important to make the top plate out of a light springy wood, like spruce. The more the top plate can vibrate, the louder the sound it produces
will be. It is also important to keep the top plate flat, so a series of braces are located on the inside to strengthen it. Without these braces the top plate would bend and crack under the large
stress created by the tension in the strings. This would also affect the magnitude of the sound being transmitted. The warped plate would not be able to "push" air very efficiently. A good experiment
to try, in order to see how important this part of the guitar is in the amplification process, is as follows:
1. Start with an ordinary rubber band, a large bowl, adhesive tape, and plastic wrap.
2. Stretch the rubber band and pluck it a few times to get a good sense for how loud it is.
3. Stretch the plastic wrap over the bowl to form a sort of drum.
4. Tape down one end of the rubber band to the plastic wrap.
5. Stretch the rubber band and pluck it a few times.
6. The sound should be much louder than before.
The Air
The final part of the guitar is the air inside the body. This is very important for the lower range of the instrument. The air just inside the sound hole oscillates, compressing and expanding the air
inside the body. This is just like blowing across the top of a bottle and listening to the tone it produces. This forms what is called a Helmholtz resonator. For more information on Helmholtz
resonators go to Helmholtz Resonance. This link also shows the correlation to acoustic guitars in great detail. The acoustic guitar makers often tune these resonators to have a resonance frequency
between F#2 and A2 (92.5 to 110.0 Hz)(Hz stands for Hertz). Having such a low resonance frequency is what aids the amplification of the lower frequency strings. To demonstrate the importance of the
air in the cavity, simply play an open A on the guitar (the second string). Now, as the string is vibrating, place a piece of cardboard over the sound hole. The sound level is reduced dramatically.
This is because you've stopped the vibration of the air mass just inside the sound hole, causing only the top plate to vibrate. Although the top plate still vibrates and transmits sound, it isn't as
effective at transmitting lower frequency waves, thus the need for the Helmholtz resonator.
Specific application-automobile muffler
General information about Automobile muffler
A muffler is a part of the exhaust system on an automobile that plays a vital role. It needs to have modes that are located away from the frequencies that the engine operates at, whether the engine
be idling or running at the maximum amount of revolutions per second.A muffler that affects an automobile in a negative way is one that causes noise or discomfort while the car engine is
running.Inside a muffler, you'll find a deceptively simple set of tubes with some holes in them. These tubes and chambers are actually as finely tuned as a musical instrument. They are designed to
reflect the sound waves produced by the engine in such a way that they partially cancel themselves out.( cited from www.howstuffworks.com )
It is very important to have it on the automobile. The legal limit for exhaust noise in the state of California is 95 dB (A) - CA. V.C. 27151 .Without a muffler the typical car exhaust noise would
exceed 110 dB.A conventional car muffler is capable of limiting noise to about 90 dB. The active-noise canceling muffler enables cancellation of exhaust noise to a wide range of frequencies.
The Configuration of A automobile muffler
How Does automobile muffler function?
General Concept
The simple and main part of designing the automobile muffler is to use the low-pass filter. It typically makes use of the change of the cross section area which can be made as a chamber to filter or
reduce the sound wave which the engine produced.
Low-Pass Filter
A low-pass filter is a circuit that passes low frequency signals but stops the high frequency signals. Once the low pass filter is set by the user at a specific cutoff frequency, all frequencies
lower than that will be passed through the filter, while higher frequencies will be attenuated in amplitude. This circuit is made up of passive components (resistor, capacitors and inductors) capable
of accomplishing this objective. File:Inductive law pass filter.jpg
the formula to be used:
Human ear sound reaction feature
When these pressure pulses reach your ear, the eardrum vibrates back and forth. Your brain interprets this motion as sound. Two main characteristics of the wave determine how we perceive the sound:
1.sound wave frequency. 2.air wave pressure amplitude.
It turns out that it is possible to add two or more sound waves together and get less sound.
Description of the muffler to cancel the noise
The key thing about sound waves is that the result at your ear is the sum of all the sound waves hitting your ear at that time. If you are listening to a band, even though you may hear several
distinct sources of sound, the pressure waves hitting your ear drum all add together, so your ear drum only feels one pressure at any given moment. Now comes the cool part: It is possible to produce
a sound wave that is exactly the opposite of another wave. This is the basis for those noise-canceling headphones you may have seen. Take a look at the figure below. The wave on top and the second
wave are both pure tones. If the two waves are in phase, they add up to a wave with the same frequency but twice the amplitude. This is called constructive interference. But, if they are exactly out
of phase, they add up to zero. This is called destructive interference. At the time when the first wave is at its maximum pressure, the second wave is at its minimum. If both of these waves hit your
ear drum at the same time, you would not hear anything because the two waves always add up to zero.
Benefits of an Active Noise-Canceling Muffler
1. By using an active muffler the exhaust noise can be easily tuned, amplified, or nearly eliminated.
2. The backpressure of a conventional muffler can be essentially eliminated, thus increasing engine performance and efficiency.
3. By increasing engine efficiency and performance, less fuel will be used and the emissions will be reduced.
Absorptive muffler
Lined ducts
It can be regarded as simplest form of absorptive muffler. Attach absorptive material to the bare walls of the duct.( in car that is the exhaustion tube) The attenuation performance improves with the
thickness of absorptive material.
The attenuation curves like a skewed bell. Increase the thickness of the wall will get the lower maximum attenuation frequency. For higher frequency though, thinner absorbent layers are effective,
but the large gap allows noise to pass directly along. Thin layers and narrow passages are therefore more effective at high frequencies. For good absorption over the widest frequency range, thick
absorbent layers and narrow passages are best.
Parallel and block-line-of-sight baffles
Divide the duct into several channels or turn the | {"url":"https://en.m.wikibooks.org/wiki/Engineering_Acoustics/Print_version","timestamp":"2024-11-10T11:13:03Z","content_type":"text/html","content_length":"1050051","record_id":"<urn:uuid:79a7e9ca-8e9f-46b4-87d8-1de34ee85cad>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00165.warc.gz"} |
American Mathematical Society
When representing the coefficients and knots of a spline using only small integers, independently rounding each infinite-precision value is not the best strategy. We show how to build an affine model
for the error expanded about the optimal full-precision free-knot or parameterized spline, then use the Lovász basis reduction algorithm to select a better rounding. The technique could be used for
other situations in which a quadratic error model can be computed. References
C. de Boor and J. R. Rice, Least squares cubic spline approximation. II: variable knots, Technical report, CSD TR 21, Purdue Univ., Lafayette, IN, 1968. J. E. Dennis, Jr., D. M. Gay, and R. E.
Welch, An adaptive nonlinear least squares algorithm, ACM Trans. Math. Software 7 (1981), 348-368, 369-383. F. N. Fritsch and G. M. Nielson, On the problem of determining the distance between
parametric curves, Technical report, Lawrence Livermore National Laboratory UCRL-JC-105408, 1990. B. A. LaMacchia, Basis reduction algorithms and subset sum problems, Master’s thesis, EECS Dept.,
Massachusetts Institute of Technology, 1991. N. L. Schryer, SSAF-smooth spline approximations to functions, Technical report, AT&T Bell Laboratories CSTR 131, 1986. TIGER/LINE census files, 1990,
technical documentation, Technical report, Bureau of the Census, U. S. Dept. of Commerce, Washington, D.C., 1991. P. van Emde Boas, Another NP-complete partition problem and the complexity of
computing short vectors in a lattice, Report 81-04, Math. Institute, Univ. of Amsterdam, 1981. K. Wall and P.-E. Danielsson, A fast sequential method for polygonal approximation of digitized
curves, Computer Vision, Graphics, and Image Processing, vol. 28, 1984, pp. 220-227.
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC: 65D07
• Retrieve articles in all journals with MSC: 65D07
Additional Information
• © Copyright 1994 American Mathematical Society
• Journal: Math. Comp. 63 (1994), 175-194
• MSC: Primary 65D07
• DOI: https://doi.org/10.1090/S0025-5718-1994-1240659-8
• MathSciNet review: 1240659 | {"url":"https://www.ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1240659-8/home.html","timestamp":"2024-11-14T22:13:54Z","content_type":"text/html","content_length":"65972","record_id":"<urn:uuid:45966051-c3c7-48c1-9abd-1aa6e36524fe>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00429.warc.gz"} |
unable to make G1000 work
Hello all, I am new to X-Plane, using version 10.51 with a phenom 100 and garmin G1000 avionics suite. I paid for the Phenom aircraft files and I contacted X-Plane tech support and they sent me the
link to download the G1000 database.
I put the Ceranado file into the root directory of x-plane 10 as instructed to to use the G1000
when I run the x-plane app the Phenom cockpit shows up but the radios are completely blank, like they are getting no power. Nothing I do seems to make a difference. I just can't turn them on.
Getting the radios to work is problem #1.
Problem #2 is that when I hover over rotary selectors, I get both a + and - rotational arrows, I can't figure out how to make the rotary selectors move. This is unlike using other aircraft where if I
am on one side of a knob I'll get a clockwise arrow and a counter-clockwise arrow on the other side of the knob. How do I get these knobs to turn?
Problem #1 is much more serious, I'll need to get the Primary and Multi-function displays working.
Any suggestions?
- Bill
Hi Bill
I'm not with Laminar Research, just a fellow simmer in the UK, but sorry to say that I also purchased the Carenado Phenom 100 for my X-Plane 11 and putting it politely, I'm very disappointed with it
Whilst the PFD/MFD are both seemingly switched on, I can't get many of the rotary controls or switches to function properly and I'm therefore finding the aircraft almost impossible to use!
To make matters worse, it appears that this aircraft does not support SIDS, STARS or ILS approach procedures in X-Plane 11 either :-(
I have therefore given this aircraft up as a complete loss and waste of money, something other simmers have also been experiencing. | {"url":"https://questions.x-plane.com/23933/unable-to-make-g1000-work","timestamp":"2024-11-04T05:07:22Z","content_type":"text/html","content_length":"19678","record_id":"<urn:uuid:f916ee56-ee67-4e05-a3c6-7b0c3f9d5abe>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00694.warc.gz"} |
The kids’ favourite festival, Lunar New Year, is approaching real soon and they’re getting crazy over lion dance, dragon dance and drums again. When a child is motivated to figure out something that
matters to him, he’d be hungry to learn. No need for any worksheets, assessment books or tests. We happened to be at […]
[Math] Solving real math problems again Read More »
[Math] Solving REAL math problems
Since attending Primary 1, Vee has little time for home practice. Nevertheless, learning takes place anytime anywhere, so here’s how he got to solve 3 real math problems yesterday: 1. Me (on the way
home): “How much water did you drink today?” Vee: “2 and a half bottles!” Me: “Wow, let’s find out how much
[Math] Solving REAL math problems Read More »
[IQ math] with chopsticks
Years ago, I bought 5 packs of wooden chopsticks from Daiso as math manipulatives. Today I challenged Vee to solve IQ math using chopsticks arrangements. E.g. 12 sticks form 4 squares in the picture.
How to move 4 sticks to form 3 squares? (In puzzle books, toothpicks are commonly used. But I find chopsticks much
[IQ math] with chopsticks Read More »
Egg science & math
A fellow mummy gave me half a tray of kampung eggs. The hospitality among Malaysians is amazing! Anyway, the eggs sparked off some rather interesting conversations & learning: – our
commercially-farmed eggs (on the left) are brown; the kampung eggs are pale. Why?!? – the kampung eggs are taller than our regular eggs. Why?!? –
Egg science & math Read More »
Math is everywhere
While spring-cleaning, I found a huge piece of quilted cloth that my mum made for Vee when he was a baby. Makes a great playmat for all of us now! One of the first things we did was to figure out how
many squares there were. 4yo Jae was really fast cos he’s been
Math is everywhere Read More »
Free flash cards – math & games (Ten Green Bottles)
Recently, the younger kids love it when I sing “Ten Green Bottles”. So I was inspired to make a set of materials on this topic. The materials cover math quantity, numerals and senses / memory games
cards. 1. Flashcards I drew one green bottle and replicated it. These are printed on A4 (or you may
Free flash cards – math & games (Ten Green Bottles) Read More »
How to teach time
Vee has been able to understand time rather well quite some time back, maybe at 4-5 years old. And I realised it’s a very useful skill for a child. Here’s what we do to learn about time: 1. Flash
cards To teach time, it’s important to use the analog clock instead of digital clock. This
Easy math with Montessori beads
Here’s more on blending right brain and Montessori math. After practising with addition and subtraction using the Hundred Board, Vee is still keen to learn more math. (He chose the math tray from the
shelves.) So I’m showing him how to solve similar equations using Montessori math beads. Some notes: I love Montessori beads because they’re very attractive
Easy math with Montessori beads Read More »
“Right brain” math at breakfast
I often get enquiries on how to teach toddlers and preschoolers math. So here’s more sharing on what we do in an easy and fun way… Usually, we do “right brain” math over breakfast. For instance,
yesterday, I rolled 10 bread bits, showed them to El and said, “This is 10.” When doing “right brain” math, there
“Right brain” math at breakfast Read More »
Right brain & Montessori math [Part 2]
Here’s another simple activity that blends right brain and Montessori math… 1. Use a thick marker pen to write numerals on small pieces of card stock. Jae and I worked on 0 to 10 first. Make 2 sets
of such cards. 2. Invite the child to stick the corresponding number of dot stickers onto the
Right brain & Montessori math [Part 2] Read More » | {"url":"https://www.mummyshomeschool.com/tag/math/page/2/","timestamp":"2024-11-12T21:54:08Z","content_type":"text/html","content_length":"183084","record_id":"<urn:uuid:d39dd13f-96ef-4221-8726-05d054af97d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00759.warc.gz"} |
Hajime Ishihara
• Japan Advanced Institute of Science and Technology (JAIST), Nomi, Ishikawa, Japan
According to our database
, Hajime Ishihara authored at least 63 papers between 1990 and 2024.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
From Proofs to Computation in Geometric Logic and Generalizations (Dagstuhl Seminar 24021).
Dagstuhl Reports, 2024
Extended Frames and Separations of Logical Principles.
Bull. Symb. Log., September, 2023
Reflexive combinatory algebras.
J. Log. Comput., July, 2023
Geometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472).
Dagstuhl Reports, 2021
Algebraic combinatory models.
CoRR, 2021
On the independence of premiss axiom and rule.
Arch. Math. Log., 2020
Equivalents of the finitary non-deterministic inductive definitions.
Ann. Pure Appl. Log., 2019
The binary expansion and the intermediate value theorem in constructive reverse mathematics.
Arch. Math. Log., 2019
Consistency of the intensional level of the Minimalist Foundation with Church's thesis and axiom of choice.
Arch. Math. Log., 2018
Preface to the special issue: Continuity, computability, constructivity: from logic to algorithms 2013.
Math. Struct. Comput. Sci., 2017
Embedding classical in minimal implicational logic.
Math. Log. Q., 2016
A note on the independence of premiss rule.
Math. Log. Q., 2016
Completeness and cocompleteness of the categories of basic pairs and concrete spaces.
Math. Struct. Comput. Sci., 2015
Generalized geometric theories and set-generated classes.
Math. Struct. Comput. Sci., 2015
Some principles weaker than Markov's principle.
Arch. Math. Log., 2015
Classical propositional logic and decidability of variables in intuitionistic propositional logic.
Log. Methods Comput. Sci., 2014
Uniformly convex Banach spaces are reflexive - constructively.
Math. Log. Q., 2013
Relating Bishop's function spaces to neighbourhood spaces.
Ann. Pure Appl. Log., 2013
The Weak Koenig Lemma, Brouwer's Fan Theorem, De Morgan's Law, and Dependent Choice.
Reports Math. Log., 2012
The uniform boundedness theorem and a boundedness principle.
Ann. Pure Appl. Log., 2012
Two subcategories of apartness spaces.
Ann. Pure Appl. Log., 2012
A predicative completion of a uniform space.
Ann. Pure Appl. Log., 2012
On the contrapositive of countable choice.
Arch. Math. Log., 2011
Apartness, compactness and nearness.
Theor. Comput. Sci., 2008
Separation properties in neighbourhood and quasi-apartness spaces.
Math. Log. Q., 2008
Editorial: Math. Log. Quart. 5/2008.
Math. Log. Q., 2008
Computability and Complexity in Analysis.
J. Univers. Comput. Sci., 2008
A continuity principle, a version of Baire's theorem and a boundedness principle.
J. Symb. Log., 2008
A VGA 30-fps Realtime Optical-Flow Processor Core for Moving Picture Recognition.
IEICE Trans. Electron., 2008
Unique Existence and Computability in Constructive Reverse Mathematics.
Proceedings of the Computation and Logic in the Real World, 2007
Binary Refinement Implies Discrete Exponentiation.
Stud Logica, 2006
Weak König's Lemma Implies Brouwer's Fan Theorem: A Direct Proof.
Notre Dame J. Formal Log., 2006
Quotient topologies in constructive set theory and type theory.
Ann. Pure Appl. Log., 2006
Quasi-apartness and neighbourhood spaces.
Ann. Pure Appl. Log., 2006
Brouwer's fan theorem and unique existence in constructive analysis.
Math. Log. Q., 2005
Constructivity, Computability, and Logic A Collection of Papers in Honour of the 60th Birthday of Douglas Bridges.
J. Univers. Comput. Sci., 2005
On constructing completions.
J. Symb. Log., 2005
Optical Manipulation of Nano Materials under Quantum Mechanical Resonance Conditions.
IEICE Trans. Electron., 2005
Strong continuity implies uniform sequential continuity.
Arch. Math. Log., 2005
Compactness in apartness spaces?
Proceedings of the Spatial Representation: Discrete vs. Continuous Computational Models, 2005
Constructive reverse mathematics: compactness properties.
Proceedings of the From sets and types to topology and analysis, 2005
Compactness under constructive scrutiny.
Math. Log. Q., 2004
Completeness of intersection and union type assignment systems for call-by-value lambda-models.
Theor. Comput. Sci., 2002
A Constructive Look at The Completeness of The Space D(R).
J. Symb. Log., 2002
Complexity of Some Infinite Games Played on Finite Graphs.
Proceedings of the Graph-Theoretic Concepts in Computer Science, 2002
Some Results on Automatic Structures.
Proceedings of the 17th IEEE Symposium on Logic in Computer Science (LICS 2002), 2002
Compactness and Continuity, Constructively Revisited.
Proceedings of the Computer Science Logic, 16th International Workshop, 2002
Coding with Minimal Programs.
Int. J. Found. Comput. Sci., 2001
Sequentially Continuity in Constructive Mathematics.
Proceedings of the Combinatorics, 2001
A Note on the Gödel-Gentzen Translation.
Math. Log. Q., 2000
A Canonical Model Construction for Substructural Logics.
J. Univers. Comput. Sci., 2000
Function algebraic characterizations of the polytime functions.
Comput. Complex., 1999
A Definitive Constructive Open Mapping Theorem?
Math. Log. Q., 1998
Computable Kripke Models and Intermediate Logics.
Inf. Comput., 1998
Decidable Kripke Models of Intuitionistic Theories.
Ann. Pure Appl. Log., 1998
Effectiveness of the Completeness Theorem for an Intermediate Logic.
J. Univers. Comput. Sci., 1997
Sequential Continuity of Linear Mappings in Constructive Mathematics.
J. Univers. Comput. Sci., 1997
Absolute Continuity and the Uniqueness of the Constructive Functional Calculus.
Math. Log. Q., 1994
Complements of Intersections in Constructive Mathematics.
Math. Log. Q., 1994
Continuity Properties in Constructive Mathematics.
J. Symb. Log., 1992
Continuity and Nondiscontinuity in Constructive Mathematics.
J. Symb. Log., 1991
Constructive Compact Operators on a Hilbert Space.
Ann. Pure Appl. Log., 1991
An omniscience principle, the König Lemma and the Hahn-Banach theorem.
Math. Log. Q., 1990 | {"url":"https://www.csauthors.net/hajime-ishihara/","timestamp":"2024-11-13T23:05:48Z","content_type":"text/html","content_length":"70350","record_id":"<urn:uuid:77aa4ff8-bf0a-4a25-a184-e6dae645b6ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00802.warc.gz"} |
Check a number is even or odd in Python | Dremendo
Check a number is even or odd in Python
if else - Question 1
In this question, we will see how to check if a number is an even or odd number in Python programming using the if else statement. To know more about if else statement click on the if else statement
Q1) Write a program in Python to input a number and check it is an even or odd number. An even number is a number which is divisible by 2 and odd number is a number which is not divisible by 2.
n=int(input('Enter a number '))
if n%2==0:
print('Even Number')
print('Odd Number')
Enter a number 18
Even Number | {"url":"https://www.dremendo.com/python-programming-tutorial/python-if-else-questions/q1-check-number-even-or-odd-in-python","timestamp":"2024-11-15T03:39:28Z","content_type":"text/html","content_length":"30288","record_id":"<urn:uuid:72cd44bd-0299-4e52-9268-c0d09528eedc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00061.warc.gz"} |
PHY 464: Quantum Mechanics I,
PHY 464: Quantum Mechanics I, Fall 2023
Instructor: Prof. Thomas Barthel
Lectures: Mondays and Wednesdays 10:05-11:20am in Biological Sciences 130
Office hours: Mon+Wed 11:30am-12:30pm in Physics 287
Turorials: t.b.a.
Teaching assistants: Jhao-Hong Peng (Physics 294),
Gyeonghun Kim (Chesterfield Bldg, Suite 400)
Grading: problem sets (40%), midterm exam (20%), final exam (40%), class participation
This course provides a systematic introduction to quantum mechanics. We will first discuss a few interesting experimental observations that lead to the development of quantum theory and then go
through the mathematical basis, the postulates of QM, and their interpretation. Considering different one-dimensional quantum systems, we will encounter phenomena like quantum tunneling and
scattering. We will see further how higher-dimensional and composite systems can be described using tensor products. This allows us to understand the essential concept of entanglement as well as Bell
inequalities. An important tool for the study of quantum systems is perturbation theory. We will discuss this technique for static problems and its convergence properties. A discussion of symmetries
will lead us to angular momentum and spin operators, and it will allow us to understand the electronic states of the hydrogen atom. Time permitting, we will learn perturbative and non-perturbative
methods to study the dynamics of quantum systems.
Some knowledge of linear algebra at the level of Math 216, 218 or 221 is needed. Please check back with the instructors if you had no prior exposure to QM (PHY 264). We will keep the course as
self-contained as possible.
Lecture Notes
[Are provided on the Sakai site PHYSICS.464D.01D.F23.]
You are encouraged to discuss homework assignments with fellow students and to ask questions on the Sakai Forum or by email. But the written part of the homework must be done individually and cannot
be a copy of another student's solution. (See the Duke Community Standard.)
Homework due dates are strict (for the good of all), i.e., late submissions are not accepted. If there are grave reasons, you can ask for an extension early enough before the due date.
[Problem sets are provided through Gradescope on the Sakai site PHYSICS.464D.01D.F23.]
The primary reading resource for the course is the textbook
• Shankar "Principles of Quantum Mechanics" 2nd Edition, Plenum Press (1994)
Further recommended textbooks on quantum mechanics:
• Cohen-Tannoudji, Diu, Laloe "Quantum Mechanics", Wiley (1991, 1992)
• Sakurai "Modern Quantum Mechanics", Addison Wesley (1993)
• Le Bellac "Quantum Physics", Cambridge University Press (2006)
• Ballentine "Quantum Mechanics", World Scientific (1998)
• Merzbacher "Quantum Mechanics" 3rd Edition, Wiley (1998)
• Schwabl "Quantum Mechanics" 4th Edition, Springer (2007)
• Gasiorowicz "Quantum Physics" 3rd Edition, Wiley (2003)
• Galindo, Pascual "Quantum Mechanics I & II", Springer (1991) | {"url":"http://webhome.phy.duke.edu/~barthel/L2023-08_QM1_phy464/","timestamp":"2024-11-03T23:19:16Z","content_type":"text/html","content_length":"6105","record_id":"<urn:uuid:f1b7e118-d08a-4866-a23f-a259e23ee975>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00767.warc.gz"} |
Catch Up Growth Calculator - Online Calculators
Enter the values e.g future value, present value, number of periods etc, to use our basic and advanced Catch Up Growth Calculator!
The Catch Up Growth Calculator is built to calculate the catch-up growth rate. This rate indicates the percentage of growth required to reach a desired future value from a current value over a
specified number of periods.
The formula is:
$\text{CG} = \frac{\text{FV} - \text{PV}}{n \times \text{PV}}$
Variable Meaning
CG Catch-Up Growth (the rate of growth needed to catch up)
FV Future Value (the targeted or desired value)
PV Present Value (the current value or starting point)
n Time period (the number of periods over which the growth is measured)
How to Calculate ?
Firstly, decide the Future Value (FV), which represents the targeted value to be reached. After that, find the Present Value (PV), which is the starting point or current value. After that, identify
the number of time periods (n) over which the growth is expected to occur. Finally, subtract the present value from the future value, then divide the result by the product of the present value and
the number of time periods to calculate the catch-up growth (CG).
Solved Calculation:
Example 1:
• Future Value (FV) = 1000
• Present Value (PV) = 800
• Time period (n) = 5 years
Calculation Instructions
Step 1: CG = $\frac{\text{FV} - \text{PV}}{n \times \text{PV}}$ Start with the formula.
Step 2: CG = $\frac{1000 - 800}{5 \times 800}$ Replace FV with 1000, PV with 800, and n with 5.
Step 3: CG = $\frac{200}{4000}$ Subtract 800 from 1000 to get 200, and multiply 5 by 800.
Step 4: CG = 0.05 Divide 200 by 4000 to get the catch-up growth.
The catch-up growth is 5% per period.
Example 2:
• Future Value (FV) = 1500
• Present Value (PV) = 1200
• Time period (n) = 3 years
Calculation Instructions
Step 1: CG = $\frac{\text{FV} - \text{PV}}{n \times \text{PV}}$ Start with the formula.
Step 2: CG = $\frac{1500 - 1200}{3 \times 1200}$ Replace FV with 1500, PV with 1200, and n with 3.
Step 3: CG = $\frac{300}{3600}$ Subtract 1200 from 1500 to get 300, and multiply 3 by 1200.
Step 4: CG = 0.0833 Divide 300 by 3600 to get the catch-up growth.
The catch-up growth is 8.33% per period.
What is Catch Up Growth:
Catch-up growth refers to a period when a child grows at a faster rate than expected to compensate for a previous growth delay, often seen in infants and young children recovering from malnutrition,
illness, or being born prematurely. A Catch-up Growth Calculator helps you see for the additional nutritional needs required for this accelerated growth, using factors like the child’s current
weight, height, and age. By calculating daily caloric and protein requirements (usually measured in kcal/kg/day), caregivers and healthcare providers can ensure that infants receive the right amount
of nutrients to support their growth.
The Catch-up Growth Calculator is particularly helpful for tracking the progress of infants diagnosed with failure to thrive, or those who have experienced slowed growth due to intrauterine growth
restriction (IUGR). By inputting the child's current measurements and expected growth targets, the calculator provides a an effective nutritional plan, following guidelines set by the World Health
Organization (WHO) and other pediatric standards.
Final Words:
Catch-up growth occurs when a child experiences faster-than-normal growth after a period of delay. A Catch-up Growth Calculator helps to calculate the nutritional needs necessary for required growth.
Thereby, it ensures that infants receive the right amount of calories and nutrients. | {"url":"https://areacalculators.com/catch-up-growth-calculator/","timestamp":"2024-11-04T00:57:42Z","content_type":"text/html","content_length":"110563","record_id":"<urn:uuid:af5ce4ce-a6a2-48ea-916f-0b22b4ab9e93>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00351.warc.gz"} |
System and method for controlling multi-zone vapor compression system
System and method for controlling multi-zone vapor compression system
A system controls a multi-zone vapor compression system (MZ-VCS). The system includes a controller to control a vapor compression cycle of the MZ-VCS using a set of control inputs determined by
optimizing a cost function including a set of control parameters. The optimizing is subject to constraints, and wherein the cost function is optimized over a prediction horizon. The system also
includes a memory to store an optimization function parameterized by a configuration of the MZ-VCS defining active or inactive modes of each heat exchanger, the optimization function modifies,
according to a current configuration, values of the control parameters of the cost function determined for a full configuration that includes all heat exchangers in the active mode. The system also
includes a processor to determine the current configuration of the MZ-VCS and to update the cost function by submitting the current configuration to the optimization function.
This invention relates to vapor compression systems, and more particularly to a system and a method for controlling a multiple-zone vapor compression system.
Vapor compression systems (VCS) move thermal energy between a low temperature environment and a high temperature environment in order to perform cooling or heating operations so that the comfort of
the occupants in an indoor space can be maintained or improved. For example, heat can be moved from an indoor space to an outdoor space in order to lower the indoor temperature or mitigate the effect
of thermal energy infiltrating an indoor space in a cooling operation. Conversely, heat can be moved from an outdoor space to an indoor space in order to raise the indoor temperature or mitigate the
effect of thermal energy exfiltration an indoor space in a heating operation.
A multi-zone vapor compression system (MZ-VCS) includes at least a single compressor and single outdoor heat exchanger connected to multiple indoor heat exchangers arranged in one or more indoor
zones. Refrigerant flow is split among the heat exchangers and modulated with flow metering valves arranged between the indoor heat exchangers and outdoor heat exchanger. These flow metering valves
can also serve as the main pressure reducing device required to lower the refrigerant temperature and pressure in order to complete the vapor compression cycle. Depending on the state of a four-way
valve connected to the compressor, high pressure refrigerant can flow from the compressor to the outdoor unit (in which case the outdoor unit heat exchanger is a condenser and the indoor heat
exchangers are evaporators) or refrigerant can flow from the compressor to the heat exchangers and the roles of the indoor and outdoor heat exchangers are reversed.
Recent advancements in power electronics and low cost micro-controllers have led to variable speed compressors, electronically controlled valves, and variable speed fans. The control of these
actuators must be coordinated to achieve zone temperature regulation, minimize energy consumption and enforce machine limitations such as a maximum safe pressure of the refrigerant or a maximum safe
temperature of a system component.
There is a need to control the overall operations of the MZ-VCS such that various constraints are enforced. For example, certain maximum or minimum temperatures and pressures should not be violated
for equipment safety. Some controllers enforce the constraints reactively, i.e., corrective action is taken once a dangerous situation is detected. In this strategy, the violations of the constraints
can occur for some period of time while the controller issues corrective actions, and therefore the threshold at which corrective action is initiated is selected conservatively to account for the
violations that are likely to occur. And since the operating regime of the highest system performance is often near the constraints, controllers with reactive constraint management that are designed
to operate away from the the constraints sacrifice the regions of highest performance, see, e.g., EP2469201.
One important requirement specific to multi-zone systems is the ability to deactivate one or more heat exchangers while remaining heat exchangers continue to provide service. An inactive heat
exchanger is characterized by an associated expansion valve that is closed, which ceases refrigerant flow through the heat exchanger thereby preventing heat exchange with the corresponding zone.
Additionally, the control objective of regulating the air temperature to a setpoint is not applicable in zones wherein the heat exchanger is inactive. The specific combination of active and inactive
heat exchangers is called the system configuration or just a configuration. In commercial MZ-VCS it is common to have 50 heat exchangers connected to an outdoor unit, creating 2^50=1.1×10^15 possible
configurations. When a heat exchanger changes from an active state to an inactive state, the MZ-VCS is said to have been reconfigured, and a system that permits reconfiguration is said to be
Accordingly, there is a need in the art for a system and a method to control every possible configuration of a reconfigurable MZ-VCS that is subject to constraints.
It is an object of some embodiments of the invention to provide a system and a method for controlling operations of a multi-zone vapor compression system (MZ-VCS). It is another object of some
embodiments of the invention to provide a system and method for controlling the vapor compression system predictively using a model of the system dynamics to determine and solve an optimization
problem such that constraints on the operation of the MZ-VCS are enforced. It is another object of some embodiments to control the operation of a MZ-VCS where zones are permitted to become active or
inactive. Further, it is an object of some embodiments that the controller can be modified online to adapt to the specific machine configuration, that is, the specific combination of heat exchangers
that are active and inactive.
Predictive control, e.g., a model predictive control (MPC), is based on an iterative, finite horizon optimization of a cost function that describes the operation of the controlled system and has the
ability to predict the MZ-VCS response to current conditions and take appropriate control actions. Further, constraints can be included in the formulation of this optimization problem. Some
embodiments of the invention are based on recognition that MPC offers attractive properties for vapor compression system control including guaranteed enforcement of constraints. Because constraint
enforcement can be guaranteed, selection of more aggressive constraints can lead to higher performance such as faster room temperature responses or safe operation over a wider range of outdoor air
MPC solves an optimization problem that encodes information about how changes in every zone affect the control objectives. Because deactivating a zone fundamentally changes the structure of the
optimization problem, different optimization problems specific to every system configuration need to be specified, but manually specifying an optimization problem for every configuration is not
practical for the large number of possible configurations. Further, the sets of different controller parameters encoding the different optimization problems would all need to be available at runtime,
requiring significantly more memory for parameter storage than is typically available for embedded hardware.
However, it is realized that a structured model describing the dynamics of a MZ-VCS can be obtained that reveals the specific coupling inherent to MZ-VCS. Specifically, some embodiments are based on
understanding that while the changes due to the outdoor unit components affect every heat exchanger, and each heat exchanger affects the outdoor unit, the specific heat exchangers largely do not
affect each other. This type of coupling results in a dynamic model that exhibits a particular structure—that is, the system of equations describing the MZ-VCS dynamics from control inputs to
measurements, when collected in matrix form, results in a specific pattern of zero-valued and non-zero-valued elements within the matrices.
It is further realized that by exploiting this pattern, an optimization problem can be formulated and parameterized by the system configuration, and, given the system configuration, an optimization
problem specific to the given configuration can be automatically obtained. Further, the closed loop stability resulting from the use of any specific optimization problem can be guaranteed by further
exploiting the model structure to compute structured controller parameters. In this manner, a reconfigurable control system is developed that retains the constraint enforcement advantages of MPC,
stable for any configuration, and does so without the burden of manually specifying different optimization problems for every system configuration.
Accordingly, one embodiment discloses a system for controlling a multi-zone vapor compression system (MZ-VCS) including a compressor connected to a set of heat exchangers for controlling environments
in a set of zones. The system comprises a controller to control a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing a cost function including a set of
control parameters, wherein the optimizing is subject to constraints, and wherein the cost function is optimized over a prediction horizon; a memory to store an optimization function parameterized by
a configuration of the MZ-VCS defining active or inactive modes of each heat exchanger, wherein the optimization function modifies values of the control parameters of the cost function according to
the configuration; and a processor to determine a current configuration of the MZ-VCS and to update the cost function by submitting the current configuration to the optimization function.
Another embodiment discloses a method for controlling a multi-zone vapor compression system (MZ-VCS) including a compressor connected to a set of heat exchangers for controlling environments in a set
of zones. The method includes determining a current configuration of the MZ-VCS defining active or inactive mode of each heat exchanger in the MZ-VCS; updating at least some values of control
parameters in a cost function by submitting the current configuration to an optimization function parameterized by a configuration of the MZ-VCS, wherein the optimization function modifies values of
the control parameters of the cost function according to the current configuration; and controlling a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing the
cost function subject to constraints. Steps of the method are performed using a processor.
Yet another embodiment discloses a non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method. The method includes determining a
current configuration of the MZ-VCS defining active or inactive mode of each heat exchanger in the MZ-VCS; updating at least some values of control parameters in a cost function by submitting the
current configuration to an optimization function parameterized by a configuration of the MZ-VCS, wherein the optimization function modifies values of the control parameters of the cost function
according to the current configuration; and controlling a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing the cost function subject to constraints.
In describing embodiments of the invention, the following definitions are applicable throughout (including above).
A “computer” refers to any apparatus that is capable of accepting a structured input, processing the structured input according to prescribed rules, and producing results of the processing as output.
Examples of a computer include a general-purpose computer; a supercomputer; a mainframe; a super mini-computer; a mini-computer; a workstation; a microcomputer; a server; an interactive television; a
hybrid combination of a computer and an interactive television; and application-specific hardware to emulate a computer and/or software. A computer can have a single processor or multiple processors,
which can operate in parallel and/or not in parallel.
A computer also refers to two or more computers connected together via a network for transmitting or receiving information between the computers. An example of such a computer includes a distributed
computer system for processing information via computers linked by a network.
A “central processing unit (CPU)” or a “processor” refers to a computer or a component of a computer that reads and executes software instructions.
A “memory” or a “computer-readable medium” refers to any storage for storing data accessible by a computer. Examples include a magnetic hard disk; a floppy disk; an optical disk, like a CD-ROM or a
DVD; a magnetic tape; a memory chip; and a carrier wave used to carry computer-readable electronic data, such as those used in transmitting and receiving e-mail or in accessing a network, and a
computer memory, e.g., random-access memory (RAM).
“Software” refers to prescribed rules to operate a computer. Examples of software include software; code segments; instructions; computer programs; and programmed logic. Software of intelligent
systems may be capable of self-learning.
A “module” or a “unit” refers to a basic component in a computer that performs a task or part of a task. It can be implemented by either software or hardware.
A “control system” refers to a device or a set of devices to manage, command, direct or regulate the behavior of other devices or systems. The control system can be implemented by either software or
hardware, and can include one or several modules.
A “computer system” refers to a system having a computer, where the computer comprises computer-readable medium embodying software to operate the computer.
A “network” refers to a number of computers and associated devices that are connected by communication facilities. A network involves permanent connections such as cables, temporary connections such
as those made through telephone or other communication links, and/or wireless connections. Examples of a network include an internet, such as the Internet; an intranet; a local area network (LAN); a
wide area network (WAN); and a combination of networks, such as an internet and an intranet.
A “vapor compression system” refers to a system that uses a vapor compression cycle to move refrigerant through components of the system based on principles of thermodynamics, fluid mechanics, and/or
heat transfer.
An “HVAC” system refers to any heating, ventilating, and air-conditioning (HVAC) system implementing the vapor compression cycle. HVAC systems span a very broad set of systems, ranging from systems
which supply only outdoor air to the occupants of a building, to systems which only control the temperature of a building, to systems which control the temperature and humidity.
“Components of a vapor compression system” refer to any components of the vapor compression system having an operation controllable by the control systems. The components include, but are not limited
to, a compressor having a variable speed for compressing and pumping the refrigerant through the system; an expansion valve for providing an adjustable pressure drop between the high-pressure and the
low-pressure portions of the system, and an evaporating heat exchanger and a condensing heat exchanger, each of which may incorporate a variable speed fan for adjusting the air-flow rate through the
heat exchanger.
An “evaporator” refers to a heat exchanger in the vapor compression system in which the refrigerant passing through the heat exchanger evaporates over the length of the heat exchanger, so that the
specific enthalpy of the refrigerant at the outlet of the heat exchanger is higher than the specific enthalpy of the refrigerant at the inlet of the heat exchanger, and the refrigerant generally
changes from a liquid to a gas. There may be one or more evaporators in the vapor-compression system.
A “condenser” refers to a heat exchanger in the vapor compression system in which the refrigerant passing through the heat exchanger condenses over the length of the heat exchanger, so that the
specific enthalpy of the refrigerant at the outlet of the heat exchanger is lower than the specific enthalpy of the refrigerant at the inlet of the heat exchanger, and the refrigerant generally
changes from a gas to a liquid. There may be one or more condensers in a vapor-compression system.
A “setpoint” refers to a target value the system, such as the vapor compression system, aims to reach and maintain as a result of the operation. The term setpoint is applied to any particular value
of a specific set of control signals and thermodynamic and environmental parameters.
“Heat load” refers to the thermal energy rate moved from a low temperature zone to a high temperature zone by the vapor compression system. The units typically associated with this signal are Joules
per second or Watts or British Thermal Units per hour (BTUs/hr).
“Thermal capacity” refers to the energy rate absorbed by a heat exchanger in a vapor compression system. The units typically associated with this signal are Joules per second or Watts or British
Thermal Units per hour (BTUs/hr).
“System configuration” or a “configuration” refers to the specific combination of activated heat exchangers and inactivated heat exchangers in a multi-zone vapor compression system.
An “active” heat exchanger is a heat exchanger for which the associated expansion valve is opened, allowing refrigerant to enter the heat exchanger. Conversely, an “inactive” heat exchanger is a heat
exchanger for which the associated expansion valve is closed, preventing refrigerant from entering the heat exchanger.
FIGS. 1A and 1B are block diagrams of a multi-zone vapor compression system (MZ-VCS) controlled according to principles employed by some embodiments of an invention;
FIG. 1C is a block diagram of a method for controlling a multi-zone vapor compression system (MZ-VCS) according to some embodiments of the invention;
FIG. 1D is an exemplar structure of a reconfigurable controller according to some embodiments of the invention;
FIG. 2A is a block diagram of a method for controlling the MZ-VCS of FIG. 1A or 1B according to one embodiment of the invention;
FIG. 2B is a signal diagram for the method of FIG. 2A;
FIG. 3A is a block diagram of a reconfigurable controller for controlling the MZ-VCS according to some embodiments of the invention;
FIGS. 3B and 3C are flow charts of methods for determining control parameters appropriate for an example configuration according to one embodiment of the invention; and
FIG. 4 is a flow chart of a method for model predictive control according to one embodiment of the invention.
A multi-zone vapor compression system (MZ-VCS) of some embodiments of the invention includes an ability to deactivate one or more heat exchangers while the remaining heat exchangers continue to
provide service. For instance, an occupant may anticipate that a zone in a space is unoccupied and can shut off the heat exchanger in order to reduce energy consumption by not conditioning the air in
the occupied space. In this case, the decision to deactivate a zone and the corresponding heat exchanger is determined by a source external (the occupant) to the MZ-VCS controller.
Additionally or alternatively, in one embodiment, the MZ-VCS controller can determine that the local heating or cooling loads in a particular zone are lower than the minimum continuously available
amount of heating or cooling provided by the heat exchanger and can automatically deactivate the heat exchanger. In this case, the MZ-VCS controller itself has determined that a particular zone is to
be deactivated. In either case, a deactivated heat exchanger is characterized by an associated expansion valve that is closed, and therefore no refrigerant flows through the heat exchanger.
Additionally, the control objective of regulating the air temperature to a setpoint is no longer applicable in zones wherein the heat exchanger has become deactivated.
To that end, various embodiments describe a system and method for controlling the operations of a multi-zone vapor compression system where individual zones are permitted to be activated or
deactivated. In some embodiments, a controller for determining the actuator commands and/or setpoints to inner feedback capacity controllers is implemented according to the principles of model
predictive control (MPC) wherein determining the actuator commands involves solving a receding horizon constrained optimization problem. The optimization problem includes a prediction model of the
dynamics of the MZ-VCS and a cost function that is to be optimized. The cost function includes penalty matrices that encode the desired closed loop performance of the system and guarantee dynamic
A configuration of the MZ-VCS defines active or inactive modes of each heat exchanger. Deactivating zones changes the configuration and implies that the control inputs in the associated deactivated
zone are not to be used and control objectives in the associated deactivated zone are not to be considered. Such a removal of the control inputs and change in the control objective fundamentally
modifies the relevant optimization problem. Preparing an appropriate optimization problem for a system that undergoes such fundamental structural changes is achieved with one or a combination of an
offline preparation of the control parameters of the cost function to be optimized and online modification of the control parameters in response to a change of the configuration of the MZ-VCS.
FIGS. 1A and 1B show block diagrams of a multi-zone vapor compression system (MZ-VCS) 100 controlled by a controller 101 according to principles employed by some embodiments of the invention. The
MZ-VCS includes a compressor and a set of heat exchangers configured for controlling environments in a set of zones. There is at least one heat exchanger for each zone. For example, in one embodiment
of FIG. 1A, each zone 125 or 135 corresponds to a room in a building enabling the MZ-VCS to provide cooling or heating to multiple zones simultaneously. In an alternative embodiment shown in FIG. 1B,
multiple heat exchangers are placed in one room or zone 137 in a building enabling the MZ-VCS to provide cooling or heating to different sections of the room.
In this disclosure, a two-zone MZ-VCS is depicted and described for clarity, but it should be understood that any number of zones can be used, subject to the physical limitations of refrigerant line
lengths, capacity and pumping power of the compressor, and building codes. If the zone is an indoor zone, such as a room or a portion of the room, the heat exchangers are indoor heat exchangers.
A compressor 110 receives a low-pressure refrigerant in a vapor state and performs mechanical work to increase the pressure and temperature of the refrigerant. Depending on the configuration of a
four-way valve 109, the high temperature refrigerant can be routed to either an outdoor heat exchanger (in which case the system moves heat to the outside environment and is providing useful cooling
and is said to operate in cooling mode) or to an indoor heat exchanger (in which case the system moves heat to one or more indoor zones and is providing useful heating and is said to operate in
heating mode).
For clarity and in order to simplify the subsequent description, a cooling mode is generally considered, i.e., the compressor is connected to the rest of the vapor compression system as shown as
solid lines of the four-way valve 109, but it should be understood that analogous statements can be made about the system operating in heating mode with appropriate substitutions, e.g., condenser for
evaporator, condensing temperature for evaporating temperature, etc.
In cooling mode, the high-temperature, high-pressure refrigerant moves to an outdoor heat exchanger 115 and in the case of an air-source vapor compression system, an associated optional fan 116 blows
air across the heat exchanger, where the air acts as a heat source or sink as shown in FIG. 1A or 1B. In the case of a ground-source vapor compression system, components of outdoor heat exchanger may
be buried underground or otherwise in direct contact with earth or water, and in that case, the ground environment acts as a heat source or sink. Heat is transferred from the refrigerant to the
environmental heat source or sink, causing the refrigerant in the outdoor heat exchanger to condense from a vapor to a liquid.
The phase change process wherein vapor refrigerant condenses from saturated vapor to a two-phase mixture of both liquid and vapor to saturated liquid is isothermal in ideal descriptions of the vapor
compression cycle, that is, the phase change process occurs at a constant temperature and therefore without a sensible change in temperature. However, if further heat is removed from the saturated
liquid, the temperature of the saturated liquid then decreases by some amount and the refrigerant is termed “subcooled.” The subcool temperature is the temperature difference between the subcooled
refrigerant and the calculated saturated liquid refrigerant temperature at the same pressure.
Liquid high temperature refrigerant exits the outdoor heat exchanger and is split by a manifold 117 in order to distribute the refrigerant between the subsequently connected indoor zones 125, 135 or
137. Separate expansion valves 126, 136 are connected to the inlet manifold. These expansion valves are restriction elements and cause the pressure of the refrigerant to be substantially reduced.
Since the pressure is reduced without substantial heat exchange in the valve, the temperature of the refrigerant is substantially reduced, termed “adiabatic” in ideal descriptions of the vapor
compression cycle. The resulting refrigerant exiting the valves is a low pressure, low temperature two-phase mixture of liquid and vapor.
Two-phase refrigerant enters the indoor heat exchangers 120, 130 where associated fans 121, 131 move air across the heat exchangers. Heat 122, 132 representing the thermal loads from the indoor
spaces is transferred from the zones to the refrigerant, causing the refrigerant to evaporate from a two-phase mixture of liquid and vapor to a saturated vapor state.
The phase change process wherein refrigerant evaporates from a saturated vapor to a two-phase mixture of both liquid and vapor to saturated vapor is isothermal in ideal descriptions of the vapor
compression cycle, i.e., occurs at a constant temperature and therefore is a process that occurs without a sensible change in temperature. However, if further heat is added to the saturated vapor,
the temperature of the saturated vapor then increases by some amount and the refrigerant is termed “superheated.” The superheat temperature is the difference between the superheated refrigerant vapor
and the calculated saturated vapor temperature at the same pressure.
The low pressure refrigerant vapor exiting the heat exchangers is rejoined to a common flow path at the outlet manifold 118. Finally, low pressure refrigerant vapor is returned to the compressor and
the cycle repeats.
In some embodiments of the invention, the MZ-VCS is controlled by a controller 200. For example, the controller 200 solves an optimization problem that encodes information about how changes in every
zone affect the control objectives. Because deactivating a zone fundamentally changes the structure of the optimization problem, different optimization problems specific to every system configuration
need to be specified.
The controller 200 is a predictive controller, such as MPC. Some embodiments are based on realization that it is possible to determine a structured model of the MZ-VCS describing the dynamics of the
MZ-VCS, which reveals the specific coupling among the components of the MZ-VCS. Specifically, some embodiments are based on understanding that while the changes due to the outdoor unit components
affect every heat exchanger, and each heat exchanger affects the outdoor unit, the specific heat exchangers largely do not affect each other. This type of coupling results in a dynamic model that
exhibits a particular structure—that is, the system of equations describing the MZ-VCS dynamics from control inputs to measurements, when collected in matrix form, results in a specific pattern of
zero-valued and non-zero-valued elements within the matrices. It is further realized that by exploiting this pattern, an optimization problem can be formulated and parameterized by the system
configuration, such that, given the system configuration, an optimization problem specific to the given configuration can be automatically obtained. To that end, the controller 200 is a
reconfigurable controller.
FIG. 1C shows a block diagram of a method for controlling a multi-zone vapor compression system (MZ-VCS) including a compressor connected to a set of heat exchangers for controlling environments in a
set of zones according to some embodiments of the invention. The method is performed by the controller 200. For example, the controller 200 can include a processor and a memory for performing steps
of the method.
The method determines 150 a current configuration 155 of the MZ-VCS defining active or inactive mode of each heat exchanger in the MZ-VCS and updates 160 at least some values of control parameters in
a cost function 165 by submitting the current configuration 155 to an optimization function 157 parameterized by a configuration of the MZ-VCS.
The optimization function modifies, according to a current configuration, values of the control parameters of the cost function determined for a full configuration that includes all heat exchangers
in the active mode. For example, a structure of the control parameters can correspond to a structure of a model of the MZ-VCS, such that there is a correspondence between control parameters and a
heat exchanger in the MZ-VCS. To that end, in some embodiments, the optimization function preserves the values of the control parameters if the corresponding heat exchanger is in the active mode and
modifies the values of the block if the corresponding heat exchanger is in the inactive mode.
For example, the configuration can be a binary vector having elements with a first value, e.g., a zero value, for the heat exchangers in the inactive mode and having elements with a second value,
e.g., a non-zero value, for the heat exchangers in the active mode. Such a correspondence can be established if, e.g., an index of the element in the configuration vector matches an index of a
corresponding heat exchanger.
For example, due to the coupling structure of the heat exchangers, the control parameters can be defined offline for full configuration of the MZ-VCS as a combination of the block matrices. An index
of each block on the diagonal of the matrix matches the index of the corresponding heat exchanger and values of each block on the diagonal of the matrix are determined for the corresponding heat
exchanger. For example, the block diagonal matrix can include one or a combination of a performance penalty matrix Q whose elements penalize outputs of the MZ-VCS, a control penalty matrix R whose
elements penalize control inputs to the MZ-VCS, and a terminal cost matrix P whose elements penalize terminal states of the MZ-VCS. Upon receiving the current configuration, the objective function
157 replaces the values of the blocks of the performance penalty matrix Q and the terminal cost matrix P with zeros if the corresponding heat exchanger is in the inactive mode, and wherein the
optimization function replaces the values of the block of the control penalty matrix R with values larger than initial values of the control penalty matrix if the corresponding heat exchanger is in
the inactive mode.
In various embodiments, the optimization function preserves the dimensions of the block diagonal matrix, which in turn, preserves the structure of the updated cost function 165. To that end, some
embodiments can optimize the cost function updated, i.e., configured for the specific configuration of the MZ-VCS, subject to constraints 167 to determine a set of control inputs 175 for controlling
a vapor compression cycle of the MZ-VCS. For example, the control inputs can be the inputs to one or combination of the compressor 110, the outdoor heat exchanger fan 116, the indoor heat exchanger
fans 121, 131 and the expansion valves 126, 136.
FIG. 1D shows an exemplar structure of the reconfigurable controller 200. The controller 200 can include a controller 180, such as one or combination of a supervisory controller described below and a
solver for optimizing the cost function 165, to control a vapor compression cycle of the MZ-VCS using the control inputs 175. The controller can be implemented, e.g., using a microprocessor or any
other programmable electronic device which accepts digital or binary data as input, processes the input according to instructions stored in its memory, and provides results as output.
Additionally or alternatively, the reconfigurable controller 200 can include a memory 190 to store the optimization function parameterized by a configuration of the MZ-VCS defining active or inactive
modes of each heat exchanger, and a processor 185 to determine the current configuration of the MZ-VCS and to update the cost function by submitting the current configuration to the optimization
function. In some embodiments, the controller, the memory, and the processor are interconnected to facilitate the operation of the controller 200. For example, the processor 185 can be used to
implement some of the functionality of the controller 180. Similarly, the memory 190 can include a non-transitory computer readable storage medium embodied thereon a program executable by a processor
for performing the method of FIG. 1C.
FIG. 2A is a block diagram of a method for controlling the MZ-VCS of FIG. 1A or 1B according to one embodiment of the invention. FIG. 2B is a signal diagram for the method of FIG. 2A. The MZ-VCS 100
is controlled by the reconfigurable controller 200 that determines control inputs forming commands subsequently issued to the actuators of the MZ-VCS. The commands can include a compressor speed
command 250, an outdoor unit fan speed command 251, or heat exchanger fan speed commands 252, 253. The heat exchanger fan speed commands may alternatively be determined by the occupants as described
below. The reconfigurable controller 200 receives sensor information 271 from sensors 270 arranged at various locations on the system. The spatial arrangement of sensors are not depicted in FIG. 2A
for clarity and simplicity, and their precise locations within the system are not pertinent to the invention. Additionally, the controller receives setpoint information 231 from an external source
such as an input interface 230 that allows an occupant to enter entering the desired zone temperatures.
In some embodiments, the compressor speed command 250 can be fixed to one or more predetermined settings or can be varied continuously. Similarly, the outdoor heat exchanger fans 116 can operate at
fixed speeds or the speeds can be varied continuously. In some configurations, an indoor heat exchanger fan 121, 131 can be determined by the MZ-VCS controller 200, or its speed can be determined by
an occupant when the occupant wishes to directly control indoor airflow. In the case an indoor fan speed is determined by the controller, the fan speed is treated by the controller as a control input
for manipulating the operation of the system. In the case an indoor fan speed is specified by an occupant, the fan speed is treated by the controller as measured disturbance acting on the system. The
expansion valves 126, 136 are controlled by the controller and can vary from a fully closed to a fully open position, including one or more intermediate positions.
In some embodiments, the MZ-VCS replaces electronically-controlled expansion valves with a series combination of a solenoid valve for on/off control, and a separate variable opening valve for precise
flowrate control. The control inputs associated with these actuators are the compressor rotational frequency (CF) command 250, the outdoor fan speed (ODF) command 251, and each electronic expansion
valve opening position (EEV[i]) command 211, 221.
Additional disturbances acting on the ME-VCS include the heat load 122, 132 associated with each zone and the outdoor air temperature (OAT). Heat loads are the amount of thermal energy moved from the
heat exchangers to the outdoor unit per unit time. The total heat is then rejected to the atmosphere at the outdoor heat exchanger temperature, which is determined by both the OAT (a disturbance
signal) and the state of the machine actuators.
The available sensors 270 can include temperature sensors that measure the evaporating temperature Te, the condensing temperature Tc, the compressor discharge temperature Td, and the air temperature
Tr[i ]in each zone, labeled 271 in FIGS. 2A and 2B, or that measure other temperatures, pressures, or flow rates. Additionally, each heat exchanger may include heat exchanger coil temperature sensors
(HX coil) that measure the refrigerant temperature at various locations along the heat exchanger, labeled 272 in FIGS. 2A and 2B.
Some embodiments include a reconfigurable controller, such as MPC, and a set of N capacity controllers, as shown in FIGS. 2A and 2B. The capacity controllers 210 receive commands 202 from the MPC
that indicate a desired reference cooling capacity, which is the proportional to the desired amount of heat removed from the zone by each evaporator per unit time. The capacity controller 210
determines a command 211 for the EEV position to produce the desired cooling capacity, based on measurements of the coil temperatures (HX coil) 272. These capacity controllers account for the fact
that the effect of EEV positions on zone temperatures is nonlinear. The cooling capacity controllers linearize the responses from the reference cooling capacity 202 of each zone CCC[i ]to the
associated zone temperature Tr[i].
The combination of the ME-VCS 100 plus the set of capacity controllers 210, 220 is referred herein as the augmented system. When viewed from the perspective of the reconfigurable controller 200, the
augmented system is linear and exhibits structure that is exploited for computing MPC controllers for each configuration. Using this approach, the reconfigurable controller is responsible for
determining some actuator commands directly, and determines other commands that may be interpreted as setpoints for the capacity controllers.
A heat exchanger associated with an opened or partially opened valve is said to be “active.” For valves that are closed, no refrigerant enters the associated heat exchanger and the evaporator is said
to be “inactive.” As referred herein, the configuration of the MZ-VCS is the combination of heat exchangers that are active and inactive. More formally, for an N-heat exchanger MZ-VCS, using the
notation (x,y):=[x^T y^T]^T, the configuration ζ(t):=(ζ[0](t), . . . , ζ[N](t)) as a vector of binary-valued elements that indicate whether zone i is active (ζ[i](t)=1) or inactive (ζ[i](t)=0) at
time t.
The control objectives can include the regulation of each zone temperature Tr[i ]to an associated reference temperature Tr[iref ]provided by an external source such as an occupant while rejecting
disturbances in heat load and outdoor air temperature. Further, one or more machine temperatures indicative of the vapor compression cycle performance may be driven to associated setpoint(s). For
example, in some embodiments the compressor discharge temperature is to be driven to a reference Td[ref ]that has been determined for optimal energy efficiency. In other embodiments, evaporator
superheat temperature(s) Tesh are to be driven to references Tesh[ref ]that have been determined for optimal energy efficiency. Alternate variables may also be selected for performance.
In some embodiments, constraints 167 that can be enforced on control inputs including maximum and minimum actuator values (CF[max ]and CF[min], ODF[max ]and ODF[min], etc.) and actuator rate limits
(ΔCF[max]/s, ΔODF[max]/s, etc.). Constraints on plant outputs may also be enforced, including maximum compressor discharge temperature Td[max], minimum evaporating temperature Te[min], and maximum
condensing temperature Tc[max], etc. Alternate variables or combinations thereof may also be used for constraints.
The reconfigurable controller 200 employing the principles of different embodiments stabilizes and achieves these objectives for each configuration of the system, and thus stability, reference
tracking, disturbance rejection and constraint enforcement can occur for every combination of heat exchangers that are active or inactive. To achieve these control objectives, a controller is
developed based on a realized structure of a model of the MZ-VCS. This structure in the model leads to a structured formulation of a constrained optimization problem that can be parameterized by the
system configuration ζ and used to automatically generate optimization problems specific to the system configuration. The structured plant model is described next.
Structure of MZ-VCS Model
Some embodiments of the invention are based on appreciation of the physics governing the operation of the MZ-VCS that reveals a chain of causality leading to a particular structure in the model
equations. Specifically, each zone temperature depends on the local heat load and the temperature of the corresponding heat exchanger. In addition, the central components of the MZ-VCS that include
the compressor and outdoor unit heat exchanger affect each of the heat exchangers. However, heat exchangers are not mutually coupled. That is, changes in one heat exchanger do not directly affect
another heat exchanger.
When the set of differential equations describing this system from the control inputs to the measurements are written in matrix form, the representation reveals a particular pattern of zero-valued
and non-zero-valued elements that create an advantageous structure. Specifically, this disclosure uses subscript 0 to denote non-repeated components of the vapor compression system (e.g., the
compressor, outdoor unit heat exchanger and associated fan), which is referred to as the “centralized subsystem” and can be described a linear time-invariant (LTI) model:
Also, the disclosure uses the subscript i∈{1, . . . , N} to denote i-th zone dynamics (principally the dynamics associated with each heat exchanger and associated zone air, including the linearizing
effect of the capacity controllers), which is referred to as the “decentralized subsystems” and can be described as a set of LTI models:
z[e][i](t)=E[e][i]x[e][i](t),∀i=1, . . . ,N,(4)
$x e i ∈ n e i , u e i ∈ m e i , z e i ∈ p e i ,$
for i∈{0, 1, . . . , N} represent the states, control inputs and performance outputs, respectively and
$y e 0 ∈ s e 0$
represents me constrained outputs of the centralized system.
As follows from the model equations (1) and (3), the evolution of the decentralized subsystems depends on the state of the centralized dynamics. On the other hand, the evolution of the centralized
dynamics is independent of the states of the decentralized subsystems. This structure reflects the physical interactions between the vapor compression system and the air temperatures in local zones:
each zone temperature depends on the local heat load and the states of the corresponding heat exchanger. On the other hand, due to the negligible impact of air temperature on the local heat
exchanger, the centralized states are independent from the decentralized ones. As a result of this structure, the composite A[e ]matrix of the system
$A e = [ A e 00 A e 10 A e 11 ⋮ ⋱ A e N 0 A e NN ] ,$
is lower block triangular with (i,j)-th block A[e][i,j]=0 when i≠j and i>0.
The evolution of both the centralized and decentralized dynamics are affected by each of the inputs. The centralized control inputs (CF and ODF) influence the cooling capacities (CCC[i]) and hence
the temperature dynamics in each zone, while the decentralized control inputs (CCC[i]) affect the centralized dynamics of the refrigerant systems. Due to this coupling the B[e ]matrix of the system
$B e = [ B e 00 B e 01 … B e 0 N B e 10 B e 11 … B e 1 N ⋮ ⋮ ⋱ ⋮ B e N 0 B e N 1 … B e NN ] ,$
does not have any particular structure. The present invention exploits this model structure to formulate an optimization problem using control parameters that can be parameterized by the
configuration signal ζ. Then, given a particular configuration, an optimization problem suitable for any instance of heat exchangers that are active or inactive can be automatically obtained by
suitable modifications to the control parameters. The structured optimization problem and modifications performed to the control parameters are described below.
Formulating the Prediction Model
Some embodiments augment the model (1) and (3) to formulate a prediction model that incorporates disturbances, additional constraints and reference set-points into the recursive prediction and
optimization. First, the model can be augmented with auxiliary states so that the prediction model accurately predicts the effect of control decisions on the constrained and performance outputs,
$y e 0 ( t ) = C e 0 x e 0 ( t ) + C w e 0 w e 0 ( t ) , ( 5 ) z e i ( t ) = E e i x e i ( t ) + E w e i w e i ( t ) , ∀ i = 0 , … , N , ( 6 )$
where w[e][i ]denotes the auxiliary offset states for each subsystem that are constant over the prediction horizon, w[e][i](t+1)=w[e][i](t). The inclusion of these offset states accounts for
unmeasured disturbances and modeling errors in the prediction model.
A second augmentation involves expressing the input as a change from a previous value:
x[u][i](t+1)=x[u][i](t)+Δu[i](t),∀i=0, . . . ,N,(7)
where x[u][i](t):=u[e][i](t−1). This change of variables enables input constraints to be placed on the rate of change of the control input Δu[i ]and on the actuator positions x[u][i]. Moreover, the
second augmentation can help to ensure that the steady state input Δu[i ]is zero when tracking a constant reference under constant disturbances.
Additionally, a state vector may be augmented with the reference signals, i.e., the setpoints for the compressor discharge temperature and the zone temperatures. In particular, the set-point is
obtained from an exogenous source and assumed to be constant over the prediction horizon, i.e., r[i](t+1)=r[i](t), i=0, . . . , N. Also, integrators may be included on the zone temperature tracking
ξ[i](t+1)=ξ[i](t)+T[s](r[i](t)−z[e][i](t)),∀i=1, . . . ,N,(8)
to achieve zero steady state tracking error in the presence of uncertainties in zone volume and heat loads. Adding integrators to the prediction model, and including them as part of the performance
outputs penalized in the cost function provides an opportunity for tuning the associated entries in the control parameters to achieve faster offset-free zone temperature responses.
By augmenting the prediction model in the manner described, the cost function is designed to minimize the tracking error and integrated error between the measured and desired values of the
performance outputs, thus the performance output is redefined as z[0]:=r[0]−z[e][0 ]for the centralized subsystem. Moreover, the constrained output is augmented as y[0]:=(y[e][0], x[u][0], Δu[0]) to
account for the limits on the control input and actuator rate. Further, define the exogenous input w[0]:=(w[e][0], r[0]) and the augmented state x[0]:=(x[e][0], x[u][0]), and the prediction model of
the centralized subsystem can be written as
Similarly, define w[i]:=(w[e][i],r[i]), x[i]:=(x[e][i],x[u][i]), z[i]:=(r[i]−z[e][i],ξ[i]) and y[i]:=(x[u][i], Δu[i]) as the exogenous inputs, states, performance and constrained outputs for the
decentralized subsystems respectively, and the prediction model of the decentralized subsystems is written as
Although the actuator positions x[u][i ]are a subset of the augmented state x[i], the state x[u][i ]has been pulled out of (9)-(13) and can be expressed as x[u][i]=Ω[i]x[i]. As described later, this
allows for monitoring the actuator positions separately as the system is reconfigured, hence maintaining the overall model structure. Finally, the subsystem models are combined by defining w:=(w[0],
. . . , w[N]), x:=(x[0], . . . , x[N]), x[u]:=(x[u][0], . . . , x[u][N]), Δu:=(Δu[0], . . . , Δu[N]), z=(z[0], . . . , z[N]) and y=(y[0], . . . , y[N]), resulting in a prediction model for the
overall system:
$[ w ( t + 1 ) x ( t + 1 ) ] = [ I 0 G A ] ︸ A a [ w ( t ) x ( t ) ] + [ 0 B ] ︸ B a Δ u ( t ) , ( 17 ) y ( t ) = [ C w C ] ︸ C a [ w ( t ) x ( t ) ] + D Δ u (
t ) , ( 18 ) z ( t ) = [ E w E ] ︸ E a [ w ( t ) x ( t ) ] . ( 19 )$
where w∈^q, x∈^n, Δu∈^m, z∈^p, y∈^w are such that q:=Σ[i=0]^n q[i], n:=Σ[i=0]^n n[i], m:=Σ[i=0]^n m[i], p:=Σ[i=0]^n p[i], w:=Σ[i=0]^n w[i], and x[a](t):=(w(t), x(t)) defines the overall state of the
prediction model, where w(t) represent the exogenous signals (i.e., reference, disturbance and so on) that are not controllable.
Moreover, the augmented model (A, B) is controllable if the original plant model (A[e], B[e]) is controllable. The composite system matrices can be calculated from (9) and (13), and have the
following form:
$A = [ A 00 A 10 A 11 ⋮ ⋱ A N 0 A NN ] + [ B 00 B 01 … B 0 N B 10 B 11 … B 1 N ⋮ ⋮ ⋱ ⋮ B N 0 B N 1 … B NN ] [ Ω 0 Ω 1 ⋱ Ω N ] , G = [ 0 G 1 ⋱ G N ] ( 20 ) B = [ B 00 … B 0 N
⋮ ⋱ ⋮ B N 0 … B NN ] , E = [ E 0 ⋱ E N ] , E w = [ E w 0 ⋱ E w N ] ( 21 ) C = [ C 0 ⋱ C N ] , C w = [ C w 0 ⋱ C w N ] , D = [ D 0 ⋱ D N ] , ( 22 )$
Although the composite state matrix A is not lower block triangular, the composite state matrix A has the structure A:=A[0]+BΩ where A[0 ]is lower block triangular and Ω is block diagonal. Some
embodiments exploit this structure to design the reconfigurable controller 200.
Structured Control Formulation
An optimization problem solved by a controller designed according to the principles of MPC determines the actuator commands that minimize a cost function subject to the system dynamics and
constraints. From the formulation of this optimization problem, a transformation is applied to generate an expression of this problem that is suitable for online execution. In the case where the cost
function includes only quadratic penalties on the states (or outputs) and inputs, and the constraints depend linearly on the states, outputs and/or inputs, then the transformation results in a
“quadratic program” for which well-known algorithms exist. Some embodiments of the present invention solve a quadratic program in order to compute actuator commands that minimize the cost and enforce
For the MZ-VCS allowing the heat exchangers to activate or deactivate, the number of inputs and outputs change for each configuration, requiring a different optimization problem for each
configuration. However, by exploiting the model structure of the MZ-VCS previously described, a single formulation of the optimization problem can be obtained wherein the control parameters in cost
function are created to have a structure that corresponds to the structure of the model of the MZ-VCS.
Specifically, consider the MPC problem formulation given by
$min U _ ( k ) x a ( N m | k ) ′ T ′ PTx a ( N m | k ) + ∑ i = 0 N m - 1 z ( i | k ) ′ Qz ( i | k ) + Δ u ( i | k ) ′ R Δ u ( i | k ) ( 23 ) s . t . x a (
i + 1 | k ) = A a x a ( i | k ) + B a Δ u ( i | k ) ( 24 ) y ( i | k ) = C a x a ( i | k ) + D Δ u ( i | k ) ( 25 ) z ( i | k ) = E a x a ( i | k ) ( 26 ) Δ
u min ≤ Δ u ( i | k ) ≤ Δ u max ( 27 ) y min ≤ y ( i | k ) ≤ y max ( 28 ) x a ( 0 | k ) = x a ( k ) . ( 29 )$
The optimization problem is formulated in discrete time with a sample period T[s], and at every timestep k, the solution to this problem is a sequence of control inputs Ū(k) over the next N[m ]steps,
called the prediction horizon. In a typical MPC approach, the first action Ū(0) encoded in this solution is applied to the MZ-VCS, and after the sampling period has elapsed, the optimization problem
is recomputed using a new prediction horizon of the same length shifted in time by one step. In this manner, MPC is said to be a receding-horizon optimal controller.
At timestep k, the state of the MZ-VCS is obtained, providing the initial condition for the optimization problem x[a](0|k). A prediction model (24)-(26) is created based on (17) and used to encode
the MZ-VCS dynamics into the optimization problem, provide a set of performance outputs z to be penalized in the cost function (23) and a set of constrained outputs y to be constrained as part of the
optimization problem. The performance outputs can include error signals indicative of the difference between a measured zone temperature and a zone temperature setpoint. The constrained outputs may
be measurements, actuator values, or virtual signals created from these performance outputs.
In one embodiments, the cost function (23) includes quadratic penalties z′Qz on the performance outputs (where z∈^p is a vector of performance outputs, Q is a diagonal matrix of dimension p×p whose
elements penalize the corresponding performance outputs, and where the quadratic term z′Qz results in a scalar value). Similarly, the cost includes quadratic penalties u′Ru on the control inputs
(where u∈^m is a vector of performance outputs, R is a diagonal matrix of dimension m×m whose elements penalize the corresponding control inputs, and where the quadratic term u′Ru results in a scalar
value). These performance output and control input penalties are computed at each timestep i over the prediction horizon. Additionally, a so-called terminal cost (applied only at the end of the
prediction horizon, i=N[m]) is included and penalizes the predicted terminal state of the MZ-VCS. The terminal cost is also a quadratic penalty consisting of the predicted state x[a]∈^n+q at timestep
N[m ]multiplied by a (n+q)×(n+q) terminal penalty matrix T′PT, where T is a transformation matrix of dimension n×(n+q) such that Tx[a ]shifts the states from steady state solution and P is a diagonal
matrix of dimension n×n whose elements penalize the corresponding states. Linear constraints may also be included on the control inputs (27) or on the constrained outputs (28).
The desired transient performance of the closed loop system is encoded by using the elements of the controller parameters Q and R as penalties that indicate the relative importance of tracking a
particular performance output or using a particular control input to achieve the control objectives. Consequently, determining the entries of the penalty matrices are critical to the machine
performance and must typically be obtained by a trial-and-error tuning process. The entries of the controller parameter P is computed to ensure that the resulting closed loop system is stable, which
supports the design of the reconfigurable MPC.
When the MZ-VCS is reconfigured, the numbers of inputs u, performance outputs z, and states x are changed, requiring a new formulation of the optimization problem. However, by exploiting the model
structure previously described, a cost function can be obtained that permits automatic reformulation to the appropriate configuration by manipulating the controller parameters Q, R, and P in the cost
function. Recall that the system configuration is formally defined as ζ(k):=(ζ[0](k), . . . , ζ[N](k)) which is a vector that indicates whether zone i is active (ζ[i](k)=1) or inactive (ζ[i](k)=0) at
timestep k. Since the centralized subsystem is always on unless the entire machine is turned off, we assign ζ[0](t):→{1} for consistent notation.
By arranging the model so that performance outputs, control inputs and states are grouped according to the associated heat exchangers using equations (1) and (3), a corresponding structure may be
created in the performance penalty Q, the control penalty R, and the terminal cost P. These structured control parameters are then modified based on the given system configuration ζ(k) as described
in the next section.
Reconfigurable MPC Using Quadratic Program
FIG. 3A shows a block diagram of a reconfigurable controller 200 for an MZ-VCS 100 according to some embodiments of the invention that use quadratic program (QP) matrices for determining the control
inputs consistent with a reconfigurable MPC approach.
A configuration supervisor module 301 uses sensor information 271 from the MZ-VCS and signals 231 from occupants indicative of desired heat exchanger activation and zone temperature setpoints and
determines the appropriate system configuration ζ(k) 311 at timestep k. This system configuration is provided to a module configured to determine a set QP matrices 380 appropriate for the particular
system configuration, where the QP matrices are associated with a constrained optimization problem. The QP matrices are provided to a QP solver module 306 configured to solve a quadratic program. The
QP solver module also receives a signal 307 indicative of a state of the MZ-VCS and determined by a state estimator module 304. The state estimator module receives sensor information from the MZ-VCS
and the current set of actuator commands 308 to determine the state estimate.
FIG. 3B shows a flow chart of a method for determining QP matrices 380 according to some embodiments. The steps of the method can be performed by a processor, such as the processor 185. Referring to
FIG. 3B, the system configuration is monitored 305 for changes, and if a change in configuration has been determined, the new configuration is read 310. The system configuration ζ(k) is provided to a
module that modifies the reconfigurable controller parameters 320. The reconfigurable control parameters are the structured performance penalty matrix Q 350, the structured control penalty matrix R
351, and the structured terminal cost matrix P 352. These matrices are computed before any reconfiguration has occurred, and may be computed offline as part of a controller design and tuning process.
Determining the values of these reconfigurable control parameters will be described in a subsequent section.
FIG. 3C shows a flow chart of a method for modifying reconfigurable parameters labeled as a box 320 in FIG. 3B. Referring FIG. 3C, the configuration signal ζ(k) is used to modify the reconfigurable
controller parameters Q, R and P to obtain the modified controller parameters Q[ζ], R[ζ], and P[ζ] 375. For the heat exchangers that are deactivated (indicated by a zero-valued element in the
corresponding entry of ζ(k)), the corresponding performance variable(s) 355 should not be considered in the instantiated optimal control problem to be created. Therefore, the penalty corresponding to
this performance variable 360 is replaced with a zero, and therefore the resulting controller has no incentive to reduce the associated error signal, hence it is effectively removed from the
optimization problem. In the case that multiple performance variables are associated with a heat exchanger (for example, it may be desired to use both the zone temperature tracking error and the
integral of the zone temperature tracking error for each zone) then there are multiple entries in Q associated with a single heat exchanger and these entries are replaced with a block of zeros of
appropriate dimension. After replacing the associated entries of Q with zeros for every heat exchanger that is deactivated in the specific instance of the configuration signal, the instance of the
performance penalty matrix Q[ζ] is obtained. The subscript ζ indicates a particular instance of a reconfigurable parameter or signal after modification that corresponds to the particular system
configuration ζ(k).
Similarly, the reconfigurable control penalty matrix R is modified using the configuration signal. However, in this case, entires in R that correspond to control inputs associated with a deactivated
zone 361 are replaced with very large values. The entry 361 in FIG. 3C indicates that R[1 ]is replaced with ∞. This should be interpreted in practice as a very large penalty relative to the other
entries in R. A large value in the corresponding entry of R indicates that the controller should not consider using the corresponding control input as an available degree-of-freedom with which to
manipulate the MZ-VCS. Therefore a very large penalty in the corresponding entry in R effectively removes the control input associated with the deactivated heat exchanger from the optimization
problem. For example, in one embodiment, the optimization function replaces the values of the block of the control penalty matrix R with values larger than a threshold if the corresponding heat
exchanger is in the inactive mode. For example, the threshold can be any number larger than the values initially determined for the control penalty matrix. For example, the threshold can be any
number larger than the Hessian used in the optimization problem. For example, the threshold can be any very large number permitted by the memory and approaching ∞.
In some embodiments, there may be more than one control input associated with a heat exchanger (for example both the capacity command (CCC[i]) and the heat exchanger fan speed (IDF[i]) may be control
inputs associated with a zone). In this case, dimensions R and associated diagonal blocks are determined for compatibility. After replacing the associated entries of R with very large values for
every heat exchanger that is deactivated in the specific instance of the configuration signal, the instance of the control penalty matrix R[ζ] is obtained.
Finally, the reconfigurable terminal cost matrix P is similarly modified. In this case, entires of P that correspond to states associated with a deactivated zone 362 are replaced with zero-valued
elements. Note that the dimension of the state associated with each heat exchanger may be unity or greater, and the corresponding blocks in P will be of suitable dimension to maintain conformability.
A zero-valued block in P, indicates that the predicted terminal states associated with a deactivated zone 357 should not be considered in the optimization problem when computing a terminal state that
guarantees stability. After replacing the associated entries of P with zeros for every heat exchanger that is deactivated in the specific instance of the configuration signal, the instance of the
terminal cost matrix P[ζ] is obtained.
Solving the Instantiated Optimal Control Problem
Referring back to FIG. 3B, the set of the instantiated control parameters Q[ζ], R[ζ], and P[ζ] 375 obtained after modification are then used in conjunction with fixed parameters 376 stored in memory
and retrieved 325 to formulate the instantiated optimal control problem 330. The instantiated optimal control problem is the set of equations in (23)-(29) where the instantiated control parameters Q
[ζ], R[ζ], and P[ζ] are used in place of the reconfigurable control parameters Q, R, and P. In various embodiments, the modifications performed to the reconfigurable control parameters do not alter
their dimensions, i.e., elements within the matrices are replaced with zero-valued terms or very large terms, retaining their original sizes. Because the reconfigurable control parameters retain
their dimension, and because the other parameters required to specify the optimal control problem are fixed, every instance of the optimal control problem has a fixed and predetermined dimension.
This feature of the invention enables automatic reconfiguration because the optimal control problem does not need to be reformulated—penalties in the cost are modified to produce the effect of
removing subsystems while maintaining stability and without formulating a new problem.
One embodiment, in order to compute the solution to the instantiated optimal control problem, a transformation is applied 335 to obtain a set of matrices 380 that represent a quadratic program (QP),
and these matrices are sent 340 to a module configured to solve QPs for online execution.
The MPC optimal control problem (23)-(29) can be formulated as a quadratic programming problem
$min U U ′ Q p U + 2 x ′ C p U + x ′ Ω p x s . t . G p U ≤ S p x + Wp ( 30 )$
where a Hessian cost matrix Q[p], a linear cost matrix C[p], a state cost matrix Ω[p], a constraint matrix G[p], a state constraint matrix S[p], and constraints vector W[p ]are computed from the
parameters of Equations (23)-(29).
For example, one embodiment determines the matrices of Equation (30) by first computing the “batch” dynamics over the N[m]-step prediction horizon
where X=[x[a](0|k), . . . , x[a](N[m]|k]′ is the predicted augmented state, U=[Δu(0|k), . . . , Δu(N[m]−1|k)]′ is the predicted change in control input, Y=[y(0|k), . . . , y(N[m]|k)]′ is the
predicted constrained output over the N[m]-step horizon, x[a]=x[a](0|k) is the current augmented state, and the batch matrices are given by
$A b = [ I A 1 A 2 ⋮ A m N ] B b = [ 0 0 … 0 B 0 … 0 AB B ⋱ ⋮ ⋮ ⋱ ⋱ 0 A N m - 1 B … AB B ]$ $C b = [ C a ⋱ C a ] D b = [ D ⋱ D ]$
The batch dynamics matrices A[b], B[b], C[b], and D[b ]do not depend on the system configuration ζ. The cost (23) of the MPC optimal control problem can be written in terms of the batch dynamics
matrices A[b], B[b], C[b], and D[b ]according to
where Q[b ]and R[b ]are the batch cost matrices
$Q b = [ E a ′ Q ς E a ⋱ E a ′ Q ς E a T ′ P ς T ] R b = [ R ς ⋱ R ς ]$
where Q[ζ], R[ζ] and P[ζ] are the modified controller parameters 375 corresponding to the configuration ζ. Then the cost matrices of the quadratic programming problem (30) are
Q[p]=R[b]+B[b′]Q[b]B[b ]
C[p]=2A[b′]Q[b]B[b ]
The constraint matrices G[p], S[p], and W[p ]for the quadratic programming problem (30) are given by
$G p = [ I - I C b B b + D b - C b B b - D b ] , S p = [ 0 0 - C b A b C b A b ] , W p = [ Δ u max 1 N m - Δ u min 1 N m y max 1 N m - y min 1 N m ]$
where I[N][m]∈^N^m^×N^m is an identity matrix and 1[N][m]∈^N^m is a vector of ones.
Some embodiments of the invention are based on the observation that for convex quadratic programming problems, the solution U can be found by solving the dual problem
$min λ λ ′ Q d λ + 2 x ′ C d λ + 2 C d 0 ′ λ + x ′ Ω d x s . t . λ ≥ 0 ( 33 )$
where the dual cost Hessian Q[d], the dual state linear cost matrix C[d], the dual linear cost vector C[d0], and the dual state cost matrix Ω[d ]are computed from the parameters Q[p], C[p], Ω[p], G
[p], S[p], and W[p ]in Equation (30) according to
C[d]=S[p]+G[p]Q[p]^−1C[p ]
C[d0]=W[p ]
The solution of Equation (30) is generated from the solution λ of Equation (33) according to
where the transformation matrices Φ and Ψ are computed from Q[p], C[p], and G[p ]according to
Determining the Reconfigurable Control Parameters
This section describes how matrices Q, R, and P are determined by some embodiments of the invention. In general, the process for determining these reconfigurable control parameters is performed in
offline calculations and stored in memory accessible by a processor during online execution.
The reconfigurable performance penalty matrix Q and the reconfigurable control penalty matrix R, are determined in a tuning or calibration process. Procedures for tuning these penalty matrices are
well known in the field of optimal control and standard approaches may be used here. It is important to note here that the tuning process for determining the entries of Q and R are conducted under
the assumption that all heat exchangers are active. That is, the desired transient performance of the closed loop controller is specified through the entires in the penalty matrices for an N-unit
MZ-VCS where all zones are active. The automatic reconfiguration process previously described is then applied to modify these matrices for any other configuration.
While it is uncomplicated to create Q and R that are structured corresponding to the MZ-VCS model structure, determining the terminal state penalty matrix is not obvious. Typical methods for
computing a terminal penalty matrix produce an unstructured matrix, that is, a matrix with no discernible pattern of elements, and therefore no obvious means are available to modify P so that a
stable feedback system is achieved when heat exchangers are deactivated.
Some embodiments are based on realization of a formulation of a linear matrix inequality (LMI) problem that produces a terminal penalty matrix with the desired block diagonal structure that can be
subsequently modified in the online reconfiguration process 320. By formulating an appropriate LMI, a structured terminal penalty matrix is created with the desired diagonal structure where the
diagonal entries can be associated with particular heat exchangers and replaced with zeros when the associated heat exchangers are deactivated. In this manner, a stable constrained optimal controller
can be automatically created for every possible configuration of the MZ-VCS. Details of the LMI problem use to create the structured terminal state penalty matrix are described in the remainder of
this section.
Some embodiments of the invention construct the terminal cost x[a′]PTx[a]=x[a′]T′PTx[a ]and the structured terminal control Δu=KTx[a ]with the following form
$T = [ - Π I q ] , ( 35 ) P = [ P 0 P 1 ⋱ P N ] , K = [ K 00 K 01 … K 0 N K 11 ⋱ K NN , ] , ( 36 )$
where T∈^n×(n+q) characterizes a parameterized steady-state solution x[s]=Πw[s ]for a given constant exogenous input w[s], and Π∈^n×q is obtained by solving the following matrix equation:
$[ I n - A - E ] Π = [ G E w ] ( 37 )$
Equation (37) is solvable if rank
$[ A - I n B E 0 ] = n + p .$
The terminal control matrix K features a structure such that the centralized control input Δu[0 ]feeds back the state information from all subsystems, whereas conversely, the decentralized control
input Δu[i], ∀i∈ only feeds back its own state information. The proposed structure will allow blocks of the terminal cost and terminal controller to be zeroed when the corresponding subsystem is
turned off.
The terminal cost matrix P and controller matrix K can be determined offline by solving a linear matrix inequality for a master problem when all the decentralized subsystems are active. Some
embodiments express the terminal cost matrix P and terminal control matrix K as P=^−1 and K=P, where ∈^n×n and ∈^m×n are of the following form
$= [ S 0 S 1 ⋱ S N ] , = [ L 00 L 01 … L 0 N L 11 ⋱ L NN ] , ( 38 )$
and determined by solving the following linear matrix inequality
$[ ( AS + B ) T ( Q 1 2 E ) T ( R 1 2 ) T ( A + B ) S 0 0 Q 1 2 E 0 I 0 R 1 2 0 0 I ] ± 0 ( 39 ) ς ≻ 0. ( 40 )$
The above-described embodiments of a configuration-dependent block diagonal terminal cost and structured terminal control design enable the user to design P and K by solving linear matrix inequality
offline in a computer, deploy the controller parameters into a microprocessor, and reconfigure the controller parameters online through simple matrix operation based on reading the configuration ζ of
the system. Moreover, some embodiments guarantee the reconfigured MPC problem is locally asymptotically stable for any configuration ζ of the system, and that the modified terminal cost P[ζ] and
modified terminal controller K[ζ] satisfy the following matrix inequality
where A[ζ] and B[ζ] represent the composite system matrices (20) corresponding to configuration ζ, and are calculated by eliminating the columns in input matrix B corresponding to inactive actuators,
that is,
$A ς = A o + B ς Ω , ( 42 ) B ς = [ B 00 B 01 … B 0 N B 10 B 11 … B 1 N ⋮ ⋮ ⋱ ⋮ B N 0 B N 1 … B NN ] [ ς 0 I m 0 ς 1 I m 1 ⋱ ς N I m N ] , ( 43 )$
and K[ζ] is a modified terminal control in which elements corresponding to inactive zones are replaced with zeros and expressed as
$K ς = [ ς 0 K 00 ς 0 K 01 … ς 0 K 0 N ς 1 K 11 ⋱ ς N K NN , ] , ( 44 )$
Note that the use of K is for analysis purposes and used to calculate a corresponding terminal cost matrix P that exhibits a particular advantageous structure as shown in Equation (35). However,
formulating the instantiated optimal control problem 330 does not require the control parameter K, and therefore a configuration-dependent modification of K is not required. However, the structured
cost matrix corresponding to the terminal controller is modified 320 online as previously described.
Configuration Supervisor
Referring to FIG. 3A, a configuration supervisor module 309 determines the appropriate system configuration, that is, the set of heat exchangers that are active and inactive. The configuration
supervisor receives signals 231 from occupants that are indicative of the desired active heat exchanger and their respective zone setpoint temperatures. Using this information and with sensor
information 271 indicative of the measured zone temperature, the configuration supervisor determines which heat exchangers should be activated so that the zone temperature may be driven toward the
zone temperature setpoint.
For example, an occupant may use a user interface module 230 to indicate that a particular zone should be turned on and operate with a particular zone setpoint temperature. Then the configuration
supervisor may compare the measured zone temperature with the desired zone temperature in order to determine if the associated heat exchanger should be activated. It may be that the zone is colder
than the setpoint temperature and therefore the configuration supervisor may decide to deactivate the heat exchanger. Or, it may be that the zone is warmer than the setpoint temperature and therefore
the configuration supervisor may decide to activate the heat exchanger.
A configuration supervisor may deactivate a zone in one of two ways: (1) it may decide that the local conditions are such that the zone no longer requires conditioning, or (2) the occupant may
specify that the zone is to be shut off. If the zone is to be shut off while one or more of the other zones remain in service, then the indicated zone is deactivated by the configuration supervisor.
FIG. 4 shows a flow chart of a method for model predictive control of the VCS according to one embodiment of the invention. Some embodiments determine 401 the measured outputs, e.g., receives
information from the sensors of the MZ-VCS, and estimates 402 the state and configuration of the MZ-VCS. Next, the method solves 403 the constrained finite time optimization problem, applies 404 the
first step of that solution to the MZ-VCS and/or capacity controllers, and transitions 405 to the next control cycle.
The above-described embodiments of the present invention can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software or a combination thereof.
When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.
Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component. Though, a processor may be implemented using circuitry in any suitable
Also, the various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms.
Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code
or intermediate code that is executed on a framework or virtual machine. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.
Also, the embodiments of the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly,
embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in
illustrative embodiments.
Use of ordinal terms such as “first,” “second,” in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal
order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal
term) to distinguish the claim elements.
Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope
of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.
1. A system for controlling a multi-zone vapor compression system (MZ-VCS) including a compressor connected to a set of heat exchangers for controlling environments in a set of zones, comprising:
a controller configured to control a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing a cost function including a set of control parameters, wherein
the optimizing is subject to constraints, and wherein the cost function is optimized over a prediction horizon;
a memory configured to store an optimization function parameterized by a configuration of the MZ-VCS defining active or inactive modes of each heat exchanger, wherein the optimization function
modifies, according to a current configuration, values of the control parameters of the cost function determined for a full configuration that includes all heat exchangers in the active mode,
wherein the configuration is a vector having elements with first values for the heat exchangers in the inactive mode and having elements with second values for the heat exchangers in the active
mode, wherein an index of the element in the configuration vector matches an index of a corresponding heat exchanger; and
a processor configured to determine the current configuration of the MZ-VCS and to update the cost function by submitting the current configuration to the optimization function.
2. The system of claim 1, wherein a structure of the control parameters corresponds to a structure of a model of the MZ-VCS, such that there is a correspondence between control parameters and a heat
exchanger in the MZ-VCS, and wherein the optimization function preserves the values of the control parameters if the corresponding heat exchanger is in the active mode and modifies the values of the
block if the corresponding heat exchanger is in the inactive mode.
3. The system of claim 1, wherein the control parameters include at least one block diagonal matrix, an index of each block on the diagonal of the matrix matches the index of the corresponding heat
exchanger and values of each block on the diagonal of the matrix are determined for the corresponding heat exchanger, wherein the optimization function preserves the values of the block if the
corresponding heat exchanger is in the active mode and modifies the values of the block if the corresponding heat exchanger is in the inactive mode.
4. The system of claim 3, wherein the at least one block diagonal matrix include one or a combination of a performance penalty matrix Q whose elements penalize outputs of the MZ-VCS, a control
penalty matrix R whose elements penalize control inputs to the MZ-VCS, and a terminal cost matrix P whose elements penalize terminal states of the MZ-VCS.
5. The system of claim 4, wherein the optimization function replaces the values of the blocks of the performance penalty matrix Q and the terminal cost matrix P with zeros if the corresponding heat
exchanger is in the inactive mode, and wherein the optimization function replaces the values of the block of the control penalty matrix R with values larger than initial values of the control penalty
matrix if the corresponding heat exchanger is in the inactive mode.
6. The system of claim 3, wherein modification of the values of the control parameters preserves the dimension of the block diagonal matrix.
7. The system of claim 1, further comprising:
a set of capacity controllers corresponding to the set of heat exchangers for transforming the set of control parameters into position of valves in the heat exchangers.
8. The system of claim 1, further comprising:
at least one input interface for accepting values of the modes for each heat exchanger in the MZ-VCS, wherein the processor determines the current configuration based on the values of the modes
received from the input interface.
9. The system of claim 1, further comprising:
a set of sensors for measuring temperature in the corresponding zones controlled by the MZ-VCS; and
at least one input device for setting desired temperature in the corresponding zones, wherein the processors determines the current configuration based on the measurements from the set of sensors
and values of the desired temperature.
10. A method for controlling a multi-zone vapor compression system (MZ-VCS) including a compressor connected to a set of heat exchangers for controlling environments in a set of zones, comprising:
determining a current configuration of the MZ-VCS defining active or inactive mode of each heat exchanger in the MZ-VCS;
updating at least some values of control parameters in a cost function by submitting the current configuration to an optimization function parameterized by a configuration of the MZ-VCS, wherein
the optimization function modifies values of the control parameters of the cost function according to the current configuration, wherein the control parameters include at least one block diagonal
matrix, an index of each block on the diagonal of the matrix matches an index of a corresponding heat exchanger and values of each block on the diagonal of the matrix are determined for the
corresponding heat exchanger, wherein the optimization function preserves the values of the block if the corresponding heat exchanger is in the active mode and modifies the values of the block if
the corresponding heat exchanger is in the inactive mode; and
controlling a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing the cost function subject to constraints, wherein steps of the method are performed
using a processor.
11. The method of claim 10, wherein the configuration is a vector having elements with first values for the heat exchangers in the inactive mode and having elements with second values for the heat
exchangers in the active mode, wherein an index of the element in the configuration vector matches an index of a corresponding heat exchanger.
12. The method of claim 10, wherein the values of the control parameters are initialized for a full configuration that includes all heat exchangers in the active mode.
13. The method of claim 10, wherein the at least one block diagonal matrix include one or a combination of a performance penalty matrix Q whose elements penalize outputs of the MZ-VCS, a control
penalty matrix R whose elements penalize control inputs to the MZ-VCS, and a terminal cost matrix P whose elements penalize states of the MZ-VCS.
14. The method of claim 13, wherein the optimization function replaces the values of the block of the performance penalty matrix Q with zeros when the corresponding heat exchanger is in the inactive
mode, wherein the optimization function replaces the values of the block of the terminal cost matrix P with zeros when the corresponding heat exchanger is in the inactive mode, and wherein the
optimization function replaces the values of the block of the control penalty matrix R with values larger than other values of the control penalty matrix when the corresponding heat exchanger is in
the inactive mode.
15. A non-transitory computer readable storage medium having embodied thereon a program executable by a processor for performing a method, the method comprising:
determining a current configuration of an MZ-VCS defining an active or inactive mode of each heat exchanger in the MZ-VCS;
updating at least some values of control parameters in a cost function by submitting the current configuration to an optimization function parameterized by a configuration of the MZ-VCS, wherein
the optimization function modifies values of the control parameters of the cost function according to the current configuration, wherein the configuration is a vector having elements with zero
values for the heat exchangers in the inactive mode and having elements with non-zero values for the heat exchangers in the active mode, wherein an index of the element in the configuration
vector matches an index of a corresponding heat exchanger, wherein the values of the control parameters are initialized for a full configuration that includes all heat exchangers in the active
mode; and
controlling a vapor compression cycle of the MZ-VCS using a set of control inputs determined by optimizing the cost function subject to constraints.
16. The medium of claim 15, wherein the control parameters include at least one block diagonal matrix, an index of each block on the diagonal of the matrix matches the index of the corresponding heat
exchanger and values of each block on the diagonal of the matrix are determined for the corresponding heat exchanger, wherein the optimization function preserves the values of the block if the
corresponding heat exchanger is in the active mode and modifies the values of the block if the corresponding heat exchanger is in the inactive mode.
17. The medium of claim 16, wherein the at least one block diagonal matrix include one or combination of a performance penalty matrix Q whose elements penalize outputs of the MZ-VCS, a control
penalty matrix R whose elements penalize control inputs to the MZ-VCS, and a terminal cost matrix P whose elements penalize states of the MZ-VCS, wherein the optimization function replaces the values
of the block of the performance penalty matrix Q with zeros when the corresponding heat exchanger is in the inactive mode, wherein the optimization function replaces the values of the block of the
terminal cost matrix P with zeros when the corresponding heat exchanger is in the inactive mode, and wherein the optimization function replaces the values of the block of the control penalty matrix R
with values greater than a threshold when the corresponding heat exchanger is in the inactive mode.
Referenced Cited
U.S. Patent Documents
5007245 April 16, 1991 Defenbaugh et al.
7331191 February 19, 2008 He et al.
8376242 February 19, 2013 Uselton
20130085614 April 4, 2013 Wenzel
20150267932 September 24, 2015 Kim
20150378373 December 31, 2015 Sprinkle et al.
Foreign Patent Documents
2469201 January 2013 EP
2014172374 October 2014 WO
2016025802 February 2016 WO
2017018497 February 2017 WO
Patent History
Patent number
: 10094598
: Jun 6, 2016
Date of Patent
: Oct 9, 2018
Patent Publication Number
20170350625 Assignee
Mitsubishi Electric Research Laboratories, Inc.
(Cambridge, MA)
Daniel J Burns
(Wakefield, MA),
Junqiang Zhou
(Columbus, OH),
Claus Danielson
(Somerville, MA),
Stefano Di Cairano
(Somerville, MA)
Primary Examiner
Jonathan Bradford Application Number
: 15/174,377 | {"url":"https://patents.justia.com/patent/10094598","timestamp":"2024-11-13T05:24:53Z","content_type":"text/html","content_length":"212389","record_id":"<urn:uuid:a48c7fe2-470f-45f8-ad2a-4c7e76384c6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00206.warc.gz"} |
vb.net not equal | Code Ease
vb.net not equal
Answered on: Wednesday 15 May, 2024 / Duration: 20 min read
Programming Language: VBA , Popularity : 10/10
Solution 1:
In VB.NET, the "not equal" operator is represented by "<>" or "!=". This operator is used to compare two values and returns true if they are not equal.
Here is an example code snippet in VB.NET:
Dim num1 As Integer = 10
Dim num2 As Integer = 5
If num1 <> num2 Then
Console.WriteLine("num1 is not equal to num2")
Console.WriteLine("num1 is equal to num2")
End If
num1 is not equal to num2
In VBA (Visual Basic for Applications), the "not equal" operator is represented by "<>". Here is an example code snippet in VBA:
Dim num1 As Integer
Dim num2 As Integer
num1 = 10
num2 = 5
If num1 <> num2 Then
MsgBox "num1 is not equal to num2"
MsgBox "num1 is equal to num2"
End If
A message box will display "num1 is not equal to num2"
Solution 2:
vb.net NotEqual Operator
The NotEqual operator in VB.NET, represented by the "<>" symbol, checks if two expressions are not equal. It returns a Boolean value: True if the expressions are not equal, and False if they are
expression1 <> expression2
Dim a As Integer = 10
Dim b As Integer = 15
If a <> b Then
Console.WriteLine("a and b are not equal")
End If
a and b are not equal
vba NotEqual Operator
The NotEqual operator in VBA, also represented by the "<>" symbol, has the same functionality as in VB.NET. It checks if two expressions are not equal and returns a Boolean value.
expression1 <> expression2
Dim a As Integer
Dim b As Integer
a = 10
b = 15
If a <> b Then
MsgBox "a and b are not equal"
End If
a and b are not equal
Key Differences
The main difference between the NotEqual operator in VB.NET and VBA is in the way they handle null values.
* VB.NET: If either expression is null, the NotEqual operation returns Null.
* VBA: If either expression is null, the NotEqual operation returns True.
Example with Null Values
Dim a As String = Nothing
Dim b As String = String.Empty
If a <> b Then
Console.WriteLine("a and b are not equal")
End If
Output (VB.NET)
a and b are not equal
Dim a As String
Dim b As String
a = Nothing
b = ""
If a <> b Then
MsgBox "a and b are not equal"
End If
Output (VBA)
a and b are not equal
The NotEqual operator in VB.NET and VBA performs the same operation of checking for inequality between expressions. However, VB.NET's handling of null values with the NotEqual operator differs from
Solution 3:
In VB.NET, you can use the "Not" and "=" operators to test for inequality, whereas in VBA, you can use the "Not" and ">" or "<" operators for the same purpose. Here are some code examples and outputs
in both languages:
VB.NET Example:
Dim num1 As Integer = 5
Dim num2 As Integer = 10
If num1 <> num2 Then
Console.WriteLine("{0} is not equal to {1}", num1, num2)
End If
' Output:
' 5 is not equal to 10
VBA Example:
Dim num1 As Integer: num1 = 5
Dim num2 As Integer: num2 = 10
If num1 <> num2 Then
Debug.Print num1 & " is not equal to " & num2
End If
' Output:
' 5 is not equal to 10
VB.NET uses the "Not" operator to negate a boolean expression. For example, to test if a variable is not equal to a value, you could use the following code:
If Not num1 = num2 Then
Console.WriteLine("{0} is not equal to {1}", num1, num2)
End If
' Output:
' 5 is not equal to 10
In VBA, you can use the "Not" operator to test if a variable is not equal to a value, but you need to use the ">" or "<" operator instead of the "=" operator. Here's an example:
If Not num1 = num2 Then
Debug.Print num1 & " is not equal to " & num2
End If
' Output:
' 5 is not equal to 10
Note that the above VBA code will throw a compile error because you can't directly compare two variables using the "=" operator. So you need to use the ">" or "<" operator to compare the variables.
Here's an example:
If num1 <> num2 Then
Debug.Print num1 & " is not equal to " & num2
End If
' Output:
' 5 is not equal to 10
In summary, to test for inequality in VB.NET, you can use the "Not" and "=" operators. In VBA, you can use the "Not" and ">" or "<" operators for the same purpose. | {"url":"https://codeease.net/programming/vba/vb-net-not-equal","timestamp":"2024-11-02T07:56:18Z","content_type":"text/html","content_length":"53669","record_id":"<urn:uuid:3e86a212-1feb-484b-98b1-eedee58ff5e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00646.warc.gz"} |
\(r(G)\) is bounded by \(r(\chi(G))\) (1)
For two graphs \( G\) and \( H\), let \( r(G,H)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in red and blue, then there
is either a subgraph isomorphic to \( G\) with all red edges or a subgraph isomorphic to \( H\) with all blue edges. The classical Ramsey numbers are those for the complete graphs and are denoted by
\( r(s,t)= r(K_s, K_t)\). If \( s=t\), we write \( r(s)=r(s, s)\).
For a graph \( G\), the chromatic number \( \chi(G)\) is the least integer \( k\) such that the vertices of \( G\) can be colored in \( k\) colors so that adjacent vertices have different colors. If
\( \chi(G) \leq k\), we say that \( G\) is \( k\)-colorable. The following problem relates Ramsey numbers to chromatic numbers.
A problem on \(k\)-chromatic graphs [1]
Let \(G\) denote a graph on \(n\) vertices with chromatic number \(k\). Is it true that \[ r(G) > (1-\epsilon)^k r(k) \] holds for any fixed \(\epsilon\), \(0 < \epsilon < 1\), provided \(n\) is
large enough?
1 P. Erdös, Some of my favourite problems in number theory, combinatorics, and geometry, Combinatorics Week (Portuguese) (São Paulo, 1994), Resenhas 2 (1995), 165-186. | {"url":"https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/RGLowerBoundByChromaticNumber1.html","timestamp":"2024-11-02T12:27:16Z","content_type":"text/html","content_length":"4791","record_id":"<urn:uuid:a1108441-6d16-4a96-87d4-0160efc89655>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00809.warc.gz"} |
Using Complex Numbers To Simplify Expressions Worksheet
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Simplifying EXPRESSIONS With Complex Numbers Examples YouTube
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Imaginary And Complex Numbers Metropolitan Community
Imaginary and Complex Numbers Practice Simplify 4 2i 3 5i 3 4i 5 2i 8 7i 5 4i 3 2i 5 4i 2 3 4i 2 3 2i 5 4i 3 4i 3 7 i Write in standard form 5 3 i Simplify i925 Simplify i460 4 i Write in standard
form 5 2 i 16 8 6 6 4 25 6 8 15 2
Simplifying rational expressions Multiplying dividing rational expressions Adding subtracting rational expressions Complex fractions Solving rational equations
Imaginary and Complex Numbers Practice Simplify 4 2i 3 5i 3 4i 5 2i 8 7i 5 4i 3 2i 5 4i 2 3 4i 2 3 2i 5 4i 3 4i 3 7 i Write in standard form 5 3 i Simplify i925 Simplify i460 4 i Write in standard
form 5 2 i 16 8 6 6 4 25 6 8 15 2
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Simplify Expression Worksheet For 7th 10th Grade Lesson Planet
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Complex Numbers Worksheet For 10th 12th Grade Lesson Planet | {"url":"https://szukarka.net/using-complex-numbers-to-simplify-expressions-worksheet","timestamp":"2024-11-08T11:50:19Z","content_type":"text/html","content_length":"25292","record_id":"<urn:uuid:54a0440b-d64e-46e8-aa02-856c23b78fd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00477.warc.gz"} |
Skyscrapers rules
Following the footsteps of Sudoku, Kakuro and other Number Logic puzzles, Skyscrapers is one more family of addictive easy to learn logic puzzles. Using pure logic and requiring no math to solve,
these fascinating puzzles offer endless fun and intellectual entertainment to puzzle fans of all skills and ages.
Skyscrapers is a building arranging puzzle. Unlike other logic puzzles, Skyscrapers are solved by placing buildings in a grid so the number of visible buildings, as viewed from the direction of each
clue, is equal to the value of the clue.
Skyscrapers puzzles come in many sizes and range from very easy to extremely difficult taking anything from five minutes to several hours to solve. However, make one mistake and you’ll find yourself
stuck later on as you get closer to the solution...
If you like Sudoku, Kakuro and other logic puzzles, you will love Conceptis Skyscrapers as well!
Classic Skyscrapers
Each puzzle consists of an NxN grid with some clues along its sides. The object is to place a skyscraper in each square, with a height between 1 and N, so that no two skyscrapers in a row or column
have the same number of floors. In addition, the number of visible skyscrapers, as viewed from the direction of each clue, is equal to the value of the clue. Note that higher skyscrapers block the
view of lower skyscrapers located behind them.
Below is a 3D diagram of what a puzzle would look like when viewed from an airplane. The blocks are city skyscrapers and the clues indicate how many of them are visible when viewed from that
direction. With this diagram, it is clear how lower skyscrapers are hidden by the higher ones. | {"url":"https://www.conceptispuzzles.com/?uri=puzzle/skyscrapers/rules","timestamp":"2024-11-09T10:19:11Z","content_type":"application/xhtml+xml","content_length":"20704","record_id":"<urn:uuid:a18f1d31-5c60-4336-9e9d-c4bb09f59c26>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00171.warc.gz"} |
Convergence Tests | Brilliant Math & Science Wiki
Recall that the sum of an infinite series \( \sum\limits_{n=1}^\infty a_n \) is defined to be the limit \( \lim\limits_{k\to\infty} s_k \), where \( s_k = \sum\limits_{n=1}^k a_n \). If the limit
exists, the series converges; otherwise it diverges.
Many important series do not admit an easy closed-form formula for \( s_k \). In this situation, one can often determine whether a given series converges or diverges without explicitly calculating \(
\lim\limits_{k\to\infty} s_k \), via one of the following tests for convergence.
The first and simplest test is not a convergence test.
Divergence test:
If \( \lim\limits_{n\to\infty} a_n \) does not exist, or exists and is nonzero, then \( \sum\limits_{n=1}^\infty a_n \) diverges.
The proof is easy: if the series converges, the partial sums \( s_k \) approach a limit \( L \). Then
\[\lim_{n\to\infty} a_n = \lim_{n\to\infty} (s_n-s_{n-1}) = L-L = 0.\]
The series \( \sum\limits_{n=1}^\infty \sin n \) diverges, because \( \lim\limits_{n\to\infty} \sin n \) does not exist.
The divergence test does not apply to the harmonic series \( \sum\limits_{n=1}^\infty \frac1{n} \), because \( \lim\limits_{n\to\infty} \frac1{n} = 0 \). In this case, the divergence test gives
no information.
It is a common misconception that the "converse" of the divergence test holds, i.e. if the terms go to \( 0 \) then the sum converges. In fact, this is false, and the harmonic series is a
counterexample--it diverges (as will be shown in a later section).
The intuition for the next two tests is the geometric series \( \sum ar^n\), which converges if and only if \( |r|<1 \). The precise statement of the test requires a concept that is used quite often
in the study of infinite series.
A series \( \sum\limits_{n=1}^\infty a_n \) is absolutely convergent if \( \sum\limits_{n=1}^\infty |a_n|\) converges. If a series is convergent but not absolutely convergent, it is called
conditionally convergent.
Ratio test:
Suppose \( \lim\limits_{n\to\infty} \left| \dfrac{a_{n+1}}{a_n} \right| = r \). If \( r<1 \), the series \( \sum a_n \) converges absolutely. If \( r>1 \), the series diverges. If \( r = 1 \) (or
the limit does not exist), the test gives no information.
Consider the series \( \sum\limits_{n=0}^{\infty} \binom{2n}{n} x^n \). For which values of \( x\) does this series converge?
Partial Solution:
The ratio \( \left| \dfrac{a_{n+1}}{a_n} \right| \) is
\[ \frac{\binom{2n+2}{n+1} |x|^{n+1}}{\binom{2n}{n}|x|^n} = \frac{(2n+2)(2n+1)}{(n+1)^2} |x| &= \frac{\left(2+\frac2n\right)\left(2+\frac1n\right)}{\left(1+\frac1n\right)^2} |x|, \]
which approaches \(4|x|\) as \(n\to\infty.\)
So, if \( 4|x|<1 \), the series converges absolutely, and if \( 4|x|>1, \) the series diverges. For \( x = \pm \frac14, \) the question is more delicate. It turns out that the series converges
for \( x=-\frac14 \) but not for \( x=\frac14.\) Hence the answer is \( x \in \left[-\frac14, \frac14\right). \) \(_\square\)
The ratio test is quite useful for determining the interval of convergence of power series, along the lines of the above example. Note that at the endpoints of the interval, the ratio test fails.
The root test works whenever the ratio test does, but the limit involved in the root test is often more difficult from a practical perspective.
Root test:
Suppose \( \limsup\limits_{n\to\infty} \sqrt[n]{|a_n|} = r\). Then if \( r<1 \), the series \( \sum a_n \) converges absolutely; if \( r>1 \), it diverges; if \( r=1 \), the test is inconclusive.
Here, \( \limsup \) denotes the limit of the supremum of a sequence, \( \lim\limits_{n\to\infty} \sup\limits_{m\ge n} \sqrt[m]{|a_m|} \) in this case. If we allow \( r \) to be \( \infty\) \((\)which
is taken to be \( >1 \) for purposes of the test\(),\) the \( \limsup \) always exists (while the limit might not); if the limit exists, then it equals the \( \limsup \). In practice, using the root
test usually involves computing the limit.
A fact that is often useful in applications of the root test is that \( \lim\limits_{n\to\infty} n^{1/n} = 1. \) \(\big(\)This follows because the limit of the natural log, \( \frac{\ln n}{n}, \) is
\( 0 \) by L'Hopital's rule.\(\big)\)
Does \( \sum\limits_{n=1}^{\infty} \frac{2^n n^{n^2+1}}{(n+1)^{n^2}} \) converge or diverge?
Take \( |a_n|^{1/n} \) and get
\[ \frac{2 n^{n+1/n}}{(n+1)^n} &= \frac{2 n^{1/n}}{\left(1+\frac1n\right)^n} \to \frac{2\cdot 1}{e} < 1, \]
so the series converges (absolutely). \(_\square\)
Often the series \( a_n \) can be extended to a nice function \( f(x) \), and the integral of \( f(x) \) is "close" to the sum.
Integral test:
If \( f(x) \) is a nonnegative, continuous, decreasing function on \( [1,\infty) \), then the series \( \sum\limits_{n=1}^\infty f(n) \) converges if and only if the improper integral \( \int_1^\
infty f(x) \, dx \) converges.
Note that it is important that \( f(x) \) is decreasing and continuous, as otherwise it is conceivable that the values of \( f \) at integers might be unrelated to its values everywhere else \((\)
e.g. imagine an \( f\) that is 0 except very near integers, where it spikes to \( 1 \); such an \( f \) might have a convergent integral, but the series will diverge\().\)
The \( p \)-series \( \sum\limits_{n=1}^\infty \frac1{n^p} \) are defined for any real number \( p \). For which \( p \) does the associated \( p\)-series converge?
For \( p\le 0\), the series diverges by the divergence test. For \( p > 0 \), \( f(x) = \frac1{x^p} \) is a nonnegative decreasing function on \( [1,\infty) \). For \( p \ne 1 \),
\[\int_1^\infty \frac1{x^p} \, dx = \frac1{1-p} x^{1-p} \biggr\rvert_1^\infty,\]
which diverges for \( p < 1 \) and converges to \( \frac1{p-1} \) for \( p > 1 \). So the same is true of the associated series.
The case \( p = 1 \) is the harmonic series, which diverges because the associated integral
\[\int_1^\infty \frac1{x} \, dx = \ln x\biggr\rvert_1^\infty\]
diverges. So the answer is that the \( p \)-series converges if and only if \( p>1 \). \(_\square\)
This test can determine that a series converges by comparing it to a (simpler) convergent series.
Comparison test:
If \( \sum b_n \) is absolutely convergent and \( |a_n|\le |b_n|\) for sufficiently large \( n \), then \( \sum a_n \) is absolutely convergent.
Note that it only makes sense to compare nonnegative terms, so this test will never help with conditionally convergent series.
Does \( \sum\limits_{n=1}^\infty \frac1{n^2+n+1} \) converge or diverge?
Since \( \frac1{n^2+n+1} < \frac1{n^2} \), and \( \sum \frac1{n^2} \) converges by the integral test \((\)it is a \( p\)-series with \(p>1), \) the series converges by the comparison test. \(_\
The comparison test can also determine that a series diverges:
Does \( \sum\limits_{n=1}^\infty \frac1{2n-1} \) converge or diverge?
Since \( \frac1{2n} \le \frac1{2n-1} \), if the series converges, then so does \( \sum\limits_{n=1}^\infty \frac1{2n} = \frac12 \sum\limits_{n=1}^\infty\frac1n \). But the harmonic series
diverges, so the original series must diverge as well. \(_\square\)
The comparison test is useful, but intuitively it feels limited. For instance, \( \frac1{n^2-n+1} \) is not \( \le \frac1{n^2} \), and yet the series \( \sum \frac1{n^2-n+1}\) ought to converge
because the terms "behave like" \( \frac1{n^2} \) for large \( n \). A refinement of the comparison test, described in the next section, will handle series like this.
Instead of comparing to a convergent series using an inequality, it is more flexible to compare to a convergent series using behavior of the terms in the limit.
Limit comparison test:
If \( \sum b_n \) converges absolutely and \( \lim\limits_{n\to\infty} \left| \frac{a_n}{b_n} \right| = c \) exists (and is finite), then \( \sum a_n \) converges absolutely.
More symmetrically, if \( x_n,y_n > 0 \) and \( \lim\limits_{n\to\infty} \frac{x_n}{y_n} \) exists and is nonzero, then \( \sum x_n \) and \( \sum y_n \) both converge or both diverge.
\( \sum\limits_{n=1}^{\infty} \frac1{n^2-n+1} \) converges, because \( \sum\limits_{n=1}^{\infty} \frac1{n^2} \) does and
\[ \lim_{n\to\infty} \frac{\hspace{2mm} \frac1{n^2-n+1}\hspace{2mm} }{\frac1{n^2}} &= \lim_{n\to\infty} \frac{n^2}{n^2-n+1} \\ &= \lim_{n\to\infty} \frac1{1-\frac1n+\frac{1}{n^2}} \\&= 1. \]
Comparing to \( p\)-series is often the right strategy.
Does \( \sum\limits_{n=1}^\infty \left(\sqrt[n]{2}-1\right) \) converge or diverge?
Apply the limit comparison test with \( \frac1n \) and use L'Hopital's rule, since the derivative of \( 2^x \) is \( 2^x \ln 2:\)
\[ \lim_{n\to\infty} \frac{2^{1/n}-1}{\frac1n} &= \lim_{n\to\infty} \frac{2^{1/n} \ln 2 \cdot -\frac1{n^2}}{-\frac1{n^2}}\\\\ &= \lim_{n\to\infty} \left(2^{1/n} \ln 2\right) \\\\&= \ln 2. \]
So the series diverges by limit comparison with the harmonic series. \(_\square\)
Alternating series arises naturally in many common situations, including evaluations of Taylor series at negative arguments. They furnish simple examples of conditionally convergent series as well.
There is a special test for alternating series that detects conditional convergence:
Alternating series test:
If \( a_n \) is a decreasing sequence of positive integers such that \( \lim\limits_{n\to\infty} a_n = 0 \), then \( \sum\limits_{n=1}^\infty (-1)^n a_n \) converges.
If \( a_n = \frac1n \), the test immediately shows that the alternating harmonic series \( \sum\limits_{n=1}^\infty \frac{(-1)^n}n \) is (conditionally) convergent.
Note that it is enough for the \( a_n \) to be eventually decreasing \((\)i.e. \( a_{n+1} \le a_n \) for sufficiently large \( n).\)
Show that \( \sum\limits_{n=1}^\infty (-1)^n \frac{n}{n^2+25} \) converges.
This follows directly from the alternating series test, if we can show that \( \frac{n}{n^2+25} \) is eventually decreasing. The easiest way to do this is to consider the function \( f(x) = \frac
{x}{x^2+25} \) and take its derivative:
\[f'(x) = \frac{(x^2+25)-x(2x)}{(x^2+25)^2} = \frac{25-x^2}{(x^2+25)^2}.\]
So \( f'(x) < 0 \) for \( x>5 \), which implies the sequence \( f(n) \) is decreasing for \( n > 5 \). \(_\square\)
One interesting fact about the alternating series test is that it gives an effective error bound as well:
Let \( \sum (-1)^n a_n \) be a series that satisfies the conditions of the alternating series test, and suppose that \( a_n \) is decreasing for \( n \ge 1 \) (not just eventually decreasing). If
the sum of the series is \( L \) and the \( k^\text{th}\) partial sum is denoted \( s_k \), then
\[|L-s_k| \le a_{k+1}.\]
Give an upper bound for the error in the estimate \( \pi \approx 4-\frac43+\frac45-\cdots -\frac4{399} \).
Assuming that \( \pi \) is the sum of the series \( \sum\limits_{n=0}^\infty (-1)^n \frac4{2n+1} \), the alternating series test says that this error is at most \( \frac4{401},\) which is roughly
\( 0.01\). In fact, the sum is \( 3.13659\ldots\), so the error is almost exactly half that, or \( 0.005 \). \(_\square\)
\(\big(\)To show that \( \pi \) is, in fact, the sum of the series, one possibility is to derive the Taylor series \( \arctan x = x-\frac{x^3}3+\frac{x^5}5-\cdots,\) which is valid on \( (-1,1)
\), and then to use a theorem of Abel which shows that the identity can be extended to the endpoint \( 1 \) of the interval.\(\big)\)
The alternating series test is actually a special case of the Dirichlet test for convergence, presented in the next section.
Dirichlet test:
Suppose \( a_n,b_n \) are sequences and \(M\) is a constant, and
(1) \(a_n \) is a decreasing sequence,
(2) \( \lim\limits_{n\to\infty} a_n = 0 \),
(3) if \( s_k \) is the \(k^\text{th}\) partial sum of the \( b_n\), then \( |s_k|\le M \) for all \( k \).
Then \( \sum\limits_{n=1}^\infty a_nb_n\) converges.
The alternating series test is the special case where \( b_n = (-1)^n \) \((\)and \( M = 1).\)
Let \(a_n\) be a decreasing sequence of real numbers such that \( \lim\limits_{n\to\infty} a_n = 0 \). Show that
\[\sum_{n=1}^\infty a_n \sin nx\]
converges for all real numbers \( x \) which are not integer multiples of \( 2\pi\). (This is useful in the theory of Fourier series.)
This follows from the Dirichlet test and the identity
\[\sin x+\sin 2x+\cdots+\sin nx = \frac{\sin \frac{n}2 x \sin \frac{n+1}2 x}{\sin \frac12 x}\]
because the absolute value of the quantity on the right is \( \le \frac1{\big|\sin \frac12 x\big|},\) which is a constant real number as long as the denominator is not \( 0 \). \((\)This is why
we had to assume that \( x \) was not an integer multiple of \( 2\pi.)\) \(_\square\)
Abel's test is similar to Dirichlet's test, and is most useful for conditionally convergent series.
Abel test:
Suppose \( a_n,b_n\) are sequences and \( M \) is a constant, and
(1) \( \sum a_n \) converges,
(2) \( b_n \) is a monotone (increasing or decreasing) sequence,
(3) \( |b_n|<M \) for all \( n \).
Then \( \sum a_nb_n \) converges.
Note that if \( a_n \) is positive \(\big(\)or \( \sum a_n \) is absolutely convergent\(\big),\) this follows immediately from the comparison test \((\)without assumption (2)\().\) So the interesting
series to which this applies is conditionally convergent.
The series \( \sum\limits_{n=1}^\infty (-1)^n \frac{\arctan n}n \) converges by Abel's test \(\big(\)take \(b_n = \arctan n\), which is increasing and bounded above by \( \frac{\pi}2\big).\) | {"url":"https://brilliant.org/wiki/convergence-tests/?subtopic=sequences-and-limits&chapter=sequences-and-series","timestamp":"2024-11-09T08:06:34Z","content_type":"text/html","content_length":"61698","record_id":"<urn:uuid:7fcccd95-2685-4fc5-9db0-ad725f32f911>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00731.warc.gz"} |
Re: Strange behaviour of NMinimize when doing curve fitting
• To: mathgroup at smc.vnet.net
• Subject: [mg120717] Re: Strange behaviour of NMinimize when doing curve fitting
• From: Peter Pein <petsie at dordos.net>
• Date: Sun, 7 Aug 2011 06:15:03 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j1im4n$cni$1@smc.vnet.net>
Am 06.08.2011 08:14, schrieb Hovhannes Babayan:
> Hello,
> I want to fit a set of points with some curve, which form I know, but coefficients have to be determined to make the best possible fit. Also, for the each point a weight (a number ranging from 0 to 1) is defined. Point with greater weight is more important and point with lower weight is not so important. Point with weight=0 doesn't matter at all, point with weight=1 is extremely important.
> The code below uses NMinimize and draw the best fit graphic.
> The function f[r_, z_, u_, a_, b_] is a fitting function, where z,u,a,b are parameters.
> The question is following. Notice that I've used a/110 and b/100, instead of just a and b. When I use just a and b, produced curve doesn't fit at all my points. When I use a/50 and b/50 result is better, with a/70 and b/70 even better, and so on. The best seems to be a/110 and b/100 combination. Why it is so important for NMinimize??
>>From theoretical point of view it absolutely doesn't matter what is used: a/100 or a/1000 or just a. Parameter "a" should be found in every case.
> Please, if you have met such behavior, let me know, because currently I am doing NMinimize of NMinimize just to find good values to divide a and b, and it has no sense :)
> In order to try the code below, just copy/paste it into Mathematica, it should print best fit result (first element is error, second element is array with values of parameters z,u,a,b), should plot points and fitting curve. Use a/1 and b/1 to have the worst result and a/110 and b/100 to have good result.
> Thanks forehand for any suggestions, remarks and/or ideas.
as you can see in this notebook [1], there is no more difference in the
results, when rationalizing the input and restricting u to be positive
and the other variables to be nonnegative.
[1] http://dl.dropbox.com/u/3030567/Mathematica/hovhannes.nb | {"url":"https://forums.wolfram.com/mathgroup/archive/2011/Aug/msg00082.html","timestamp":"2024-11-13T19:26:27Z","content_type":"text/html","content_length":"32198","record_id":"<urn:uuid:9f45f16c-123e-46de-bec0-718f1698c9e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00488.warc.gz"} |
Unscramble HABITABLY
How Many Words are in HABITABLY Unscramble?
By unscrambling letters habitably, our Word Unscrambler aka Scrabble Word Finder easily found 89 playable words in virtually every word scramble game!
Letter / Tile Values for HABITABLY
Below are the values for each of the letters/tiles in Scrabble. The letters in habitably combine for a total of 21 points (not including bonus squares)
• H [4]
• A [1]
• B [3]
• I [1]
• T [3]
• A [1]
• B [3]
• L [1]
• Y [4]
What do the Letters habitably Unscrambled Mean?
The unscrambled words with the most letters from HABITABLY word or letters are below along with the definitions.
• habit (n.) - The usual condition or state of a person or thing, either natural or acquired, regarded as something had, possessed, and firmly retained; as, a religious habit; his habit is morose;
elms have a spreading habit; esp., physical temperament or constitution; as, a full habit of body. | {"url":"https://www.scrabblewordfind.com/unscramble-habitably","timestamp":"2024-11-06T23:09:21Z","content_type":"text/html","content_length":"54644","record_id":"<urn:uuid:5e8f4e9c-ab6a-4424-9cbd-99be1cb83bce>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00497.warc.gz"} |
Choosing between and evaluating ARIMA / ESM / UCM for a model with inputs.
My problem is outlined as follows:
I have a time series which I am trying to forecast (let's call this series OUTPUT), let's say through the end of 2018. The seasonal aspect of this value may be present, but if it is, is VERY slight.
I also have two other time series (let's call them INPUT1 and INPUT2) that are being used to predict the OUTPUT series. I have values for these two series through the end of 2018 and would like to
use their relationship with the OUTPUT in my forecast.
My attempts thus far have been using PROC ARIMA with an estimate statement that looks something like:
estimate p=1 input=( / (1) INPUT1 / (1) INPUT2)
But I'm unsure if this is correct. I've been unable to find any documentation anywhere on how to determine the proper differentiation or format of the inputs to use in a PROC ARIMA. It's also
possible that PROC ESM or USM are more useful for this and I've been neglecting them. Any insight on how to choose and fit a forecasting model for a time series with inputs would be great. Thanks in
07-27-2016 11:27 AM | {"url":"https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/Choosing-between-and-evaluating-ARIMA-ESM-UCM-for-a-model-with/m-p/287503","timestamp":"2024-11-05T19:57:31Z","content_type":"text/html","content_length":"207709","record_id":"<urn:uuid:a46474c8-0a29-4067-8577-f9e32e8ee857>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00149.warc.gz"} |
The emergency it represents in the clinical setting of : Hypertension Author : - Dr. Edward Tsang (registered Chinese Herbalist & Acupuncturist ) Wu Zhu Metaphysician Hypertension is quite common and
popular for modern society nowadays due to people’s daily diet. Patients with the symptom of diastolic blood pressure over 120 mmHg is defined as hypertensive crisis (Cameron et al | {"url":"http://medicinemodern.com/m/mate.polimi.it1.html","timestamp":"2024-11-11T14:51:36Z","content_type":"text/html","content_length":"28082","record_id":"<urn:uuid:a4fa0b38-c861-46ab-9061-b7498a320e5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00184.warc.gz"} |
Cycle statistics and comparisons
Cycle statistics and comparisons#
Here we will use the ‘cycle’ submodule of EMD to identify and analyse individual cycles of an oscillatory signal
Simulating a noisy signal#
Firstly we will import emd and simulate a signal
import emd
import numpy as np
import matplotlib.pyplot as plt
from scipy import ndimage
# Define and simulate a simple signal
peak_freq = 15
sample_rate = 256
seconds = 10
noise_std = .4
x = emd.simulate.ar_oscillator(peak_freq, sample_rate, seconds,
noise_std=noise_std, random_seed=42, r=.96)[:, 0]
t = np.linspace(0, seconds, seconds*sample_rate)
# Plot the first 5 seconds of data
plt.figure(figsize=(10, 2))
plt.plot(t[:sample_rate*4], x[:sample_rate*4], 'k')
# sphinx_gallery_thumbnail_number = 5
[<matplotlib.lines.Line2D object at 0x7f0eb911ff10>]
Extract IMFs & find cycles#
We next run a mask sift with the default parameters to isolate the 15Hz oscillation. There is only one clear oscillatory signal in this simulation. This is extracted in IMF-3 whilst the remaining
IMFs contain low-amplitude noise.
# Run a mask sift
imf = emd.sift.mask_sift(x, max_imfs=5)
# Visualise the IMFs
emd.plotting.plot_imfs(imf[:sample_rate*4, :])
<Axes: xlabel='Time (samples)'>
Next we locate the cycle indices from the instantaneous phase of our IMFs. We do this twice, once to identify all cycles and a second time to identify only ‘good’ cycles based on the cycle validation
check from the previous tutorial.
# Extract frequency information
IP, IF, IA = emd.spectra.frequency_transform(imf, sample_rate, 'nht')
# Extract cycle locations
all_cycles = emd.cycles.get_cycle_vector(IP, return_good=False)
good_cycles = emd.cycles.get_cycle_vector(IP, return_good=True)
We can customise the parts of the signal in which we look for cycles by defining a mask. This is a binary vector indicating which samples in a time-series should be included in the cycle detection.
This could be useful for several reasons, we can mask our sections of signal with artefacts, limit cycle detection to a specific period during a task or limit cycle detection to periods where there
is a high amplitude oscillation.
Here we will apply a low amplitude threshold to identify good cycles which have amplitude values strictly above the 33th percentile of amplitude values in the dataset - excluding the lowest amplitude
Note that the whole cycle must be in the valid part of the mask to be included, a cycle will be excluded if a single sample within it is masked out.
thresh = np.percentile(IA[:, 2], 33)
mask = IA[:, 2] > thresh
mask_cycles = emd.cycles.get_cycle_vector(IP, return_good=True, mask=mask)
We can compute a variety of metric from our cycles using the emd.cycles.get_cycle_stat function. This is a simple helper function which takes in a set of cycle timings (the output from
emd.cycles.get_cycle_vector) and any time-series of interest (such as instaneous amplitude or frequency). The function then computes a metric from the time-series within each cycle.
The computed metric is defined by the func argument, this can be any function which takes a vector input and returns a single-number. Often we will use se the numpy built-in functions to compute
simple metrics (such as np.max or np.mean) but we can use a custom user-defined function as well.
Finally we can define whether to return the result in full or compressed format. The full form returns a vector of the same length as the input vector in which the indices for each cycle contains the
its cycle-stat whilst, the compressed form returns a vector containing single values for each cycle in turn.
For instance, the following example computes the maximum instantaneous amplitude for all detected cycles in IMF-3 and returns the result in the full-vector format.
cycle_amp = emd.cycles.get_cycle_stat(all_cycles[:, 2], IA[:, 2], out='samples', func=np.max)
# Make a summary figure
plt.figure(figsize=(10, 4))
plt.plot(t[:sample_rate*4], imf[:sample_rate*4, 2], 'k')
plt.plot(t[:sample_rate*4], IA[:sample_rate*4, 2], 'b')
plt.plot(t[:sample_rate*4], cycle_amp[:sample_rate*4], 'r')
plt.legend(['IMF-3', 'Instantaneous Amplitude', 'Cycle-max Amplitude'])
<matplotlib.legend.Legend object at 0x7f0eb91eb550>
We can see that the original IMF in black and its instantaneous amplitude in blue. The red line is then the full-format output containing the cycle maximum amplitude. This nicely corresponds to the
peak amplitude for each cycle as seen in blue.
The next section computes the average instantaneous frequency within each cycle, again returning the result in full format.
cycle_freq = emd.cycles.get_cycle_stat(all_cycles[:, 2], IF[:, 2], out='samples', func=np.mean)
# Make a summary figure
plt.figure(figsize=(10, 4))
plt.plot(t[:sample_rate*4], IF[:sample_rate*4, 2], 'b')
plt.plot(t[:sample_rate*4], cycle_freq[:sample_rate*4], 'r')
plt.legend(['Instantaneous Frequency', 'Cycle-mean frequency'])
<matplotlib.legend.Legend object at 0x7f0eb9299310>
We can get a nice visualisation of cycle-average frequency by overlaying the full stat vector onto the Hilbert-Huang transform. This is similar to the plot above but now we can see the signal
amplitude values in the colour-scale of the HHT (hotter colours show higher amplitudes). Here we plot the cycle-average frequency for cycles above our amplitude thresholdover the HHT
# Compute cycle freq using amplitude masked-cycle indices
cycle_freq = emd.cycles.get_cycle_stat(mask_cycles[:, 2], IF[:, 2], out='samples', func=np.mean)
# Carrier frequency histogram definition
freq_range = (3, 25, 64)
# Compute the 2d Hilbert-Huang transform (power over time x carrier frequency)
f, hht = emd.spectra.hilberthuang(IF, IA, freq_range, mode='amplitude', sum_time=False)
# Add a little smoothing to help visualisation
shht = ndimage.gaussian_filter(hht, 1)
# Make a summary plot
plt.figure(figsize=(10, 7))
plt.plot(t[:sample_rate*4], imf[:sample_rate*4, 2], 'k')
plt.plot((0, 4), (thresh, thresh), 'k:')
plt.xlim(0, 4)
plt.pcolormesh(t[:sample_rate*4], f, shht[:, :sample_rate*4], cmap='hot_r', vmin=0)
plt.plot(t[:sample_rate*4], cycle_freq[:sample_rate*4], 'k')
plt.title('Hilbert-Huang Transform')
plt.xlabel('Time (seconds)')
plt.ylabel('Frequency (Hz)')
Text(92.09722222222221, 0.5, 'Frequency (Hz)')
Compressed cycle stats#
The full-format output is useful for visualisation and validation, but often we only want to deal with a single number summarising each cycle. The compressed format provides this simplified output.
Note that the first value of the compressed format contains the average for missing cycles in the analysis (where the value in the cycles vector equals zero) we will discard this for the following
analyses as we are focusing on the properties of well formed oscillatory cycles.
For a first example, we compute the average frequency and amplitude of all cycles. We then make a scatter plot to explore any relationship between amplitude and frequency.
# Compute cycle average frequency for all cycles and masked cycles
all_cycle_freq = emd.cycles.get_cycle_stat(all_cycles[:, 2], IF[:, 2], func=np.mean)
mask_cycle_freq = emd.cycles.get_cycle_stat(mask_cycles[:, 2], IF[:, 2], func=np.mean)
# Compute cycle frequency range for all cycles and for masked cycles
all_cycle_amp = emd.cycles.get_cycle_stat(all_cycles[:, 2], IA[:, 2], func=np.mean)
mask_cycle_amp = emd.cycles.get_cycle_stat(mask_cycles[:, 2], IA[:, 2], func=np.mean)
# Make a summary figures
plt.plot(all_cycle_freq, all_cycle_amp, 'o')
plt.plot(mask_cycle_freq, mask_cycle_amp, 'o')
plt.xlabel('Cycle average frequency (Hz)')
plt.ylabel('Cycle average amplitude')
plt.plot((9, 22), (thresh, thresh), 'k:')
plt.legend(['All-cycles', 'Masked-cycles', 'Amp thresh'])
<matplotlib.legend.Legend object at 0x7f0ebb0e6d10>
We see that high amplitude cycles are closely clustered around 15Hz - the peak frequency of our simulated oscillation. Lower amplitude cycles are noisier and have a wider frequency distribution. The
rejected bad-cycles tend to have low amplitudes and come from a wide frequency distribution.
A small number of cycles pass the amplitude threshold but are rejected by the cycle quality checks. These cycles may have phase distortions or other artefacts which have lead to
emd.cycles.get_cycle_vector to remove them from the set of good cycles.
We can include more complex user-defined functions to generate cycle stats. Here we compute a range of cycle stats in compressed format (discarding the first value in the output). We compute the
cycle average frequency and cycle-max amplitude for all cycles and again for only the good cycles. We can then make a scatter plot to explore any relationship between amplitude and frequency.
We can include more complicated metrics in user-specified functions. Here we compute the Degree of Non-linearity (DoN; https://doi.org/10.1371/journal.pone.0168108) of each cycle as an indication of
the extent to which a cycle contains non-sinudoisal content.
Note that the original DoN uses the zero-crossing frequency rather than mean frequency as a normalising factor. These factors are highly correlated so, for simplicity, we use the mean here.
Here we compute the degree of non-linearity for all cycles and good cycles separately and plot the results as a function of cycle average frequency
# Compute cycle average frequency for all cycles and masked cycles
all_cycle_freq = emd.cycles.get_cycle_stat(all_cycles[:, 2], IF[:, 2], func=np.mean)
mask_cycle_freq = emd.cycles.get_cycle_stat(mask_cycles[:, 2], IF[:, 2], func=np.mean)
# Define a simple function to compute the range of a set of values
def degree_nonlinearity(x):
return np.std((x - x.mean()) / x.mean())
# Compute cycle freuquency range for all cycles and for masked cycles
all_cycle_freq_don = emd.cycles.get_cycle_stat(all_cycles[:, 2], IF[:, 2],
cycle_freq_don = emd.cycles.get_cycle_stat(mask_cycles[:, 2], IF[:, 2],
# Make a summary figures
plt.plot(all_cycle_freq, all_cycle_freq_don, 'o')
plt.plot(mask_cycle_freq, cycle_freq_don, 'o')
plt.xlabel('Cycle average frequency (Hz)')
plt.ylabel('Cycle IF don (Hz)')
plt.legend(['All-cycles', 'Masked-cycles'])
<matplotlib.legend.Legend object at 0x7f0eb98c0750>
The majority of cycles with very high degree of non-linearity in this simulation have been rejected by either the amplitude threshold or the cycle quality checks. The surviving cycles (in orange) are
tightly clustered around 15Hz peak frequency with a relatively low degree of non-linearity. We have not defined any non-linearity in this simulation.
Further Reading & References#
Andrew J. Quinn, Vítor Lopes-dos-Santos, Norden Huang, Wei-Kuang Liang, Chi-Hung Juan, Jia-Rong Yeh, Anna C. Nobre, David Dupret, and Mark W. Woolrich (2001) Within-cycle instantaneous frequency
profiles report oscillatory waveform dynamics Journal of Neurophysiology 126:4, 1190-1208 https://doi.org/10.1152/jn.00201.2021
Total running time of the script: (0 minutes 1.274 seconds) | {"url":"https://emd.readthedocs.io/en/stable/emd_tutorials/03_cycle_ananlysis/emd_tutorial_03_cycle_02_statistics.html","timestamp":"2024-11-02T22:05:53Z","content_type":"text/html","content_length":"64035","record_id":"<urn:uuid:94da3267-ec65-4c1a-96f9-f994f2e0413c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00400.warc.gz"} |
Understanding the Bubble Sort Algorithm - Enozom
Bubble sort Algorithm is one of the simplest and most well-known sorting algorithms in computer science. Despite its simplicity, it plays a fundamental role in the understanding of sorting techniques
and algorithmic thinking. This article will delve into what bubble sort is, how it works, its advantages and disadvantages, and provide a step-by-step guide to implementing it.
Bubble sort is a comparison-based algorithm used to sort a list of elements in either ascending or descending order. It repeatedly steps through the list, compares adjacent elements, and swaps them
if they are in the wrong order. This process is repeated until the list is sorted.
How Does Bubble Sort Work?
The bubble sort algorithm works by repeatedly swapping adjacent elements if they are in the wrong order. Here’s a step-by-step breakdown:
• Starting Point: Begin at the start of the list.
• Comparison: Compare the first two adjacent elements.
• Swapping: If the first element is greater than the second, swap them.
• Proceeding: Move to the next pair of adjacent elements.
• Repeating: Repeat the process for each pair of adjacent elements until the end of the list is reached.
• Iteration: Repeat the entire process for the whole list until no more swaps are needed.
The algorithm gets its name because smaller elements “bubble” to the top of the list while larger elements sink to the bottom.
Consider an unsorted list: [5, 3, 8, 4, 2]
• First pass:
□ Compare 5 and 3: Swap -> [3, 5, 8, 4, 2]
□ Compare 5 and 8: No Swap -> [3, 5, 8, 4, 2]
□ Compare 8 and 4: Swap -> [3, 5, 4, 8, 2]
□ Compare 8 and 2: Swap -> [3, 5, 4, 2, 8]
• Second pass:
□ Compare 3 and 5: No Swap -> [3, 5, 4, 2, 8]
□ Compare 5 and 4: Swap -> [3, 4, 5, 2, 8]
□ Compare 5 and 2: Swap -> [3, 4, 2, 5, 8]
□ Compare 5 and 8: No Swap -> [3, 4, 2, 5, 8]
• Third pass:
□ Compare 3 and 4: No Swap -> [3, 4, 2, 5, 8]
□ Compare 4 and 2: Swap -> [3, 2, 4, 5, 8]
□ Compare 4 and 5: No Swap -> [3, 2, 4, 5, 8]
□ Compare 5 and 8: No Swap -> [3, 2, 4, 5, 8]
• Fourth pass:
□ Compare 3 and 2: Swap -> [2, 3, 4, 5, 8]
□ Compare 3 and 4: No Swap -> [2, 3, 4, 5, 8]
□ Compare 4 and 5: No Swap -> [2, 3, 4, 5, 8]
□ Compare 5 and 8: No Swap -> [2, 3, 4, 5, 8]
• Fifth pass:
□ No swaps are needed, indicating the list is sorted.
Advantages of Bubble Sort
• Simplicity: Bubble sort is easy to understand and implement.
• Educational Value: It is a great algorithm for teaching the basics of sorting algorithms.
• Space Complexity: Bubble sort requires only a constant amount of additional memory space (O(1)).
Disadvantages of Bubble Sort
• Inefficiency: Bubble sort is not efficient for large lists. It has a time complexity of O(n^2), where n is the number of elements in the list.
• Performance: It performs poorly compared to more advanced sorting algorithms like quicksort, mergesort, and heapsort.
Implementing Bubble Sort in Python
Here’s how you can implement the bubble sort algorithm in Python:
def bubble_sort(arr):
n = len(arr)
for i in range(n):
# Track whether any swaps were made in this pass
swapped = False
# Last i elements are already sorted
for j in range(0, n-i-1):
# Swap if the element found is greater than the next element
if arr[j] > arr[j+1]:
arr[j], arr[j+1] = arr[j+1], arr[j]
swapped = True
# If no swaps were made, the list is sorted
if not swapped:
return arr
# Example usage
unsorted_list = [5, 3, 8, 4, 2]
sorted_list = bubble_sort(unsorted_list)
print(“Sorted list:”, sorted_list)
FAQ: Common Questions About Bubble Sort Algorithm
1. What is the best-case time complexity of Bubble Sort?
The best-case time complexity of bubble sort is O(n). This occurs when the input list is already sorted. In this case, the algorithm only needs to make one pass through the list to confirm that no
swaps are necessary.
2. Is Bubble Sort a stable sorting algorithm?
Yes, bubble sort is a stable sorting algorithm. Stability means that two equal elements will maintain their relative order in the sorted list as they were in the input list.
3. Can Bubble Sort be optimized?
Yes, bubble sort can be optimized by stopping the algorithm if no swaps are made during a pass through the list. This indicates that the list is already sorted and further passes are unnecessary.
4. How does Bubble Sort compare with other sorting algorithms?
Bubble sort is generally less efficient compared to more advanced algorithms like quicksort, mergesort, and heapsort. These algorithms have better average and worst-case time complexities, making
them more suitable for larger datasets. However, bubble sort’s simplicity and ease of understanding make it useful for educational purposes and small datasets.
5. Can Bubble Sort be used for linked lists?
Yes, bubble sort can be adapted to sort linked lists. However, its performance on linked lists is generally poor compared to other sorting algorithms designed specifically for linked lists, such as
merge sort.
6. How can Bubble Sort be adapted for descending order sorting?
To adapt bubble sort for descending order sorting, simply change the comparison condition from greater-than to less-than. Here’s an example:
def bubble_sort_descending(arr):
n = len(arr)
for i in range(n):
swapped = False
for j in range(0, n-i-1):
if arr[j] < arr[j+1]:
arr[j], arr[j+1] = arr[j+1], arr[j]
swapped = True
if not swapped:
return arr
# Example usage
unsorted_list = [5, 3, 8, 4, 2]
sorted_list = bubble_sort_descending(unsorted_list)
print(“Sorted list in descending order:”, sorted_list)
7. What are the primary use cases for Bubble Sort?
Bubble sort is mainly used for educational purposes to teach the basic concepts of sorting algorithms and algorithmic thinking. It is also useful for small datasets where its inefficiency is not a
significant drawback.
8. Can Bubble Sort handle duplicate values in the list?
Yes, bubble sort can handle duplicate values. It treats each comparison independently and swaps adjacent elements based on the defined condition (either greater-than or less-than), ensuring that
duplicates are placed correctly in the sorted order.
9. What is the difference between Bubble Sort and Selection Sort?
Bubble sort and selection sort are both simple, comparison-based sorting algorithms with a time complexity of O(n^2). The key difference is in their approach:
• Bubble Sort: Repeatedly compares and swaps adjacent elements.
• Selection Sort: Finds the minimum (or maximum) element and places it in its correct position in each iteration.
10. How does Bubble Sort perform on nearly sorted lists?
Bubble sort performs relatively well on nearly sorted lists. Its best-case time complexity of O(n) makes it efficient when only a few elements are out of order, as it can quickly detect that the list
is nearly sorted and stop early if no swaps are needed.
Bubble sort, with its simplicity and educational value, serves as an excellent introduction to the world of sorting algorithms. However, its inefficiency and poor performance compared to more
advanced algorithms limit its practical use, especially for large datasets. Understanding bubble sort provides a foundation for learning more efficient algorithms, highlighting the importance of
algorithmic optimization and the trade-offs between simplicity and performance. | {"url":"https://enozom.com/blog/understanding-the-bubble-sort-algorithm/","timestamp":"2024-11-03T00:01:52Z","content_type":"text/html","content_length":"166591","record_id":"<urn:uuid:ff0a4f60-494a-4f6b-82e4-d499d941af24>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00618.warc.gz"} |
From New World Encyclopedia
A compass (or mariner's compass) is a navigational instrument for finding directions on the earth. It consists of a magnetized pointer free to align itself accurately with Earth's magnetic field,
which is of great assistance in navigation. The cardinal points are north, south, east and west. A compass can be used in conjunction with a chronometer and a sextant to provide a very accurate
navigation capability. This device greatly improved maritime trade by making travel safer and more efficient. An early form of compass was invented in China in 271 C.E. and is one of four great
inventions of ancient China. The familiar mariner's compass was invented in Europe around 1300.
More technically, a compass is a magnetic device using a needle to indicate the direction of the magnetic north of a planet's magnetosphere. Any instrument with a magnetized bar or needle turning
freely upon a pivot and pointing in a northerly and southerly direction can be considered a compass. A compass dial is a small pocket compass with a sundial. A variation compass, a specific
instrument with a delicate construction, is used by observing variations of the needle. A gyrocompass or astrocompass can also be used to ascertain true north.
Prior to the introduction of the compass, directions at sea were determined primarily by the position of celestial bodies. Navigation was supplemented in some places by the use of soundings.
Difficulties arose where the sea was too deep for soundings and conditions were continually overcast or foggy. Thus the compass was not of the same utility everywhere. For example, the Arabs could
generally rely on clear skies in navigating the Persian Gulf and the Indian Ocean (as well as the predictable nature of the monsoons). This may explain in part their relatively late adoption of the
compass. Mariners in the relatively shallow Baltic made extensive use of soundings.
Developments in Chinese
Due to the place of its first appearance, most scholars credit at present the invention of the compass to China. Since there has been frequently confusion as to when a compass was introduced for the
first time, it may be appropriate to list the important events leading up to its invention in chronological order:
• The earliest Chinese literary reference to magnetism lies in a fourth century B.C.E. book called Book of the Devil Valley Master (鬼谷子): "The lodestone makes iron come or it attracts it."^[1]
• The first mention of the magnetic attraction of a needle is to be found in a Chinese work composed between 20 and 100 C.E. (Louen-heng): "A lodestone attracts a needle."^[2]
• The earliest reference to a magnetic device as a direction finder is recorded in a Song dynasty book dated to 1040-1044. Here we find a description of an iron "south-pointing fish" floating in a
bowl of water, aligning itself to the south. The device is recommended as a means of orientation "in the obscurity of the night." There is, however, no mention of a use for navigation, nor how
the fish was magnetized.^[3]
• The first incontestable reference to a magnetized needle in Chinese literature appears as late as 1086.^[4] The Dream Pool Essay written by Song Dynasty scholar Shen Kua contained a detailed
description of how geomancers magnetized a needle by rubbing its tip with lodestone, and hung the magnetic needle with one single strain of silk with a bit of wax attached to the center of the
needle. Shen Kua pointed out that a needle prepared this way sometimes pointed south, sometimes north.
• The earliest recorded actual use of a magnetized needle for navigational purposes then is to be found in Zhu Yu's book Pingzhou Table Talks (Pingzhou Ke Tan) of 1117 C.E.: "The navigator knows
the geography, he watches the stars at night, watches the sun at day; when it is dark and cloudy, he watches the compass."
• A pilot's compass handbook titled Shun Feng Xiang Song (Fair Winds for Escort) in the Oxford Bodleian Library contains great details about the use of compass in navigation.
• "Earliest records show a spoon shaped compass made of lodestone or magnetite ore, referred to as a "South-pointer" dating back to sometime during the Han Dynasty (2nd century B.C.E. to 2nd
century CE). The spoon-shaped instrument was placed on a cast bronze plate called a "heaven-plate" or diviner's board that had the eight trigrams (Pa Gua) of the I Ching, as well as the 24
directions (based on the constellations), and the 28 lunar mansions (based on the constellations dividing the Equator). Often, the Big Dipper (Great Bear) was drawn within the center disc. The
square symbolized earth and the circular disc symbolized heaven. Upon these were inscribed the azimuthal points relating to the constellations. Its primary use was that of geomancy
(prognostication) to determine the best location and time for such things as burials. In a culture that placed extreme importance on reverence for ancestors, this remained an important tool well
into the nineteenth century. Even in modern times there are those who use this divination concepts of Feng Shui (literally, of wind and water) for locating buildings or fortuitous times and
locations for almost any enterprise. There is a story that the first Chin emperor used the divining board and compass in court to affirm his right to the throne. Primarily, the compass was used
for geomancy for a long time before it was used for navigation." ^[5]
Question of Diffusion
There is much debate on what happened to the compass after its first appearance with the Chinese. Different theories include:
• Travel of the compass from China to the Middle East via the Silk Road, and then to Europe
• Direct transfer of the compass from China to Europe, and then later from Europe to the Middle East
• Independent creation of the compass in the Europe and then its transfer thereafter to the Middle East.
The latter two are supported by evidence of the earlier mentioning of the compass in European works rather than Arabic. The first European mention of a magnetized needle and its use among sailors
occurs in Alexander Neckam's De naturis rerum (On the Natures of Things), probably written in Paris in 1190.^[6] Other evidence for this includes the Arabic word for "Compass" (al-konbas), possibly
being a derivation of the old Italian word for compass.
In the Arab world, the earliest reference comes in The Book of the Merchants' Treasure, written by one Baylak al-Kibjaki in Cairo about 1282.^[7] Since the author describes having witnessed the use
of a compass on a ship trip some forty years earlier, some scholars are inclined to antedate its first appearance accordingly. There is also a slightly earlier non-Mediterranean Muslim reference to
an iron fish-like compass in a Persian talebook from 1232.^[8]
Question of independent European invention
There have been various arguments put forward whether the European compass was an independent invention or not:
Arguments that support independent invention:
• The navigational needle in Europe points invariably north, whereas always south in China.
• The European compass showed from the beginning sixteen basic divisions, not twenty-four as in China.
• The apparent failure of the Arabs to function as possible intermediaries between East and West due to the earlier recorded appearance of the compass in Europe (1190) than in the Muslim world
(1232, 1242, or 1282).
Arguments against independent invention:
• The temporal priority of the Chinese navigational compass (1117) as opposed to the European compass (1190).
Impact in the Mediterranean
In the Mediterranean the practice from ancient times had been to curtail sea travel between October and April, due in part to the lack of dependable clear skies during the Mediterranean winter (and
much of the sea is too deep for soundings). With improvements in dead reckoning methods, and the development of better charts, this changed during the second half of the thirteenth century. By around
1290 the sailing season could start in late January or February, and end in December. The additional few months were of considerable economic importance; it enabled Venetian convoys, for instance, to
make two round trips a year to the eastern Mediterranean, instead of one.
Around the time Europeans learned of the compass, traffic between the Mediterranean and northern Europe increased, and one factor may be that the compass made traversal of the Bay of Biscay safer and
Modern liquid-filled compass
In 1936 Tuomas Vohlonen of Finland invented and patented the first successful portable liquid-filled compass designed for individual use.^[9]
Construction of a simple compass
A magnetic rod is required when constructing a compass. This can be created by aligning an iron or steel rod with Earth's magnetic field and then tempering or striking it. However, this method
produces only a weak magnet so other methods are preferred. This magnetised rod (or magnetic needle) is then placed on a low friction surface to allow it to freely pivot to align itself with the
magnetic field. It is then labeled so the user can distinguish the north-pointing from the south-pointing end; in modern convention the north end is typically marked in some way, often by being
painted red.
Flavio Gioja (fl. 1302), an Italian marine pilot, is sometimes credited with perfecting the sailor's compass by suspending its needle over a fleur-de-lis design, which pointed north. He also enclosed
the needle in a little box with a glass cover.
Modern hand-held navigational compasses use a magnetized needle or dial inside a fluid-filled (oil, kerosene, or alcohol is common) capsule; the fluid causes the needle to stop quickly rather than
oscillate back and forth around magnetic north. Most modern recreational and military compasses integrate a protractor with the compass, using a separate magnetized needle. In this design the
rotating capsule containing the magnetized needle is fitted with orienting lines and an outlined orienting arrow, then mounted in a transparent baseplate containing a direction-of-travel (DOT)
indicator for use in taking bearings directly from a map. Other features found on some modern handheld compasses are map and romer scales for measuring distances and plotting positions on maps,
luminous markings or bezels for use at night or poor light, various sighting mechanisms (mirror, prism, etc.) for taking bearings of distant objects with greater precision, 'global' needles for use
in differing hemispheres, adjustable declination for obtaining instant true bearings without resort to arithmetic, and devices such as inclinometers for measuring gradients.
The military forces of a few nations, notably the United States Army, continue to utilize older lensatic card compass designs with magnetized compass dials instead of needles. A lensatic card compass
permits reading the bearing off of the compass card with only a slight downward glance from the sights (see photo), but requires a separate protractor for use with a map. The official U.S. military
lensatic compass does not use fluid to dampen needle swing, but rather electromagnetic induction. A 'deep-well' design is used to allow the compass to be used globally with little or no effect in
accuracy caused by a tilting compass dial. As induction forces provide less damping than fluid-filled designs, a needle lock is fitted to the compass to reduce wear, operated by the folding action of
the rear sight/lens holder. The use of air-filled induction compasses has declined over the years, as they may become inoperative or inaccurate in freezing temperatures or humid environments.
Other specialty compasses include the optical or prismatic hand-bearing compass, often used by surveyors, cave explorers, or mariners. This compass uses an oil-filled capsule and magnetized compass
dial with an integral optical or prismatic sight, often fitted with built-in photoluminescent or battery-powered illumination. Using the optical or prism sight, such compasses can be read with
extreme accuracy when taking bearings to an object, often to fractions of a degree. Most of these compasses are designed for heavy-duty use, with solid metal housings, and many are fitted for tripod
mounting for additional accuracy.
Mariner's compasses can have two or more magnetic needles permanently attached to a compass card. These move freely on a pivot. A lubber line, which can be a marking on the compass bowl or a small
fixed needle indicates the ship's heading on the compass card.
Traditionally the card is divided into thirty-two points (known as rhumbs), although modern compasses are marked in degrees rather than cardinal points. The glass-covered box (or bowl) contains a
suspended gimbal within a binnacle. This preserves the horizontal position.
Large ships typically rely on a gyrocompass, using the more reliable magnetic compass for back-up. Increasingly electronic fluxgate compasses are used on smaller vessels.
Some modern military compases, like the [SandY-183 http://www.orau.org/PTP/collection/radioluminescent/armycompass.htm](the one pictured) contains the radioactive material Tritium (3H) and a
combination of Phosphorous. The SandY-183 contained 120mCi (millicuries) of tritium. The name SandY-183 is derived from the name of the company, Stocker and Yale (SandY).
Solid state compasses
Small compasses found in clocks, cell phones (e.g. the Nokia 5140i) and other electronic gear are Solid-state electronics usually built out of two or three magnetic field sensors that provide data
for a microprocessor. Using trigonometry the correct heading relative to the compass is calculated.
Often, the device is a discrete component which outputs either a digital or analog signal proportional to its orientation. This signal is interpreted by a controller or microprocessor and used either
internally, or sent to a display unit. An example implementation, including parts list and circuit schematics, shows one design of such electronics. The sensor uses precision magnetics and highly
calibrated internal electronics to measure the response of the device to the Earth's magnetic field. The electrical signal is then processed or digitized.
Bearing compass
A bearing compass is a magnetic compass mounted in such a way that it allows the taking of bearings of objects by aligning them with the lubber line of the bearing compass.^[10]
Compass correction
Like any magnetic device, compasses are affected by nearby ferrous materials as well as by strong local electromagnetic forces. Compasses used for wilderness land navigation should never be used in
close proximity to ferrous metal objects or electromagnetic fields (batteries, car bonnets, engines, steel pitons, wristwatches, and so forth.)
Compasses used in or near trucks, cars or other mechanized vehicles are particularly difficult to use accurately, even when corrected for deviation by the use of built-in magnets or other devices.
Large amounts of ferrous metal combined with the on-and-off electrical fields caused by the vehicle's ignition and charging systems generally result in significant compass errors.
At sea, a ship's compass must also be corrected for errors, called compass deviation, caused by iron and steel in its structure and equipment. The ship is swung, that is rotated about a fixed point
while its heading is noted by alignment with fixed points on the shore. A compass deviation card is prepared so that the navigator can convert between compass and magnetic headings. The compass can
be corrected in three ways. First the lubber line can be adjusted so that it is aligned with the direction in which the ship travels, then the effects of permanent magnets can be corrected for by
small magnets fitted within the case of the compass. The effect of ferromagnetic materials in the compass's environment can be corrected by two iron balls mounted on either side of the compass
binacle. The coefficient ${\displaystyle a_{0}}$ representing the error in the lubber line, while ${\displaystyle a_{1},b_{1}}$ the ferromagnetic effects and ${\displaystyle a_{2},b_{2}}$ the
non-ferromagnetic component.
Fluxgate compasses can be calibrated automatically, and can also be programmed with the correct local compass variation so as to indicate the true heading.
Using a compass
The simplest way of using a compass is to know that the arrow always points in the same direction, magnetic North, which is roughly similar to true north. Except in areas of extreme magnetic
declination variance (20 degrees or more), this is enough to protect from walking in a substantially different or even opposite direction than expected over short distances, provided the terrain is
fairly flat and visibility is not impaired. In fact, by carefully recording distances (time or paces) and magnetic bearings traveled, one can plot a course and a return to one's starting point using
the compass alone.
However, compass navigation used in conjunction with a map (terrain association) requires a different compass method. To take a map bearing or true bearing (a bearing taken in reference to true, not
magnetic north) to a destination with a protractor compass, the edge of the compass is placed on the map so that it connects the current location with the desired destination (some sources recommend
physically drawing a line). The orienting lines in the base of the compass dial are then rotated to align with actual or true north by aligning them with a marked line of longitude (or the vertical
margin of the map), ignoring the compass needle entirely. The resulting true bearing or map bearing may then be read at the degree indicator or direction-of-travel (DOT) line, which may be followed
as an azimuth (course) to the destination. If a magnetic north bearing or compass bearing is desired, the compass must be adjusted by the amount of magnetic declination before using the bearing so
that both map and compass are in agreement. In the given example, the large mountain in the second photo was selected as the target destination on the map.
The modern hand-held protractor compass always has an additional direction-of-travel (DOT) arrow or indicator inscribed on the baseplate. To check one's progress along a course or azimuth, or to
ensure that the object in view is indeed the destination, a new compass reading may be taken to the target if visible (here, the large mountain). After pointing the DOT arrow on the baseplate at the
target, the compass is oriented so that the needle is superimposed over the orienting arrow in the capsule. The resulting bearing indicated is the magnetic bearing to the target. Again, if one is
using 'true' or map bearings, and the compass does not have preset, pre-adjusted declination, one must additionally add or subtract magnetic declination to convert the magnetic bearing into a true
bearing. The exact value of the magnetic declination is place-dependent and varies over time, though declination is frequently given on the map itself or obtainable on-line from various sites. If
not, any local walker club should know it. If the hiker has been following the correct path, the compass' corrected (true) indicated bearing should closely correspond to the true bearing previously
obtained from the map.
This method is sometimes known as the Silva 1-2-3 System, after Silva Compass, manufacturers of the first protractor compasses.^[11] ^[12]
Compass balancing
Because the Earth's magnetic field varies at different latitudes, compasses are often balanced during manufacture. Most manufacturers balance their compass needles for one of five zones, ranging from
zone 1, covering most of the Northern Hemisphere, to zone 5 covering Australia and the southern oceans. This balancing prevents excessive dipping of one end of the needle which can cause the compass
card to stick and give false readings. Suunto has recently introduced two-zone compasses that can be used in one entire hemisphere, and to a limited extent in another without significant loss of
Points of the compass
Originally, many compasses were marked only as to the direction of magnetic north, or to the four cardinal points (north, south, east, west). Later, mariners divided the compass card into 32 equally
spaced points divided from the cardinal points.
The 360-degree system later took hold, which is still in use today for civilian navigators. The degree dial spaces the compass markings with 360 equidistant points. Other nations adopted the 'grad'
system, which spaces the dial into 400 grads or points.
Most military defense forces have adopted the 'mil' system, in which the compass dial is spaced into 6400 units (some nations use 6000) or 'mils' for additional precision when measuring angles,
laying artillery, and so forth.
Some different compass systems:
See also
• Azimuth
• Coordinates
• Global positioning system
• Inertial navigation system
• Radio compass
ISBN links support NWE through referral fees
• Aczel, Amir. 2002. The Riddle of the Compass: The Invention that Changed the World. Fort Washington, PA: Harvest Books. ISBN 0156007533.
• Gurney, Alan. 2004. Compass: A Story of Exploration and Innovation. New York, NY: W.W. Norton. ISBN 0393327132.
• Kreutz, Barbara M. 1973. "Mediterranean Contributions to the Medieval Mariner's Compass." In Technology and Culture 14 (3): 367-383.
• Lane, Frederic C. 1963. "The Economic Meaning of the Invention of the Compass." In The American Historical Review 68 (3) (Apr., 1963): 605-617.
• Li Shu-hua. 1954. "Origine de la Boussole 11. Aimant et Boussole." In Isis 45 (2) (Jul., 1954): 175-196.
• Needham, Joseph. Science and Civilization in China, Vol. 4, part 1: Physics. Cambridge Univ. Press, 1962.
• Needham, Joseph, and Colin A. Ronan. Chapter 1, "Magnetism and Electricity." The Shorter Science & Civilisation in China. Vol 3.
• Williams, J.E.D. 1992. From Sails to Satellites. New York, NY: Oxford University Press. ISBN 0198563876.
External links
All links retrieved January 7, 2024.
• Evening Lecture To The British Association At The Southampton Meeting by Sir William Thomson (Lord Kelvin) on Friday, August 25, 1882 The Tides. Refers to compass correction by Fourier series.
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C M_pi_2: C ExplainedC M_pi_2: C Explained
C M_pi_2 is a mathematical constant representing the ratio of a circle’s circumference to its diameter. It is roughly equal to 3.14159 and is often written as the symbol PI. It is an important
concept in mathematics and its applications can be found throughout the sciences and engineering. In this article, we’ll explain what C M_pi_2 is, its advantages, how to use it, what common mistakes
to avoid, best practices, troubleshooting tips, and alternatives.
What is C M_pi_2?
C M_pi_2 is an irrational number, which means it has an infinite number of decimal places and cannot be expressed as a simple fraction. It is often approximated by the decimal number 3.14159. C
M_pi_2 can be calculated using several methods, such as a numerical approximation or by drawing a circle and measuring its circumference and diameter. It has many practical applications in everyday
mathematics and science, ranging from calculating the area of a circle to analyzing complex data sets.
C M_pi_2 is also used in trigonometry, where it is used to calculate the length of a side of a triangle when the angles and one side are known. It is also used in physics to calculate the force of
gravity between two objects, and in engineering to calculate the volume of a cylinder. C M_pi_2 is an important number in mathematics and science, and its applications are far-reaching.
What are the Advantages of C M_pi_2?
C M_pi_2 is a fundamental concept in mathematics and has many useful applications in daily life. It can be used to calculate the area of a circle, the circumference of a circle or other curved
shapes, the gradient of a line, the distance between two points, the volume of a sphere, and other geometric calculations. Additionally, C M_pi_2 can be used to make complex calculations in physics,
engineering, economics, and other areas.
C M_pi_2 is also used to calculate the angles of a triangle, the area of a triangle, the area of a trapezoid, and the area of a polygon. It can also be used to calculate the volume of a cone, the
volume of a cylinder, and the volume of a pyramid. Furthermore, C M_pi_2 can be used to calculate the surface area of a sphere, the surface area of a cube, and the surface area of a cylinder.
How to Use C M_pi_2
C M_pi_2 can be used in a variety of calculations in mathematics and science. In basic calculations, you can use the approximate value 3.14159 or you can use a calculator to use the more precise
value. To calculate the circumference of a circle, for example, you can simply multiply the diameter by C M_pi_2. In more complex calculations, you can use numerical approximations or computer
libraries and software to calculate the exact value of C M_pi_2 for the given application.
In addition, C M_pi_2 can be used to calculate the area of a circle by multiplying the square of the radius by C M_pi_2. It can also be used to calculate the volume of a sphere by multiplying the
cube of the radius by C M_pi_2 and then multiplying the result by four-thirds. C M_pi_2 is an essential tool for many mathematical and scientific calculations.
Common Mistakes to Avoid When Using C M_pi_2
When using C M_pi_2 for calculations, it is important to make sure you are using the correct value for the application. Using an inexact value such as 3 may be close enough for some calculations, but
it is important to use a precise value when accuracy is critical. Additionally, it is important to remember that C M_pi_2 is an irrational number and cannot be expressed as an exact fraction.
It is also important to remember that C M_pi_2 is a constant and cannot be changed. If you need to use a different value for a calculation, you will need to use a different constant or calculate the
value yourself. Additionally, it is important to be aware of the limitations of C M_pi_2 when using it for calculations. For example, it is not suitable for calculations involving angles greater than
180 degrees.
Best Practices for Working with C M_pi_2
When working with C M_pi_2, it is important to use the most accurate value for the application. If accuracy is not critical, a good practice is to use an approximation such as 3.14159. In situations
where accuracy is critical, it is best to use a numerical approximation or software library to calculate the exact value of C M_pi_2. Additionally, it is important to remember that C M_pi_2 is an
irrational number and cannot be expressed as an exact fraction.
When working with C M_pi_2, it is also important to consider the precision of the value. For example, if the application requires a value with six decimal places, it is best to use a numerical
approximation or software library to calculate the exact value of C M_pi_2 with the desired precision. Additionally, it is important to remember that C M_pi_2 is an irrational number and cannot be
expressed as an exact fraction with a finite number of decimal places.
Troubleshooting Tips for C M_pi_2
If you are having trouble using C M_pi_2 in your calculations, there are several simple troubleshooting steps you can take. Make sure that you are using the exact value of C M_pi_2 for your
application, as an approximation may not be accurate enough for all situations. Additionally, if you are having trouble with numerical approximations, try using a software library or calculator to
calculate the exact value of C M_pi_2 for your application.
If you are still having difficulty, you may need to review the mathematical principles behind C M_pi_2 and make sure you understand how to use it correctly. Additionally, you can consult online
resources or textbooks for more information on the topic. Finally, if you are still having trouble, you can reach out to a professional for help.
Alternatives to C M_pi_2
C M_pi_2 is an important concept in mathematics and engineering but there are several alternatives that can be used in certain applications. The arc-length formula can be used to calculate the length
of an arc on a circle if you only know the measure of its angle in radians. The area-of-a-circle formula can also be used if you only know its radius or diameter. Additionally, formulas such as
Heron’s formula or the Shoelace algorithm can be used to calculate areas or perimeters of complex shapes without using C M_pi_2. | {"url":"https://bito.ai/resources/c-m_pi_2-c-explained/","timestamp":"2024-11-13T11:40:30Z","content_type":"text/html","content_length":"375892","record_id":"<urn:uuid:6f76fdff-ca79-4334-9bc1-d64f8ba5af95>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00639.warc.gz"} |
Backpropagation and convex programming in MRAS systems
The backpropagation algorithm is very useful for general optimization tasks, particularly in neural network function approximators and deep learning applications. Great progress in nonlinear function
approximation has been made due to the effectiveness of the backprop algorithm. Whereas in traditional control applications, we typically use feedback regulation to stabilize the states of the
system, in model reference adaptive control systems, we want to specify an index of performance to determine the “goodness” of our adaptation. An auxiliary dynamic system called the reference model
is used in generating this index of performance (IP). The reference model specifies in terms of the input and states of the model a given index of performance and a comparison check determines
appropriate control laws by comparing the given IP and measured IP based on the outputs of the adjustable system to that of the reference model system. This is called the error state space.
Nonlinear Model Reference Adaptive Systems
With nonlinear systems, the unknown nonlinearity, say \(f(.)\), is usually approaximated with a function approximator such as a single hidden layer neural network. To date, the state-of-the-art used
in adjusting the weights of a neural network is the backpropagation algorithm. The optimization in classical backprop is unrolled end-to-end so that the complexity of the network increases when we
want to add an argmin differentiation layer before the final neural network layer. The final layer determines the controller parameters or generates the control laws used in adjusting the plant
behavior. Fitting the control laws into actuator constraints such as model predictive control schemes allow is not explicitly formulated when using the backprop algorithm; ideally, we would want to
fit a quadratic convex layer to compute controller parameters exactly. We cannot easily fit a convex optimization layer into the backprop algorithm using classical gradient descent because the
explicit Jacobians of the gradients of the system’s energy function with respect to system parameters is not exactly formulated (but rather are ordered derivatives which fluctuate about the global/
local minimum when the weights of the network converge).
The final layer determines the controller parameters or generates the control laws used in adjusting the plant behavior. Fitting the control laws into actuator constraints such as model
predictive control schemes allow is not explicitly formulated when using the backprop algorithm; ideally, we would want to fit a quadratic convex layer to compute controller parameters exactly.
To generate control laws such as torques to control a motor arm in a multi-dof robot-arm for example, we would want to define a quadratic programming layer as the last layer of our neural network
optimization algorithm so that effective control laws that exactly fit into actuator saturation limits are generated. Doing this requires a bit of tweaking of the backprop algorithm on our part.
Solving Quadratic Programming in a Backprop setting
When trying to construct a controller for a regulator, or an MRAS system, we may imagine that the control law determination is a search process for a control scheme that takes an arbitrary nonzero
initial state to a zero state, ideally in a short amount of time. If the system is controllable, then we may require the controller taking the system, from state \(x(t_0)\) to the zero state at time
\(T\). If \(T\) is closer to \(t_0\) than not, more control effort would be required to bear states to \(t_0\). This would ensure the transfer of states. In most engineering systems, an upper bound
is set on the magnitudes of the variables for pragmatic purposes. It therefore becomes impossible to take \(T\) to \(0\) without exceeding the control bounds. Unless we are ready to tolerate high
gain terms in the controller parameters, the control is not feasible for finite T. So what do we do? To meet the practical bounds manufacturers place on physical actuators, it suffices to manually
formulate these bounds as constraints into the control design objectives.
Model predictive controllers have explicit ways of incorporating these constraints into the control design. There are no rules for tuning the parameters of an MRAC system so that the control laws
generated in our adjustment mechanism are scaled into the bounds of the underlying actuator.
Since most controller hardware constraints are specified in terms of lower and upper bounded saturation, the QP problem formulated below is limited to inequality constraints. For equality-constrained
QP problems, Mattingley and Boyd, Vanderberghe’s CVX Optimization, or Brandon Amos’ ICML submission offer good treatments.
There are no rules for tuning the parameters of an MRAC system so that the control laws generated in our adjustment mechanism are scaled into the bounds of the underlying actuator.
We define the standard QP canonical form problem with inequality contraints thus:
\text{minimize} \quad \frac{1}{2}x^TQx + q^Tx \label{eq:orig}
subject to
G x \le h \nonumber
where \(Q \succeq \mathbb{S}^n_+ \) (i.e. a symmetric, positive semi-definite matrix) \(\in \mathbb{R}^n, q \in \mathbb{R}^n, G \in \mathbb{R}^{p \times n}, \text{ and } h \in \mathbb{R}^p \).
Suppose we have our convex quadratic optimization problem in canonical form, we can use primal-dual interior point methods (PDIPM) to find an optimal solution to such a problem. PDIPMs are the
state-of-the-art in solving such problems. Primal-dual methods with Mehrota predictor-corrector are consistent for reliably solving QP embedded optimization problems within 5-25iterations, without
warm-start (Boyd and Mattingley, 2012).
Slack Variables
Given \eqref{eq:orig}, one can introduce slack variables, \(s \in \mathbb{R}^p\) as follows,
\text{minimize} \quad \frac{1}{2}x^TQx + q^Tx \label{eq:orig1}
subject to
\quad G x + s = h, \qquad s \ge 0 \nonumber
where \(x \in \mathbb{R}^n, s \in \mathbb{R}^p\). If we let a dual variable \(z \in \mathbb{R}^p \) be associated with the inequality constraint, then we can define the KKT conditions for \eqref
{eq:orig1} as
\[Gx + s = h, \quad s \ge 0 \\ z \ge 0 \\ Qx + q + G^T = 0 \\ \\ z_i s_i = 0, i = 1, \ldots, p.\]
More formally, if we write the Lagrangian of system \eqref{eq:orig} as
L(z, \lambda) = \frac{1}{2}x^TQx + q^Tx +\lambda^T(Gz -h) \label{eq:Lagrangian}
it follows that the KKT for stationarity, primal feasibility and complementary slackness are,
Q x^\ast + q + G^T \lambda^\ast = 0 , \label{eq:KKTLagrangian}
\[K \left(\lambda^\ast\right) \left(G x^\ast - h\right) = 0\]
where \(K(\cdot) = \textbf{diag}(k) \) is an operator that creates a matrix diagonal of the entries of the vector \(k\). Computing the time-derivative of \eqref{eq:KKTLagrangian}, we find that
dQ x^* + Q dx + dq + dG^T \lambda^* + G^T d\lambda = 0 \label{eq:KKTDiff}
\[K(\lambda^*)\left(G x^* - h\right) = 0\]
QP Layer as the last layer in backpropagation
Vectorizing \eqref{eq:KKTDiff}, we find
\[\begin{bmatrix} Q & G^T \\ K(\lambda^\ast) G & K(dGx^\ast - h) \\ \end{bmatrix} \begin{bmatrix} dx \\ d\lambda \\ \end{bmatrix} = \begin{bmatrix} -dQ x^\ast - dq - dG^T \lambda^\ast \\ -K(\lambda^\
ast) dG x^\ast + DK(\lambda^\ast) dh \\ \end{bmatrix}\]
so that the Jacobians of the variables to be optimized can be formed with respect to the states of the system. Finding \(\dfrac{\partial J}{\partial h^*}\), for example, would involve passing \(dh\)
as identity and setting other terms on the rhs in the equation above to zero. After solving the equation, the desired Jacobian would be \(dz\). With backpropagation, however, the explicit Jacobians
are useless since the gradients of the network parameters are computed using chain rule for ordered derivatives i.e.
\[\dfrac{\partial ^+ J}{ \partial h_i} = \dfrac{\partial J}{ \partial h_i} + \sum_{j > i} \dfrac{\partial ^+ J}{\partial h_j} \dfrac{ {\partial} h_j}{ \partial h_i}\]
where the derivatives with superscripts denote ordered derivatives and those with subscripts denote ordinary partial derivatives. The simple partial derivatives denote the direct effect of \(h_i\) on
\(h_j\) through the linear set of equations that determine \(h_j\). To illustrate further, suppose that we have a system of equations given by
\[x_2 = 3 x_1 \\ x_3 = 5 x_1 + 8 x_2\]
The ordinary partial derivatives of \(x_3\) with respect to \(x_1\) would be \(5\). However, the ordered derivative of \(x_3\) with respect to \(x_1\) would be \(29\) (because of the indirect effect
by way of \(x_2\)).
So with the backprop algorithm, we would form the left matrix-vector product with a previous backward pass vector, \(\frac{\partial J}{\partial x^\ast} \in \mathbb{R}^n \); this is mathematically
equivalent to \(\frac{\partial J}{ \partial x^\ast} \cdot \frac{\partial x^\ast}{ \partial h} \). Therefore, computing the solution for the derivatives of the optimization variables \(dx, d\lambda\),
we have through the matrix inversion of \eqref{eq:KKTDiff},
\[\begin{bmatrix} dx \\ d\lambda \end{bmatrix} = \begin{bmatrix} Q & G^T K(\lambda^\ast) \\ G & K(Gx^\ast - h) \end{bmatrix}^{-1} = \begin{bmatrix} {\dfrac{dJ}{dx^\ast}}^T \\ 0 \end{bmatrix}.\]
The relevant gradients with respect to every QP paramter is given by
\[\dfrac{\partial J}{\partial q} = d_x, \qquad \dfrac{\partial J}{ \partial h} = -K(\lambda^\ast) d_\lambda \\ \dfrac{\partial J}{\partial Q} = \frac{1}{2}(d_x x^T + x d_x^T), \qquad \dfrac{\partial
J}{\partial G} = K(\lambda^\ast)(d_\lambda z^T + \lambda d_z^T )\]
QP Initialization
For the primal problem,
\[\text{minimize} \quad \frac{1}{2}x^T Q x + p^T x + (\frac{1}{2}\|s\|^2_2) \\ \text{ subject to } \quad Gx + s = h \\\]
with \(x\) and \(s\) as variables to be optimized, the corresponding dual problem is,
\[\text{maximize} \quad -\frac{1}{2}w^T Q w - h^T z + (\frac{1}{2}\|z\|^2_2) \\ \text{ subject to } \quad Qw + G^T z + q = 0 \\\]
with variables \(w\) and \(z\) to be optimized.
Optimization Steps
• When the primal and dual starting points \(\hat{x}, \hat{s}, \hat{y}, \hat{z} \) are unknown, they can be initialized as proposed by Vanderberghe in cvxopt namely, we solve the following linear
\[\begin{bmatrix} G & -I \\ Q & G^T \end{bmatrix} \begin{bmatrix} z \\ x \\ \end{bmatrix} = \begin{bmatrix} h \\ -q \\ \end{bmatrix}\]
with the assumption that \(\hat{x} = x,\hat{y} = y\).
The initial value of \(\hat{s}\) is computed from the residual \(h - Gx = -z\), as
\[\hat{s} = \begin{cases} -z \qquad \text{ if } \alpha_p < 0 \qquad else \\ -z + (1+\alpha_p)\textbf{e} \end{cases}\]
for \(\alpha_p = \text{ inf } { \alpha | -z + \alpha \textbf{e} \succeq 0 } \).
Similarly, \(z\) at the first iteration is computed as follows
\[\hat{z} = \begin{cases} z \qquad \text{ if } \alpha_d < 0 \qquad else \\ z + (1+\alpha_d)\textbf{e} \end{cases}\]
for \(\alpha_d = \text{ inf } { \alpha | z + \alpha \textbf{e} \succeq 0 } \).
Note \(\textbf{e}\) is identity.
• Following Boyd and Mattingley’s convention, we can compute the afiine scaling directions by solving the system,
\[\begin{bmatrix} G &I &0\\ 0 &K(z) & K(s) \\ Q &0 &G^T \end{bmatrix} \begin{bmatrix} \Delta z^{aff} \\ \Delta s^{aff} \\ \Delta x^{aff} \end{bmatrix} = \begin{bmatrix} -Gx - s + h \\ -K(s)z \\
-G^Tz + Qx + q \end{bmatrix}\]
with \( K(s) \text{ as } \textbf{diag}(s) \text{ and } K(z) \text{ as } \textbf{diag(z)} \)
• The centering-plus-corrector directions can be used to efficiently compute the primal and sualvariables by solving
\[\begin{bmatrix} G &I &0\\ 0 &K(z) & K(s) \\ Q &0 &G^T \end{bmatrix} \begin{bmatrix} \Delta z^{cc} \\ \Delta s^{cc} \\ \Delta x^{cc} \end{bmatrix} = \begin{bmatrix} 0 \\ \sigma \mu \textbf{e} -
K(\Delta s^{aff}) \Delta z^{aff} \\ 0 \end{bmatrix}\]
\alpha = \left(\dfrac{(s+ \alpha \Delta s^{aff})^T(z + \alpha \Delta z^{aff})}{s^Tz}\right)^3 \nonumber
and the step size \(\alpha = \text{sup} {\alpha \in [0, 1] | s + \alpha \Delta s^{aff} \ge 0, \, z + \alpha \Delta z^{aff} \ge 0}. \)
• Finding the primal and dual variables is then a question of composing the two updates in the foregoing to yield
\[x \leftarrow x + \alpha \Delta x, \\ s \leftarrow s + \alpha \Delta s, \\ z \leftarrow z + \alpha \Delta z.\]
Example code
An example implementation of this algorithm in the PyTorch Library is available on my github page.
I would like to thank Brandon Amos of the CMU Locus Lab for his generosity in answering my questions while using his qpth OptNET framework.
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How do you solve and write the following in interval notation: 3x – 2 <7 and –3x <= 15? | HIX Tutor
How do you solve and write the following in interval notation: # 3x – 2 <7# and #–3x <= 15#?
Answer 1
Se a solution process below:
Solve First Equation For x:
#3x - 2 < 7#
#3x - 2 + color(red)(2) < 7 + color(red)(2)#
#3x - 0 < 9#
#3x < 9#
#(3x)/color(red)(3) < 9/color(red)(3)#
#(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) < 3#
#x < 3#
Solve second Equation For x:
#-3x <= 15#
#(-3x)/color(blue)(-3) color(red)(>=) 15/color(blue)(-3)#
#(color(blue)(cancel(color(black)(-3)))x)/cancel(color(blue)(-3)) color(red)(>=) -5#
#x color(red)(>=) -5#
#x >= -5# and #x < 3#
Or, in interval notation:
#[-5, 3)#
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Answer 2
To solve the system of inequalities 3x - 2 < 7 and -3x ≤ 15, you first solve each inequality separately.
For the first inequality, 3x - 2 < 7, you add 2 to both sides, resulting in 3x < 9. Then, divide both sides by 3, giving x < 3.
For the second inequality, -3x ≤ 15, you divide both sides by -3, remembering to reverse the inequality sign when dividing by a negative number. This yields x ≥ -5.
In interval notation, the solution to the system of inequalities is (-∞, 3) for x < 3 and [-5, ∞) for x ≥ -5.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Key Concepts & Glossary
Key Equations
Horizontal ellipse, center at origin [latex]\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center at origin [latex]\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Horizontal ellipse, center [latex]\left(h,k\right)[/latex] [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center [latex]\left(h,k\right)[/latex] [latex]\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Key Concepts
• An ellipse is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus
(plural: foci).
• When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order
to graph the ellipse.
• When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.
• Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.
center of an ellipse
the midpoint of both the major and minor axes
conic section
any shape resulting from the intersection of a right circular cone with a plane
the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant
plural of focus
focus (of an ellipse)
one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\left(x,y\right)[/latex] on the ellipse is a constant
major axis
the longer of the two axes of an ellipse
minor axis
the shorter of the two axes of an ellipse | {"url":"https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/key-concepts-glossary-25/","timestamp":"2024-11-07T14:19:38Z","content_type":"text/html","content_length":"48621","record_id":"<urn:uuid:689bbd9e-7b24-4eb4-8799-863372494894>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00147.warc.gz"} |
How to find all possible paths from point A to B in any direction i...
Commented: Bruno Luong on 20 Sep 2021
I have a MXN matrix and I select two given points A and B. How do I find and store all the possible unique paths form A to B? There are no constraint on which direction I can go from the current
point, it can be up, down, left, right, or diagonal (in all four directions).
14 Comments
23 views (last 30 days)
How to find all possible paths from point A to B in any direction in a matrix?
It would require that M and N be very small. Even for a 10x10 matrix, it would require about 10 GB RAM.
Walter Roberson on 27 Sep 2020
MATLAB permits recursive functions, using the same syntax as most other programming languages -- which is to say that you just place a call to the function you are in the middle of defining.
The limitation on recursion in MATLAB is that by default only 500 levels of recursion are permitted. However, you can change that by using
where N is your new depth limit. Be warned that if you do this, then there is a risk of crashing the computation by running out of stack space, as each call takes up memory (a copy of all local
variables must be saved.)
John D'Errico on 29 Sep 2020
The important point to reconize is just the sheer enormity of the number of all possilbe paths, even for a small matrix.
Almost always there are better ways to solve a problem than a complete sampling of the space you wish to investigate. This is why optimization methods exist, to help you to avoid brute force sampling
Accepted Answer
Edited: Bruno Luong on 20 Nov 2020
Tiny matrix of size 4 x 3.
All paths of two opposite corners:
• 38 paths for 4-connectivity,
• 2922 paths for 8-connectivity
close all
m=4; n=3;
% Adjacent matrix of a graph of 4-connected grid of size m x n
[X,Y] = meshgrid(1:n,1:m);
mxn = numel(X);
I = sub2ind(size(X),Y(1:end-1,:),X(1:end-1,:));
J = I+1;
A = sparse(I,J,1,mxn,mxn);
I = sub2ind(size(X),Y(:,1:end-1),X(:,1:end-1));
J = I+size(X,1);
A = A + sparse(I,J,1,mxn,mxn);
A4 = A + A';
% Adjacent matrix of a graph of 8-connected grid of size m x n
I = sub2ind(size(X),Y(1:end-1,1:end-1),X(1:end-1,1:end-1));
J = I+size(X,1)+1;
A = A + sparse(I,J,1,mxn,mxn);
I = sub2ind(size(X),Y(2:end,1:end-1),X(2:end,1:end-1));
J = I+size(X,1)-1;
A = A + sparse(I,J,1,mxn,mxn);
A8 = A + A';
% source and destination
is = 1; js = 1;
id = m; jd = n;
s = sub2ind([m,n],is,js);
d = sub2ind([m,n],id,jd);
allp4 = AllPath(A4, s, d);
PlotandAnimation(4, A4, allp4, [m,n]);
allp8 = AllPath(A8, s, d);
PlotandAnimation(8, A8, allp8, [m,n]);
function PlotandAnimation(nc, A, allp, sz)
fprintf('%d-connected %d x %d\n', nc, sz);
% Plot and animation
[i,j] = ind2sub(sz,1:prod(sz));
nodenames = arrayfun(@(i,j) sprintf('(%d,%d)', i, j), i, j, 'unif', 0);
G = graph(A);
h = plot(G);
labelnode(h, 1:prod(sz), nodenames)
th = title('');
colormap([0.6; 0]*[1 1 1]);
E = table2array(G.Edges);
E = sort(E(:,1:2),2);
np = length(allp);
for k=1:np
pk = allp{k};
pkstr = nodenames(pk);
s = sprintf('%s -> ',pkstr{:});
s(end-3:end) = [];
fprintf('%s\n', s);
Ek = sort([pk(1:end-1); pk(2:end)],1)';
b = ismember(E, Ek, 'rows');
set(h, 'EdgeCData', b, 'LineWidth', 0.5+1.5*b);
set(th, 'String', sprintf('%d-connected, path %d/%d', nc, k, np));
% EDIT: better code available in the comment
function p = AllPath(A, s, t)
% Find all paths from node #s to node #t
% INPUTS:
% A is (n x n) symmetric ajadcent matrix
% s, t are node number, in (1:n)
% OUTPUT
% p is M x 1 cell array, each contains array of
% nodes of the path, (it starts with s ends with t)
% nodes are visited at most once.
if s == t
p = {s};
p = {};
As = A(:,s)';
As(s) = 0;
neig = find(As);
if isempty(neig)
A(:,s) = [];
A(s,:) = [];
neig = neig-(neig>=s);
t = t-(t>=s);
for n=neig
p = [p; AllPath(A,n,t)]; %#ok
p = cellfun(@(a) [s, a+(a>=s)], p, 'unif', 0);
end %AllPath
4 Comments
Walter Roberson on 18 Sep 2021
Jagan, read about:
• breadth-first search (bfs() in MATLAB)
• A* algorithm (less common)
• Dijkstra's algorithm (very common approach)
If what you need is the "cost" of the shortest path and not the particular edges, then there is an algorithm involving matrix multiplication.
Bruno Luong on 20 Sep 2021
There are plenty implementations on file exchange
More Answers (1)
With this script you got all possible paths, but it is very slow so you have to optimize it (shouldnt be that hard but dont have time for it anymore).
The script tries every path by going in any of the 8 directions at every step until it reaches its goal position.
a=[1 2 3; 4 5 6];
for i=10^(numel(a)-1):10^numel(a)-1
for n=1:numel(j)
if j(n)=='1'
if ~isnan(abackup(pos1-1,pos2))
if pos1==goal1 && pos2==goal2
if j(n)=='2'
if ~isnan(abackup(pos1-1,pos2+1))
if pos1==goal1 && pos2==goal2
if j(n)=='3'
if ~isnan(abackup(pos1,pos2+1))
if pos1==goal1 && pos2==goal2
if j(n)=='4'
if ~isnan(abackup(pos1+1,pos2+1))
if pos1==goal1 && pos2==goal2
if j(n)=='5'
if ~isnan(abackup(pos1+1,pos2))
if pos1==goal1 && pos2==goal2
if j(n)=='6'
if ~isnan(abackup(pos1+1,pos2-1))
if pos1==goal1 && pos2==goal2
if j(n)=='7'
if ~isnan(abackup(pos1,pos2-1))
if pos1==goal1 && pos2==goal2
if j(n)=='8'
if ~isnan(abackup(pos1-1,pos2-1))
if pos1==goal1 && pos2==goal2
0 Comments | {"url":"https://www.mathworks.com/matlabcentral/answers/600589-how-to-find-all-possible-paths-from-point-a-to-b-in-any-direction-in-a-matrix","timestamp":"2024-11-02T05:21:15Z","content_type":"text/html","content_length":"328152","record_id":"<urn:uuid:8e6dc26e-37df-44b2-9ce3-8f6ac5c331d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00292.warc.gz"} |
water saturation calculator
Water Saturation Calculator
Calculating water saturation is crucial in various fields, particularly in geology and hydrology. It helps determine the amount of water present in a substance relative to its maximum capacity. With
the advent of technology, online calculators make this process more accessible and efficient. This article presents a functional water saturation calculator along with detailed instructions on its
How to Use
To utilize the water saturation calculator, follow these simple steps:
1. Input the required parameters, including porosity, water content, and bulk density, into the designated fields.
2. Click the “Calculate” button to obtain the water saturation percentage.
The formula used for calculating water saturation (S_w) is as follows:
• Sw is the water saturation percentage.
• Vw is the volume of water.
• Vt is the total volume.
Example Solve
Let’s consider an example:
• Porosity (Φ): 0.25
• Water Content (Vw): 10 (in cubic meters)
• Bulk Density (ρb): 1500 (in kilograms per cubic meter)
Using the formula:
Frequently Asked Questions (FAQs)
What is water saturation?
Water saturation refers to the percentage of pore space filled with water in a substance, such as soil or rock.
Why is water saturation important?
Water saturation is crucial in various industries, including agriculture, oil and gas exploration, and environmental science, as it provides insights into groundwater availability, oil reservoirs,
and soil fertility.
Can this calculator be used for different units?
Yes, this calculator accepts input in various units, including cubic meters, cubic feet, kilograms per cubic meter, and pounds per cubic foot.
Is the formula used in this calculator accurate?
Yes, the formula used in this calculator provides an accurate estimation of water saturation based on the provided inputs.
The water saturation calculator presented in this article offers a convenient tool for determining water saturation percentages in different substances. By following the provided instructions and
using the accurate formula, users can efficiently analyze water content in various materials. | {"url":"https://calculatordoc.com/water-saturation-calculator/","timestamp":"2024-11-12T05:17:58Z","content_type":"text/html","content_length":"92578","record_id":"<urn:uuid:a58cdda3-6f7d-4de1-845a-fb56eb370fa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00574.warc.gz"} |
The Stacks project
Definition 111.40.1. Let $X$ be a locally ringed space. An invertible ${\mathcal O}_ X$-module on $X$ is a sheaf of ${\mathcal O}_ X$-modules ${\mathcal L}$ such that every point has an open
neighbourhood $U \subset X$ such that ${\mathcal L}|_ U$ is isomorphic to ${\mathcal O}_ U$ as ${\mathcal O}_ U$-module. We say that ${\mathcal L}$ is trivial if it is isomorphic to ${\mathcal O}_ X$
as a ${\mathcal O}_ X$-module.
Comments (2)
Comment #7166 by Harun KIR on
In Definition 110.40.7, there is an assertion that for a ring $R$ we set $Pic(R)=Pic(Spec(R)).$ Yes, it is really an assertion. So it should be out of the definition. You may claim that you are
just defining $Pic(R)$ as you did. But it highly causes to some issues in the common literature. By the way, in the left side you have a group structure for a ring. Is the same thing true for the
left side for a ring if you think $\Pic(R)$ what really it is. At any rate, we first should define $Pic(R)$ as usual, then we should prove that they are really the same. As your previous works,
it follows from Bourbaki, Commutative Algebra.
Comment #7306 by Johan on
This is the chapter on exercises, so I am going to be a bit sloppy here. But yes in principle I agree with you. The Picard group of a ring is "officially" defined in Section 15.117 (it doesn't
have a formal definition environment, so you could complain on that page if you like).
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Computational Technology Resources - CCP - Paper
Civil-Comp Proceedings
ISSN 1759-3433
CCP: 75
Edited by: B.H.V. Topping and Z. Bittnar
Paper 69
Hyper-Elastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the Seth-Hill Strain Measures
K. Farahani and H. Bahai
Department of System Engineering, Brunel University, Uxbridge, Middlesex, United Kingdom
K. Farahani, H. Bahai, "Hyper-Elastic Constitutive Equations of Conjugate Stresses and Strain Tensors for the Seth-Hill Strain Measures", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the
Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 69, 2002. doi:10.4203/ccp.75.69
Keywords: constitutive equations, energy conjugate stresses, hyper-elastic.
The concept of energy conjugacy first presented by Hill [
] states that a stress measure is said to be conjugate to a strain measure if represents power or rate of change of internal energy per unit reference volume. Based on this definition, Guo and Man [
] derived explicit tensorial formulations for conjugate stress for , while earlier, the stress measure conjugate to logarithmic strain tensor , had been derived by Hoger [
]. Also, following the Hill's principal axis method and energy conjugacy notion, a relation between the components of two Seth-Hill conjugate stress tensors in the principal axes was derived in [
]. Some well known relations of the Seth-Hill strain measures with their conjugate stresses are givien in [
]. Also basic relations were derived for the rate of logarithmic strain [
Use of hypo-elastic constitutive equations for large strains in nonlinear finite element applications usually needs special considerations [8]. For example, the strain doesn't vanish in some elastic
loading and unloading closed cycles, and they need objective rate tensors, and incrementally objective algorithm for numerical application and integration. Some of them may fluctuate under excessive
shear deformation [8]. Hyper-elastic constitutive equation in comparison, don't need these considerations. However, their behaviour for large elastic strains is important, and may differ in tension
and compression. In the present work, Hyper-elastic constitutive equations for the Seth-Hill strains and their conjugate stresses are explored as a natural generalisation of Hook's law for finite
elastic deformations. Based on the uniaxial and simple shear tests, the response of the material for different constitutive equations is examined. Some required kinematic and tensor equations are
used from [6,10]. Together with an objective rate model, the effect of different constitutive laws on Cauchy stress components is compared. It is shown that the logarithmic strain and its conjugate
stress give answers closer to that of the rate model. In addition, use of Biot stress-strain pairs for a bar element, results in an elastic spring which obeys the hook's law even for large
deformations and behaves the same in tension and compression. The volume change of material is another factor which is noticed.
R. Hill, "On constitutive inequalities for simple materials", Int. J. Mech. Phys. Solids, 16, 229-242, 1968. doi:10.1016/0022-5096(68)90031-8
Z.-H. Guo, C.-S. Man, "Conjugate stress and tensor equation doi:10.1016/0020-7683(92)90194-X
A. Hoger, "The stress conjugate to logarithmic strain", Int. Journal of Solids and Structures, 23 (12), 1645-1656, 1987. doi:10.1016/0020-7683(87)90115-6
M.E. Gurtin, K. Spear, "On the relationship between the logarithmic strain rate and the stretching tensor", Int. Journal of Solids and Structures 19, 437-444, 1983. doi:10.1016/0020-7683(83)
R.W. Ogden, "Nonlinear elastic deformations", Ellis Harwood, 1984.
Z.-H. Guo, R.N. Dubey, "Basic aspects of Hill's method in solid mechanics", SM Arch, 9, 353-380, 1984.
R. Hill, "Aspects of invariance in solid mechanics", Advances in Applied Mechanics 18, 1-75, 1978. doi:10.1016/S0065-2156(08)70264-3
J.K. Dienes, "On the Analysis of Rotation and Stress Rate in Deforming Bodies", Acta Mechanica, 32, 217-232, 1979. doi:10.1007/BF01379008
A. Hoger, D.E. Carlson, "Determination of stretch and rotation in the polar decomposition of deformation gradient", Quart. App. Math. 10, 1984.
K. Farahani, R. Naghdabadi, "Conjugate stresses of the Seth-Hill strain tensors", International Journal of Solids and Structures, 37, 5247-5255, 2000. doi:10.1016/S0020-7683(99)00209-7
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purchase this book (price £125 +P&P) | {"url":"https://www.ctresources.info/ccp/paper.html?id=2961","timestamp":"2024-11-02T08:02:16Z","content_type":"text/html","content_length":"10547","record_id":"<urn:uuid:7c083a81-0400-44b7-97f6-b6ec0425811e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00409.warc.gz"} |
Transactions Online
Kiyoshi NISHIYAMA, Masahiro SUNOHARA, Nobuhiko HIRUMA, "A New Formula to Compute the NLMS Algorithm at a Computational Complexity of O(2N)" in IEICE TRANSACTIONS on Fundamentals, vol. E102-A, no. 11,
pp. 1545-1549, November 2019, doi: 10.1587/transfun.E102.A.1545.
Abstract: The least mean squares (LMS) algorithm has been widely used for adaptive filtering because of easily implementing at a computational complexity of O(2N) where N is the number of taps. The
drawback of the LMS algorithm is that its performance is sensitive to the scaling of the input. The normalized LMS (NLMS) algorithm solves this problem on the LMS algorithm by normalizing with the
sliding-window power of the input; however, this normalization increases the computational cost to O(3N) per iteration. In this work, we derive a new formula to strictly perform the NLMS algorithm at
a computational complexity of O(2N), that is referred to as the C-NLMS algorithm. The derivation of the C-NLMS algorithm uses the H[∞] framework presented previously by one of the authors for
creating a unified view of adaptive filtering algorithms. The validity of the C-NLMS algorithm is verified using simulations.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1545/_p
author={Kiyoshi NISHIYAMA, Masahiro SUNOHARA, Nobuhiko HIRUMA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A New Formula to Compute the NLMS Algorithm at a Computational Complexity of O(2N)},
abstract={The least mean squares (LMS) algorithm has been widely used for adaptive filtering because of easily implementing at a computational complexity of O(2N) where N is the number of taps. The
drawback of the LMS algorithm is that its performance is sensitive to the scaling of the input. The normalized LMS (NLMS) algorithm solves this problem on the LMS algorithm by normalizing with the
sliding-window power of the input; however, this normalization increases the computational cost to O(3N) per iteration. In this work, we derive a new formula to strictly perform the NLMS algorithm at
a computational complexity of O(2N), that is referred to as the C-NLMS algorithm. The derivation of the C-NLMS algorithm uses the H[∞] framework presented previously by one of the authors for
creating a unified view of adaptive filtering algorithms. The validity of the C-NLMS algorithm is verified using simulations.},
TY - JOUR
TI - A New Formula to Compute the NLMS Algorithm at a Computational Complexity of O(2N)
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1545
EP - 1549
AU - Kiyoshi NISHIYAMA
AU - Masahiro SUNOHARA
AU - Nobuhiko HIRUMA
PY - 2019
DO - 10.1587/transfun.E102.A.1545
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 11
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - November 2019
AB - The least mean squares (LMS) algorithm has been widely used for adaptive filtering because of easily implementing at a computational complexity of O(2N) where N is the number of taps. The
drawback of the LMS algorithm is that its performance is sensitive to the scaling of the input. The normalized LMS (NLMS) algorithm solves this problem on the LMS algorithm by normalizing with the
sliding-window power of the input; however, this normalization increases the computational cost to O(3N) per iteration. In this work, we derive a new formula to strictly perform the NLMS algorithm at
a computational complexity of O(2N), that is referred to as the C-NLMS algorithm. The derivation of the C-NLMS algorithm uses the H[∞] framework presented previously by one of the authors for
creating a unified view of adaptive filtering algorithms. The validity of the C-NLMS algorithm is verified using simulations.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1545/_p","timestamp":"2024-11-01T22:57:57Z","content_type":"text/html","content_length":"62558","record_id":"<urn:uuid:b5ba6dca-3e8d-4f1f-992f-65076e0fd3e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00668.warc.gz"} |
Eclipse Qrisp
qcla(a, b, radix_base=2, radix_exponent=1, t_depth_reduction=True, ctrl=None)[source]#
Implementation of the higher radix quantum carry lookahead adder (QCLA) as described here. This adder stands out for having logarithmic T-depth like Drapers QCLA. Compared to Drapers QCLA, the
higher radix QCLA allows a more dynamic structure and the use of customizable “sub-adders” which enables this adder to beat Drapers QCLA in terms of speed (ie. T-depth).
In Python syntax, this function performs the inplace addition:
Apart from the two quantum arguments, this function supports the specification of the adder-radix R. The adder-radix can be specified in the form of an exponential of integers:
\[R = r^k\]
Where \(r\) is the radix base and \(k\) is the radix exponent.
Calling qcla with radix base \(r\) and radix exponent \(k\), will precalculate the carry values using the Brent-Kung tree with carry-radix \(r\) and cancel the recursion \(k\) layers before
An additional compilation option is given with the t_depth_reduction keyword. This compilation option modifies the way the carry values are uncomputed. If t_depth_reduction is set to True the
carry values will be uncomputed using the intermediate result of the sub-adder - if set to False they will be uncomputed using the automatic uncomputation algorithm.
The advantage of the automated version is that, both T-depth and CNOT-depth are scaling with the the logarithm of the input size. For t_depth_reduction = True the T-depth is significantly reduced
(and still logarithmic) however the CNOT depth becomes linear.
aQuantumFloat or List[Qubit] or int
The value that is added.
bQuantumFloat or List[Qubit]
The value that is operated on.
radix_baseinteger, optional
The radix of the Brent-Kung tree. The default is 2.
radix_exponentinteger, optional
The cancellation treshold for the Brent-Kung recursion. The adder-Radix is then \(R = r^k\). The default is 1.
t_depth_reductionbool, optional
A compilation option that reduces T-depth but in turn weakens CNOT depth to linear scaling. The default is True.
Tried to add QuantumFloat of higher precision onto QuantumFloat of lower precision.
We try out several constellations of parameters:
>>> from qrisp import QuantumFloat, qcla
>>> a = QuantumFloat(8)
>>> b = QuantumFloat(8)
>>> a[:] = 4
>>> b[:] = 15
>>> qcla(a, b)
>>> print(b)
{19: 1.0}
We now measure the T-depth. To get the optimal result, we need to tell the compiler that we only care about T-gates. This can be achieved with the gate_speed keyword of the compile method. This
keyword allows you to specify a function of Operation objects, which returns the speed of that Operation. For more information check out the compile documentation page.
For T-depth, there is already a pre-coded function: T-depth.
>>> from qrisp import t_depth_indicator
>>> gate_speed = lambda x : t_depth_indicator(x, epsilon = 2**-10)
>>> qc = b.qs.compile(gate_speed = gate_speed, compile_mcm = True)
>>> qc.t_depth()
This function contains many allocations/deallocations that can be leveraged into parallelism, implying it can profit a lot from additional workspace:
>>> qc = b.qs.compile(workspace = 10, gate_speed = gate_speed, compile_mcm = True)
>>> qc.t_depth()
We can verify the logarithmic behavior by comparing to the Gidney-adder:
>>> from qrisp import gidney_adder
>>> a = QuantumFloat(40)
>>> b = QuantumFloat(40)
>>> gidney_adder(a, b)
>>> qc = b.qs.compile(gate_speed = gate_speed, compile_mcm = True)
>>> qc.t_depth()
>>> a = QuantumFloat(40)
>>> b = QuantumFloat(40)
>>> qcla(a, b)
>>> qc = b.qs.compile(workspace = 50, gate_speed = gate_speed, compile_mcm = True)
>>> qc.t_depth()
The function can also be used to perform semi-classical in-place addition
>>> b = QuantumFloat(10)
>>> b[:] = 20
>>> qcla(22, b)
>>> print(b)
{42: 1.0} | {"url":"https://www.qrisp.eu/reference/Primitives/generated/qrisp.qcla.html","timestamp":"2024-11-05T00:12:35Z","content_type":"text/html","content_length":"39589","record_id":"<urn:uuid:560c7c79-92bb-481d-9d61-e0062de910b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00697.warc.gz"} |
If f(θ)=∣∣cos2θcosθsinθsinθcosθsinθsin2θ−cosθ−sinθcosθ0∣∣.... | Filo
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Question Text If . Then, for all
Updated On May 4, 2023
Topic Determinants
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 2
Upvotes 300
Avg. Video Duration 12 min | {"url":"https://askfilo.com/math-question-answers/if-f-theta-left-begin-array-ccc-cos-2-theta-cos-theta-sin","timestamp":"2024-11-08T19:31:49Z","content_type":"text/html","content_length":"307334","record_id":"<urn:uuid:6cb472b9-4d91-4cec-bf44-73621a6573da>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00404.warc.gz"} |
KOS.FS - faculty addon
Mathematics I. (E011067)
Departments: ústav technické matematiky (12101)
Abbreviation: Approved: 19.04.2017
Valid until: ?? Range: 4P+4C
Semestr: Credits: 6
Completion: Z,ZK Language: EN
In the course, greater emphasis is placed on the theoretical basis of the concepts discussed and on the derivation of basic relationships and connections between concepts. Students will also get to
know the procedures for solving problems with parametric input. In addition, students will gain extended knowledge in some thematic areas: eigennumbers and eigenvectors of a matrix, Taylor
polynomial, integral as a limit function, integration of some special functions.
doc. Ing. Tomáš Bodnár Ph.D.
Zimní 2023/2024
doc. Ing. Tomáš Bodnár Ph.D.
Zimní 2022/2023
doc. Ing. Tomáš Bodnár Ph.D.
Zimní 2021/2022
1. Basics of linear algebra – vectors, vector spaces, linear independence of vectors, dimensions, bases.
2. Matrix, operation, rank. Determinant. Regular and singular matrices, inverse matrices.
3. Systems of linear equations, Frobenian theorem, Gaussian elimination method.
4. Eigennumbers and eigenvectors of a matrix.
5. Differential calculus of real functions of one variable. Sequence, monotony, limit.
6. Limit and continuity of a function. Derivation, geometric and physical meaning.
7. Monotonicity of a function, local and absolute extrema, convexity, inflection point. Asymptotes, graph of the function.
8. Taylor polynomial, remainder after n-th power. Approximate solution of the equation f(x)=0.
9. Integral calculus of real functions of one variable – indefinite integral, integration by parts, integration by substitution.
10. Definite integral, its calculation.
11. Application of a definite integral: surface area, volume of a rotating body, length of a curve, application in mechanics.
12. Numerical calculation of the integral.
13. Improper integral.
Structure of tutorial
The same as lectures.
Neustupa, J.: Mathematics I, CTU Publishing House, Prague, 1996,
Finney, R. L., Thomas, G.B.: Calculus, Addison-Wesley, New York, Ontario, Sydney, 1994
Knowledge of high school mathematics in the range of a real gymnasium. | {"url":"https://kos.fs.cvut.cz/synopsis/E011067/en/printA4wide","timestamp":"2024-11-08T17:38:00Z","content_type":"text/html","content_length":"5928","record_id":"<urn:uuid:44ab50fb-ff6e-442b-b13d-f67c7d8364f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00807.warc.gz"} |
On multiflow lexicographics
Given an undirected Eulerian network with the terminal-set {s} U T, we call a vector ξ = (ξ(t): t ∈ T) feasible if there exists an integer maximum multiflow having exactly ξ(t) (s, t)-paths for each
t ∈ T. This paper contributes to describing the set Ξ of feasible vectors. First, the feasible vectors are shown to be bases of a polymatroid (T, p) forming a proper part of the polytope defined by
the supply-demand conditions; p(V) = max{ξ(V): ξ ∈ Ξ}, V ⊆ T is described by a max-min theorem. The question whether Ξ contains all the integer bases, thereby admitting a polyhedral description,
remains open. Second, the lexicographically minimum (and thereby maximum) feasible vector is found, for an arbitrary ordering of T. The results are based on the integrality theorem of A. Karzanov and
Y. Manoussakis (Minimum (2, r)-metrics and integer multiflows, Europ. J. Combinatorics (1996) 17, 223-232) but we develop an original approach, also providing an alternative proof to this theorem.
ASJC Scopus subject areas
• Discrete Mathematics and Combinatorics
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How To Draw A Number 3
How To Draw A Number 3 - Web there's techniques used to deal with big multiplication like 100 by using patterns found in multiplying by 10s and 1s. All draw game prizes must be claimed at a florida
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equation that could represent this picture. Enter number 1 enter number 2 enter number 3 enter number 4 enter number 5 enter number 6.
1 drawing to win her prize. Here is a video for you. Generate a random number between any two numbers, or simulate a coin flip or dice roll online. Web september 3, 2023. Enter number 1 enter number
2 enter number 3 enter number 4 enter number 5 enter number 6. Here’s a look at thursday, dec. On the final line you practice writing the letter all on your own.
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Enter number 1 enter number 2 enter number 3 enter number 4 enter number 5 enter number 6. All draw game prizes must be claimed at a florida lottery retailer or florida lottery office on.
Hand drawn sketch number three Royalty Free Vector Image
To draw a winner among a set of participants. 1 drawing to win her prize. I hope you will like this video. On the final line you practice writing the letter all on your own..
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Web september 3, 2023. Learn how to draw numberblock three and her crown with three points. You just drew the fractions 1/4 and 3/4 on a number line! The winning numbers for tuesday night's drawing.
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Draw a number line, plotting the multiples of y starting from 0 and mark the dividend x on the number line. Of course, that is just the beginning. Web september 3, 2023. So, count 3.
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See the number three on a number line, five frame, ten frame, numeral, word,. The ohio lottery says the winning powerball numbers for the $543 million jackpot on monday, december 18. Web the number
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Each sheet also features the number word spelled. Web by steve wyborney | february 6, 2022 | 0 are you looking for an activity for 2.22.2022? The numerator shows the parts that we have. Starting.
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How to Draw a Cartoon Number Three 3 🎨🎨 ️Easy Step by Step Draw a
Web hello friends, in this video i will show you how you can draw fish with number 3. Can be used to pick a number for giveaways, sweepstakes, charity lotteries, etc. Fill the number with.
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Learn how to draw numberblock three and her crown with three points. This means that the fraction 3/4 is 3 out of 4 equal parts. To draw a winner among a set of participants. Each.
How to Draw Number 3 Drawing Number Three in 3D on Paper with Pencil
Web learn how to draw numberblock three with this brilliant drawing tutorial for kids! On the final line you practice writing the letter all on your own. Of course, that is just the beginning. Draw.
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Several-Variable Calculus [ARCHIVED CATALOG] College Bulletin 2023-2024
MATH 034. Several-Variable Calculus
Same topics as MATH 033 except in more depth using the concepts of linear algebra. The department strongly recommends that students take linear algebra first so that they are eligible for this
course. Students may take only one of MATH 033 , MATH 034, and MATH 035 for credit.
Prerequisite: Credit for, or placement out of, MATH 025 or Math 026 and also MATH 027 or MATH 028 , along with a grade of C or better in at least one of the two previously mentioned math courses.
Natural sciences and engineering.
1 credit.
Eligible for COGS
Fall 2023. Einstein, Reinhart
Spring 2024. Einstein, Talvacchia.
Fall 2024. Staff.
Spring 2025. Staff.
Fall 2025. Staff.
Spring 2026. Staff.
Catalog chapter: Mathematics and Statistics
Department website: http://www.swarthmore.edu/mathematics-statistics
Access the class schedule to search for sections.
Close Window
All © 2024 Swarthmore College.
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Outline of the Method of Conducting a Trigonometrical Survey, for the Formation of Geographical and Topographical Maps and Plans
towns, villages, &c., carefully noted, and care taken to insure their correct orthography, and to quote the authority upon which it rests when different from that sanctioned by custom.
In measuring long lines between conspicuous objects, marks should be left, to be connected by check lines, or on which to base smaller triangles; where impeded by a house or any obstacle, the means
of avoiding it and returning again to the measured line are to be found further on.
Irregular inclosures and roads, even where triangles cannot be measured, can still be surveyed by the chain alone, but of course not so accurately as with the aid of the theodolite.
This method of "traversing" is managed as follows:-Suppose
A B the first line, and B C the direction in which the next is required to be measured, prolong A B to E, make B F equal to B E, and measure the cord E F, from which data the direction of B C can be
laid down.
The dimensions in the field-book may be kept either between two parallel lines running up the page, with the offsets written on the right and left of these lines as in the example facing page 36, or
on a species of diagram bearing some sort of resemblance to the outline of the ground to be surveyed, which latter method is supposed to assist in the plotting; but if references to the starting
points of the different lines, and their junctions with each other, are entered in the field-book kept according to the first system, and the angles forward written on the right or left of the ruled
lines. according to the direction of the next forward station, there can never be any difficulty in plotting the work, even after a con
atmosphere, it is a good precaution to divide the scale for laying off distances from the fieldbook, on the paper upon which the plot is to be made, as it will then always expand and contract with
the outline of the survey; and also to mount the paper before commencing plotting, or not at all.
siderable lapse of time, which however should not be delayed longer than is absolutely necessary. It is customary for land surveyors to compute their work from the plot, adding up the contents of
each inclosure for the general total, which is perhaps checked by the calculation of two or three large triangles ruled in pencil so as to correspond nearly to the extreme boundaries, whose lengths
are taken from the scale; but if the rigid mode of computing everything from the field-book is deemed too troublesome, still the areas of the large triangles, measured on the ground, should be
calculated from their dimensions taken from the field-book, and the contents of the irregular boundaries added to or subtracted from this amount, which constitutes a far more accurate check upon the
sum of the contents of the various inclosures than the method in general use. The calculation of irregular portions outside these triangles is much facilitated by the well-known method of reducing
irregular polygons to triangles having equivalent areas.
When the contents of fields are to be calculated from the plot, the scale should not be less than twenty, and may be as much as three or four chains to one inch. The former of these two last scales
is that on which all plans for railroads submitted to the House of Commons are required to be drawn, and the latter is used for plans of estates, &c.
To return to the second division of this subject, viz. the filling up of the interior, partly by measurement and partly by sketching, which is generally the mode adopted in the construction of
topographical maps.
The roads, with occasional check lines, are measured as already described, the field-book being kept in the same method as when the entire county is to be laid down by measurement, excepting that all
conspicuous objects some distance to the right and left of the lines are to be fixed by intersections with the theodolite, either from the extremities of these lines or from such intermediate points
as appear best adapted for determining their positions. These points when plotted, together with the offsets* from the field-book,
* Mr. Holtzapfell's "Engine-divided Scales,” engraved on pasteboard, will be found very useful, and their low price is an additional recommendation. Marquois scales are also adapted for plotting and
drawing parallel lines at measured intervals, as well as for other purposes. The offset and plotting scales, introduced by Major Robe on the Ordnance Survey, are as
present so many known fixed stations between the measured lines, and of course facilitate the operation of sketching the boundaries of fields, &c., and also render the work more correct, as the
errors inseparable from sketching will be confined within very narrow limits.
In all cases where the compass is used to assist in filling-in the interior (and it should never be trusted in any more important part of the work), it becomes of course necessary to ascertain its
variation by one of the methods which will be hereafter explained. Independent of the annual change in its deviation, the horizontal needle is subject to a small daily variation, which is greatest in
summer, and least in winter, varying from 15′ to 7'. Its maximum on any day is attained to the eastward about 7 A.M., from which time it continues moving west till between 2 and 3 P.M., when it
returns again towards the east*; but this oscillation is too small to be appreciable, as the prismatic compass used in the field cannot be read to within one-half, or at the nearest one-quarter, of a
degree of the truth. Portions of the work, as plotted from the fieldbook, are then transferred to card-board or drawing-paper, or traced off on thin bank post paper, which latter has the advantage of
being capable of folding over a piece of Bristol board fitting into the portfolio, and from its large size, containing on the same sheet distant trigonometrical points which may constantly be of use.
It can be folded over the pasteboard, so as to expose any portion that may be required, and when the work is drawing near to the edge, it is only necessary to alter its position. In moist weather,
prepared paper, commonly termed asses' skin, is the only thing that can be used, as the rain runs off it immediately, without producing any effect on the sketch.
The portable instruments generally used in sketching between
convenient as any that have been contrived. The plotting scale has one bevelled edge; and the scale, whatever it may be, engraved on each side, is numbered each way from a zero line. The offset scale
is separate, and slides along the other, its zero coinciding with the line representing the measured distance; the dimensions are marked on the bevelled edge of this short scale to the right and left
of zero, so that offsets on either side of the line can be plotted without moving the scales; and from the two being separate, there is no chance of their being injured, as in those contrivances
where the plotting and offset scales are united. * See Colonel Beaufoy's experiments on the variation of the needle. Also the article Observatory (Magnetical), Aide Mémoire.
measured lines and fixed points in the interior, as well as in military sketches made in the exigency of the moment without any measurement whatever, are a small 4-inch, or box sextant (or some small
reflecting instrument * as a substitute for it), and the azimuth prismatic compass. Any reflecting instrument is certainly capable of observing angles between objects nearly in the same horizontal
plane, with more accuracy than the compass; and from its observations being instantaneous, and not affected by the movement of the hand, it is better adapted for use on horseback, but it is not so
generally useful in filling up between roads, or in sketching the course of a ravine or stream, or any continuous line.
Whichever of these instruments is preferred, of course a scale of chains, yards, or paces, and a protractor, are required, for laying off linear and angular distances in the field.
A very convenient method of using the latter for protracting bearings observed with the azimuth compass, is to have lines engraved transversely across the face of the protractor, at about a quarter
of an inch apart. The paper upon which the sketch is to be made must also be ruled faintly across in pencil at short unequal distances, at right angles to the meridian, with which lines one or more
of those on the protractor can be made to correspond, by merely turning it round on its zero as a pivot, this point being kept in coincidence with the station from whence the bearing is to be drawn.
The bevelled edge of the protractor is thus evidently parallel to the meridian, and the observed bearing being marked
* In using reflecting instruments, avoid very acute angles, and do not select any object for observation which is close, on account of the parallax of the instrument. The brightest and best defined
of the two objects should be the reflected one; and if they form a very obtuse angle, it is measured more correctly by dividing it into two portions, and observing the angle each of them makes with
some intermediate point. Also, if the objects are situated in a plane very oblique to the horizon, an approximation to their horizontal angular distance is obtained by observing each of them with
reference to some distant mark considerably to the right or left, and taking the difference of these angles for the one required.
The index error of a sextant must also be frequently ascertained. The measure of the diameter of the sun is the most correct method; but for a box sextant, such as is used for sketching, it is
sufficient to bring the direct and reflected image of any well-defined line, such as the angle of a building (not very near) into coincidence—the reading of the graduated line is then the index
error. For the adjustment of the box sextant, see Simms on Mathematical Instruments. The less the glasses are moved about the better.
and ruled from this point, is the angle made by the object with the meridian.
For instance, the bearing of a distant object upon which it is required to place, was observed from D to be 30°. The protractor in the sketch is shown in the proper position for laying off this
angle, and the dotted line DE is the direction required.
In fixing the position of any point with the compass, by bearings taken from that point to two or three surrounding stations whose places are marked on the paper, the zero of the protractor is made
to coincide with one of these stations, and its position being adjusted by means of the lines ruled across its face and on the paper, the observed angle is protracted from this station, and produced
through it. The same operation being repeated at the other points, the intersection of these lines gives the required place of observation.
Instead of the above system of ruling east and west lines across the paper, lines may be drawn parallel to the meridian for adjusting the place of the protractor. Thus, suppose from the point D any
observed bearing, say 40°, is to be laid down. By placing the zero C of the protractor on any convenient meridian, and turning it upon this point as a pivot until the required angle of 40° at E
coincides. also with the same meridian NS, it is only
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Important Qualities to Look for in an Algebra Tutor - Signage Standards
Tutors can help students study in a one-on-one learning environment. They can review homework or help prepare for an exam. They can also help students develop strong study habits.
Some online tutoring services offer flexible hours and pay-as-you-go plans, while others provide a full range of academic resources. Learner offers a streamlined process for selecting a tutor and has
a robust virtual learning platform.
Often the first high school math course students take, Algebra 1 introduces students to variables and number symbols that represent mathematical concepts like equations and inequalities. This is the
foundational math course that helps students develop a fundamental command of algebra topics and prepares them for more advanced courses like geometry and calculus.
Teachers say that routines core to their instruction, such as repeated practice and checking for understanding, can be harder to do with virtual learners. And that means that students who don’t grasp
the basics may have a hard time keeping up when the class moves on to algebra topics like polynomials and radicals.
Experts recommend targeted tutoring support for students who need it, rather than remediation — which can be demotivating and push students who struggle to grade-level content even further behind. To
help your student get the most out of tutoring sessions, make sure they’re in a quiet place and free of distractions (turn off televisions, phones, music, etc.).
As the name implies, Algebra 2 is an extension of the fundamental concepts introduced in algebra 1. It builds on some topics from previous math classes like geometry and introduces new topics like
functions. Students learn how to simplify rational exponents, solve linear and quadratic equations, and graph complex exponential and logarithmic functions. They also learn how to model real world
problems using growth and decay models.
In addition, students explore polynomial equations and functions to build a strong foundation for the study of rational functions, trigonometric functions, and sequences and series. This course will
help to prepare students for future courses in calculus, physics, and biology.
Many students find Algebra 2 to be difficult because it often builds on the basics that they learned in their previous math classes, including Algebra 1, and introduces new topics that are more
abstract than concrete. However, the class can be made easier by working extra practice, staying up to date with homework, and seeking out online resources that can help reinforce concepts.
The study of Abstract Algebra is a branch of mathematics that focuses on the algebraic structure of various sets rather than their individual elements. It includes advanced topics such as groups,
rings, fields, modules, and vector spaces. Mathematicians specializing in this subject are called algebraists.
Like other branches of algebra, it incorporates concepts from set theory and algebraic structures such as permutation groups and composition series in group theory; polynomial rings and ideals in
ring theory; and Galois theory in field theory. In addition, it involves the study of algebraic operations in vector spaces such as linear transformations and matrices.
Students who learn Abstract Algebra face unique challenges that can be difficult to understand, even for the most seasoned student. The goal of tutoring sessions with Troy Algebra Tutors is to help
students develop their understanding of this subject and how it intersects with other subjects, such as Linear Algebra. A recent article by Durand-Guerrier, Hausberger, and Spitalas highlights the
role of research on learning Abstract Algebra, focusing specifically on students’ understanding and use of mathematical definitions.
Trigonometry is a branch of mathematics that deals with the relationships between sides and angles of right-angled triangles. It provides the formulas, functions, and identities to find missing or
unknown angles or sides of a given triangle.
The word “trigonometry” is derived from the Greek words for triangle and measure (trigonon and metron). It emerged in the Hellenistic world around the 3rd century BCE, from applications of geometry
to astronomical studies.
There are six basic trigonometric functions that can be applied to a given angle: sine (sin), cosine (cos), tangent (tan), secant (sec), and cosecant (csc). The value of each function depends on the
length of the opposite side of the triangle, or hypotenuse. Each of these ratios are used in a variety of ways in science, engineering, and video games to help answer questions that would be
impossible to find using other methods. The most common application is finding the height of a tree from its base, or distance from the ground to the top of the tree. If you are looking for a good
Algebra tutor in Troy, consider tutors from The Turing Center. | {"url":"https://www.cam-tyler.com/important-qualities-to-look-for-in-an-algebra-tutor/","timestamp":"2024-11-03T18:19:36Z","content_type":"text/html","content_length":"32061","record_id":"<urn:uuid:72079d6a-7701-4651-bd8c-33933020d8f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00178.warc.gz"} |
Problem 456: intervalxor2
Time limit: 1s Memory limit: 256MB Input: Output:
You became fond of the problem Interval XOR from the third round, so we decided to give you a more challenging version.
You are given an array of length $n$, which has positive integers, and $q$ queries.
For each query, we will give the values $l$ and $r$ and we are required to solve the problem Interval XOR for the values in the range [$l, r$] in the array. In other words, you will need to find the
maximum XOR we can get by removing exactly one value from that interval.
The first line of the input will contain $n$ and $q$ ($1 \le n, q \le 2 * 10^5$), the number of values in the array and the number of query operations your program will have to perform.
The next line of the array will have $n$ values, the starting values of the array ($0 \le v_i < 2^{30}$).
The next $q$ lines of the array will contain $2$ values each, respecting the format given earlier in the statement ($1 \le l, r \le n$).
For tests worth $14$ points, $1 \le n, q \le 2 * 10^3$.
For tests worth $24$ more points, $l = 1$ for all the queries.
The output will contain a line for each operation given in the input.
Example 1 | {"url":"https://kilonova.ro/contests/4/problems/456","timestamp":"2024-11-14T10:56:20Z","content_type":"text/html","content_length":"31889","record_id":"<urn:uuid:28d99b49-8901-4043-83c7-8fe5aa67121e>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00568.warc.gz"} |
Product of Numbers Calculator
Welcome to the Product of Numbers Calculator. Here you can enter two numbers and we will not only calculate the Product of those two numbers, but also explain what it means to get the Product of
those two numbers.
Please enter your numbers below to get the product.
Product of
Here are some examples of what our Product of Numbers Calculator can answer:
What is the Product of 2 and 3? What is the Product of 3 and 4? What is the Product of 5 and 7? What is the Product of 10 and 12? What is the Product of 23 and 45? Copyright
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Travel Time · CellularPotts.jl
This example demonstrates how to record and analyze outcomes from a cellular potts simulation.
Here we will track the mean displacement of cell centers with and without active migration. From this data, we can calculate mean square displacement as a function of lag time.
$$\[ msd(\tau) = <\Delta r(\tau)^2> = <(r(t+\tau) - r(t))^2>\]$$
Here r(t) represents the current position of the cell and the brackets indicate an average over time. If a cell is moving randomly, this function will be linear. Cells with directional motion will
deflect the curve upward. Let's see if we can replicate these theoretic results.
using CellularPotts, Plots
using Random, Statistics
#Set a random seed for reproducibility
#Models have same space and cell initializations
space = CellSpace(100,100)
#Cells will have same volume, the moving cell will be in the center and the stationary cell will be placed in the corner
initialCellState = CellState(
names = [:StationaryCell, :MovingCell],
volumes = [200, 200],
counts = [1,1],
positions = [(20,20),(50,50)])
#Add a migration penalty to one cell encourage cell movement
penalties = [
AdhesionPenalty([0 30 30;
30 30 0]),
VolumePenalty([30, 30]),
PerimeterPenalty([0, 10]),
MigrationPenalty(40, [0, 40], size(space)) # 0 means no migration
#Generate the model
cpm = CellPotts(space, initialCellState, penalties)
#Record model iterations
cpm.recordHistory = true
#Simulate the model
for _ in 1:1000
Given a CellSpace and a cell ID, calculate the cell's center.
function meanPosition(space, n)
totalPixels = 0
avePos = zeros(2) #x and y positions
#Loop over all points in space
for i in axes(space,1)
for j in axes(space,2)
#Save positions that match cell ID
if space[i,j] == n
avePos[1] += i
avePos[2] += j
totalPixels += 1
return avePos ./ totalPixels
#Plot trajectories
p1 = visualize(cpm)
trajectoryRandom = zeros(2,cpm.step.counter)
trajectoryDirected = zeros(2,cpm.step.counter)
for i in 1:cpm.step.counter
trajectoryRandom[:,i] .= meanPosition(cpm(i).space.nodeIDs, 1)
trajectoryDirected[:,i] .= meanPosition(cpm(i).space.nodeIDs, 2)
plot!(p1, trajectoryRandom[1,:], trajectoryRandom[2,:]; color=:grey20)
plot!(p1, trajectoryDirected[1,:], trajectoryDirected[2,:]; color=:grey20)
MeanSqDis(cpm, τ, id) = mean([sum(abs2, meanPosition(cpm(i+τ).space.nodeIDs, id) - meanPosition(cpm(i).space.nodeIDs, id)) for i in 1:τ:cpm.step.counter-τ])
MeanSqDis (generic function with 1 method)
Here we choose a lag time of 50 steps (arbitrary)
scatter([MeanSqDis(cpm, τ, 1) for τ in 1:50],
labels = "Random")
scatter!([MeanSqDis(cpm, τ, 2) for τ in 1:50],
title = "Random vs Directed Motion",
xlabel = "Lag Time",
ylabel = "Mean Squared Displacement",
framestyle = :box,
labels = "Directed")
Here we see that, as a function of lag time, mean squared displacement for random motion does produce a linear relationship whereas directed motion is deflected upward.
This page was generated using Literate.jl. | {"url":"https://robertgregg.github.io/CellularPotts.jl/dev/ExampleGallery/TravelTime/TravelTime/","timestamp":"2024-11-08T02:22:26Z","content_type":"text/html","content_length":"196820","record_id":"<urn:uuid:f182648f-1a39-419a-8a53-87607055de4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00808.warc.gz"} |
Storing Projected Values
I have a graph setup giving me daily data. (e.g. sales). I've used as part of the graph "Multi-Period Projection" and set it for 5 days out. What I want to know is how accurate this is to the actual.
Today is 1/31/2023 and sales was $1M, and the prejection for 2/3/2024 is $1.3M, I want to know how close the projection was to the actual for that day. To do this, I need to know what the project was
today and then compare to the actual. How do I store/retain the projecte amount?
Best Answers
• Short answer: you can't.
Long answer: Projected values update in real time as new data becomes available. You could store the projected values in a beastmode or in a webform and join them back to your data, but at what
point would you consider them locked in? 1 day out? 3 days out? It sounds like perhaps this is just a curiosity thing in which case I would just jot them down and see how well they perform. If
you actually need to measure the variance to the projected values I would recommend projecting your own values using linear regression with scripting tiles (python, R) in magic ETL or in a
Jupyter Workspace.
If I solved your problem, please select "yes" above
• What would be functionally equivalent is to retroactively calculate what the projections would have been for your entire data set, so you can see historically how your actuals compare to your
forecasts. If you're using linear regression, I played with an approach where I tool @marcel_luthi 's data set-up from this thread on rolling averages:
And create a scatter where the x-axis is pre-existing date, the y-axis is pre-existing value, and the series is the date. Then you can see all the past regressions (and filter for which
regression you want):
Disadvantage, is ideally I would be able to connect these points with a line graph, but as far as I can tell, you can't have multiple regression lines on any of the line graph options.
Please 💡/💖/👍/😊 this post if you read it and found it helpful.
Please accept the answer if it solved your problem.
• Short answer: you can't.
Long answer: Projected values update in real time as new data becomes available. You could store the projected values in a beastmode or in a webform and join them back to your data, but at what
point would you consider them locked in? 1 day out? 3 days out? It sounds like perhaps this is just a curiosity thing in which case I would just jot them down and see how well they perform. If
you actually need to measure the variance to the projected values I would recommend projecting your own values using linear regression with scripting tiles (python, R) in magic ETL or in a
Jupyter Workspace.
If I solved your problem, please select "yes" above
• I understand what you are stating. Just hoping for some easy way to do this without someone remembering to write them down. I would say we could do it for a week or two and then just do a check
every other month for about a week.
Thank you for your help.
• What would be functionally equivalent is to retroactively calculate what the projections would have been for your entire data set, so you can see historically how your actuals compare to your
forecasts. If you're using linear regression, I played with an approach where I tool @marcel_luthi 's data set-up from this thread on rolling averages:
And create a scatter where the x-axis is pre-existing date, the y-axis is pre-existing value, and the series is the date. Then you can see all the past regressions (and filter for which
regression you want):
Disadvantage, is ideally I would be able to connect these points with a line graph, but as far as I can tell, you can't have multiple regression lines on any of the line graph options.
Please 💡/💖/👍/😊 this post if you read it and found it helpful.
Please accept the answer if it solved your problem.
• David,
This gives me some other approaches and thought. I really appreciate it.
Thank you,
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How SHA256 works
SHA-256 has the input message size < 2^64-bits. Block size is 512-bits and has a word size of 32-bits. The output is a 256-bit digest.
The compression function processes a 512-bit message block and a 256-bit intermediate hash value. There are two main components of this function: compression function and a message schedule.
The algorithm works as follows:
1. Padding of the message, which is used to make the length of a block to 512-bits if it is smaller than the required block size of 512-bits.
2. Parsing the message into message blocks that ensure that the message and its padding is divided into equal blocks of 512-bits.
3. Setting up the initial hash value, which is the eight 32-bit words obtained by taking the first 32-bits of the fractional parts of the square roots of the first eight prime numbers. These initial
values are randomly chosen to initialize the process and gives a level of confidence that no backdoor exists in the algorithm.
1. Each message block is processed in a sequence and requires 64 rounds to compute the full hash output. Each round uses slightly different constants to ensure that no two rounds are the same.
2. First, the message schedule is prepared.
3. Then, eight working variables are initialized.
4. Then, the intermediate hash value is calculated.
5. Finally, the message is processed, and the output hash is produced:
one round of SHA256 compression function
In the preceding diagram,
, and
are the registers.
are applied bitwise. performs the bitwise rotation. Round constants are
, which are
mod 2^32
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Find maximum average subarray of k length - TutorialCup
Find maximum average subarray of k length
Difficulty Level Easy
Frequently asked in Amazon
Views 1779
Problem Statement
You are given an array of integers and a number k. The problem statement asks to find maximum average subarray of k length. Subarray is nothing but an array composed from a contiguous block of the
original array’s elements
arr[] = {1,3,12,34,76,10}
[2, 4]
Explanation: Array starting from index 2 to index 4 holds the maximum average. Thus [2,4] is maximum average subarray of length 2.
arr[] = {1,-2,8,3,-6,9}
[1, 3]
Explanation: Array starting from index 1 to index 3 holds the maximum average.
Algorithm to find maximum average subarray of k length
1. Declare a currentSum.
2. Set the currentSum to arr[0].
3. Traverse the array from i=1 to less than the length of n.
Store and update the addition of currentSum and arr[i] into the currentSum.
4. Set maximumSum to currentSum.
5. Set END to k-1.
6. Traverse the array from i=k to k to i < n.
7. Set sum to sum + arr[i] - arr[i - k].
Check if the sum is greater than maximumSum.
1. Set sum to maximumSum.
2. Set END to i.
8. Return END – k + 1.
We are given an array of integers and a number k. We have to find out the maximum average of sub-array of size k. This means the average we find should be of k contiguous numbers of the array. So we
are going to find the average of k numbers within the sub-array. We have declared the condition if k is greater than n,. Return -1 if it is true. It means that if the value of k is greater than the
value of n, which means we have not a valid value of k, it is not possible that k is greater than the size of the array.
We have already created the currentSum. Copy the first value of the given array to the currentSum. Then we will be traversing the array from index 1 because we already performed an operation on the
value of currentSum. So we will be traversing the array and get the sum of previous element of currentSum and arr[i] and store it into the currentSum.
We have set the value of maximumSum to (k-1)th position of currentSum and END to k-1. Traverse the array from kth position till the last element of the array. We are going to get the difference arr
[i] and arr[i – k] and we add it to the value of sum and store it to sum, and also we need not worry about the negative value we have started from the position k, so we won’t get any differences in
there. If the sum is greater than the maximumSum, set the maximumSum from sum value and end value to i. We are going to repeat the same process until the length of an array is reached. Then we will
return END – k + 1.
This technique is also referred to as Sliding Window Technique.
Code to find maximum average subarray of k length
C++ Code
using namespace std;
int getMaxAvgSubArray(int arr[], int n, int k)
if (k > n)
return -1;
int currentSum;
currentSum = arr[0];
for (int i = 1; i < n; i++)
currentSum = currentSum + arr[i];
int maximumSum = currentSum;
int END = k - 1;
for (int i = k; i < n; i++)
currentSum = currentSum + arr[i] - arr[i-k];
if (currentSum > maximumSum)
maximumSum = currentSum;
END = i;
return END - k + 1;
int main()
int arr[] = {1,3,12,34,76,10};
int k = 3;
int n = sizeof(arr)/sizeof(arr[0]);
int start=getMaxAvgSubArray(arr, n, k);
cout<<"Maximum average subarray: ["<< start<<" "<< (start + k - 1)<<"]";
return 0;
Maximum average subarray: [2 4]
Java Code
class MaximumAverageSubarray
public static int getMaxAvgSubArray(int []arr,int n, int k)
if (k > n)
return -1;
int currentSum = arr[0];
for (int i = 1; i < k; i++)
currentSum = currentSum + arr[i];
int maximumSum = currentSum;
int END = k - 1;
for (int i = k; i < n; i++)
currentSum = currentSum + arr[i] - arr[i-k];
if (currentSum > maximumSum)
maximumSum = currentSum;
END = i;
return END - k + 1;
public static void main (String[] args)
int []arr = {1,3,12,34,76,10};
int k = 3;
int n = arr.length;
int start=getMaxAvgSubArray(arr, n, k);
System.out.println("Maximum average subarray: ["+start+" "+ (start + k - 1)+"]");
Maximum average subarray: [2 4]
Time Complexity to find maximum average subarray of k length
Since we have traversed through the array once, we have linear time complexity. O(n) where “n” is the number of elements in the array.
Space Complexity to find maximum average subarray of k length
Since we used an array for storing the input, we have linear space complexity. O(n) where “n” is the number of elements in the array. | {"url":"https://tutorialcup.com/interview/array/find-maximum-average-subarray-of-k-length.htm","timestamp":"2024-11-14T18:49:27Z","content_type":"text/html","content_length":"107365","record_id":"<urn:uuid:a4ad6d57-ea46-4a59-a32e-2c5ac83c6006>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00358.warc.gz"} |
Jason Sachs ●
November 7, 2017
This is a bit of a side tangent from my normal at-least-vaguely-embedded-related articles, but I wanted to share a moment of enlightenment I had recently about descriptors in Python. The easiest way
to explain a descriptor is a way to outsource attribute lookup and modification.
Python has a bunch of “magic” methods that are hooks into various object-oriented mechanisms that let you do all sorts of ridiculously clever things. Whether or not they’re a good idea is another...
Jason Sachs ●
October 18, 2017
●1 comment
The last two articles were on discrete logarithms in finite fields — in practical terms, how to take the state \( S \) of an LFSR and its characteristic polynomial \( p(x) \) and figure out how many
shift steps are required to go from the state 000...001 to \( S \). If we consider \( S \) as a polynomial bit vector such that \( S = x^k \bmod p(x) \), then this is equivalent to the task of
figuring out \( k \) from \( S \) and \( p(x) \).
This time we’re tackling something...
Jason Sachs ●
October 1, 2017
Last time we talked about discrete logarithms which are easy when the group in question has an order which is a smooth number, namely the product of small prime factors. Just as a reminder, the goal
here is to find \( k \) if you are given some finite multiplicative group (or a finite field, since it has a multiplicative group) with elements \( y \) and \( g \), and you know you can express \( y
= g^k \) for some unknown integer \( k \). The value \( k \) is the discrete logarithm of \( y \)...
Jason Sachs ●
September 16, 2017
●4 comments
Last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with Blankinship’s algorithm.
Discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg.
What is a Discrete Logarithm, Anyway?
Regular logarithms are something that you’re probably familiar with: let’s say you have some number \( y = b^x \) and you know \( y \) and \( b \) but...
Jason Sachs ●
September 9, 2017
Last time we talked about basic arithmetic operations in the finite field \( GF(2)[x]/p(x) \) — addition, multiplication, raising to a power, shift-left and shift-right — as well as how to determine
whether a polynomial \( p(x) \) is primitive. If a polynomial \( p(x) \) is primitive, it can be used to define an LFSR with coefficients that correspond to the 1 terms in \( p(x) \), that has
maximal length of \( 2^N-1 \), covering all bit patterns except the all-zero...
Jason Sachs ●
July 17, 2017
Last time, we looked at the basics of LFSRs and finite fields formed by the quotient ring \( GF(2)[x]/p(x) \).
LFSRs can be described by a list of binary coefficients, sometimes referred as the polynomial, since they correspond directly to the characteristic polynomial of the quotient ring.
Today we’re going to look at how to perform certain practical calculations in these finite fields. I maintain a Python library called libgf2,...
Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.
— Évariste Galois, May 29, 1832
I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a
decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical...
Other articles in this series:
This article is mainly an excuse to scribble down some cryptic-looking mathematics — Don’t panic! Close your eyes and scroll down if you feel nauseous — and...
This article is about something profound that a brilliant young professor at Stanford wrote nearly 45 years ago, and now we’re all stuck with it.
The idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision
to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...
Jason Sachs ●
February 27, 2017
●1 comment
I’ve been wasting far too much of my free time lately on this stupid addicting game called the Kittens Game. It starts so innocently. You are a kitten in a catnip forest. Gather catnip.
And you click on Gather catnip and off you go. Soon you’re hunting unicorns and building Huts and studying Mathematics and Theology and so on. AND IT’S JUST A TEXT GAME! HTML and Javascript, that’s
it, no pictures. It’s an example of an
Jason Sachs ●
November 11, 2015
●9 comments
Today we will be drifting back into the topic of numerical methods, and look at an algorithm that takes in a series of discretely-sampled data points, and estimates the maximum value of the waveform
they were sampled from.
Jason Sachs ●
December 23, 2018
Today I’m going to talk a little bit about three-terminal linear passive networks. These generally come in two flavors, wye and delta.
Why Wye?
The town of Why, Arizona has a strange name that comes from the shape of the original road junction of Arizona State Highways 85 and 86, which was shaped like the letter Y. This is no longer the
case, because the state highway department reconfigured the intersection
Jason Sachs ●
September 1, 2011
●9 comments
A recent electronics.StackExchange question brings up a good topic for discussion. Let's say you have a power supply and a 2-wire load you want to be able to switch on and off from the power supply
using a MOSFET. How do you choose which circuit topology to choose? You basically have four options, shown below:
From left to right, these are:
High-side switch, N-channel MOSFET High-side switch, P-channel MOSFET Low-side switch, N-channel...
This blog needs some short posts to balance out the long ones, so I thought I’d cover some of the algorithms I’ve used over the years. Like the Euclidean algorithm and Extended Euclidean algorithm
and Newton’s method — except those you should know already, and if not, you should be locked in a room until you do. Someday one of them may save your life. Well, you never know.
Other articles in this series:
Jason Sachs ●
April 25, 2013
I recently faced a little "asterisk" problem, which looks like it can be solved with some interesting ICs.
I needed to plan out some test instrumentation to capture voltage and current information over a short period of time. Nothing too fancy, 10 or 20kHz sampling rate, about a half-dozen channels
sampled simultaneously or near simultaneously, for maybe 5 or 10 seconds.
Here's the "asterisk": Oh, by the way, because the system in question was tied to the AC mains, I needed some...
In part 1 we talked about the use of a MOSFET for a power switch. Here's a different circuit that also uses a MOSFET, this time as a switch for signals:
We have a thermistor Rth that is located somewhere in an assembly, that connects to a circuit board. This acts as a variable resistor that changes with temperature. If we use it in a voltage divider,
the midpoint of the voltage divider has a voltage that depends on temperature. Resistors R3 and R4 form our reference resistance; when...
Jason Sachs ●
December 24, 2023
In this article we explore the use of continued fractions to approximate any particular real number, with practical applications.
Warning: In the interest of maintaining a coherent stream of consciousness, I’m lowering the setting on my profanity filter for this post. Just wanted to let you know ahead of time.
I’ve been a user of Stack Overflow since December of 2008. And I say “user” both in the software sense, and in the drug-addict sense. I’m Jason S, user #44330, and I’m a programming addict. (Hi,
Jason S.) The Gravatar, in case you were wondering, is a screen...
This is a short article about how to analyze the diode in some gate drive circuits when figuring out turn-off characteristics --- specifically, determining the relationship between gate drive current
and gate voltage during turn-off of a power transistor.
Jason Sachs ●
September 29, 2019
AS-SALT, JORDAN — Dr. Reza Al-Faisal once had a job offer from Google to work on cutting-edge voice recognition projects. He turned it down. The 37-year-old Stanford-trained professor of engineering
at Al-Balqa’ Applied University now leads a small cadre of graduate students in a government-sponsored program to keep Jordanian society secure from what has now become an overwhelming influx of
refugees from the Palestinian-controlled West Bank. “Sometimes they visit relatives...
Jason Sachs ●
October 30, 2013
●1 comment
It's that time again to review all the oddball goodies available in electronic components. These are things you should have in your bag of tricks when you need to design a circuit board. If you read
my previous posts and were looking forward to more, this article's for you!
1. Bus switches
I can't believe I haven't mentioned bus switches before. What is a bus switch?
There are lots of different options for switches:
• mechanical switch / relay: All purpose, two...
Jason Sachs ●
June 19, 2018
Last time, we talked about error correction and detection, covering some basics like Hamming distance, CRCs, and Hamming codes. If you are new to this topic, I would strongly suggest going back to
read that article before this one.
This time we are going to cover Reed-Solomon codes. (I had meant to cover this topic in Part XV, but the article was getting to be too long, so I’ve split it roughly in half.) These are one of the
workhorses of error-correction, and they are used in...
I’ve already written about the unexciting (but useful) 1st-order system, and about slew-rate limiting. So now it’s time to cover second-order systems.
The most common second-order systems are RLC circuits and spring-mass-damper systems.
Spring-mass-damper systems are fairly common; you’ve seen these before, whether you realize it or not. One household example of these is the spring doorstop (BOING!!):
(For what it’s worth: the spring...
This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get
bored and move on to something else. That’s fine by me.
Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a \( p=126, q=130 \) Padé approximation of a time delay:
Impressed? Maybe you should be. This...
I got a bad feeling yesterday when I had to include reference information about a 16-bit CRC in a serial protocol document I was writing. And I knew it wasn’t going to end well.
The last time I looked into CRC algorithms was about five years ago. And the time before that… sometime back in 2004 or 2005? It seems like it comes up periodically, like the seventeen-year locust or
sunspots or El Niño,...
Все счастли́вые се́мьи похо́жи друг на дру́га, ка́ждая несчастли́вая семья́ несчастли́ва по-сво́ему.
— Лев Николаевич Толстой, Анна Каренина
Happy families are all alike; every unhappy family is unhappy in its own way.
— Lev Nicholaevich Tolstoy, Anna Karenina
I was going to write an article about second-order systems, but then realized that it would be...
My software team recently finished a round of code reviews for some of our motor controller code. I learned a lot from the experience, most notably why you would want to have code reviews in the
first place.
My background is originally from the medical device industry. In the United States, software in medical devices gets a lot of scrutiny from the Food and Drug Administration, and for good reason; it’s
a place for complexity to hide latent bugs. (Can you say “
Jason Sachs ●
September 7, 2013
●6 comments
When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't
you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between
encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...
This article is about something profound that a brilliant young professor at Stanford wrote nearly 45 years ago, and now we’re all stuck with it.
The idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision
to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...
Jason Sachs ●
January 28, 2014
●1 comment
I was recently using the publish() function in MATLAB to develop some documentation, and I ran into a problem caused by a bad hash function.
In a resource-limited embedded system, you aren't likely to run into hash functions. They have three major applications: cryptography, data integrity, and data structures. In all these cases, hash
functions are used to take some type of data, and deterministically boil it down to a fixed-size "fingerprint" or "hash" of the original data, such that... | {"url":"https://www.embeddedrelated.com/blogs-5/nf/Jason_Sachs/all.php","timestamp":"2024-11-05T07:19:13Z","content_type":"text/html","content_length":"71530","record_id":"<urn:uuid:5a7211ff-9b08-454f-86d3-aaf0d96b8d8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00409.warc.gz"} |
7 Important Ratios to consider while selecting a right Mutual Fund
7 Important Ratios to consider while selecting a right Mutual Fund
Investing in Mutual Fund is inevitable. We have talked a lot about selecting right category of Mutual funds according to your financial goals, time horizon and risk appetite in our earlier blogs.
However, after selecting right category of Mutual Funds, how to select the right Mutual Fund scheme? How to compare two funds within the same category?
In this blog, we are elaborating few ratios for you. The information on these ratios will help you find better scheme for you.
1. Alpha & Beta
First of the ratios, Alpha is a measure of how much returns the Mutual Fund scheme has delivered over the market. It measures a fund’s performance to its benchmark. In simple terms, Alpha is the
excess returns generated by the fund compared to benchmark.
Alpha = Actual Rate of Return – Risk-Free Rate – Beta * Market Risk Premium
Where: (Market Risk Premium = Market Return – Risk-Free Rate)
Example: Let us take the example of a Portfolio with a Beta of 1.5 that generated an Actual Return of 10% during last year. If the current Market Return is 6% and the Risk-Free Rate is 4%, then
calculate the Alpha of the Portfolio.
Alpha = 10% – 4% – 1.5 * 2%
Alpha = 3%
The higher a fund’s alpha, the better it is. Alpha of 3% is indicates that it is a very good fund and has delivered much better returns than its benchmark.
Beta is a volatility parameter relative to its benchmark. A beta above 1 indicates that the fund can perform better on uptrends and fall more in market corrections. In short, a fund probably will
gain more and fall more than the average. A beta below 1 indicates the reverse. High-beta funds that don’t perform well during rising market indicate that there’s something inappropriate in the
fund’s strategy. High-beta funds are suitable for risk-taking investors.
The formula for beta is little complicated.
However, you do not need to remember any of the formula mentioned in this blog. All the important formulas are mentioned on the research websites. You just need to interpret those formulas and make a
wise decision with the help of these ratios.
2. Sharpe Ratio
Sharpe Ratio measures the risk-adjusted returns of two or more funds within a same category. This ratio shows how much return an investor is earning in connection to the level of underlying risk.
Sharpe Ratio is calculated by taking the difference between the returns of the investment (Mutual Fund) and the risk-free return, divided by the standard deviation of the asset.
Sharpe Ratio = (Fund Return – Risk-free Return)/Standard Deviation of the Fund
Investors will get an idea regarding the degree of risk that a fund took to generate extra returns over risk-free instruments, such as 10-year G-Sec bonds.
The higher the Sharpe Ratio, the better is the fund’s ability to reward investors with higher risk-adjusted returns.
Example: Let’s compare two Mutual Fund Schemes as under.
‘A’ Fund ‘B’ Fund ‘C’ Fund ‘D’ Fund
Returns 10% 12% 12% 14%
Risk Free Rate 6% 6% 6% 5%
Standard Deviation 12% 12% 13% 8%
Sharpe Ratio 0.33 0.50 0.46 1.13
As you can see, Sharpe Ratio increases with the returns.
In short, you can interpret Sharpe Ratio as,
• Less than 1: Bad
• 1 – 1.99: Adequate/good
• 2 – 2.99: Very good
• Greater than 3: Excellent
3. Standard Deviation:
Standard deviation is a arithmetic tool that measures the deviation or dispersion of the data (here returns) from the mean or average. When it comes to Mutual Funds, it states that how much returns
from your mutual fund portfolio is drifting from the expected return, based on the fund’s historical performance.
For example, if the portfolio A has a standard deviation of 6% and average return of 14%, it means that it has a tendency of deviating by 8% from its expected average return and may give returns
between 7% to 22%.
Standard deviation is directly proportional to the volatility of the portfolio.
4. Sortino Ratio
Alpha, Sharpe Ratios are measures of excess Returns. However, Sortino Ratio is the one that captures downside risk of the fund. It measures a fund’s ability to cover the downside risk, especially
during adverse market conditions.
It calculates the risk instead of the total volatility of the portfolio. Here, he downside risk signifies returns that fall below a minimum expected rate such as risk-free returns and/or negative
For instance, a fund has generated returns of 10%, 15%, 20%, 2%, -5%, -3%, and 4% respectively, in the last seven years.
Assuming that the risk-free rate is 6%, the returns below this limit will be included. In this case 2%, -5%, -3%, and 4% returns will be considered as downside deviation because they are below
risk free rate of 6%.
Sortino Ratio = (Fund Return – Risk-free Return)/Downside Deviation
All mutual funds have a possible downside risk as the returns are market linked. However, some schemes have a better ability to manage it. Thus, the Sortino Ratio is an important ratio to measure
risk-adjusted returns.
‘A’ Fund ‘B’ Fund ‘C’ Fund ‘D’ Fund
Returns 15% 12% 14% 12%
Risk Free Rate 6% 6% 6% 6%
Downside Deviation 12% 7% 9% 5%
Sortino Ratio 0.75 0.86 0.89 1.20
Lower the downside deviation, higher the Sortino Ratio.
It is similar to Sharpe Ratio. However, Sharpe ratio accounts for risk-adjustments in investments with both positive and negative returns. On the contrary, the Sortino ratio examines risk-adjusted
returns, but it only considers the downside risks.
The higher the Sortino Ratio, the better is the fund’s ability of earning higher returns by not taking unjustified risk.
5. Treynor Ratio
By the definition, the Treynor Ratio is used for determining the excess returns earned per unit for a given level of systemic risk.
Treynor Ratio = (Fund Return – Risk-free Return)/Beta of the Fund
While the Sharpe Ratio considers standard deviation for calculating risk-adjusted returns, the Treynor Ratio considers the ‘Beta’ of the Mutual fund (a measure of systemic risk).
Investors invests in Mutual Funds is to reduce risk by diversification. However, systemic risk (market risk) cannot be mitigated by diversification. Hence, mutual funds should compensate investors
by efficiently managing the portfolio to generate a risk premium.
The Beta of a mutual fund scheme is its volatility relative to its benchmark index.
As the purpose of mutual funds is to outperform the underlying market index, the Treynor Ratio is a useful ratio for examining the scheme’s performance.
‘A’ Fund ‘B’ Fund ‘C’ Fund
Returns 17% 17% 17%
Risk Free Rate 6% 6% 6%
Beta 1.1 0.9 0.8
Treynor Ratio 0.10 0.12 0.14
Lower the Beta, Higher the Treynor Ratio.
In the above table, all the schemes have generated similar returns.
However, Fund A’s higher return has come from investing in highly volatile stocks. A Fund with a higher Treynor Ratio is better.
This ratio also helps you to compare different Mutual Fund schemes and you can pick the one most suitable for your risk profile.
6. Portfolio Turnover Ratio
A Mutual Fund scheme is basket of stocks. A Fund manager selects different types of stocks as per the objective of the fund and his expertise. However, it is not a one-time activity. A Fund Manager
reviews stocks in a portfolio and buy and sell it according the market conditions and opportunities.
The portfolio turnover ratio is the rate of which stocks in a fund are bought and sold by the fund managers. In simple words, the ratio refers to the percentage (%) change of the assets (stocks in
case of Mutual Funds) in a fund over a one-year period.
• Here securities are stocks in Mutual Fund context. Minimum of securities bought or sold refers to the total amount of new securities purchased or the total amount of securities sold (whichever is
less) over a one-year period.
• Average net assets refer to the monthly average amount of net assets in the fund.
Higher Portfolio turnover ratio often mean the scheme is more actively managed, which leads to higher costs and taxes i.e. expense ratio.
7. Expense Ratio
Last one of the seven Ratios. By reading itself, you must have understood the meaning of this ratio. Managing the Mutual fund incurs may expenses. These can be marketing and advertising expenses,
administrative expenses, transaction costs, investment management fees, registrar fees, custodian fees, audit fees etc.
All costs for running and managing a mutual fund scheme are collectively referred to as Expense Ratio. It is calculated as a percentage (%) of the Mutual Scheme’s average Net Asset Value (NAV).
Investors doesn’t pay anything out of this or her pocket, however, it already reflects in daily NAV of a Mutual Fund Scheme. The daily NAV of a mutual fund is disclosed after deducting the expenses.
Lower the expenses ratio, the better can be the returns generated by the fund. However, this is not the case with every category or every fund.
Equity Category generally has higher expenses ratio as it requires active management and review of the portfolio as compared to Debt Mutual Fund schemes.
At times, the scheme’s performance can justify the higher expenses ratio. Only expenses ratio cannot be the criteria for judging the fund.
Don’t get worried by all the complicated formulas. You don’t have to calculate actually.
All above ratios are readily available on various websites.
Investors just need to know how to interpret them and select a right Mutual Fund scheme.
Important – Financial Advisor in Nashik, Financial Advisor in Thane, Financial Advisor in Jalgaon, Financial Advisor in Pune
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Equation of a circle calculator
This circle equation calculator displays a standard form equation of a circle, a parametric form equation of a circle, and a general form equation of a circle given the center and radius of the
circle. Formulas can be found below the calculator.
Equation of a circle
An equation of a circle is an algebraic way to define all points that lie on the circumference of the circle. That is, if the point satisfies the equation of the circle, it lies on the circle's
circumference. There are different forms of the equation of a circle:
• general form
• standard form
• parametric form
• polar form.
General Form Equation of a Circle
The general equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
With general form, it is difficult to reason about the circle's properties, namely the center and the radius. But it can easily be converted into standard form, which is much easier to understand.
Standard Form Equation of a Circle
The standard equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
$(x^2-x_0) + (y^2-y_0)=r^2$
You can convert general form to standard form using the technique known as Completing the square. From this circle equation, you can easily tell the coordinates of the center and the radius of the
Parametric Form Equation of a Circle
The parametric equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
$x=r cos \theta + x_0\\y=r sin \theta + y_0$
This equation is called "parametric" because the angle theta is referred to as a "parameter". This is a variable which can take any value (but of course it should be the same in both equations). It
is based on the definitions of sine and cosine in a right triangle.
Polar Form Equation of a Circle
The polar form looks somewhat similar to the standard form, but it requires the center of the circle to be in polar coordinates $(r_0, \phi)$ from the origin. In this case, the polar coordinates on a
point on the circumference $(r, \theta)$ must satisfy the following equation
$r^2-2r r_0 cos(\theta - \phi)+r^2_0=a^2$,
where a is the radius of the circle.
PLANETCALC, Equation of a circle calculator | {"url":"https://planetcalc.com/8115/?thanks=1","timestamp":"2024-11-14T16:44:49Z","content_type":"text/html","content_length":"37566","record_id":"<urn:uuid:d378fc04-e91f-4986-9ccb-ef99d0afc70c>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00733.warc.gz"} |
Home | mysite
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Nezhla Aghaei
Section de mathématiques
24, rue du Général Dufour
Case postale 64
1211 Geneva 4, Switzerland
Centre for Quantum Mathematics
Department of Mathematics and Computer Science (IMADA)
University of Southern Denmark
Campusvej 55
5230 Odense M, Denmark
linkedin: Nezhla Aghaei
Welcome to my website.
I am a mathematical physicist and currently at Mari Curie Fellow at QM center at South Denmark University (SDU)
and the university of Geneva.
Mari Curie Fellow title: Super Andersen-Kashaev TQFT
Before, I was a postdoc at Max Planck Institute in Bonn and
in the Albert Einstein Centre at the University of Bern in the group of Prof. Reffert.
I was a PhD student in Research Training Group (RTG), Mathematics inspires by
string theory and quantum field theory, in the mathematics department
at the University of Hamburg and in the string theory group at DESY, Germany,
under the supervision of Prof. Jörg Teschner.
My research interests are mathematical structure underline the exact results in
supersymmetric gauge theory, including:
Research interests in Mathematics:
• The mathematics of quantum Teichmüller theory and TQFT:
Relation to quantum group, Representation theory and Knot theory
• The mathematics of: conformal field theories and integrable models
• Topological recursion
Research interests in String Theory:
• 4d-2d and 3d-3d correspondence
• Non-perturbative physics of supersymmetric gauge theories
• Conformal field theories (CFT) and Integrability
• Complex Chern-Simons theory
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Measure Liquid by Cans
This article will discuss a famous interview puzzle in which we have to draw every quantities of Liquid up to 40 liters using cans of different sizes from a drum full of Liquid. We can have a maximum
of 4 cans; our task is to determine the capacity of cans so that every quantity between 1-and 40 can be drawn out using these four cans.
Problem Statement
The problem states that we have a huge drum of Liquid, and we need to draw some liquid from it. To be precise, we need to measure every quantities of Liquid, between 1 - 40 liters using 4 cans. We
need to determine the dimensions of these cans so that every quantity between 1 and 40 can be measured. Note: We can use every can only once.
Suppose I need to measure 38 liters. I can have cans of sizes 1,3,9,27. The right way to measure 38 liters using these cans are 27 + 9 + 3 - 1 = 38. | {"url":"https://www.naukri.com/code360/library/measure-liquid-by-cans","timestamp":"2024-11-09T23:09:49Z","content_type":"text/html","content_length":"359389","record_id":"<urn:uuid:b9b8bf7b-13dc-4776-8897-5859d94a774c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00159.warc.gz"} |
class springcraft.GNM(atoms, force_field, masses=None, use_cell_list=True)¶
Bases: object
This class represents a Gaussian Network Model.
atomsAtomArray, shape=(n,) or ndarray, shape=(n,3), dtype=float
The atoms or their coordinates that are part of the model. It usually contains only CA atoms.
force_fieldForceField, natoms=n
The ForceField that defines the force constants between the given atoms.
massesbool or ndarray, shape=(n,), dtype=float, optional
If an array is given, the Kirchhoff matrix is weighted with the inverse square root of the given masses. If set to true, these masses are automatically inferred from the res_name
annotation of atoms, instead. This requires atoms to be an AtomArray. By default no mass-weighting is applied.
use_cell_listbool, optional
If true, a cell list is used to find atoms within cutoff distance instead of checking all pairwise atom distances. This significantly increases the performance for large number of atoms,
but is slower for very small systems. If the force_field does not provide a cutoff, no cell list is used regardless.
kirchhoffndarray, shape=(n,n), dtype=float
The Kirchhoff matrix for this model. This is not a copy: Create a copy before modifying this matrix.
covariancendarray, shape=(n*3,n*3), dtype=float
The covariance matrix for this model, i.e. the inverted Kirchhofff matrix. This is not a copy: Create a copy before modifying this matrix.
bfactor(mode_subset=None, tem=None, tem_factors=1.380649e-23)¶
Computes the isotropic B-factors/temperature factors/ Deby-Waller factors for atoms/coarse-grained beads using the mean-square fluctuation.
These can be used to relate results obtained from ENMs to experimental results.
mode_subsetndarray, shape=(n,), dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first trivial mode (0) are included.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with K_B as preset).
bfac_valuesndarray, shape=(n,), dtype=float
B-factors of C-alpha atoms.
dcc(mode_subset=None, norm=True, tem=None, tem_factors=1.380649e-23)¶
Computes the normalized dynamic cross-correlation between nodes of the GNM.
The DCC is a measure for the correlation in fluctuations exhibited by a given pair of nodes. If normalized, pairs with correlated fluctuations (same phase and period), anticorrelated
fluctuations (opposite phase, same period) and non-correlated fluctuations are assigned (normalized) DCC values of 1, -1 and 0 respectively. For results consistent with MSFs,
temperature-weighted absolute values can be computed (only relevant if results are not normalized).
mode_subsetndarray, shape=(n,), dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first trivial mode (0) are included.
normbool, optional
Normalize the DCC using the MSFs of interacting nodes.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with \(k_B\) as preset).
dccndarray, shape=(n, n), dtype=float
DCC values for ENM nodes.
The DCC for a nodepair \(ij\) is computed as:
\[DCC_{ij} = \frac{3 k_B T}{\gamma} \sum_k^L \left[ \frac{\vec{u}_k \cdot \vec{u}_k^T}{\lambda_k} \right]_{ij}\]
with \(\lambda\) and \(\vec{u}\) as Eigenvalues and Eigenvectors corresponding to mode \(k\) of the modeset \(L\).
DCCs can be normalized to MSFs exhibited by two compared nodes following:
\[nDCC_{ij} = \frac{DCC_{ij}}{[DCC_{ii} DCC_{jj}]^{1/2}}\]
Compute the Eigenvalues and Eigenvectors of the Kirchhoff matrix.
eig_valuesndarray, shape=(k,), dtype=float
Eigenvalues of the Kirchhoff matrix in ascending order.
eig_vectorsndarray, shape=(k,n), dtype=float
Eigenvectors of the Kirchhoff matrix. eig_values[i] corresponds to eigenvectors[i].
Compute the oscillation frequencies of the model.
The first mode corresponds to rigid-body translations/rotations and is omitted in the return value. The returned units are arbitrary and should only be compared relative to each other.
frequenciesndarray, shape=(n,), dtype=float
Oscillation frequencies of the model in in ascending order. NaN values mark frequencies corresponding to translations or rotations.
mean_square_fluctuation(mode_subset=None, tem=None, tem_factors=1.380649e-23)¶
Compute the mean square fluctuation for the atoms according to the GNM. This is equal to the sum of the diagonal of of the GNM covariance matrix, if all k-1 non-trivial modes are considered.
mode_subsetndarray, shape=(n,), dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first trivial mode (0) are included.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with K_B as preset).
msqfndarray, shape=(n,), dtype=float
The mean square fluctuations for each atom in the model.
class springcraft.ANM(atoms, force_field, masses=None, use_cell_list=True)¶
Bases: object
This class represents an Anisotropic Network Model.
atomsAtomArray, shape=(n,) or ndarray, shape=(n,3), dtype=float
The atoms or their coordinates that are part of the model. It usually contains only CA atoms.
force_fieldForceField, natoms=n
The ForceField that defines the force constants between the given atoms.
massesbool or ndarray, shape=(n,), dtype=float, optional
If an array is given, the Hessian is weighted with the inverse square root of the given masses. If set to true, these masses are automatically inferred from the res_name annotation of
atoms, instead. This requires atoms to be an AtomArray. By default no mass-weighting is applied.
use_cell_listbool, optional
If true, a cell list is used to find atoms within cutoff distance instead of checking all pairwise atom distances. This significantly increases the performance for large number of atoms,
but is slower for very small systems. If the force_field does not provide a cutoff, no cell list is used regardless.
hessianndarray, shape=(n*3,n*3), dtype=float
The Hessian matrix for this model. Each dimension is partitioned in the form [x1, y1, z1, ... xn, yn, zn]. This is not a copy: Create a copy before modifying this matrix.
covariancendarray, shape=(n*3,n*3), dtype=float
The covariance matrix for this model, i.e. the inverted Hessian. This is not a copy: Create a copy before modifying this matrix.
massesNone or ndarray, shape=(n,), dtype=float
The mass for each atom, None if no mass weighting is applied.
bfactor(mode_subset=None, tem=None, tem_factors=1.380649e-23)¶
Computes the isotropic B-factors/temperature factors/ Deby-Waller factors for atoms/coarse-grained beads using the mean-square fluctuation. These can be used to relate results obtained from
ENMs to experimental results.
mode_subsetndarray, shape=(n,), dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first six trivial modes (0-5) are included.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with K_B as preset).
bfac_valuesndarray, shape=(n,), dtype=float
B-factors of C-alpha atoms.
dcc(mode_subset=None, norm=True, tem=None, tem_factors=1.380649e-23)¶
Computes the normalized dynamic cross-correlation between nodes of the ANM.
The DCC is a measure for the correlation in fluctuations exhibited by a given pair of nodes. If normalized, pairs with correlated fluctuations (same phase and period), anticorrelated
fluctuations (opposite phase, same period) and non-correlated fluctuations are assigned (normalized) DCC values of 1, -1 and 0 respectively. For results consistent with MSFs,
temperature-weighted absolute values can be computed (only relevant if results are not normalized).
mode_subsetndarray, shape=(n,), dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first six trivial modes (0-5) are included.
normbool, optional
Normalize the DCC using the MSFs of interacting nodes.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with \(k_B\) as preset).
dccndarray, shape=(n, n), dtype=float
DCC values for ENM nodes.
The DCC for a nodepair \(ij\) is computed as:
\[DCC_{ij} = \frac{3 k_B T}{\gamma} \sum_k^L \left[ \frac{\vec{u}_k \cdot \vec{u}_k^T}{\lambda_k} \right]_{ij}\]
with \(\lambda\) and \(\vec{u}\) as Eigenvalues and Eigenvectors corresponding to mode \(k\) of the modeset \(L\).
DCCs can be normalized to MSFs exhibited by two compared nodes following:
\[nDCC_{ij} = \frac{DCC_{ij}}{[DCC_{ii} DCC_{jj}]^{1/2}}\]
Compute the Eigenvalues and Eigenvectors of the Hessian matrix.
The first six Eigenvalues/Eigenvectors correspond to trivial modes (translations/rotations) and are usually omitted in normal mode analysis.
eig_valuesndarray, shape=(k,), dtype=float
Eigenvalues of the Hessian matrix in ascending order.
eig_vectorsndarray, shape=(k,n), dtype=float
Eigenvectors of the Hessian matrix. eig_values[i] corresponds to eig_vectors[i].
Computes the frequency associated with each mode.
The first six modes correspond to rigid-body translations/ rotations and are omitted in the return value.
The returned units are arbitrary and should only be compared relative to each other.
freqndarray, shape=(n,), dtype=float
The frequency in ascending order of the associated modes’ Eigenvalues.
Compute the atom displacement induced by the given force using Linear Response Theory. [1]
forcendarray, shape=(n,3) or shape=(n*3,), dtype=float
The force that is applied to the atoms of the model. The first dimension gives the atom the force is applied on, the second dimension gives the three spatial dimensions.
Alternatively, a flattened array in the form [x1, y1, z1, ... xn, yn, zn] can be given.
displacementndarray, shape=(n,3), dtype=float
The vector of displacement induced by the given force. The first dimension represents the atom index, the second dimension represents spatial dimension.
M Ikeguchi, J Ueno, M Sato, A Kidera, “Protein Structural Change Upon Ligand Binding: Linear Response Theory.” Phys Rev Lett. 94, 7, 078102 (2005).
mean_square_fluctuation(mode_subset=None, tem=None, tem_factors=1.380649e-23)¶
Compute the mean square fluctuation for the atoms according to the ANM. This is equal to the sum of the diagonal of each 3x3 superelement of the ANM covariance matrix, if all k-6 non-trivial
modes are considered.
mode_subsetndarray, dtype=int, optional
Specifies the subset of modes considered in the MSF computation. Only non-trivial modes can be selected. The first mode is counted as 0 in accordance with Python conventions. If
mode_subset is None, all modes except the first six trivial modes (0-5) are included.
temint, float, None, optional
Temperature in Kelvin to compute the temperature scaling factor by multiplying with the Boltzmann constant. If tem is None, no temperature scaling is conducted.
tem_factorsint, float, optional
Factors included in temperature weighting (with K_B as preset).
msqfndarray, shape=(n,), dtype=float
The mean square fluctuations for each atom in the model.
normal_mode(index, amplitude, frames, movement='sine')¶
Create displacements for a trajectory depicting the given normal mode.
This is especially useful for molecular animations of the chosen oscillation mode.
Note, that the first six modes correspond to rigid-body translations/ rotations and are usually omitted in normal mode analysis.
The index of the oscillation. The index refers to the Eigenvalues obtained from eigen(): Increasing indices refer to oscillations with increasing frequency. The first 6 modes
represent tigid body movements (rotations and translations).
The oscillation amplitude is scaled so that the maximum value for an atom is the given value.
The number of frames (models) per oscillation.
movement{‘sinusoidal’, ‘triangle’}
Defines how to depict the oscillation. If set to 'sine' the atom movement is sinusoidal. If set to 'triangle' the atom movement is linear with sharp amplitude.
displacementndarray, shape=(m,n,3), dtype=float
Atom displacements that depict a single oscillation. m is the number of frames.
class springcraft.ForceField¶
Bases: object
Subclasses of this abstract base class define the force constants of the modeled springs between atoms in a Elastic network model.
cutoff_distancefloat or None
The interaction of two atoms is only considered, if the distance between them is smaller or equal to this value. If None, the interaction between all atoms is considered.
natomsint or None
The number of atoms in the model. If a ForceField does not depend on the respective atoms, i.e. atom_i and atom_j is unused in force_constant(), this attribute is None instead.
contact_shutdownndarray, shape=(n,), dtype=float, optional
Indices that point to atoms, whose contacts to all other atoms are artificially switched off. If None, no contacts are switched off.
contact_pair_offndarray, shape=(n,2), dtype=int, optional
Indices that point to pairs of atoms, whose contacts are artificially switched off. If None, no contacts are switched off.
contact_pair_onndarray, shape=(n,2), dtype=int, optional
Indices that point to pairs of atoms, whose contacts are are established in any case. If None, no contacts are artificially switched on.
abstract force_constant(atom_i, atom_j, sq_distance)¶
Get the force constant for the interaction of the given atoms.
ABSTRACT: Override when inheriting.
atom_i, atom_jndarray, shape=(n,), dtype=int
The indices to the first and second atoms in each interacting atom pair.
sq_distancendarray, shape=(n,), dtype=float
The distance between the atoms indicated by atom_i and atom_j.
Implementations of this method do not need to check whether two atoms are within the cutoff distance of the ForceField: The given pairs of atoms are limited to pairs within cutoff distance of
each other. However, if cutoff_distance is None, the atom indices contain the Cartesian product of all atom indices, i.e. each possible combination.
class springcraft.PatchedForceField(force_field, contact_shutdown=None, contact_pair_off=None, contact_pair_on=None, force_constants=None)¶
Bases: ForceField
This force field wraps another force field and applies custom changes to selected pairs of atoms.
The base force field. For all atoms pairs, that are not patched, the force constant from the base force field is taken
contact_shutdownndarray, shape=(n,), dtype=float, optional
Indices that point to atoms, whose contacts to all other atoms are artificially switched off.
contact_pair_offndarray, shape=(n,2), dtype=int, optional
Indices that point to pairs of atoms, whose contacts are artificially switched off.
contact_pair_onndarray, shape=(n,2), dtype=int, optional
Indices that point to pairs of atoms, whose contacts are are artificially established.
force_constantsndarray, shape=(n,), dtype=float, optional
Individual force constants for artificially established contacts. Must be given, if contact_pair_on is set.
class springcraft.InvariantForceField(cutoff_distance)¶
Bases: ForceField
This force field treats every interaction with the same force constant.
The interaction of two atoms is only considered, if the distance between them is smaller or equal to this value.
class springcraft.HinsenForceField(cutoff_distance=None)¶
Bases: ForceField
The Hinsen force field was parametrized using the Amber94 force field for a local energy minimum, with crambin as template. In a strict distance-dependent manner, contacts are subdivided into
nearest-neighbour pairs along the backbone (r < 4 Å) and mid-/far-range pair interactions (r >= 4 Å). Force constants for these interactions are computed with two distinct formulas. 2.9 Å is the
lowest accepted distance between CA atoms. Values below that threshold are set to 2.9 Å.
cutoff_distancefloat, optional
The interaction of two atoms is only considered, if the distance between them is smaller or equal to this value. By default all interactions are included.
K Hinsen et al., “Harmonicity in small proteins.” Chemical Physics 261(1-2): 25-37 (2000).
class springcraft.ParameterFreeForceField(cutoff_distance=None)¶
Bases: ForceField
The “parameter free ANM” (pfENM) method is an extension of the original implementation of the original ANM forcefield with homogenous parametrization from the Jernigan lab. Unlike in other ANMs,
neither distance cutoffs nor distance-dependent spring constants are used. Instead, the residue-pair superelement of the Hessian matrix is weighted by the inverse of the squared distance between
residue pairs.
cutoff_distancefloat, optional
The interaction of two atoms is only considered, if the distance between them is smaller or equal to this value. By default all interactions are included.
L Yanga, G Songa, and R L Jernigan, “Protein elastic network models and the ranges of cooperativity.” PNAS. 106, 30, 12347-12352 (2009).
class springcraft.TabulatedForceField(atoms, bonded, intra_chain, inter_chain, cutoff_distance)¶
Bases: ForceField
This force field uses tabulated force constants for interactions between atoms, based on amino acid type and distance between the atoms.
The distances are separated into bins. A value is within bin[i], if value <= cutoff_distance[i].
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
bonded, intra_chain, inter_chainfloat or ndarray, shape=(k,) or
shape=(20, 20) or shape=(20, 20, k), dtype=float The force constants for interactions between each combination of amino acid type and for each distance bin. The order of amino acids is
alphabetically with respect to the one-letter code, i.e. 'ALA', 'CYS', 'ASP', 'GLU', 'PHE', 'GLY', 'HIS', 'ILE', 'LYS', 'LEU', 'MET', 'ASN', 'PRO', 'GLN', 'ARG', 'SER', 'THR', 'VAL',
'TRP' and 'TYR'. bonded gives values for bonded amino acids, intra_chain gives values for non-bonded interactions within the same peptide chain and inter_chain gives values for non-bonded
interactions for amino acids in different chains. The possible shapes are:
■ Scalar value: Same value for all amino acid types and distances.
■ 1-dim array: Individual value for each distance bin.
■ 2-dim array: Individual value for each pair of amino acid types. Note the alphabetical order shown above.
■ 3-dim array: Individual value for each distance bin and pair of amino acid types.
cutoff_distancefloat or None or ndarray, shape=(k), dtype=float
If no distance dependent values are given for bonded, intra_chain and inter_chain, this parameter accepts a float, that represents the general cutoff distance, below which interactions
between atoms are considered (or None for no cutoff distance). Otherwise, an array of monotonically increasing distance bin edges must be given. The edges represent the right edge of each
bin. All interactions at distances above the last edge are not considered.
natomsint or None
The number of atoms in the model.
interaction_matrixndarray, shape=(n, n, k), dtype=float
Force constants between the atoms in atoms. If the tabulated force constants are distance dependent, k is the number of distance bins. Otherwise, k = 1. This is not a copy, modifications
on this array affect the force field.
static s_enm_10(atoms)¶
The sENM10 forcefield by Dehouck and Mikhailov was parametrized by statisctical analysis of a NMR conformational ensemble dataset. Non-bonded interactions between amino acid species are
parametrized in an amino acid type-specific manner, with a cutoff distance of 1 nm. Bonded interactions are evaluated with \(10 \, RT/\text{Å}^2\), corresponding to the tenfold mean of all
amino acid species interactions at a distance of 3.5 nm.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
Force field tailored to the sENM10 parameter set.
Y Dehouck, A S Mikhailov, “Effective Harmonic Potentials: Insights into the Internal Cooperativity and Sequence-Specificity of Protein Dynamics.” PLOS Computational Biology 9(8): e1003209
static s_enm_13(atoms)¶
The sENM13 forcefield by Dehouck and Mikhailov was parametrized by statisctical analysis of a NMR conformational ensemble dataset. Non-bonded interactions between amino acid species are
parametrized in an amino acid type-specific manner, with a cutoff distance of 1.3 nm. Bonded interactions are evaluated with \(10 \, RT/\text{Å}^2\), corresponding to the tenfold mean of all
amino acid species interactions at a distance of 3.5 nm.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
Force field tailored to the sENM13 parameter set.
Y Dehouck, A S Mikhailov, “Effective Harmonic Potentials: Insights into the Internal Cooperativity and Sequence-Specificity of Protein Dynamics.” PLOS Computational Biology 9(8): e1003209
static d_enm(atoms)¶
The dENM forcefield by Dehouck and Mikhailov was parametrized by statisctical analysis of a NMR conformational ensemble dataset. Non-bonded amino acid interactions are solely assigned
depending on the spatial pair distance, ignorant towards interacting amino acid species. Spatial distances are divided into 27 bins. Bonded interactions are evaluated with \(46.83 \, RT/\text
{Å}^2\), corresponding to the tenfold mean of all amino acid species interactions at a distance of 3.5 nm.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
Force field tailored to the dENM parameter set.
Y Dehouck, A S Mikhailov, “Effective Harmonic Potentials: Insights into the Internal Cooperativity and Sequence-Specificity of Protein Dynamics.” PLOS Computational Biology 9(8): e1003209
static sd_enm(atoms)¶
The sdENM forcefield by Dehouck and Mikhailov was parametrized by statistical analysis of a NMR conformational ensemble dataset. Effective harmonic potentials for non-bonded interactions
between amino acid pairs are evaluated according to interacting amino acid species as well as the spatial distance between them. Spatial distances are divided into 27 bins, with amino acid
specific interaction tables for each distance bin. Bonded interactions are evaluated with \(43.52 \, RT/\text{Å}^2\), corresponding to the tenfold mean of all amino acid species interactions
at a distance of 3.5 nm.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
Force field tailored to the sdENM parameter set.
Y Dehouck, A S Mikhailov, “Effective Harmonic Potentials: Insights into the Internal Cooperativity and Sequence-Specificity of Protein Dynamics.” PLOS Computational Biology 9(8): e1003209
static e_anm(atoms, nonbonded_mean=False)¶
The “extended ANM” (eANM) method discriminates between non-bonded interactions of amino acids within a single polypeptide chain (intrachain) and those present in different chains (interchain)
in a residue-specific manner: the former are described by Miyazawa-Jernigan parameters, the latter by Keskin parameters, which are both derived by mean-force statistical analysis of protein
structures resolved by X-ray crystallography. Bonded interactions are evaluated with \(82 \, RT/\text{Å}^2\). For noncovalent interactions the cut-off is set to 13 Å.
By averaging over all non-bonded residue-specific parameters, an eANM variant with homogenous parametrization of non-bonded interactions can be derived.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
nonbonded_meanBooleam (optional)
If True, the average of nonbonded interaction tables is computed and used for nonbonded interactions, which yields an homogenous, amino acid-species ignorant parametrization of
non-bonded contacts.
Force field tailored to the eANM method.
K Hamacher, J A McCammon, “Computing the Amino Acid Specificity of Fluctuations in Biomolecular Systems.” J Chem. Theory Comput. 2, 3, 873–878 (2006).
S Miyazawa, R L Jernigan, “Residue – Residue Potentials with a Favorable Contact Pair Term and an Unfavorable High Packing Density Term, for Simulation and Threading.” J Mol Biol., 256(3)
623-44 (1996).
O Keskin, I Bahar, R L Jernigan, A Y Badretdinov, O B Ptitsyn, “Empirical solvent-mediated potentials hold for both intra-molecular and inter-molecular inter-residue interactions.” Protein
Science, 7 2578-2586 (1998)
In this variant of the “extended ANM” (eANM) method, non-bonded interactions between amino acids are parametrized in a residue-specific manner using solely Miyazawa-Jernigan (MJ) parameters.
MJ parameters were derived from contact numbers between amino acids in a X-ray structure dataset of globular proteins using the Bethe approximation. Bonded interactions are evaluated with \
(82 \, RT/\text{Å}^2\). For noncovalent interactions the cut-off is set to 13 Å.
Averaging over all non-bonded residue-specific parameters yields a homogenous description of non-bonded interactions.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
nonbonded_meanBooleam (optional)
If True, the average of nonbonded interaction tables is computed and used for nonbonded interactions, which yields an homogenous, amino acid-species ignorant parametrization of
non-bonded contacts.
Force field tailored to the eANM method in the MJ parameterset variant.
S Miyazawa, R L Jernigan, “Residue - Residue Potentials with a Favorable Contact Pair Term and an Unfavorable High Packing Density Term, for Simulation and Threading.” J Mol Biol., 256(3)
623-44 (1996).
@TODO Additional citations
For this variant of the “extended ANM” (eANM), non-bonded interactions between amino-acid pairs are parametrized in a residue-specific manner using Keskin parameters. This parameterset was
derived from contact frequencies between different protein monomers using the methodology established by Miyazawa-Jernigan. Bonded interactions are evaluated with \(82 \, RT/\text{Å}^2\). For
noncovalent interactions, the cut-off is set to 13 Å.
Averaging over all non-bonded residue-specific parameters yields a homogenous description of non-bonded interactions.
atomsAtomArray, shape=(n,)
The atoms in the model. Must contain only CA atoms and only canonic amino acids. CA atoms with the same chain ID and adjacent residue IDs are treated as bonded.
nonbonded_meanBooleam (optional)
If True, the average of nonbonded interaction tables is computed and used for nonbonded interactions, which yields an homogenous, amino acid-species ignorant parametrization of
non-bonded contacts.
Force field tailored to the eANM method in the Keskin parameterset variant.
O Keskin, I Bahar, R L Jernigan, A Y Badretdinov, O B Ptitsyn, “Empirical solvent-mediated potentials hold for both intra-molecular and inter-molecular inter-residue interactions.” Protein
Science, 7 2578-2586 (1998)
springcraft.compute_kirchhoff(coord, force_field, use_cell_list=True)¶
Compute the Kirchhoff matrix for atoms with given coordinates and the chosen force field.
coordndarray, shape=(n,3), dtype=float
The coordinates.
force_fieldForceField, natoms=n
The ForceField that defines the force constants.
use_cell_listbool, optional
If true, a cell list is used to find atoms within cutoff distance instead of checking all pairwise atom distances. This significantly increases the performance for large number of atoms,
but is slower for very small systems. If the force_field does not provide a cutoff, no cell list is used regardless.
kirchhoffndarray, shape=(n,n), dtype=float
The computed Kirchhoff matrix.
pairsndarray, shape=(k,2), dtype=int
Indices for interacting atoms, i.e. atoms within cutoff_distance.
springcraft.compute_hessian(coord, force_field, use_cell_list=True)¶
Compute the Hessian matrix for atoms with given coordinates and the chosen force field.
coordndarray, shape=(n,3), dtype=float
The coordinates.
force_fieldForceField, natoms=n
The ForceField that defines the force constants.
use_cell_listbool, optional
If true, a cell list is used to find atoms within cutoff distance instead of checking all pairwise atom distances. This significantly increases the performance for large number of atoms,
but is slower for very small systems. If the force_field does not provide a cutoff, no cell list is used regardless.
hessianndarray, shape=(n*3,n*3), dtype=float
The computed Hessian matrix. Each dimension is partitioned in the form [x1, y1, z1, ... xn, yn, zn].
pairsndarray, shape=(k,2), dtype=int
Indices for interacting atoms, i.e. atoms within cutoff_distance. | {"url":"https://springcraft.biotite-python.org/apidoc.html","timestamp":"2024-11-14T14:50:02Z","content_type":"text/html","content_length":"85594","record_id":"<urn:uuid:7c1ed468-1635-4b2a-bb6e-42d83b1f872c>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00072.warc.gz"} |
What is: Variance Ratio
What is Variance Ratio?
The Variance Ratio is a statistical measure that compares the variance of two or more datasets to determine if they exhibit similar variability. It is particularly useful in the fields of statistics,
data analysis, and data science, where understanding the dispersion of data points is crucial for making informed decisions. By calculating the Variance Ratio, analysts can assess the relative
stability of different datasets, which can be essential for hypothesis testing and model validation.
Understanding Variance
Variance itself is a measure of how far a set of numbers is spread out from their average value. In mathematical terms, it is calculated as the average of the squared differences from the mean. A
higher variance indicates that the data points are more spread out, while a lower variance suggests that they are closer to the mean. The Variance Ratio leverages this concept to compare the
variances of multiple datasets, providing insights into their relative variability.
Calculating Variance Ratio
The calculation of the Variance Ratio involves dividing the variance of one dataset by the variance of another. This ratio can be expressed as VR = Var1 / Var2, where Var1 is the variance of the
first dataset and Var2 is the variance of the second dataset. A Variance Ratio close to 1 indicates that the two datasets have similar variability, while a ratio significantly greater or less than 1
suggests a disparity in their variances.
Applications of Variance Ratio
The Variance Ratio is widely used in various applications, including finance, quality control, and experimental research. In finance, it can help investors compare the volatility of different assets,
aiding in portfolio diversification. In quality control, manufacturers can use the Variance Ratio to assess the consistency of production processes. In experimental research, it assists in evaluating
the reliability of different measurement methods or treatments.
Variance Ratio in Hypothesis Testing
In hypothesis testing, the Variance Ratio plays a critical role, particularly in ANOVA (Analysis of Variance). ANOVA tests the hypothesis that the means of several groups are equal, and the Variance
Ratio is used to determine whether the observed variances among the groups are statistically significant. A significant Variance Ratio indicates that at least one group differs from the others,
prompting further investigation.
Limitations of Variance Ratio
While the Variance Ratio is a powerful tool, it has its limitations. It assumes that the datasets being compared are independent and normally distributed. If these assumptions are violated, the
results may be misleading. Additionally, the Variance Ratio does not provide information about the direction of the difference in variances, only the magnitude of the difference.
Interpreting Variance Ratio Results
Interpreting the results of the Variance Ratio requires careful consideration of the context. A ratio significantly greater than 1 suggests that the first dataset has greater variability, while a
ratio less than 1 indicates that the second dataset is more variable. Analysts must also consider the sample sizes and the potential impact of outliers on the variance calculations, as these factors
can skew the results.
Variance Ratio vs. Other Statistical Measures
The Variance Ratio is often compared to other statistical measures, such as the F-test and the Levene’s test. While the F-test also assesses the equality of variances, it is more sensitive to
deviations from normality. Levene’s test, on the other hand, is robust against non-normal distributions and is preferred when the assumption of normality cannot be met. Understanding these
differences is essential for selecting the appropriate statistical method for a given analysis.
Conclusion on Variance Ratio Usage
In summary, the Variance Ratio is a vital statistical tool that provides insights into the relative variability of datasets. Its applications span various fields, making it an essential concept for
statisticians, data analysts, and data scientists. By understanding how to calculate and interpret the Variance Ratio, professionals can make more informed decisions based on the variability of their | {"url":"https://statisticseasily.com/glossario/what-is-variance-ratio-explained-in-detail/","timestamp":"2024-11-06T11:38:55Z","content_type":"text/html","content_length":"137856","record_id":"<urn:uuid:866f67fb-baea-4667-b533-9ca8ac756e38>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00229.warc.gz"} |
Software but Quantum - Part 2
Well hello there.
We'll be picking up on the first part of this series, and get into the fundamental idea of superposition, which is super important to learn when it comes to Quantum Computers solving real-life
Recall in the last post, we talked about Wave-Particle Duality (WPD) and Wavefunctions. These two are different things. The Wave that light behaves as is real, but the Wavefunction is a mathematical
idea that we use to determine where the electron might be in an atom.
Now, a question you might have is why we can't just say where the electron will be. Why are we using a Wavefunction to tell us where it probably will be?
This comes down to something called the Heisenberg Uncertainty Principle (HUP). The HUP tells us that it's impossible to accurately know the position and speed of a particle at the same time.
For example, we know that an electron behaves as a wave, because of the Double-Slit Experiment (read the last post if you don't know what this is). If the electron behaves as a wave (the real kind),
the probability that we'll find the electron is highest where the wave has peaks.
This might seem kind of arbitrary but it makes sense if you think about it. A wave represents energy travelling through space. The peaks represent where the highest amount of energy will occur. We
know an electron or photon or any particle for that matter has energy. So that's were we'll probably find the particle.
So the HUP tells us that if we look at a wave, we'd know the speed of the particle
Let v be the velocity
Let λ be the wavelength
Let p be the momentum of the particle
Let h be Planck's constant
p = h / λ
p = mv
v = p / m
v = h / λm
So, we don't know where the particle is (due to the multiple peaks) but we do know the speed of the particle.
Alright, let's say we do look at the wave and try to figure out where the particle is. This is where things start to get weird. The wavefunction collapses entirely, leaving us with a single position
of a particle.
For example in the below picture, you know where the rollercoaster is but not how fast it's going.
This is called Superposition and Wavefunction Collapse (WFC). Nobody really knows how WFC works, or how the Wave knows where to put a particle when someone looks at it. This is also the idea behind
Erwin Schrodinger's thought experiment involving a cat.
The next part of this series will talk about how superposition let's us do math beyond the regular bits of 1 and 0 that we use in classical computers.
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Contrastive Model: Instance-Instance
It was discovered that the success of [[mutual information based contrastive learning]] Contrastive Model: Context-Instance In contrastive methods, we can manipulate the data to create data entries
and infer the changes using a model. These methods are models that “predict relative position”1. Common tricks are shuffling image sections like jigsaw, and rotate the image. We can also adjust the
model to discriminate the similarities and differences. For example, to generate contrast, we can also use [[Mutual Information]] Mutual Information Mutual information is defined as $$ I(X;Y) = \
mathbb E_{p_{XY}} \ln … is more related to the encoder architecture and the negative sampling strategy^1. Instance-instance method is more direct in solving the contrastive problem. It take the
instance itself directly and make comparisons for discrimination.
Cluster Discrimination
Illustration of Deep Cluster based on Liu2020.
Instance Discrimination
There are two interesting models under the umbrella of instance discrimination,
Illustration of MoCo based on Liu2020.
Illustration of SimCLR based on Liu2020.
Planted: by L Ma;
L Ma (2021). 'Contrastive Model: Instance-Instance', Datumorphism, 08 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/contrastive-models/instance-instance/. | {"url":"https://datumorphism.leima.is/wiki/machine-learning/contrastive-models/instance-instance/?ref=footer","timestamp":"2024-11-02T13:54:35Z","content_type":"text/html","content_length":"115118","record_id":"<urn:uuid:55d3d167-9ea5-47e7-8dd9-ec308918de04>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00222.warc.gz"} |
taking business calc and prin of finance class should i buy calculator in body
i simply want superior advantage in the class and im taking it strictly online so no one will interfere
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Sharpe index interpretation
The Sharpe ratio is simply the return per unit of risk (represented by variability). The higher the Sharpe ratio, the better the combined performance of "risk" and The Sharpe ratio is simply the
return per unit of risk (represented by variability). In the classic case, the unit of risk is the standard deviation of the returns. The Sharpe ratio is a measure of risk-adjusted return. It
describes how much excess return you receive for the volatility of holding a riskier asset.
In finance, the Sharpe ratio measures the performance of an investment compared to a risk-free Because it is a dimensionless ratio, laypeople find it difficult to interpret Sharpe ratios of different
investments. For example, how much better is May 17, 2019 The Sharpe ratio is calculated by subtracting the risk-free rate from the return of the portfolio and dividing that result by the standard
deviation of Jun 21, 2019 Most finance people understand how to calculate the Sharpe ratio and what it represents. The ratio describes how much excess return you The Sharpe ratio uses standard
deviation to measure a fund's risk-adjusted returns. The higher a fund's Sharpe ratio, the better a fund's returns have been William Sharpe devised the Sharpe ratio in 1966 to measure this risk/
return relationship, and it has been one of the most-used investment ratios ever since. Here, The Sharpe Ratio formula is calculated by dividing the difference of the best available risk free rate
of return and the average rate of return by the standard
Jun 21, 2019 Most finance people understand how to calculate the Sharpe ratio and what it represents. The ratio describes how much excess return you
Description: Sharpe ratio is a measure of excess portfolio return over the risk-free rate relative to its standard deviation. Normally, the 90-day Treasury bill rate is Define d, the differential
return, as: Let d-bar be the expected value of d and sigmad be the predicted standard deviation of d. The ex ante Sharpe Jun 6, 2019 The higher the Sharpe ratio is, the more return the investor is
getting per unit of risk. The lower the Sharpe ratio is, the more risk the investor is In this lesson, you will learn the definition of a measure for calculating risk- adjusted return called Sharpe
Ratio, its formula, examples, and its
Sharpe Ratio Definition. This online Sharpe Ratio Calculator makes it ultra easy to calculate the Sharpe Ratio. The Sharpe Ratio is a commonly used investment ratio that is often used to measure the
added performance that a fund manager is said to account for.
Nov 27, 2019 Sharpe Ratio is used to evaluate the risk-adjusted performance of a mutual fund. Basically, this ratio tells an investor how much extra return he Apr 1, 2015 he Sharpe ratio is one of
the most frequently used risk adjusted ratios, and calculates return per unit of risk as measured by volatility.
The Sharpe ratio is simply the return per unit of risk (represented by variability). The higher the Sharpe ratio, the better the combined performance of "risk" and
Moreover, in case of negative returns, the m2 measure continues to hold its meaning, while the Sharpe ratio very hard to interpret. M2 measure calculation. The Aug 4, 2016 There's no easy solution
for interpretation, but it's useful to look at SRs in context. The chart above is one example. But it would also be helpful to The Sharpe ratio is simply the return per unit of risk (represented by
variability). The higher the Sharpe ratio, the better the combined performance of "risk" and The Sharpe ratio is simply the return per unit of risk (represented by variability). In the classic case,
the unit of risk is the standard deviation of the returns. The Sharpe ratio is a measure of risk-adjusted return. It describes how much excess return you receive for the volatility of holding a
riskier asset.
Jul 24, 2013 The Sharpe ratio definition (or reward to variability ratio) is the excess The .8 can be interpreted as meaning that for every unit of risk that you
The Sharpe Ratio (or Sharpe Index) is commonly used to gauge the performance of an investment by adjusting for its risk., which adjusts return with the standard deviation of the portfolio, the
Treynor Ratio uses the Portfolio Beta, which is a measure of systematic risk. Definition: The Sharpe ratio is an investment measurement that is used to calculate the average return beyond the risk
free rate of volatility per unit. In other words, it’s a calculation that measures the actual return of an investment adjusted for the riskiness of the investment.
The Sharpe ratio is a measure of risk-adjusted return. It describes how much excess return you receive for the volatility of holding a riskier asset. The Sharpe ratio can also help explain whether a
portfolio's excess returns are due to smart investment decisions or a result of too much risk. Although one portfolio or fund can enjoy higher returns than its peers, it is only a good investment if
those higher returns do not come with an excess of additional risk. In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the
performance of an investment (e.g., a security or portfolio) compared to a risk-free asset, after adjusting for its risk. The Sharpe Ratio (or Sharpe Index) is commonly used to gauge the performance
of an investment by adjusting for its risk., which adjusts return with the standard deviation of the portfolio, the Treynor Ratio uses the Portfolio Beta, which is a measure of systematic risk. | {"url":"https://digoptionewsomez.netlify.app/tonsall25071cymu/sharpe-index-interpretation-468.html","timestamp":"2024-11-13T05:28:13Z","content_type":"text/html","content_length":"29868","record_id":"<urn:uuid:1167b5c4-bd55-4f65-9f8a-3ef0f5e1c3e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00210.warc.gz"} |
Principal stress vs Bending stress - Difference [Explained]
Principal stress vs Bending stress – Difference [Explained]
The key difference between principal stress vs bending stress is that principal stress indicates maximum and minimum normal stress on the object while bending stress indicates stress arises due to
the bending load.
Before moving on to our main topic, let’s have a quick look at each of them.
In this article, we’re going to discuss:
• Principal stress:
• Bending stress:
• Principal stress vs Bending stress:
• FAQ:
Principal stress:
Principal stress is the stress acting on the principal plane whereas the principal plane is the plane that possesses only normal stress and no shear stress.
Look at the member shown in below Fig-A. It experiences axial stresses in x and y direction (`\sigma_{x}, \sigma_{y}`) and complementary shear stresses (`\tau_{xy}, \tau_{yx}`).
Each plane in above figure-A is experiencing normal force and shear force. Other than these planes, the member also has a number of planes.
As shown in figure-B, a specific plane inclined at angle `\theta_{P}` has no shear stress and has only normal stress acting on it. Such a plane is known as a principal plane and the normal stress on
that plane is known as the principal stress.
The `sigma_{1}` and `sigma_{2}` in the above figure are the major and minor principal stresses respectively.
Bending stress:
Bending stress is the internal resistance developed by the object to resist the deformation caused by the bending moment.
The above figure-A shows the cantilever beam subjected to the bending load and fig-B indicates the distribution of bending stress over the cross-section at a location a – a.
The above figure-B indicates that the bending stress never gets uniformly distributed over the cross-section. It is negligible at the neutral axis position and maximum at the upper and lower
The bending stresses on either side of the neutral axis are opposite in nature. It means that if the bending stress above the neutral axis is tensile in nature then the bending stress below the
neutral axis is compressive.
For the cross-section shown in above figure-B, the bending stress `\sigma_{b}` is given by,
`\sigma_{b} = \frac{M}{I} \times y`
M = Bending moment at section a-a
I = Moment of inertia about the neutral axis (N.A.)
y = Distance between N.A. and location where `\sigma_{b}` has to find
Principal stress vs Bending stress:
Sr. No. Principal stress Bending stress
1 The normal stresses acting on the principal planes are known as principal stresses. It is stress developed by an object to resist deformation due to bending load.
2 It arises due to the application of axial forces and complementary shear forces. It arises due to the application of bending moment.
3 It is used as a criterion for theories of failure. It is not used in theories of failure.
4 It is useful for the design of brittle materials. It is used for the analysis of the object subjected to the bending load.
5 It is calculated at the principal planes. It is calculated at the plane perpendicular to the neutral axis of the beam.
The above table indicates the difference between the principal stress and bending stress.
Why principal stress and bending stress are important?
The principal stress is an important failure criterion for the design of brittle materials, whereas the bending stress is important in the design of a member subjected to a bending moment.
Related articles:
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Pivoting transposes rows to columns in order to group and aggregate data display. This is useful, for example, in datasets where you want to use just a subset of values from a column dimension, view
different aggregations of the same column and the same aggregation of different columns, or simply to select more than one measure to easily group and compare values. Pivot tables can be generated
from samples and by querying against a complete dataset. The pivot inspector is interactive, so once a pivot sheet is created you can test and update your results iteratively.
For example, in a workbook that includes height, weight, age, gender, and state of residence for a set of individuals living in the United States, you could choose just certain states as the first
column grouping, gender as the second column aggregation, then weight as the measure with average as the operator to yield the average weight by gender for the selected states.
Depending on the measure being used, supported aggregations are:
• ANY - returns any value in the group
• AVG - average of the values in a group for numeric values
• COUNT - number of elements in a group
• MAX - maximum of the values in a group for numeric values
• MIN - minimum of the values in a group for numeric values
• STDEV - statistical standard deviation inside the group for numeric values
• SUM - sum of the values in a group for numeric values
• VAR - statistical variance in the group for numeric values
First and last are special functions that compute the first or last element inside a group given a specific order. For example, given the grouping by state and gender, you could select weight as
measure with first as the operator and height as the OrderBy criteria to yield the weight of the tallest person per state by gender.
Creating a Pivot
To create a pivot:
1. In the active sheet, select Pivot from the Edit menu, or click the Pivot icon on the toolbar.
2. A new sheet will open in the pivot view with an Explorer at the top for visual display of the results, and a Preview table at the bottom.
3. In the Pivot Sheet dialog to the right, select one or more Columns on which to pivot and aggregate. Aggregation is hierarchical in the order of columns selected.
4. Select a column in the first Columns field then click to select the values by which you want to group data. Select All is an option but will not always yield useful results. It is wiser to choose
to aggregate by values with lower cardinality so the results will be comprehensible.
5. In the Measure fields, select the data characteristics you want to view. Only the columns other than those already selected will be available.
You must also select an operation. Functions are filtered depending on the data type selected as the measure. FIRST and LAST require an additional OrderBy value.
6. Click Pivot to process.
The Explorer will display the pivot aggregation hierarchically with color coding for columns (purple) and measure (orange).
The Preview contains the sheet generated by the pivot transformation. The sheet column headers display the actual yielded pivot aggregation and values.
Note that these panes scroll independently.
7. Once a Pivot sheet is created you can refine the results dynamically by changing values or introducing an additional pivot by Rows.
The column value you choose in the Rows field becomes a row label in the resulting pivot table. Spitting by rows lets you view and categorize data from an independent angle.
Typically you would split by rows to view characteristics with fewer values, or to simplify very large datasets with numerous enough values to populate your columns with a high density of | {"url":"https://datameer.atlassian.net/wiki/spaces/DAS110/pages/9645138006/Pivoting+Data?atl_f=content-tree","timestamp":"2024-11-13T11:20:30Z","content_type":"text/html","content_length":"930606","record_id":"<urn:uuid:e7449a75-8c5b-4d7a-ba3f-e2774e738415>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00066.warc.gz"} |
Buoyant Force Calculator - Weight of Displaced Liquid
This online tool allows you to calculate the buoyant force and weight of the liquid displaced by a submerged object in water.
Using the Calculator
To utilize the Buoyant Force Calculator, simply enter the volume of the body submerged in water (in cubic meters), the density of the liquid (in kilograms per cubic meter), and optionally adjust the
acceleration of gravity to match different environmental conditions. The calculator provides default values for convenience, set to 0.01 cubic meters for the volume, 1000 kilograms per cubic meter
for the liquid density (water), and Earth's gravity as the default acceleration.
Once you've entered the required values, click the "Calculate" button. The calculator will compute two important results:
• Buoyant Force: This is the upward force exerted by the liquid on the submerged object. It is expressed in newtons (N) and represents the force necessary to counteract the weight of the liquid
displaced by the object.
• Weight of Displaced Liquid: This is the weight of the liquid displaced by the submerged object. It is calculated by multiplying the volume of the body submerged in water by the density of the
liquid. The weight is given in kilograms (kg).
These outputs provide valuable insights into the forces at play and help you understand the behavior of submerged objects in fluid environments. For more detailed information and theoretical
background, refer to the "Understanding Buoyancy with Archimedes' Principle" section below.
Understanding Buoyancy with Archimedes' Principle
At the heart of this calculator lies Archimedes' principle, which states that the buoyant force experienced by a submerged object is equal to the weight of the liquid displaced by the object.
Mathematically, it can be represented as:
$F_A=\rho_w Vg$
When an object is immersed in a liquid, it experiences two opposing forces: the gravitational force acting downward (represented by $F_g=mg$ or $F_g=\rho Vg$) and the buoyant force acting upward. The
behavior of the object depends on its density relative to the density of the liquid. If the object's density is greater, it will sink; otherwise, it will be pushed out of the liquid until the two
forces balance. The portion of the object submerged in the liquid is proportional to the ratio of the liquid density to the object's density.
Moreover, the discrepancy between the actual weight and the apparent weight of the object is equal to the weight of the liquid displaced. This weight can be calculated by multiplying the volume of
the submerged body by the density of the liquid.
Delve into the fascinating world of buoyancy and uncover the secrets of submerged objects with the Buoyant Force Calculator - Weight of Displaced Liquid. Gain a deeper understanding of the forces at
play and enhance your knowledge of fluid dynamics.
Similar calculators
PLANETCALC, Buoyant Force Calculator - Weight of Displaced Liquid | {"url":"https://embed.planetcalc.com/975/?thanks=1","timestamp":"2024-11-07T23:12:14Z","content_type":"text/html","content_length":"36371","record_id":"<urn:uuid:d7e20a74-e2c0-4099-a426-7194689ab0c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00078.warc.gz"} |
This blog is about the Cycle of Nine implemented in the Digital Root or Modulus 9-Function. The Digital Root generates many Patterns that were used in Ancient Architectures.
One of the most important Digital Root Patterns is the Vedic Square. It is the Digital Root of the Multiplication Table of the numbers 1 to 9.
This Table contains the Harmonics of the Numbers 1 to 9. These Harmonics are highly related to the Harmonic Pattern behind the Cycles in our Universe.
The first part of this Blog is about the Digital Root. It contains the patterns that are behind the Cycle of Nine.
This part is very technical but it makes it possible to show that there is a deep structure behind the Modulus-9.
This pattern has to do with just two numbers, 2 and 3. They generate the Spirals of Expansion and Compression of our Universe.
2 and 3 and their Sum 5 are also the Numbers behind the Harmonics of our Universe.
The Second Part is about the Vedic Square. It is called the Vedic Square because this Square is one of the most important tools in Ancient Vedic Mathematics.
Vedic Mathematics was used in many Ancient Cultures (China, Egypt, Greece) with different names. The Chinese art of Feng Shui was called Vaastu Shastra in India.
Pythagoras, trained in Egypt (Heliopolis), used the same principles and used the same Patterns the Ancient Vedic Scientists were using.
The last part is about the Game of Chess. This game is just like many other Ancient Games a Simulator of the Game of the Universe.
This blog contains many links to other Blogs and Resources on the Internet. These references make it possible to dig deeper into this fascinating subject.
About the Digital Root
When you divide a number X by a number N the Remainder of the division is called X Modulus N. 22 mod 7 = 1 because 22 = 3×7 + 1.
The Modulus-function N maps the Set of the Natural Numbers to the Numbers 0, 1, 2, ….,N-1.
One of the most famous and ancient Modulus-functions is called the Digital Root. The Digital Root is the Modulus 9 function.
Because 10 mod 9 = 1 every Power of 10 has a Modulus 9 of 1. Therefore (a10**X+ b10**Y+…) mod 9 = a + b, the Sum of the Digits of the Number. 62 mod 9 = 6+2 = 8.
Digital Roots have been recorded for thousands of years, formalized by Pythagoras in 530BC and even earlier in Indian Vedic Mathematics (Vaastu Shastra).
Digital Roots are used in Numerology. In Numerology Numbers have a Meaning.
In Gematria Letters and Words are transformed into Numbers which have a meaning.
In Ancient Languages like Hebrew Letters are also Numbers. Numerologists believe that Words with the same Digital Root have the same Meaning.
The numbers 0 to 9 of the Digital Root are the Points of the Tetraktys of Pythagoras.
The Modulus 9 pattern contains 2 number groups (3,6, 9) and (1,2,4, 5,7,8).
Later we will see that the last group contains 2 subgroups (1,4,7) and (2,5,8). Together with (3,6,9) we can map these 3 Triangels on the Modulus 9 Circle.
4 is the Middle of 1+7=8, 5 is the Middle of 2+8=10 =1 and 6 is the Middle of 3+9=12=3. 5 is also the Middle of the Middle.
The group (1,2,4,5,7,8 ) is called the Ring Z/9 in Mathematics. Z/9 is isomorphic with the Sequence 2**N mod 9 where N is positive and negative. The sequence 1,2,4,8,16(7),32(5),64(1),128 (2),256
(4),… repeats itself until infinity.
This Sequence is the Expansion and Compression Pattern of the Number 2.
The Ring Z/9 is part of the Tetraktys and forms a Hexagram. This Hexagram is a 2D-projection of the Cube of Space. When we combine the (3,6,9)-pattern with the Hexagon a (4×4) Triangle is created.
The number 2 is the Container, the Cube, inside the Tetraktys. That is the Reason why the Second letter in the Hebrew Alphabet Beth means Vessel or Container.
(3,6,9) is a Triangular Cycle that repeats itself until Infinity. The Number 3, the Trinity, is the Mover of the Container of 2. This Rotation moves With and Against the Clock.
This is the reason why the 3th Letter of the Hebrew Alphabet, Gimel, means Camel. The Camel of Gimel carries the Water into the 2 Containers of Beth.
The Number-2-pattern contains 3 Binary Groups (called Polar Pairs) with a Sum of Nine (1,8), (2,7), (4,5). The Number-3-Pattern contains 2 Polar Pairs (3,6) and (0,9). The Polar Pairs represent the
Lines of the Tetraktys.
(0,9) maps unto Itself and represents The Beginning and The End, The Now. (0,9) is a Point and a Line.
The Polar Pairs of the Z/9 create a Cyclic Pattern that contains two Squares, (1,2,4,0) and (5,7,8,0). Both of them Share the Zero, The Void.
The Sum of the Opposite Numbers of the Z/9, (4,8 = 12=3), (1,5 =6 ), (2,7=9) of the Tetraktys shows the 3,6,9-pattern again.
There are 8 Ternary Groups ((1,5,9), (1,6,8), (2,6,7), (2,5,8),(2,4,9),(3,4,8),(3,5,7),(4,5,6)) with a Sum of 15. This Ternary Group represents Triangels. All of them are part of the famous Lo Shu
3×3 Magic Square.
When we use the number 3 as a generator 3 Triangles are created (1,4,7), (2,5,8) and (3,6,9).
The 3 Triangles move With and Against the Clock ((1,4,7) and (7,4,1)).
It takes 3 rotations to get every Triangle back to its original position. (1,4,7) becomes (7,1,4) and (4,7,1). This means that there are 6 permutations of every Triangle.
Every addition of two Triangels produces another Triangle. An Example: (1,4,7) + (2,5,8) = (3,9,6).
When we create a Matrix to find all the combinations a new group of 9 transformations ((1,1,1),(2,2,2),(3,3,3),(4,4,4),(5,5,5),(6,6,6),(7,7,7),(8,8,8),(9,9,9)) appears. They are the Triangels that
are a Line and a Point. An Example: (1,4,7) + (1,1,1)=(2,5,8).
There are now (18 +9=27) x27 = 729=3**6 = 9**3 possibilities.
The same 27×27 Matrix appears when we Multiply the 3 Triangels. An Example: (5,8,2)x(5,8,2)= (25,64,4)= (7,1,4) and (3,6,9)x(5,8,2)=(15,48,18)= (6,3,9).
Another interesting patterns becomes visible when we look at the Opposite Numbers of 3 Triangels (1,5), (2,6), (3,7),(4,9) en (5,9) in the Picture above. They recreate the Triangels. An Example:
(5+9=5, 2+6=8 ,8+3=11=2).
About the Digital Root of the Golden Mean
The 27×27 Matrix pattern also emerges out of 24 repeating numbers (1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9) of the Digital Root of the Fibonacci Sequence (The Golden Ratio).
this solution gives the densest
lattice packing of spheres in 24 dimensio
When we group the Golden Ratio pattern in 2′s (2×12) the Polar Pairs appear. The 12 pattern has a Sum of 108 = 0 Modulus 9. 108 and 24 are related to the Gayatri Mantra.
When we group the pattern of 24 numbers (3×8) of the Golden Ration into Trinities the Triangle Pattern appears again.
1 2 3 4 -3 -2 -1 -4 (Pattern-number)
1 1 2 3 (7) 5 8 4 3 (2)
7 1 8 9 (7) 8 8 7 6 (2)
4 1 5 6 (7) 2 8 1 9 (2)
The Pattern of the Pattern is (1,2,3,4,-3,-2,-1,-4). The last part of the Pattern (-3,-2,-1,-4) can be transformed into the first part (1,2,3,4) by adding 4.
The Digital Sum of the first 3×4 numbers is 7 and the Digital Sum of the last 3×4 numbers is 2.
When we rearange the 24 cycle in 6 groups of 4 digits another pattern shows itself: (1,4,8,5), (1,3,8,6), (2,7,2,7),(3,1,6,8), (5,8,4,1), (8,9,1,9).
When we combine all the different rotations of the 3 Triangels a Cyclic Flow Pattern appears that looks like the Jitterbug of Buckminster Fuller.
The Jitterbug is a 3D projection of the 4D 24-Cell (again 24!) also called the Hyperdiamond.
The 24-cell is self-dual and is the regular polytope with no analogue among the five Platonic solids of 3-space.
About the Vedic Square
One of the Simple Structures of Numbers that contains a lot of patterns is the Vedic Square. The Vedic Square was called the Eight Mansions in China. The Vedic Square is the Digital Root of the
Multiplication Table of the numbers 1 to 9.
The Multiplication Table is a subset of the 27×27 Matrix of the 3 Triangels.
The Multiplication Table contains the Harmonics of the Numbers 1 to 9.The Vedic Square was used to build the Pyramids, create the Chinese I Ching, the Game of Chess, Dante Alighieri used it to
structure his trilogy La Divina Commedia, the Sistine Chapel was build and the frescoes and symbols were arranged according to its concepts and the first chapter of Genesis was written and imbued
with its numerous concepts graphic images.
Scholars and Artists discovered that the various lines of the Vedic Square could be used to direct a design. By selecting a line of numbers, and using a constant angle of rotation, various designs
could be produced. These designs are visible in abstract Islamic Art.
The Vedic Square is a Symmetrical Structure because AxB=BxA. This is called the Associative Property of Multiplication. The Square is a combination of Two Triangels and contains 45 distinctive
The Vedic Square repeats itself until infinity when you extend the Square to a NxN Square.
The number pattern of the diagonal of the Vedic Square, 1,4,9,7,7,9,4,1,9, is the Digital Root Pattern of the Square Roots. This patterns repeats itself until Infinity.
The Vedic Square contains the 5 Polar Pairs, the 8 Lo Shu Ternary Groups and the 3 Trinity Patterns ((1,4,7), (2,5,8), (3,6,9). It also contains the Star of David, The Zodiac, the Tree of Life and
many other Mystic Patterns.
It is possible to transform the Vedic Square to the Lo Shu Magic Square.
The patterns of the Vedic Square Rotate. The End of a Horizontal and a Vertical Pattern connects with the Beginning of the Pattern. This means that the Vedic Square is a Torus.
This Torus is called the Rodin Torus. The Rodin Torus is a Coil that produces a Uniform Electro-Magnetic Field.
The 3-6-9 and 6-3-9 Cycle in the Vedic Square can be thought of as Clockwise and Counter-Clockwise, or as Electricity and Magnetism. They are transport-channels.
The ((3,6,9),(6,3,9)-Matrix divides the Vedic Square in 9 2×2 Squares.
The 9 2×2 Squares have a Sum of 9,18 and 27 which is 1×9,2×9 and 3×9. If we leave out the (3,6,9)-Matrix and divide by 9, a 3×3 matrix results with 1,2,3 on the Outside and a Cross of 2′s in the
middle. This 3×3 matrix shows the Expansion of the 2 into the (1,2,3).
The Rows and Colums of Ring Z/9 add up to 45. The Rows and Colums of the Number 3-Pattern add up to 54 which is a Mirror of 45. The (4,5)-pattern generates the Star of David and the Zodiac.
About Indian Vastu Science
The Game of Chess originated in India. It was passed on to the medieval West through the intermediary of the Persians and the Arabs.
The form of the Chess-Board corresponds to the Vastu-Mandala, the 9×9 diagram which also constitutes the basic lay-out of a temple or a city.
Hindu mythology has it that Vaastu Purusha was born of Lord Shiva’s sweat when he fought the deadly demon Andhakasura.
Vaastu Purusha himself became uncontrollable and destructive and the heavenly gods finally subjugated him and brought him down on earth with face down, with his face in the Northeast and his feet in
the Southwest.
45 deities stayed there, 32 of them in the outer enclosure and 13 of them in the inner enclosure holding him in place at various points or locations on his body.
32 =64/2 and the Number of the 32 Paths of Wisdom of the oldest book of Hebrew Mysticms the Sepher Yesirah (the Book of Formation or Book of Creation, ספר יצירה).
64 is the Number of the I Tjing. 45 (5×9) is the Sum of the Lo Shu Magic Square and the Number of the Vedic Square.
All these Mystic Structures come from the same Source and are different Views on the same Pattern, the Tetraktys, the Triangular Numbers created by the Meru Prastara or Sri Yantra also known as the
Pascal Triangle.
The Vastu Mandala is an expansion of a Point (the Bindu) into the Line(2), The Trinity (3) and the Rotating (With the Clock and Against the Clock) and Expanding Square (4), represented by the Symbol
of the Swastika. The Swastika is a Fractal Generating Pattern.
Every Point is a generator from which the Swastika-pattern generates a new Swastika. The 2×2 Square is transformed by the Swastika Pattern into the 4×4 and the 8×8 Square.
As you can see the Vastu Jain Symbol is an Indian Version of the Tetraktys of Pythagoras.
The Swastika contains the Four Points of the last line of the Tetraktys that are related to the Tethahedron.
About the Game of Chess
The Chess-Board symbolizes the Unfolding of Space by the Number-2-pattern and it synthesizes the Complementary Cycles of Sun and Moon.
The number 64, the sum of the Black & White (Yin/Yang) Squares on the Chess-board, is a divisor of the number 25920 (25920/64=405, 25920/9= 2880/9=320/5=72), which measures the Precession of
the Equinoxes.
The Polar Pairs in the Modulo 9 Pattern are expressions of the Planets.
(1,8), the Castles, relates to the Planet Mars.
(2,7), the Bishops, relates to the Planet Venus. Venus is the Ruler of the Heart and the (2,7) is situated in the Middle of the Vedic Square.
When viewed from the Earth, the Planet Venus inscribes a near perfect five-pointed star (pentagram) around the sun every eight years. The points of a five-pointed star (pentagram) touch the circle of
a pentacle every 72 degrees. Likewise, many in Islam expect 72 virgins in heaven.
A full 360 degrees of procession takes 25,920 years, which is also seventy-two (72) 360-year cycles.
(3,6), the Knights relates to the Planet of the Messenger, Mercury. Mercury is Hermes, the Messenger God, with winged sandals. The moves of the Knights create a pattern that looks like the Swastika.
The (3,6)-number-lines are Transport-Channels (Gimel) as you can see in the Vedic Square and the Rodin Torus. The planet Mercury traces a Hexagram during its movement around the Zodiac.
(0,9) is the Planet Jupiter, the Ruler of Modulus 9 who determinates the Rules of the Game. (0,9) is the Beginning and the End of the Game and is the cause of the Rotation of the Swastika related to
the (3,6.9)-pattern.
The numbers 4 and 5 are the Moon (Queen) and the Sun (King). The Moon moves the quickest of all the planets, so does the Queen on the chessboard.
The Number 5 of the King is the Center of the 3×3 Lo Shu Magic Square and the Center of the Tetraktys.
The 8 Pawns represent the number 2 and are connected to the Planet Saturn, the 2nd Son of the Central Sun and the Trinity (1+ 2 = 3). The Pawns start to move with 2 steps and later move 1 step. The
2 is the Center (The Son of the Sun) of the Trinity.
The 2 is also the Generator of the Expansion Pattern and the Polar Companion of the 7, the Center of the 3D-version of the Square, the Cube of Space.
The Pawn (2, Saturn) promotes into a Queen (Moon, 2×2) when he has reached the Other Side.
About the Multiplication Table of 9
About the Tetraktys and the Lo Shu
About the Harmonics of the Universe
About Harmonics and Entrainment
About the Jitterbug of Buckminster Fuller
A Simulation of the Jitterbug Pattern (CUBIC WONDER)
About Plato and the Sri Yantra | {"url":"http://hans.wyrdweb.eu/tag/gematria/","timestamp":"2024-11-05T09:56:32Z","content_type":"application/xhtml+xml","content_length":"43689","record_id":"<urn:uuid:688d8867-6252-4e83-b867-0859bfcf82aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00600.warc.gz"} |
Multi-Faceted Constructs in Abnormal Psychology: Implications of the Bifactor S - 1 Model for Individual Clinical Assessment
Like in many other areas of psychological assessment the bifactor model has gained increasing interest in clinical psychology in recent years (Markon
). Although the model is more than 80 years old (Holzinger and Swineford
) it has been leading a niche existence for a long time and was rediscoverd only about ten years ago (Eid et al.
; Reise
). Over the last decade, however, it has become very popular and is widely used for analyzing multidimensional data. Its application is mainly based on the idea that dimensions that are correlated
have something in common and this common part should be represented by a general factor. According to the basic idea of the bifactor model, an observed variable (e.g., clinical scale) can be
decomposed into three parts: (1) A part shared with all other scales (represented by the general factor), (2) and a part that is only shared with the other scales representing the same facet of a
construct, but not the scales representing other facets (represented by a facet-specific factor), and (3) measurement error. Based on this decomposition, several interesting research questions can be
analyzed, for example, whether the general factor is sufficient to predict other phenomena or to which degree specific factors contribute to predicting phenomena beyond the general factor (Eid et al.
In many applications, the bifactor model is not based on a strong theoretical definition of the general factor but it is applied in a more exploratory way to find out empirically what a general
factor might mean. Unfortunately, different empirical applications of the bifactor model in the same area of research have not resulted in clear results. A prominent example in abnormal psychology is
the general factor of psychopathology (p factor; Caspi and Moffitt
; Lahey et al.
). After reviewing the research on the p factor of psychopathology over the last eight years, Watts et al. (
) concluded that “the precise nature of the p factor is not yet understood” (p. 1285). It is only recently that systematic analyses of applications of the bifactor model revealed that many
applications are affected by serious problems (e.g., negative variances of specific factors, vanishing specific factors, irregular loadings patterns; see, for example, Eid et al.
). Moreover, the results of these applications are typically not in line with the theoretical expectations. For example, Watts et al. (
) convincingly stated that from a theoretical point of view all observed variables of a bifactor model should have relatively equal loadings on the common general factor and each facet should be
represented by a specific factor. This structure, however, is often not supported (Eid et al.
; Watts et al.
). The many estimation problems and the theoretically not expected results found in many applications show that the bifactor model is not a reasonable model for analyzing multi-faceted constructs in
clinical psychology (Heinrich et al.
; Markon
; Watts et al.
). A major reason for the unsatisfactory results in clinical psychology is that the facets of clinical symptoms typically are not interchangeable as it is required for a psychometrically valid
application of a bifactor model (Eid et al.
Burns, Geiser, Servera, Becker, and Beauchaine (this issue) discuss and illustrate the problems that are related to the application of the bifactor model in abnormal psychology referring to
attention-deficit/hyperactivity disorder (ADHD) and oppositional defiant disorder (ODD) symptoms. They give a thorough overview of 22 recent applications of the bifactor model to analyzing the
structure of ADHD/ODD symptoms. They show that also in this important area of abnormal child psychology the typical application problems of the bifactor model show up. From a theoretical point of
view it is particularly problematic that the results are very inconsistent across the different applications and do not allow to give the general factor a clear meaning. The statement of Watts et al.
) that “the precise nature of the p factor is not yet understood” (p. 1285) refers in an analogous way to the general factor of ADHD/ODD symptoms.
In contrast to the many problematic applications of the bifactor model, Burns et al. (
this issue
) convincingly show that the application of the bifactor S -1 model (Eid et al.
) is not affected by these problems and leads to consistent results across different assessment methods of ADHD/ODD symptoms. In the bifactor S -1 model, the general factor is defined by the
indicators of one facet that is taken as reference facet. The indicators of the reference facets have loadings only on the general factor whereas the items of all other facets load on the general
factor as well as a group factor that is specific to all indicators of a facet (called specific residual factor by Burns et al.) (see Fig.
). The specific residual factors, can be correlated. It is a major strength of Burn et al.’s applications of the bifactor S-1 model that the reference facet is chosen based on strong theoretical
arguments. This gives the general factor as well as the specific factor a clear meaning. Moreover, the meanings of the general (reference) factor and the specific factors did not differ between the
three different rater groups analyzed by Burns et al., which was not the case when applying the original bifactor model (which is called symmetrical bifactor model by Burns et al.). Choosing this
reference factor in all future applications of the bifactor S-1 model would ensure that the meaning of the general factor does not change between studies, which was not the case in the 22 previous
application of the bifactor model to ADHD/ODD symptoms. This makes it possible to use the bifactor S-1 model – in contrast to the traditional (symmetrical model) – for individual clinical assessment.
To the best of my knowledge the implications of the bifactor S-1 model for individual clinical assessment has not be discussed so far. In my comment on the article of Burns et al. (
this issue
), I will focus on the implications of the bifactor S-1 model for individual clinical assessment. I will put this in a broader context of linking psychometric modeling of multi-faceted constructs and
individual clinical assessment, and illustrate the implications referring to the ADHD/ODD symptoms considered by Burns et al.. Finally, I will propose a strategy for how multidimensional models can
generally be used for the assessment of multi-faceted constructs.
Psychometric Modeling of Multi-Faceted Constructs: Implications for Individual Clinical Assessment
Psychometric models – like the bifactor model – have at least two functions. First, they allow researchers to test theoretical assumptions about the structure of observed variables and to estimate
parameters that are important for evaluating the quality of assessment instruments (e.g., reliability). Second, they are measurement models that allow estimating individual scores on the latent
variables that can be used for individual assessment. For example, models of item response theory are not only applied to analyze the structure of test items but also to estimate, for example,
ability scores and the precision with which this scores can be estimated (e.g., Van der Linden and Hambleton
). It is a sign of high quality when individual assessment is based on estimated scores of latent variables of a well-fitting psychometric model.
The many applications of the bifactor model pursued the first purpose of psychometric models and aimed at analyzing the structure of a multi-faceted construct. Given the many inconsistent results of
applications of the bifactor model it would not be reasonable to use this model for clinical assessment. In order to use a psychometric model for individual clinical assessments it is necessary to
show that applications of the model to different samples result in consistent findings.
It is a strong merit of the study of Burns et al. (
this issue
) that they show that applications of the bifactor S-1 model to different rater groups led to consistent results. If future applications of the bifactor S-1 model to ADHD/ODD symptoms show consistent
results this will be an important cornerstone for using this model for individual clinical assessment. The bifactor S-1 model can complement clinical assessment in an important way as it provides a
clinician with important information that is not provided by other psychometric models.
In Fig.
the factor scores of two individuals with respect to the bifactor S-1 model presented by Burns et al. (Fig.
in their paper) are presented. The figure is based on centered factors (factor means of 0) which is the default setting of many programs for confirmatory factor analysis (CFA). Individual A (black
dots) has a value of 3 on the general reference factor, that means that the hyperactivity-impulsivity score is above average. The scores on the specific factors are both −1 for individual A. This
means that the inattention and oppositional defiant scores of Individual A are lower than expected given the hyperactivity-impulsivity score of this individual. Compared to all other individuals with
the same hyperactivity-impulsivity score the inattention and oppositional defiant scores are below average, Individual A has comparably less problems with respect to inattention and oppositional
defiant disorder symptoms. Individual B, on the other hand, has a value of −3 on the general reference factor showing that this individual has a hyperactivity-impulsivity score below average. The
scores of this individual on the two specific factors, however, are 0.5 showing that this individual has inattention and oppositional defiant symptoms that are higher than expected given the
hyperactivity-impulsivity score. Compared to all other individuals with the same hyperactivity-impulsivity score the inattention and oppositional defiant problems are above average.
Fig. 1
Individual factor scores for two individuals according to (
) the bifactor S-1 model presented by Burns et al. (
this issue
) with hyperactivity-impulsivity (HI) as general reference factor (GRF-HI) and two specific reference factors for inattention (SRF-IN) and oppositional defiant disorder (SRF-OD) (see Fig.
), and (
) a multidimensional model with correlated first-order facet-specific factors (see Fig.
) for hyperactivity-impulsivity (HI), inattention (IN), and oppositional defiant disorder (OD). All factors are centered (mean of 0). The factor scores of the model in the multidimensional model are
calculated based on the regression eqs. IN=0.6 ·GRF-HI+SRF-IN and OD=0.6 ·GRF-HI+SRF-OD. The regression coefficient of 0.6 was roughly based on the results in Burns et al. (
this issue
The two examples in Fig.
show that the factor scores reveal interesting individual and interindividual differences. It is important that the diagnostic information given by the factor scores are different from the factor
scores of the multidimensional CFA model with correlated first-order factors which is a kind of natural starting model for analyzing multi-faceted construct. The basic structure of this model for the
three facets of ADHD/ODD is presented in Fig.
. In this model, there is a factor for each of the three facets and the factors can be correlated. The factor scores of the two individuals presented in Fig.
on the three first-order factors is presented in Fig.
. Comparing Figs.
reveals important differences in the diagnostic information between the two models. The factor scores in Fig.
indicate whether and to which degree the individual factor scores deviate from the mean of the factor. Because the general reference factor equals the first-order hyperactivity-impulsivity factor,
the factor scores on the first factor in Fig.
1a and b
are the same. However, the factors scores of Individual A on the first-order factors of inattention and oppositional defiant problems are positive (0.5) showing that the scores of Individual A are
above average compared to the
sample. Whereas the inattention and oppositional defiant problems of Individual A are below average compared to individuals having the same hyperactivity-impulsivity score they are above average
compared to the total sample. Both pieces of information give important insights into the symptom profile of Individual A and complement each other. For Individual B the situation is different.
Individual B has scores on all the first-order facets below average and is less affected by these problems than the average of the total sample. However, the scores of Individual B on the specific
factors are positive indicating that the severity of inattention and oppositional defiant problems are stronger than average compared to individuals having the same hyperactivity-impulsivity score.
Because the information represented by the factor scores give different insights into the severity of psychological symptoms, it is worthwhile to estimate both types of factor scores and use them for
psychological assessment.
Fig. 2
Multidimensional models for analyzing multi-faceted constructs. (
) Multidimensional model with correlated first-order facet-specific factors for hyperactivity-impulsivity (HI), inattention (IN), and oppositional defiant disorder (OD). (
) Bifactor S-1 model presented by Burns et al. (
this issue
) with hyperactivity-impulsivity (HI) as general reference factor (GRF-HI) and two specific reference factors for inattention (SRF-IN) and oppositional defiant disorder (SRF-OD). (
) Bifactor S-1 model with a directly assessed general hyperactivity-impulsivity factor (GF-HI) and three specific reference factors for hyperactivity-impulsivity assessed in specific situations
(SRF-HI1, SRF-HI2, SRF-HI3). Y
: observed variables, E
: error variables, λ
: factor loadings, i: indicator, j: facet
Because the interpretation of the factor scores (with respect to the deviation from the mean) also depend on the sample it is advisable to apply both the multidimensional model with correlated
first-order factors and the bifactor S-1 model to representative samples stemming from interesting norm populations. Because the multidimensional model with correlated first-order factors allows
interesting insights and it is more restrictive than the bifactor S-1 model (Geiser et al.
) test construction and item selection should be based on the multidimensional model with correlated first-order factors. If the latter model fits the data, then the bifactor S – 1 model will also
fit the data. As Geiser et al. have shown, the bifactor S-1 model can be restricted in such a way that it is a reformulation of the multidimensional model with correlated first-order factors and
shows the same fit. For individual clinical assessment, it is desirable that the parameters of the bifactor S-1 model do not differ between relevant subgroups to make sure that individual scores can
be compared across groups, and a measurement instrument can be broadly applied. Therefore, it is worthwhile to extend the bifactor S-1 model to a multigroup model allowing testing different types of
measurement invariance across subgroups (Millsap
). Therefore, the study of Burns et al.’s (
this issue
) can build an important cornerstone for a research program focusing on the suitability of the bifactor S-1 model for clinical assessment.
Guidelines for Analyzing Multi-Faceted Constructs with Multidimensional Models
A strong merit of Burns et al.’s (
this issue
) application of the bifactor S-1 model is that they select the reference facet in a convincing way based on theoretical assumptions and empirical studies on the development of ADHD/ODD symptoms. Not
in all areas of clinical research such an outstanding facet might exist. How can this problem be solved? How can multi-faceted constructs be generally analyzed? In the following some general
guidelines and decision rules will be presented (see Fig.
A starting point for analyzing multi-faceted construct is the multidimensional model with correlated first-order factors. This model represents the idea that there are multiple distinct
(non-overlapping) facets of a construct. This model is an appropriate model for test construction and items selection. The number and meaning of the different facets depend on the area of interest
and should be guided by theoretical assumptions about the construct under consideration.
If there is a theoretically outstanding facet – like in the analysis of Burns et al. (
this issue
) – this facet can be taken as reference facet and a bifactor S-1 can be defined with the items of the reference facet defining the general reference factor (Fig.
If there is no theoretically outstanding facet the researcher can decide whether there is a reference facet of special interest. Consider, for example, a researcher is assessing
hyperactivity-impulsivity in three different types of situations: (1) at school, (2) at home, and (3) during sports and exercise. From a theoretical point of view there might not be a class of
situations that is superordinate to other classes of situations. However, a researcher might be interested in comparing ADHD/ODD symptoms at home to ADHD/ODD symptoms in situations outside the home.
In this case, the items assessing ADHD/ODD symptoms at home would indicate the general reference factor and the scores on the specific factors would indicate to which degree an individual shows more
intense or less intense symptoms at schools and during sports and exercise compared to what one expects based on the ADHD/ODD symptoms at home.
If there is no outstanding facet but strong assumptions about a reasonable general factor, the general factor can be directly assessed (see Fig.
). The model in Fig.
is also a bifactor S-1 model but with the directly assessed general factor as reference factor. If, for example hyperactivity-impulsivity is assessed with respect to the three classes of situations
(at school, at home, during sports and exercise) one could add items assessing ADHD/ODD symptoms in general. One could, for example, present the same items with four different instructions assessing
the symptoms in general (like the scales used in the study of Burns et al.
this issue
) and assessing the symptoms in the three different situations. In such an application, the general factor also has a clear meaning (symptoms in general). Moreover, the specific factors indicate to
which degree the symptoms in the three situations differ from what can be expected given the general assessment.
If there is no theoretically outstanding facet and no facet of special interest and if the general factor cannot be directly assessed one can retain the multidimensional model with correlated
first-order factors as the most appropriate model and can present profiles of individual scores like the profiles presented in Fig.
. For example, in the study of Burns et al. (
this issue
) academic and social impairment is assessed by three factors (social impairment, academic impairment, peer rejection). In this case, the direct assessment of a general factor might not be possible
because it might not be clear what such a factor would mean and how it can be defined by appropriate indicators. If a researcher does not want one of the three impairment factors as an appropriate
comparison standard, the multidimensional model with correlated first-order factors might be appropriate.
Fig. 3
Decision flow chart for selecting an appropriate model for analyzing multi-faceted constructs
Even if a variant of the bifactor S-1 will be specified from a theoretical point of view it is advisable to also report the results of the multidimensional CFA model with correlated first-order
factors. From the perspectives of individual clinical assessment, the factor scores of both types of models (multidimensional model with correlated first-order factor, bifactor S-1 model) are
interesting and should be reported (with confidence intervals). The flow chart in Fig.
does not mean that one has to strictly follow this strategy. For example, it could be interesting to apply all three models presented in Fig.
in an empirical study combining the advantages of these models.
Summary and Discussion
Burns et al. (
this issue
) show that the application of the symmetrical bifactor model to ADHD/ODD symptoms reveals problematic and inconsistent results that indicate that the symmetrical bifactor is not an appropriate model
for analyzing this type of symptoms and can not be used as a psychometric basis for individual clinical assessment. In contrast, applications of the bifactor S-1 model do not show this problems and
reveal consistent results. Moreover, Burns et al. show that there is a theoretically outstanding reference facet that gives the factors of the bifactor S-1 a clear and interesting meaning. This makes
it possible to use the bifactor S-1 model for individual clinical assessment. The factor scores have a clear meaning. The factor scores on the general reference factor indicate the severity of
individual clinical problems compared to the total sample. The factor scores on the specific factors compare the individual clinical symptoms to the distribution of the individuals having the same
score on the general reference factor. This is important information that complement the information given by the factor scores on the first-order factors of a multidimensional CFA model. Both
diagnostic information are interesting and it is therefore advisable to report both types of factor scores.
A general decision flow chart for selecting an appropriate model for analyzing multi-faceted constructs was presented. Starting with a multidimensional model with correlated first-order factors a
researcher can decide whether a bifactor S-1 model with a general reference factor or with a directly assessed general factor would be an appropriate model. Both models (with a general reference
factor and with a directly assessed general factor) reveal important information in addition to the basic multidimensional model with correlated first-order factors, which should always be reported.
Given the superiority of the bifactor S-1 model over the bifactor model for the analysis of multi-faceted clinical symptoms (Burns et al.
this issue
; Heinrich et al.
), it is worthwhile to start a research program on the feasibility of the bifactor S-1 model for individual clinical assessment. Currently there are too few applications of this model in clinical
psychology, and more empirical studies are needed.
Compliance with Ethical Standards
Conflict of Interest
The authors of the current study have no conflicts of interest. Because this is only a coment and no empirical data is presented, no ethical approval was necessary.
Open Access
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Spearman's Correlation Calculator
Descriptive Statistics
Spearman’s Correlation Calculator
Spearman’s Correlation Calculator
What is Spearman’s Correlation Calculator?
The Spearman’s Correlation Calculator is a tool designed to compute the Spearman’s rank correlation coefficient between two sets of data. This coefficient measures the strength and direction of
association between two ranked variables. It is particularly useful when the relationship between the variables is not linear.
Application of Spearman’s Correlation
In various fields such as psychology, social sciences, and market research, understanding the relationship between variables is crucial. Spearman’s correlation helps in ranking the data and finding
whether an increase in one variable corresponds to an increase or decrease in another. For example, it can be used to see if there’s a relationship between the number of hours studied and exam
Benefits in Real-Use Cases
Spearman’s correlation is beneficial in scenarios where data do not meet the assumptions of parametric tests like Pearson’s correlation. This includes non-linear relationships or ordinal data. For
instance, in educational assessments, students’ ranks in different subjects can be compared to understand the consistency of their performance across subjects.
How the Answer is Derived
The calculation involves ranking the data points of each variable separately. Once both sets are ranked, the differences between the ranks of corresponding values are computed. The squared
differences are then summed, and the Spearman’s coefficient is calculated. This metric ranges from -1 to 1, where values closer to 1 or -1 indicate a stronger relationship, and values near 0 indicate
a weaker or no relationship.
Key Considerations
While using the Spearman’s Correlation Calculator, it’s important to ensure the data sets are of the same length and contain numerical values. This ensures accurate and reliable results. The
calculator simplifies the process, providing a quick and efficient way to determine correlation without manual computations.
Understanding the relationships between variables is essential for data analysis and decision-making. The Spearman’s Correlation Calculator offers a straightforward approach to uncover these
insights, making it a valuable tool for researchers, analysts, and students.
What is Spearman’s rank correlation?
Spearman’s rank correlation is a measure used to evaluate the strength and direction of association between two ranked variables. It is based on the ranks of the data rather than the raw data itself,
making it useful for non-linear relationships and ordinal data.
How does Spearman’s rank correlation differ from Pearson’s correlation?
Pearson’s correlation measures the linear relationship between two continuous variables, whereas Spearman’s rank correlation measures the monotonic relationship between two ranked variables. This
means Spearman’s can identify non-linear relationships that Pearson’s might miss.
Can Spearman’s correlation handle ties in data?
Yes, Spearman’s correlation can handle ties in data. When ties occur, average ranks are assigned to the tied values, which maintains the integrity of the rank-based calculation.
What type of data is suitable for Spearman’s correlation?
Spearman’s correlation works best with ordinal data or continuous data that do not necessarily follow a linear relationship. It is particularly useful when dealing with ranked data or when the
assumptions for Pearson’s correlation are not met.
How do I interpret the Spearman’s correlation coefficient?
The Spearman’s correlation coefficient ranges from -1 to 1. A value close to 1 indicates a strong positive association, meaning as one variable increases, the other variable also increases. A value
close to -1 indicates a strong negative association, meaning as one variable increases, the other decreases. Values near 0 suggest little to no association between the variables.
Are there limitations to using Spearman’s correlation?
Spearman’s correlation assumes that the relationship between the variables is monotonic. It cannot detect complex relationships that are not strictly increasing or decreasing. Additionally, it
requires data sets to have the same length and should ideally contain numerical values.
Why do I need to rank the data before calculating Spearman’s correlation?
Ranking the data transforms the values into a scale that can reveal the strength and direction of a monotonic relationship, even if the original data do not meet the assumptions of linearity. Ranking
is essential for handling non-linear and ordinal data appropriately.
How can I use the Spearman’s Correlation Calculator effectively?
Input your two data sets, ensure they are of the same length and contain numeric values. The calculator will then rank the data, compute the differences between the ranks, and calculate the
Spearman’s correlation coefficient. This allows for a quick and accurate assessment of the relationship between your variables. | {"url":"https://www.onlycalculators.com/statistics/descriptive-statistics/spearmans-correlation-calculator/","timestamp":"2024-11-13T05:56:06Z","content_type":"text/html","content_length":"244538","record_id":"<urn:uuid:749dbdaa-c2b4-4dfe-89fd-76821a9aaa86>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00118.warc.gz"} |
Eureka Math Grade 7 Module 5 Lesson 1 Answer Key
Engage NY Eureka Math 7th Grade Module 5 Lesson 1 Answer Key
Eureka Math Grade 7 Module 5 Lesson 1 Example Answer Key
Example 1: Spinner Game
Suppose you and your friend are about to play a game using the spinner shown here:
Rules of the game:
1. Decide who will go first.
2. Each person picks a color. Both players cannot pick the same color.
3. Each person takes a turn spinning the spinner and recording what color the spinner stops on. The winner is the person whose color is the first to happen 10 times.
Play the game, and remember to record the color the spinner stops on for each spin.
Students try their spinners a few times before starting the game. Before students begin to play the game, discuss who should go first. Consider, for example, having the person born earliest in the
year go first. If it is a tie, consider another option like tossing a coin. Discuss with students the following questions:
→ Will it make a difference who goes first?
The game is designed so that the spinner landing on green is more likely to occur. Therefore, if the first person selects green, this person has an advantage.
→ Who do you think will win the game?
The person selecting green has an advantage.
→ Do you think this game is fair?
No. The spinner is designed so that green will occur more often. As a result, the student who selects green will have an advantage.
→ Play the game, and remember to record the color the spinner stops on for each spin.
Example 2: What Is Probability?
Probability is a measure of how likely it is that an event will happen. A probability is indicated by a number between 0 and 1. Some events are certain to happen, while others are impossible. In most
cases, the probability of an event happening is somewhere between certain and impossible.
For example, consider a bag that contains only red cubes. If you were to select one cube from the bag, you are certain to pick a red one. We say that an event that is certain to happen has a
probability of 1. If we were to reach into the same bag of cubes, it is impossible to select a yellow cube. An impossible event has a probability of 0.
The figure below shows the probability scale.
Eureka Math Grade 7 Module 5 Lesson 1 Exercise Answer Key
Exercise 1.
Which color was the first to occur 10 times?
Answers will vary, but green is the most likely.
Exercise 2.
Do you think it makes a difference who goes first to pick a color?
Yes. The person who goes first could pick green.
Exercise 3.
Which color would you pick to give you the best chance of winning the game? Why would you pick that color?
Green would give the best chance of winning the game because it has the largest section on the spinner.
Exercise 4.
Below are three different spinners. On which spinner is the green likely to win, unlikely to win, and equally likely to win?
Green is likely to win on Spinner B, unlikely to win on Spinner C, and equally likely to win on Spinner A.
Exercise 5.
Decide where each event would be located on the scale above. Place the letter for each event in the appropriate place on the probability scale.
A. You will see a live dinosaur on the way home from school today.
B. A solid rock dropped in the water will sink.
C. A round disk with one side red and the other side yellow will land yellow side up when flipped.
D. A spinner with four equal parts numbered 1–4 will land on the 4 on the next spin.
E. Your full name will be drawn when a full name is selected randomly from a bag containing the full names of all of the students in your class.
F. A red cube will be drawn when a cube is selected from a bag that has five blue cubes and five red cubes.
G. Tomorrow the temperature outside will be -250 degrees.
Answers are noted on the probability scale above.
A. Probability is 0, or impossible, as there are no live dinosaurs.
B. Probability is 1, or certain to occur, as rocks are typically more dense than the water they displace.
C. Probability is \(\frac{1}{2}\), as there are two sides that are equally likely to land up when the disk is flipped.
D. Probability of landing on the 4 would be \(\frac{1}{4}\), regardless of what spin was made. Based on the scale provided, this would indicate a probability halfway between impossible and equally
likely, which can be classified as being unlikely to occur.
E. Probability is between impossible and equally likely to occur, assuming there are more than two students in the class. If there were two students, then the probability would be equally likely. If
there were only one student in the class, then the probability would be certain to occur. If, however, there were more than two students, the probability would be between impossible and equally
likely to occur.
F. Probability would be equally likely to occur as there are an equal number of blue and red cubes.
G. Probability is impossible, or 0, as there are no recorded temperatures at -250 degrees Fahrenheit or Celsius.
Exercise 6.
Design a spinner so that the probability of spinning a green is 1.
The spinner is all green.
Exercise 7.
Design a spinner so that the probability of spinning a green is 0.
The spinner can include any color but green.
Exercise 8.
Design a spinner with two outcomes in which it is equally likely to land on the red and green parts.
The red and green areas should be equal.
An event that is impossible has a probability of 0 and will never occur, no matter how many observations you make. This means that in a long sequence of observations, it will occur 0% of the time. An
event that is certain has a probability of 1 and will always occur. This means that in a long sequence of observations, it will occur 100% of the time.
Exercise 9.
What do you think it means for an event to have a probability of \(\frac{1}{2}\)?
In a long sequence of observations, it would occur about half the time.
Exercise 10.
What do you think it means for an event to have a probability of \(\frac{1}{4}\)?
In a long sequence of observations, it would occur about 25% of the time.
Eureka Math Grade 7 Module 5 Lesson 1 Problem Set Answer Key
Question 1.
Match each spinner below with the words impossible, unlikely, equally likely to occur or not occur, likely, and certain to describe the chance of the spinner landing on black.
Question 2.
Decide if each of the following events is impossible, unlikely, equally likely to occur or not occur, likely, or certain to occur.
a. A vowel will be picked when a letter is randomly selected from the word lieu.
b. A vowel will be picked when a letter is randomly selected from the word math.
c. A blue cube will be drawn from a bag containing only five blue and five black cubes.
d. A red cube will be drawn from a bag of 100 red cubes.
e. A red cube will be drawn from a bag of 10 red and 90 blue cubes.
a. Likely; most of the letters of the word lieu are vowels.
b. Unlikely; most of the letters of the word math are not vowels.
c. Equally likely to occur or not occur; the number of blue and black cubes in the bag is the same.
d. Certain; the only cubes in the bag are red.
e. Unlikely; most of the cubes in the bag are blue.
Question 3.
A shape will be randomly drawn from the box shown below. Decide where each event would be located on the probability scale. Then, place the letter for each event on the appropriate place on the
probability scale.
A. A circle is drawn.
B. A square is drawn.
C. A star is drawn.
D. A shape that is not a square is drawn.
Probability Scale
Question 4.
Color the squares below so that it would be equally likely to choose a blue or yellow square.
Color five squares blue and five squares yellow.
Question 5.
Color the squares below so that it would be likely but not certain to choose a blue square from the bag.
Color 6, 7, 8, or 9 squares blue and the rest any other color.
Question 6.
Color the squares below so that it would be unlikely but not impossible to choose a blue square from the bag.
Color 1, 2, 3, or 4 squares blue and the others any other color.
Question 7.
Color the squares below so that it would be impossible to choose a blue square from the bag.
Color all squares any color but blue.
Eureka Math Grade 7 Module 5 Lesson 1 Exit Ticket Answer Key
Question 1.
Decide where each of the following events would be located on the scale below. Place the letter for each event on the appropriate place on the probability scale.
The numbers from 1 to 10 are written on small pieces of paper and placed in a bag. A piece of paper will be drawn from the bag.
A. A piece of paper with a 5 is drawn from the bag.
B. A piece of paper with an even number is drawn.
C. A piece of paper with a 12 is drawn.
D. A piece of paper with a number other than 1 is drawn.
E. A piece of paper with a number divisible by 5 is drawn.
The numbers from 1 to 10 are written on small pieces of paper and placed in a bag. A piece of paper will be drawn from the bag.
A. A piece of paper with a 5 is drawn from the bag.
B. A piece of paper with an even number is drawn.
C. A piece of paper with a 12 is drawn.
D. A piece of paper with a number other than 1 is drawn.
E. A piece of paper with a number divisible by 5 is drawn.
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CBSE Class 11 Applied Mathematics 2024 Complete Details
Reader's Digest: Want to know the essentials of CBSE Applied Maths for Class 11? Read this blog to learn about CBSE Class 11 Applied Mathematics, from its core concepts to the latest syllabus
updates, recommended books & effective preparation strategies.
As the academic year unfolds, CBSE students across India are gearing up for a new addition to their curriculum - Class 11 Applied Mathematics.
This dynamic subject promises to bridge the gap between theoretical mathematics and its real-world applications, offering students a unique perspective on problem-solving and mathematical modelling.
You must take the exam of 3 hours of total 80 marks. However, 20 marks are allotted for an Internal Assessment.
Here are the key points we will be discussing in this blog:
• Meaning of CBSE Class 12 Applied Mathematics: A brief introduction to the subject and its significance in the modern educational landscape.
• Class 11 Applied Maths Syllabus: A comprehensive syllabus breakdown highlighting the topics and chapters students can expect to study.
• Applications of CBSE Applied Mathematics: Exploring the real-world applications of this subject and how it connects theoretical knowledge to practical scenarios.
• CBSE Class 11 Applied Mathematics Prep Tips: Valuable tips and strategies to excel in this subject, including study techniques and recommended resources.
What is CBSE Applied Mathematics?
Applied Mathematics combines mathematical science and specialized knowledge in different subjects. This subject extensively uses mathematics techniques in other fields like Physics, Engineering,
Medical, and more.
The practical applications of Maths have provided a way to develop Mathematical Theories, resulting in the pure study of mathematics.
CBSE has decided to add this subject to the curriculum to help students understand it in-depth if they choose Maths as their career. Also, Applied Mathematics has various applications explained in
the below section.
CBSE Class 11 Applied Mathematics Press Release
As per the Systematic Reforms, the National Curriculum Framework 2005 recommends examining Maths and English at two levels. Hence, CBSE considered this recommendation to introduce Applied
Maths is a widely used subject in every career, which can help a student's life. Also, it is observed that the current Maths syllabus goes well with the Science stream but not with Commerce and other
streams. Hence, this subject is an elective course that can help to have a great career ahead.
Main Objectives of Introducing Applied Mathematics
• To have an interconnection with other subjects.
• To help in developing logical reasoning skills and apply the same in problem-solving.
• To develop an understanding of basic mathematical and statistical tools and their applications in the field of commerce (business/finance/economics) and social sciences;
• To implement real-world experiences/problems into mathematical expressions using numerical/algebraic/graphical representation;
• To make sense of the data by organizing, representing, interpreting, analyzing, and making meaningful inferences from real-world situations;
Read More: CBSE Class 11 Applied Math Books
CBSE Class 11 Applied Maths Syllabus 2024
In the CBSE Class 11 Applied Mathematics subject, seven chapters are tabulated below. Go through the topic name and marking scheme of the CBSE 11th class Applied Mathematics Syllabus below:
No. Units No. of Periods Marks
I. Numbers, Quantification, and Numerical Applications 25 09
II. Algebra 45 15
III. Mathematical Reasoning 15 06
IV. Calculus 35 10
V. Probability 25 08
VI. Descriptive Statistics 35 12
VII Basics of Financial Mathematics 45 18
VIII Coordinate Geometry 15 05
- Total 240 80
- Internal Assessment - 20
Read More: CBSE Class 11 Applied Maths - Mathematical Reasoning
What are the Practical Applications of CBSE Class 11 Applied Mathematics?
There is a wide range of applications of Applied Mathematics in various fields. In many streams, Applied Maths plays an important role. Some of the typical applications of CBSE Class 11 Applied
Mathematics are listed below:
• Information Theory
• Engineering
• Signal Processing
• Cryptography
• Operations Research
• Numerical Analysis
• Statistics
• Wavelets
• Game Theory
What are the Best Preparation Tips for CBSE Class 11 Applied Mathematics 2024?
Preparation for any subject varies for each student; however, here are a few experts' suggested tips and tricks that can help enhance your preparation.
• Structured Study Plan: Create a well-structured study plan that covers all the topics. Allocate more time to challenging areas and revise regularly. Stick to your schedule diligently.
• Quality Resources: Choose suitable study materials and resources. Look for reference books, online courses, and video tutorials to supplement your textbooks. Quality matters more than quantity.
Refer to the best CBSE Class 11 Commerce Books per the latest pattern and syllabus.
• Practice Regularly: Mathematics is a subject that requires consistent practice. Solve a wide variety of problems to reinforce your understanding. Make use of previous years' question papers and
sample papers.
• Conceptual Clarity: Ensure you have a deep understanding of the fundamental concepts. Don't just memorize formulas; understand how and when to apply them.
• Keep Formula Notes: Maintain a separate notebook for essential formulas, theorems, and key concepts. This will be handy for quick revisions.
Difference Between Core Maths & Applied Maths
One of the most common doubts with all students is the difference between pure and applied maths. Here are the significant differences between Pure Maths and Applied Maths explained.
Aspect Core Mathematics Applied Mathematics
Emphasis Focuses on theoretical concepts and abstract mathematics. Emphasizes practical applications of mathematical concepts.
Content Includes topics like algebra, calculus, geometry, and statistics. Covers topics like financial mathematics, statistics, and optimization.
Theoretical vs. Practical Primarily theoretical, with a focus on proofs and theorems. Primarily practical, with a focus on real-world problem-solving.
Application Areas Generally applicable across various fields and sciences. Specifically designed for practical use in specific fields like business, economics, and social
Problem-Solving Approach Often involves solving complex mathematical problems and proofs. Involves solving real-world problems and making data-driven decisions.
Mathematical Rigor Emphasizes mathematical rigour and precision in solutions. Focuses on finding practical solutions, often with some degree of approximation.
Relevance in Careers Beneficial for careers in mathematics, physics, engineering, and Useful for finance, economics, data science, and social sciences careers.
Common Courses and Included in most standard mathematics curricula. Often offered as a specialized course or as part of specific degree programs.
Examples of Topics Calculus, linear algebra, number theory, abstract algebra. Financial mathematics, data analysis, operations research.
Academic Levels Taught at various academic levels, from high school to university. Typically taught at the university or college level.
Problem Complexity Problems can be highly theoretical and abstract. Problems are often practical and context-dependent.
Check Also: CBSE Class 11 Applied Maths Probability
As Class 11 students venture into the realm of CBSE Applied Mathematics, they embark on a journey that bridges the gap between abstract mathematical concepts and real-world applications. This blog
has shed light on the essentials of this dynamic subject, offering valuable insights into its significance, syllabus, and practical applications.
Key Takeaways:
• CBSE Applied Mathematics is an elective course designed to interconnect mathematics with 5ther subjects, fostering logical reasoning and real-world problem-solving skills.
• The comprehensive syllabus covers topics such as algebra, calculus, probability, statistics, and financial mathematics.
• Applied Mathematics finds practical applications in diverse fields like engineering, statistics, cryptography, and more.
• Effective preparation involves a structured study plan, quality resources, regular practice, conceptual clarity, and maintaining formula notes.
• Differentiating between Core and Applied Mathematics highlights their emphasis on theory vs. practicality and their relevance in various career paths.
Check Also : CBSE Class 11 Applied Maths Algebra | {"url":"https://www.toprankers.com/cbse-class-11-applied-mathematics","timestamp":"2024-11-14T17:57:23Z","content_type":"text/html","content_length":"518159","record_id":"<urn:uuid:81cd6d7e-3486-48b6-b6eb-472d28ba02e8>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00490.warc.gz"} |
A uniform rod of mass m and length ℓ is rotating with constant ... | Filo
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A uniform rod of mass and length is rotating with constant angular velocity about an axis which passes through its one end and perpendicular to the length of rod. The area of cross section of the rod
is and its young's modulus is . Neglect gravity. The strain at the midpoint of the rod is:
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A uniform rod of mass and length is rotating with constant angular velocity about an axis which passes through its one end and perpendicular to the length of rod. The area of cross
Question Text section of the rod is and its young's modulus is . Neglect gravity. The strain at the midpoint of the rod is:
Updated On Sep 25, 2024
Topic Gravitation and Fluid
Subject Physics
Class Class 12
Answer Type Video solution: 1
Upvotes 100
Avg. Video 6 min | {"url":"https://askfilo.com/user-question-answers-physics/a-uniform-rod-of-mass-and-length-is-rotating-with-constant-3132343931333739","timestamp":"2024-11-15T01:42:52Z","content_type":"text/html","content_length":"216340","record_id":"<urn:uuid:4e238e73-aaa2-47c9-8b06-d61c53b55045>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00585.warc.gz"} |
Scale Conver
Scale Converter
Scale Conversion for Construction and Building Projects
Scale conversion is a crucial concept in the construction and building industry, allowing professionals to accurately interpret blueprints, plans, and other scaled drawings. Blueprints are often
drawn to scale, meaning that a specific measurement on the plan represents a proportionate measurement in real life. To work effectively with these plans, contractors, architects, and builders need
to convert these scaled measurements into real-world dimensions. This process involves understanding ratios and proportions to ensure accuracy in construction projects.
What is Scale Conversion?
Scale conversion is the process of translating a measurement from a scaled drawing into its actual size. Scales are typically represented as a ratio or a statement of equivalency, such as 1:100 or 1
inch = 10 feet. The first number (or measurement) represents a unit on the drawing, while the second number (or measurement) represents the corresponding real-world measurement. For example, in a
scale of 1:100, 1 unit on the drawing equals 100 units in reality.
Basic Scale Conversion Calculations
To convert a scaled measurement to its real-world size, you need to use the scale provided. For example, if a blueprint states a scale of 1 inch = 10 feet, this means that every inch on the blueprint
represents 10 feet in the real world. If an object on the plan measures 3 inches, you can calculate the actual size using the following formula:
Real Length = Scaled Length × Scale Factor
Using the example above, the real length of an object that is 3 inches on the blueprint would be:
Real Length = 3 inches × 10 feet per inch = 30 feet
This simple multiplication provides the real-world dimension, ensuring that what is seen on the blueprint matches the intended size during construction.
Converting Between Different Units
Scale conversions often require converting between different units of measurement. For instance, if a plan uses a scale of 1 cm = 2 meters and an object measures 5 cm on the plan, the actual size can
be found as follows:
Real Length = Scaled Length × Scale Factor
Real Length = 5 cm × 2 meters per cm = 10 meters
In this case, the scale factor allows for direct conversion between centimeters and meters. However, if the measurements need to be converted into a different unit, such as feet or inches, additional
unit conversion is required. For example, if you need the result in feet, you would convert meters to feet by using the conversion factor (1 meter = 3.28084 feet):
Real Length in feet = 10 meters × 3.28084 = 32.8084 feet
Typical Use Cases in Construction and Building
Scale conversion plays a pivotal role in various stages of construction and building projects. Here are some common use cases:
• Blueprint Interpretation: Contractors and architects use scale conversions to interpret blueprints and plans accurately. For instance, when preparing to lay the foundation for a building, the
dimensions on the blueprint must be converted to real-life measurements to ensure that the structure is built correctly.
• Material Estimation: By converting scaled measurements to actual dimensions, contractors can estimate the quantity of materials required. For example, if a wall is shown as 4 inches long on a
blueprint with a scale of 1 inch = 2 feet, the actual wall length is 8 feet. This allows for accurate estimation of materials such as bricks, concrete, and lumber.
• Landscaping and Site Planning: Landscape architects use scale conversion to design and implement outdoor spaces. A plan might show a garden path as 2 centimeters long at a scale of 1 cm = 5
meters. By converting this measurement, the actual path length is determined to be 10 meters, guiding the installation process.
• Furniture and Interior Design: Interior designers often work with scaled floor plans to place furniture and fixtures. For example, if a room measures 5 inches in length on a plan with a scale of
1 inch = 4 feet, the room's actual length is 20 feet. This ensures that furniture is appropriately scaled and fits within the space.
• Structural Analysis: Engineers use scaled drawings to assess structural elements, such as beams and columns. Accurate scale conversion ensures that these elements are correctly sized to support
the loads and stresses they will encounter.
Common Scales Used in Construction
Different types of construction projects use various scales depending on the level of detail required. Some common scales include:
• Architectural Scale: Commonly used for building plans, typical architectural scales include 1/4" = 1'-0" (1:48) and 1/8" = 1'-0" (1:96). These scales provide detailed views of rooms, walls, and
other architectural features.
• Engineering Scale: Used for large-scale projects such as site plans and infrastructure. Common scales include 1" = 10' (1:120) and 1" = 100' (1:1200), which are useful for representing large
areas on a manageable-sized drawing.
• Metric Scale: Often used in countries that use the metric system. Common scales include 1:100 and 1:200, providing clear and precise measurements for a variety of construction projects.
Advanced Scale Conversion: Proportions and Ratios
While basic scale conversion involves multiplying the scaled length by the scale factor, more complex conversions might involve proportions and ratios. For example, if you have a drawing with a scale
of 1:50 and you need to find the scaled size of an object that is 200 meters in reality, you can set up a proportion:
Scaled Length / Real Length = Scale Ratio
Scaled Length / 200 meters = 1 / 50
Scaled Length = 200 meters × (1 / 50) = 4 meters
This approach can be useful when needing to reverse the process and find the scaled measurement based on real-world dimensions.
Scale conversion is an essential skill in the construction and building industry. Understanding how to accurately convert between scales and units ensures that projects are executed as intended, from
blueprint interpretation to material estimation. By mastering these calculations, contractors, architects, and builders can translate the details on paper into real-world structures with precision
and efficiency. | {"url":"https://constructcalc.com/scale-converter.php","timestamp":"2024-11-09T03:36:57Z","content_type":"text/html","content_length":"19331","record_id":"<urn:uuid:89bea378-31fc-4967-94e7-2d3f0be3f084>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00643.warc.gz"} |
nested-list-weight-sum | Leetcode
Similar Problems
Similar Problems not available
Nested List Weight Sum - Leetcode Solution
LeetCode: Nested List Weight Sum Leetcode Solution
Difficulty: Medium
Topics: depth-first-search breadth-first-search
The Nested List Weight Sum problem on leetcode asks us to find the sum of all integers in a nested list of integers, where each element can be either an integer or a list. The weight of each integer
is its depth in the nested list multiplied by its value.
For example, consider the following nested list [[1,1],2,[1,1]], the depth of the integers 1 and 1 is 2 (they are nested within a list nested within another list), and the depth of integer 2 is 1 (it
is at the top level). Therefore, the weight of the nested list is (12) + (21) + (12) + (12) = 8.
To solve this problem, we can use recursion. We can define a function nestedListWeightSum that takes in a nested list and the current depth, and recursively calculates the sum. If the element is an
integer, we simply add its weight to the total sum. If it is a list, we recursively call the function with the nested list and the current depth + 1.
Here is the detailed solution in Python:
def nestedListWeightSum(nestedList, depth=1):
totalSum = 0
for item in nestedList:
if isinstance(item, int):
totalSum += item * depth
elif isinstance(item, list):
totalSum += nestedListWeightSum(item, depth+1)
return totalSum
The function takes a nestedList and a depth (which is optional and defaults to 1). We initialize the totalSum to 0 and loop through each item in the nested list.
If the item is an integer, we add its weight to the total sum by multiplying it with the current depth. If the item is a list, we recursively call the function with the nested list and the current
depth + 1, and add the result to the total sum.
Finally, we return the total sum.
We can test this function with the example nested list [[1,1],2,[1,1]] as follows:
nestedList = [[1,1],2,[1,1]]
This should output 8 as expected.
This solution has a time complexity of O(n), where n is the total number of integers in the nested list. This is because we need to visit every integer once to calculate its weight. It also has a
space complexity of O(d), where d is the maximum depth of the nested list, because we need to store the depth for each recursive call.
Nested List Weight Sum Solution Code | {"url":"https://prepfortech.io/leetcode-solutions/nested-list-weight-sum","timestamp":"2024-11-11T23:45:17Z","content_type":"text/html","content_length":"62786","record_id":"<urn:uuid:adc3e4e6-de29-4a36-b910-ff00e078240f>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00731.warc.gz"} |
Solutions & Dilutions - UCalgary Chemistry Textbook
Making Solutions
Making juice from a concentrate or juice crystals is an example of creating a homogeneous solution. If we were to make a glass of lemonade, we probably have a set amount of crystals we like to add to
make it taste great. How much we may choose to add could differ from person to person as some way want a really strong lemon taste, while others not so much. Use this understanding to address the
first question:
A take away from the above activity is the more crystals we add, the more concentrated the solution and the darker the yellow colour. Let’s now say you made yourself some lemonade but you found the
taste too strong of lemon flavour so you decided to add more water to dilute it. In the activity below, drag and drop the glasses into the before and after box to see what has happened visually to
your lemonade solution:
A key thing to realize in the above activity is that when diluting a solution we are NOT changing the amount of a species present we are just changing the overall volume that it is present in.
Calculating Concentrations
Using your understanding of molarity (M), let’s now add some numbers to the above lemonade problem. Use the idea that each triangle represents one mole of “lemonade crystals” to answer the following
Great, now that this concept makes sense, lets add a calculation. Drag the correct molarity onto each glass (remember that each triangle represents a mole):
Making the lemonade is actually a dilution question. We start off with an initial solution concentration of our lemonade ( 7 mol/L or 7 M) and then add water and dilute the lemonade so the taste is
less intense (5 mol/L or 5 M). Usually, we cannot see the number of moles on a molecular level like we can in this example here. Instead, we usually know an initial concentration and then use a
formula called:
C[1]V[1]=C[2]V[2] or C[i]V[i]=C[f]V[f]
where C = concentration (or molarity), V= volume, the “1” or “i” is initial or before dilution and the “2” or “f” is the final numbers after dilution.
How would you rearrange the dilution calculation above to solve for the final molarity (M)?
Let’s look at the above problem one more time and pretend we cannot see the number of triangles:
Up to this point we have looked at knowing the number of moles and the volume to determine the concentration/molarity. We can also do the reverse, where we know the molarity (concentration) and the
volume and can use this to determine the number of moles.
Let’s think about what is happening with our units for the above question:
Whenever you are solving a numerical problem, try to visualize what is occurring, as this will help you realize when an equation might be set up incorrectly, as the answer may not make sense. Also
remember to think about what is happening to your units – show every step with units and make sure they cancel, since wacky units might be the first sign of a calculation mistake. | {"url":"https://chem-textbook.ucalgary.ca/version2/review-of-background-topics/measurements-and-data/measurements-in-chemistry/solutions-and-dilutions/","timestamp":"2024-11-02T18:37:47Z","content_type":"text/html","content_length":"216201","record_id":"<urn:uuid:7cf8e1c0-bc7e-4e05-9cc4-2726ad35a0e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00578.warc.gz"} |
Easter Cozonacs - PHP Exam Task
[Solved] Easter Cozonacs - PHP Exam Task
21/10/2024 1:15 pm
Topic starter
Since it’s Easter you have decided to make some cozonacs and exchange them for eggs.
Create a program that calculates how much cozonacs you can make with the budget you have.
• First, you will receive your budget.
• Then, you will receive the price for 1 kg flour.
Here is the recipe for one cozonac:
• Eggs: 1 pack
• Flour: 1 kg
• Milk: 0.250 l
• The price for 1 pack of eggs is 75% of the price for 1 kg flour.
• The price for 1l milk is 25% more than price for 1 kg flour.
• Notice, that you need 0.250l milk for one cozonac and the calculated price is for 1l.
Start cooking the cozonacs and keep making them until you have enough budget. Keep in mind that:
• For every cozonac that you make, you will receive 3 colored eggs.
• For every 3rd cozonac that you make, you will lose some of your colored eggs after you have received the usual 3 colored eggs for your cozonac. The count of eggs you will lose is calculated when
you subtract 2 from your current count of cozonacs – ({currentCozonacsCount} – 2)
In the end, print the cozonacs you made, the eggs you have gathered and the money you have left, formatted to the 2nd decimal place, in the following format:
"You made {countOfCozonacs} cozonacs! Now you have {coloredEggs} eggs and {moneyLeft}BGN left."
Input / Constraints:
On the 1st line you will receive the budget – a real number in the range [0.0…100000.0]
• On the 2nd line you will receive the price for 1 kg floor – a real number in the range [0.0…100000.0]
• The input will always be in the right format.
• You will always have a remaining budget.
• There will not be a case in which the eggs become a negative count.
In the end print the count of cozonacs you have made, the colored eggs you have gathered and the money formatted to the 2nd decimal place in the format described above.
We start by calculating the price for a pack of eggs, which is 75% of the price for 1 kg floor, which in this case is 1.25. The pack of eggs price is 0.9375.
The price for 1l milk is 25% more than the price for 1kg floor and in this case it is – 1.5625, but we need the price for 0.250ml, which is - 0.390625. The total price for one cozonac is:
1.25 + 0.9375 + 0.390625 = 2.578125.
And we start subtracting the price for a single cozonac from the budget, and for every cozonac we receive 3 eggs. So after the first subtraction we will have 17.921875 budget, 1 cozonac and 3 eggs.
After the second - 15.34375 budget, 6 eggs, and on the third - 12.765625 budget and 9 eggs and since it’s the third, we need to subtract the lost eggs, which will be 3 – 2 = 1, so we subtract 1 from
9 and our eggs become 8. We continue subtracting money from the budget until the money aren't enough for us to make a cozonac. In the end we have 2.45BGN left.
Here is my solution:
$budget = readline();
$priceFlour = readline();
$priceEggs = 0.75 * $priceFlour;
$priceMilk = $priceFlour + ($priceFlour * 0.25);
$priceMilk250 = $priceMilk / 4;
$priceCozonac = $priceFlour + $priceEggs + $priceMilk250;
$cozonacCount = 0;
$coloredEggs = 0;
while ($budget >= $priceCozonac) {
$coloredEggs += 3;
$budget -= $priceCozonac;
if ($cozonacCount % 3 == 0) {
$coloredEggs -= ($cozonacCount - 2);
printf("You made $cozonacCount cozonacs! Now you have $coloredEggs eggs and %.2fBGN left.", $budget); | {"url":"https://tutorials7.com/qa/php/easter-cozonacs-php-exam-task/","timestamp":"2024-11-08T20:48:49Z","content_type":"text/html","content_length":"223623","record_id":"<urn:uuid:117f7536-09da-4ada-b17a-7933c2ef655d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00727.warc.gz"} |
Which Table Represents Exponential Growth? A 2-column Table Has 4 Rows. The First Column Is Labeled X
Step-by-step explanation:
The table that represents exponential growth is the second table.. The second column is labeled y with entries 2, 4, 6, 8. A 2-column table has 4 rows.
What is exponential growth?
Exponential growth simply means a process that increases quantity over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.
Exponential growth is the pattern of data which shows sharper increases over time. In finance, compounding creates exponential returns and the savings accounts with a compounding interest rate can
show exponential growth.
In this case, the second table described represents an exponential function. It has a common ratio of 4, meaning that each y-value is multiplied by 4 to get to the next value. Here, every time the Xs
increase by 1, the Ys increase by a multiple of 4.
One of the best examples of exponential growth is observed in bacteria. Here, it takes bacteria roughly an hour to reproduce through prokaryotic fission. This illustrates exponential growth.
Learn more about exponential growth on:
wow a 100/10 rate mines
Community- jocelynn johnson
Community is a family,
Someone who help people when in need,
But this is not a community,
This is a place where people feel like they don’t belong,
A place where people have to look around
The corner to make sure their safe,
It is for the days that growing,
For the life that’s growing,
For the changes we made that hurt our loved ones,
That may cause stress throughout the day,
And may cause you to not
Wanna play,
U say your safe,
But the truth is you might get killed at the end of the day,
Why I say,
Cause this neighborhood’s not safe,
And your running on the streets all day,
This community has changed,
All throughout the day,
Cause kids are always joining gangs,
I just wish everything would go back to
Being the same.
Step-by-step explanation: | {"url":"https://www.cairokee.com/homework-solutions/which-table-represents-exponential-growth-a-2-column-table-h-lzhq","timestamp":"2024-11-07T11:07:36Z","content_type":"text/html","content_length":"76658","record_id":"<urn:uuid:2c44d18e-dfaf-4d2a-9514-8bb307d9e500>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00400.warc.gz"} |
V. A. Kovalev, Yu. N. Radayev, R. A. Revinsky, “Generalized cross-coupled type-III thermoelastic waves propagating via a waveguide under sidewall heat interchange”, Izv. Saratov Univ. Math. Mech. Inform., 2011, Volume 11, Issue 1,Pages 59–70
Izv. Saratov Univ. Math. Mech. Inform., 2011 Volume 11, Issue 1, Pages 59–70 (Mi isu202)
This article is cited in 6 papers Mechanics Generalized cross-coupled type-III thermoelastic waves propagating via a waveguide under sidewall heat interchange V. A. Kovalev^a
Yu. N. Radayev^b
R. A. Revinsky^c ^a Moscow City Government University of Management, Chair of Applied Mathematics ^b Institute for Problems in Mechanics RAS, Moscow ^c Saratov State University, Chair of Mathematical
Theory of Elasticity and Biomechanics Abstract:
The paper is devoted to a study of cross-coupled type-III generalized thermoelastic waves propagation via a long cylindrical waveguide. The sidewall of the waveguide is assumed free from tractions
and permeable to heat. The analysis is carried out in the framework of coupled generalized theory of GNIII-thermoelasticity consistent with the basic thermodynamic principles. The theory combines the
both possible mechanisms of heat transfer: thermodiffusion and wave. Type-III generalized thermoelasticity includes classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity
(GNII) as limiting cases. The GNII-theory can be formulated as a field theory and differential field equations are of hyperbolic analytical type. Closed solution of the coupled GNIII-thermoelasticity
equations satisfying the required boundary conditions on the surface of waveguide including convective heat interchanging condition has been obtained. The paper provides numerical analysis of
frequency equation. A scheme of frequency equation roots localization is described and wavenumbers of the coupled thermoelastic waves of the first azimuthal order are computed.
Key words: thermoelasticity, type-III thermoelasticity, frequency equation, waveguide, wavenumber, wave mode, azimuthal order. UDC:
DOI: 10.18500/1816-9791-2011-11-1-59-70
© , 2024 | {"url":"https://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=isu&paperid=202&option_lang=eng","timestamp":"2024-11-14T20:45:25Z","content_type":"text/html","content_length":"10510","record_id":"<urn:uuid:39b5d9b6-e0f9-4b4a-b8cb-cc4e1ff80e82>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00358.warc.gz"} |
Volatility Hedging with MBS
Let’s say I expect a decrease in volatility and interest rates go up, can someone explain how to hedge this and why? (i.e. dynamic hedging, selling forwards – why would I sell forwards???) Thanks.
If you expect vol to go down => sell options interest rates to go down => sell bond puts, bond futures puts, payer swaptions, etc…
lower volatility means you won’t want to hedge with options as those options aren’t worth as much now, so you’ll want to dynamically hedge with futures. if you have a long position in an underlying,
you hedge by selling futures.
you need to sell forwards. 2 reasons. First, you need to shorten duration because as iterest rates go up MBS duration increases, sou you want to shorten it. To shorten it you can *IR call options
*T-bill put options *Sell T-bill forwards which also will give Since your expected vol < implied vol, your option value will decrease as implied vol will be less then expected vol, so you dont want
to hold options in this situation So you want to Sell forwards
thanks for the input guys… really cleared things up for me…
what SS / reading is this frm again ?? is this derivatives or Fixed income ? I just realised i need to review this again
-Striker- I thought Lower volatility (current) < future, means, you would want to hedge with options because they are not as expensive? (i.e. the prepayment option isn’t as valuable?
Almo, If Expected Vol > Implied Vo (volatility implied from Option prices) it means that Options could be viewed currently as Cheap, b/c if Vol does increase the option prices will increase and thus
can be sold at a gain
And since implied vol is almost certainly greater than expected vol (because options work that way), if you expect actual vol to decrease you should almost surely be selling options.
ok, rough around the edges…but, I think I was along the same lines… i.e. the current (lower) implied volatility < Expected (future) volatility means options not as expensive (therefore cheap)
therefore use options, price goes up, and profit. thanks.
CFAI Sample #2 has this question. Their explanation was to use dynamic hedging if current implied volatility > expected. In this case, lengthen duration after a decrease in interest rates by buying
futures. I answered hedge using options and got it wrong…lame.
the second sentence is the tricky one, at least i don´t understand it ok to the volatility reason and options being cheap or expensive (regardless of the difference between implied volatility and
historical volatility), but I don´t see why buying futures when interest rates have gone down and viceversa This is one of the things I will memorize
hala_madrid Wrote: ------------------------------------------------------- > the second sentence is the tricky one, at least i > don´t understand it > > ok to the volatility reason and options being
> cheap or expensive (regardless of the difference > between implied volatility and historical > volatility), but I don´t see why buying futures > when interest rates have gone down and viceversa > >
This is one of the things I will memorize The reason is to hedge negative convexity, when IR goes down negative convexity kicks in and your duration is less then you want it to be, so you buy futures
to extend the duration and vice versa
makes sense, thanks CSK
let me see if I can give a brief synopsis; if interest rates are rising, because we are long the MBS/asset we want to short; to hedge to protect the value of the MBS. however, we do not hedge the
spread risk away. why? because that is the reason for holding the MBS…the additonal yield now, In regards to interest rates falling, to help keep in check the nonlinear (negative convexity) movement
of the MBS, we would short 2 futures. (note; one could turn out to be a long future/bond that we’re using as well to hedge) now… In regards to dynamic hedging. (and I’m not sure what the difference
is between this and a 2 bond hedge?)…i.e when to use this? other then protecting our portfolio AFTER an interest rate change… if interest rate volatility > expected interest rate volatility, the
volatility has made the prepayment option more valuable to the option. hence we would not use options to hedge, but instead futures. we would long/short futures depending on the direction we expect
of interest rates then, we are saying that if interest rate volatility is < expected interest rate volatility we would long options (because options value has not incorporated increase in volatility
in it’s price yet) we would long calls or puts depending on which direction we expect of interest rates. i.e. interest rates falling, long a call on a bond. (is this scenario considered dynamic
hedging as well?? I don’t think so?) that’s what I have so far…any revisions? | {"url":"https://www.analystforum.com/t/volatility-hedging-with-mbs/13419","timestamp":"2024-11-02T01:40:31Z","content_type":"text/html","content_length":"49136","record_id":"<urn:uuid:4a46fdec-186a-4faf-a523-eeebca4e8ef3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00712.warc.gz"} |
Center for Innovative Design & Analysis
Objective of tables
To clearly identify the measures and variables being used in a study and determine levels of confidence in reporting results.
To begin, let us look at an example of a Table 1. Typically, studies begin with a summary of the patient characteristics to show the properties of the sample being studied. This table is often
referred to as "Table 1" and shows characteristics associated with the group of participants. Continuous and categorical variables are depicted in the table.
Suppose there is a drug treatment (Drug X) designed to reduce the risk of stroke among people aged 60 years or older with isolated systolic hypertension.
Table 1: Characteristics of Hypertension Participants
Baseline After 6 Months
Characteristic Active (N=2365) Placebo (N=2371) Total (N=4736) Active (N=2330) Placebo (N=23500 Total (N=4680)
Age, mean (SD), y 71.6 (6.7) 71.5 (6.7) 71.6 (6.7) 72 (6.7) 72 (6.7) 72 (6.7)
Systolic Blood Pressure, mean (SD), mmHg 170.5 (9.5) 170.1 (9.2) 170.3 (9.4) 160.5 (11) 170.1 (9.2) 165.3 (10.1)
Current Smokers (%) 12.6 12.9 12.7 12.5 12.2 12.3
Past Smokers (%) 36.6 37.6 37.1 36.2 36.3 36.2
Never Smokers (%) 50.8 49.6 50.2 51.3 51.5 51.5
General notes
• Label the table: Give each table a number and a title that concisely describes what the table represents.
• Footnote when necessary: Provide footnotes at the bottom of the table to provide explanations of table information.
• Refer to the table in the text: Refer to the table by number ("As Table 1 indicates…").
• Test Statistics: Report test statistics (t, F, x2, p) to 2 decimal places.
• Numerical precision should be consistent throughout the paper:
Summary statistics (such as means) should not be given to more than one extra decimal place over the raw data.
• Standard deviations or standard errors may warrant more precise values.
• Regression analysis results also warrant precise values.
• Continuous variables: Summarize with means and report the standard deviation
• Categorical variables: Summarize with frequencies and percentages.
Simple graphs and correlation/association
Study type
The use of simple graphs/visuals is a great way to get started with a set of data. For example, in the set of histograms above in which the Distribution of SBP is shown by gender, it spurs the
question of why the two groups look a little different. By drawing out the linear relationship, one can start to see patterns of a positive or negative correlation between these two variables to
start teasing out additional thoughts on why this would be true.
Confidence intervals
Confidence intervals are a range of values in which we feel confident that the true parameter is contained. It is important to distinguish confidence intervals from probabilities as we cannot attach
a probability to the true value of the statistic based on a single sample of data. Confidence intervals are calculated in different ways depending on the type of statistic we are evaluating. In the
example above, Systolic Blood Pressure (SBP) is a continuous variable for which a mean and standard deviation are given for the total participants in the study.
To calculate a confidence interval: Unknown mean (μ) and known standard deviation (σ)
Note: x is the sample mean and the critical value (Z) for a 95% Confidence Interval is 1.96.
To calculate a confidence interval: Unknown mean (μ) and unknown standard deviation (σ)
Benefit of confidence intervals
• The lack of precision of a sample statistic (for example: a mean) which results from the degree of variability in the factor being investigated and the limited study size, can be shown by a
confidence interval.
The width of a confidence interval is based on the standard error and the sample size.
• For more information, visit
Measures of difference and simple graphical presentation of results
Categorical vs. continuous data
• Categorical data is data that takes on a limited number of values. As the name would imply, the data fit into discrete categories. For example procedure type = "dental" and hospital = "Children’s
Hospital" would categorical data. Categorical data can also be measured numerically. For example, ASA (an anesthesiology health score) can take on the values 1-5 and each patient is put into a
category based on their health.
• Continuous data is data that can take on many values - too many to create specific categories. For example, pediatric doctors measure each patient’s height in centimeters, which can take on many
different values up to 150 cm. So, height would be continuous data. Other examples include volume, cost, and time.
Analyzing Categorical Data
Let’s go back to our ASA example which can take on the values 1-5. You want to look at the ASA values of patients at Children’s Hospital compared to those at University of Colorado Hospital. Your
table would look something like this:
ASA Children's Hospital University of Colorado Hospital
This table describes how many patients fall into each category based on ASA value (the outcome) and hospital (the exposure). Now, you want to analyze your data—there are many ways to do this based on
your research question.
Chi-square test
The chi-square test is used for categorical variables and tests whether ASA level is associated with hospital. In our example, we would be testing if ASA level differed between the two hospitals. It
is important to pay attention to the cell counts, as the test expects at least 5 values in each cell. Since the cells coinciding with ASA values of 5 are 0 and 2, this test would not work. When the
chi-square test is not an option, one may use Fisher’s exact test.
Risk ratio (RR)
Describes the risk of a certain event happening in one group compared to another.
• Risk 1=P(Disease | Exposed to causal factor)
• Risk 2=P(Disease | Not exposed to causal factor)
• Relative Risk Ratio = Risk1/Risk2
The risk of disease for a person exposed to the causal factor is (RR) times greater than for a person who was not exposed.
• Risk Difference = Risk1-Risk2 (interpreted as the excess risk in group one vs. group two)
• Odds Ratio (OR): The odds of having an event divided by the odds of not having that event.
• Odds Event 1: those who have cancer who got the treatment/those with cancer who did not get the treatment (A/B)
• Odds Event 2: those without cancer who got the treatment/those without cancer who did not get the treatment (C/D)
• OR = (A/B)/(C/D)
Interpretation: the odds of cancer are OR times lower in the group who got the treatment compared to the group that didn’t get the treatment.
Analyzing categorical data
• There are two types of common t-tests. If you are looking to see if you sample has a different mean than a population value, use a one sample t-test. 2) If you are looking to see if two
populations have different means, use the two sample t-test. If your data is dependent (i.e. two measurements on the same person), use a paired two-sample t-test. Note: An ANOVA (analysis of
variance) is an equivalent way to compare 3 or more groups
Regression techniques
• These are used when you are interested in the relationship between an outcome and its predictors (simple). Ex: does age predict ASA scores? Regression analyses are useful when you are interested
in the effect of more than one predictor variable (multiple). Ex: do age and hospital predict ASA score?
Confidence intervals
Once you calculate an estimated mean or measure of association for categorical data, you can also calculate a confidence interval around that mean. This is important because it gives a measure of
uncertainty about where the true mean lies. Once a confidence interval is calculated, it is read as "We have a certain level (i.e., 95%) of confidence that the true population parameter is within
this interval."
Graphing data
• Scatterplot: used when graphing two continuous variables
• Bar Chart: visually compare groups
• Histogram: displays the distribution
• Box Plot: nicely displays mean, median, interquartile range, and outliers
• Line Graphs: primarily used for longitudinal data (tracking over time)
Know your assumptions. In general, we assume that the data were collected without bias, it is normally distributed for continuous variables and there are no unmeasured variables that actually explain
the difference between the two means.
Roberts, Donna. "Qualitative vs Quantitative Data." Qualitative vs Quantitative Data. 2012. Web. 13 Oct. 2015.
"Risk Differences and Rate Differences." Risk Difference. Boston University School of Public Health, 16 Sept. 2015. Web. 13 Oct. 2015.
Szumilas, Magdalena. “Explaining Odds Ratios." Journal of the Canadian Academy of Child and Adolescent Psychiatry 19.3 (2010): 227–229. Print.
"Understanding Data Concepts." Understanding Data Concepts. Canada.ca, 9 Dec. 2013. Web. 13 Oct. 2015. | {"url":"https://coloradosph.cuanschutz.edu/research-and-practice/centers-programs/cida/learning/statistics-tutorials","timestamp":"2024-11-13T07:34:43Z","content_type":"text/html","content_length":"94396","record_id":"<urn:uuid:858b0267-1486-4d8e-9c80-70c02540ebee>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00632.warc.gz"} |
This is level 5, simplify the composite functions. You can earn a trophy if you get at least 9 correct and you do this activity online.
Here are the functions to be used in the questions that follow.
\(f(x) = 2x+4\), \(g(x) = 5x^2\) and \(h(x) = 7 - 3x\)
Description of Levels
Level 1 - Describe function machines using function notation.
Level 2 - Evaluate the given functions.
Level 3 - Solve the equations given in function notation.
Level 4 - Find the inverse of the given functions.
Level 5 - Simplify the composite functions.
Level 6 - Mixed questions.
Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent.
The following notes are intended to be a reminder or revision of the concepts and are not intended to be a substitute for a teacher or good textbook.
Level 1: Describe function machines using function notation.
Function notation is quite different to the algebraic notation you have learnt involving brackets. \(f(x)\) does not mean the value of f multiplied by the value of x. In this case f is the name of
the function and you would read \(f(x) = x^2\) as "f of x equals x squared".
In terms of function machines, if the input is \(x\) then the output is \(f(x)\).
\(x \to \)\( + 3 \)\( \to \)\( \times 4 \)\( \to f(x)\)
In this case 3 is added to \(x\) and then the result is multiplied by 4 to give \(f(x)\)
\( (x+3) \times 4 = f(x) \)
\( f(x) = 4(x+3) \)
Level 2: Evaluate the given functions.
if \(f(x)=x^2 + 3\) calculate the value of \(f(6)\)
This means replace the \(x\) with a 6 in the given function to obtain the result.
\(f(6) = 6^2+3\)
\(f(6) = 39\)
Level 3: Solve the equations given in function notation.
\(f(x)=3(x+7) \) find \(x\) if \(f(x) = 30\)
\(x+7 = 10\)
\(x = 3\)
Level 4: Find the inverse of the given functions.
The inverse of a function, written as \(f^{-1}(x) \) can be thought of as a way to 'undo' the function. If the function is written as a function machine, the inverse can be thought of as working
backwards with the output becomming the input and the input becoming the output.
\( f(x) = 4(x+3) \)
\(x \to \)\( + 3 \)\( \to \)\( \times 4 \)\( \to f(x)\)
\( f^{-1}(x) \leftarrow \)\( - 3 \)\( \leftarrow \)\( \div 4 \)\( \leftarrow x \)
\( f^{-1}(x) = \frac{x}{4} - 3 \)
A quicker way of finding the inverse of \(f(x)\) is to replace the \(f(x)\) with \(x\) on the left side of the equals sign and replace the \(x\) with \( f^{-1}(x) \) on the right side of the equals
sign. Then rearrange the equation to make \( f^{-1}(x) \) the subject.
Level 5: Simplify the composite functions.
A composite function contains two functions combined into a single function. One function is applied to the result of the other function. You should evaluate the function closest to \(x\) first.
if \(f(x)=2x+7\) and \(g(x)=5x^2\) find \(fg(3)\)
\(g(3) = 5 \times 3^2\)
\(g(3) = 5 \times 9\)
\(g(3) = 45\)
\(f(45) = 2 \times 45 + 7\)
\(f(45) = 97\)
so \( fg(3) = 97\)
if \(f(x)=x+2\) and \(g(x)=3x^2\) find \(gf(x)\)
\( gf(x) = 3(x+2)^2\)
\( gf(x) = 3(x^2+4x+4) \)
\( gf(x) = 3x^2+12x+12 \)
Level 6: Mixed questions.
Find \(f(x-2)\) if \(f(x)=5x^2+3\)
\(f(x-2) =5(x-2)^2+3\)
\(f(x-2) =5(x^2-4x+4)+3\)
\(f(x-2) =5x^2-20x+20+3\)
\(f(x-2) =5x^2-20x+23\)
Did you know some calculators can apply functions?
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can
double-click the 'Check' button to make it float at the bottom of your screen.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent. | {"url":"https://transum.org/Maths/Exercise/Functions.asp?Level=5","timestamp":"2024-11-09T03:45:38Z","content_type":"text/html","content_length":"58363","record_id":"<urn:uuid:65c907dc-dc7e-45c3-9c68-03ce71c3b545>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00514.warc.gz"} |
Scramblers and their implementation in GNUradio
A scrambler is a function that is applied to a sequence of data before transmitting with the goal of making this data more “random-like”. For instance, scrambling avoids long runs of the bits 0 or 1
only, which may make the receiver loose synchronization or cause spectral concentration of the signal. The receiver applies the inverse function, which is called descrambler, to recover the original
data. The documentation for the scrambler blocks in GNUradio is not very good and one may need to take a look at the implementation of these blocks to get their parameters right. Here, I’ll try to
explain a bit of the mathematics behind scramblers and the peculiarities of their implementation in GNUradio.
There are two types of scramblers, multiplicative or self-synchronizing and additive or synchronous. Both may be explained in terms of Linear Feedback Shift Registers but I will try to do an
alternative exposition without using this concept.
Let \(M\) be a module over a ring \(R\). Usually, \(M\) and \(R\) will be \(\mathbb{Z}/2\mathbb{Z}\) in the applications (this is the ring consisting of the elements \(\{0,1\}\), with addition given
by the XOR Boolean operation and multiplication given by the AND Boolean operation). Both types of scramblers are defined by some fixed coefficients \(\alpha_1,\ldots,\alpha_l \in R\).
The multiplicative scrambler transforms the sequence \(\{x_n\}_{n\geq 0} \subset M\) into the sequence \(\{y_n\}_{n\geq 0}\) given by the recurrence\[y_n = x_n + \sum_{k=1}^l \alpha_k y_{n-k}.\]Here,
the values \(y_{-1},\ldots,y_{-l}\) have to be fixed beforehand and are known as the seed of the scrambler.
The multiplicative descrambler works as follows: it transforms a sequence \(\{t_n\}_{n\geq 0} \subset M\) into the sequence \(\{z_n\}_{n \geq 0}\) given by\[z_n = t_n – \sum_{k=1}^l \alpha_k t_{n-k}.
\]Here, the values \(t_{-1},\ldots,t_{-l}\) are fixed beforehand (these are the seed of the descrambler). To see that this function does, in fact, revert the scrambling process, assume that the
receiver starts receiving and descrambling at some point in time, so that \(t_n = y_{n+N}\) for \(n \geq 0\). Then we see that \(z_n = x_{n+N}\) for \(n \geq l\). Thus, the descrambler recovers the
stream of data (obviously shifted \(N\) units in time), except for the first \(l\) elements.
We have seen that the multiplicative descrambler can start descrambling at any time, without the need to synchronize with the stream of data. For this reason, the multiplicative scrambler/descrambler
is called self-synchronizing. Another remark is that the seeds used for the scrambler and descrambler make no effect in practice and any values can be used as a seed. The descrambler “loses” the
first \(l\) elements of the data, but this is not a problem in applications.
The additive scrambler takes the sequence \(\{x_n\}_{n\geq 0}\) and transforms it into the sequence \(\{y_n\}_{n\geq 0}\) given by\[y_n = x_n + w_n,\]where \(w_n\) is defined by the recurrence\[w_n =
\sum_{k=1}^l \alpha_k w_{n-k}.\]The values \(w_{-1},\ldots,w_{-l}\) are known as the seed and are fixed beforehand.
Now we see that, in contrast to the multiplicative scrambler, there is no way to descramble this sequence in a self-synchronizing manner. In fact, the only way possible way to descramble it is that
the scrambler and descrambler start at the same time (so the descrambler input \(\{t_n\}\) is \(t_n = y_n\)). The descrambler generates the same sequence \(\{w_n\}\) using the same seed, and takes
the sequence \(\{t_n\}\) into \(\{z_n\}\) by\[z_n = t_n – w_n.\] Then, clearly \(z_n = x_n\) for all \(n\geq 0\).
Therefore, the additive scrambler and descrambler must start at the same time and use the same seed. For this reason, this scrambler is called synchronous. In applications, an unscrambled
synchronization word is sent before the scrambled data to signal the descrambler that it has to start working. Another remark is that when the characteristic of \(R\) is 2 the additive scrambler and
descrambler are the same function.
It is usual to give the coefficients \(\alpha_k\) as the coefficients of a polynomial\[p(x) = 1 – \sum_{k=1}^{l} \alpha_k x^k.\] (The reason for this is quite interesting, but it is outside the scope
of this post). When \(M = R = \mathbb{Z}/2\mathbb{Z}\), the coefficients of \(p\) are \(0\) or \(1\), so it is usual to encode them as the digits of a binary number. However, there are several
possible ways to do so. First, there is the choice of order: whether later bits correspond to higher or lower powers of \(x\). Then, the independent term of \(p\) is always \(1\), so it is possible
to omit this coefficient in the binary representation. Also, the leading term of \(p\) is always \(1\), because we can assume that \(l\) is the degree of \(p\). Then, it is possible to omit this
coefficient as well.
The choice of the binary representation to use is ultimately tied to the implementation of the the scrambler using a shift register, as there are several possible ways to do so. For instance, this
huge list uses the notation \(1\alpha_1\ldots\alpha_{l-1}\). For instance, \(1 + x + x^4\) is is represented as 0xC.
The documentation from GNUradio suggests that the notation used in GNUradio is \(\alpha_l\ldots\alpha_11\). For instance, it says that \(x^4+x^3+1\) is represented as 0x19. However, a careful look at
the code reveals that this is not the case. The following method next_bit_scramble() implements the multiplicative scrambler.
unsigned char next_bit_scramble(unsigned char input)
unsigned char output = d_shift_register & 1;
unsigned char newbit = (popCount( d_shift_register & d_mask )%2)^(input & 1);
d_shift_register = ((d_shift_register>>1) | (newbit<<d_shift_register_length));
return output;
The function popCount() just counts the number of bits that are 1. We see that d_shift_register is used to store the bits \(0\ldots0y_{n-1}\ldots y_{n-l}\) (big-endian order). The variable d_mask
stores the polynomial. Hence, we see that the representation used for \(p\) is \(\alpha_1\ldots\alpha_l\). For instance, \(x^4+x^3+1\) would be represented as 0x3. The polynomial \(x^{17} + x^{12} +
1\), which is used in 9k6 baud FSK AX.25, is represented as 0x21. When using this representation, one should also indicate the degree of \(p\). The way to indicate this in GNUradio is by means of the
variable d_shift_register_length. This should be set to \(\operatorname{deg}(p) – 1\).
A similar notation problem happens for the seed value. Although one can use any seed for the multiplicative scrambler/descrambler, it is necessary to use the correct seed for the additive scrambler.
Moreover, above we have defined the seed as the values \(w_{-1},\ldots,w_{-l}\). It is also possible to use the values \(w_{l-1},\ldots,w_0\) instead. If \(\alpha_l\) is invertible, then one can
obtain \(w_{-1},\ldots,w_{-l}\) in terms of \(w_{l-1},\ldots,w_0\) and vice versa. Thus, the choice of using one definition of the seed or the other one depends more on how the scrambler is
In GNUradio, the seed is defined as \(w_{l-1},\ldots,w_0\). The binary representation used for the seed is similar to the notation used for polynomials: it is represented as the binary number \(w_
{l-1}\ldots w_0\).
4 comments
4. Thank you for this perfect explanation.
I have been looking at the descrambler used in GR-OUTERNET and want to create a GNU RADIO flowgraph which uses the scrambler used in it.
What should be the polynomial and the settings for multiplicative scramber block?
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ETC 14 - OpenConf Peer Review & Conference Management System
Full Program
Length scale to determine the rate of energy dissipation in turbulence
The mean rate of energy dissipation ⟨ε⟩ per unit mass of turbulence is often written in the form of ⟨ε⟩ = Cu⟨u2⟩3/2/Lu, where ⟨u2⟩1/2 is the root-mean-square fluctuation of the longitudinal velocity
u and Lu is its correlation length. However, Cu is known to depend on the large-scale configuration of the flow. We define the correlation length Lu2 of the local energy u2 and find that Cu2 = ⟨ε⟩Lu2
/⟨u2 ⟩3/2 does not depend on the flow configuration. The independence from the flow configuration is also found for the two-point velocity correlation and so on when Lu2 is used to normalize the
Hideaki Mouri
Meteorological Research Institute | {"url":"http://etc14.ens-lyon.fr/openconf/modules/request.php?module=oc_program&action=summary.php&id=32","timestamp":"2024-11-09T12:48:26Z","content_type":"application/xhtml+xml","content_length":"3412","record_id":"<urn:uuid:e576a528-94e9-4715-bd05-38953fd5df1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00511.warc.gz"} |
Understanding Bubble Sort Algorithm in C# for Integers
When it comes to sorting algorithms, the bubble sort algorithm is one of the simplest ones to understand and implement. In this blog post, we will explore how to implement the bubble sort algorithm
in C# specifically for sorting integers.
What is Bubble Sort Algorithm?
Bubble sort is a comparison-based sorting algorithm where each pair of adjacent elements is compared and the elements are swapped if they are in the wrong order. This process is repeated until the
entire array is sorted. The algorithm gets its name because smaller elements "bubble" to the top of the array during each iteration.
Let's now dive into the C# implementation of the bubble sort algorithm for sorting integers.
using System;
class BubbleSort
static void Main()
int[] array = { 64, 34, 25, 12, 22, 11, 90 };
Console.WriteLine("Unsorted Array:");
Console.WriteLine("\nSorted Array:");
static void BubbleSortArray(int[] arr)
int n = arr.Length;
for (int i = 0; i < n - 1; i++)
for (int j = 0; j < n - i - 1; j++)
if (arr[j] > arr[j + 1])
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
static void PrintArray(int[] arr)
foreach (var item in arr)
Console.Write(item + " ");
In the above C# code snippet, we define a BubbleSort class that contains the main method. We initialize an array of integers, perform the bubble sort algorithm on the array, and then print out the
sorted array.
Implementing the bubble sort algorithm in C# for sorting integers is a great way to understand sorting algorithms and improve your problem-solving skills. While bubble sort is not the most efficient
algorithm for large datasets, it is a good starting point for beginners to grasp the concept of sorting.
In this blog post, we covered the basics of the bubble sort algorithm and provided a simple C# implementation for sorting integers. Experiment with different arrays and test cases to deepen your
understanding of how the algorithm works.
Happy coding! | {"url":"https://www.webdevtutor.net/blog/c-sharp-bubble-sort-algorithm-int","timestamp":"2024-11-10T06:10:08Z","content_type":"text/html","content_length":"13124","record_id":"<urn:uuid:f9c324a7-702a-4f65-9402-619d454e413c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00615.warc.gz"} |
Problem of the Day--Pricing Contingencies
Assume that you are pricing a firm-fixed-price contract using cost analysis. The prospective contractor has included a contingency of $100,000 in its cost proposal of the type described at FAR
Those that may arise from presently known and existing conditions, the effects of which are foreseeable within reasonable limits of accuracy; e.g.,anticipated costs of rejects and defective work.
Contingencies of this category are to be included in the estimates of future costs so as to provide the best estimate of performance cost.
There is a 90% chance that this contingency will occur. If it occurs, there's a 100% chance it will cost $100,000.
The prospective contractor can take Precaution A, which will cost $50,000. If the contingency occurs, Precaution A would reduce the chance of the contingency costing $100,000 to 30% (there would be a
70% chance the contingency would cost $0).
The prospective contractor can take Precaution B, which will cost $75,000. If the contingency occurs, Precaution B would reduce the chance of the contingency costing $100,000 to 10% (there would be
a 90% chance the contingency would cost $0).
The prospective contractor is free to take Precaution A, Precaution B, or do nothing.
What amount for this contingency would you allow in the contract price?
You may ask for more facts if you'd like or ask to make an assumption. Do not fight the hypothetical. Enjoy.
8 Comments
Recommended Comments
It looks to me like an expected value problem.
In the scenario with no precaution the expected value is the chance of the contingency occurring, times the expected value if it does occur, so .9*(1*100,000)=90,000.
Precaution A gives an expected value of 50,000+(.9*(.3*100,000))=77,000
Precuation B gives an expected value of 75,000+(.9*(.1*100,000))=84,000
Since the scenario with the lowest expected value is Precaution A, I would allow the contractor to price the contingency at a maximum of $77,000.
But wait. If the contractor did not price Precaution A or B into its FFP, then those hypothetical precautions are meaningless. Under Precaution A, you should allow $77,000 + $50,000 = $127,000. You
need to allow the contractor to price in the cost of risk mitigation.
The total of $77k includes the $50k to pay for Precaution A, plus the expected value of the contingency, which is $27k (90% x 30% x $100k). It is interesting that there is no single scenario in which
the contractor will actually experience a cost of $77k. Either they will pay the $50k for precaution A and then contingency either won't occur, or will cost $0, in which case their total cost will be
only $50k, or they will pay the $50k and have bad luck anyway and still experience the $100k contingency, in which case their cost is $150k. Only if they run the scenario multiple times will they all
average out to $77k in the long run.
Have any of ye land lubbers had experience using this method? Would a contractor actually be willin' to accept eatin' the $73k doubloons in Scenario A and $91k in Scenario B? I may be cynical due to
me sole-source encounters with dodgy contractors.
5 hours ago, Contracting Pirate said:
Have any of ye land lubbers had experience using this method? Would a contractor actually be willin' to accept eatin' the $73k doubloons in Scenario A and $91k in Scenario B? I may be cynical
due to me sole-source encounters with dodgy contractors.
The contractor doesn't have to eat the loss. They could buy insurance, right?
On 12/4/2018 at 6:44 PM, Don Mansfield said:
From the seller's perspective, at a 90% probability, I'd consider it a sure thing and price it assuming 100% of the costs would be incurred.
21 hours ago, lotus said:
From the seller's perspective, at a 90% probability, I'd consider it a sure thing and price it assuming 100% of the costs would be incurred.
That's exactly what the seller did in the problem. | {"url":"https://www.wifcon.com/discussion/index.php?/blogs/entry/4002-problem-of-the-day-pricing-contingencies/","timestamp":"2024-11-02T18:09:29Z","content_type":"text/html","content_length":"133208","record_id":"<urn:uuid:d7bfce29-6513-4d8f-9695-0a09a0800264>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00394.warc.gz"} |
53rd Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics
Bulletin of the American Physical Society
53rd Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics
Volume 67, Number 7
Monday–Friday, May 30–June 3 2022; Orlando, Florida
Session M11: Focus Session: Measurement Induced Phase Transitions and Quantum Simulation of Phase Transitions Hide Abstracts
Focus Live Streamed
Chair: Steven Rolston, University of Maryland, College Park
Room: Grand Ballroom E
Wednesday, M11.00001: Measurement induced phase transition in ground states
June 1, Invited Speaker: Ehud Altman
2:00PM -
An external observer can have a profound impact on the quantum state that it is observing. Recent studies of random quantum circuits have shown that if the object of observation is a large
many-body system, then mere observation can induce a phase transition in the large scale quantum correlations of the system. In this talk I will argue that observing a quantum ground state
can similarly lead to a phase transition in the structure of long range correlations in the state. As a concrete example I will consider the impact of weak measurement of the density of a
one dimensional quantum liquid applied for a finite time.
Wednesday, M11.00002: Observing a purification phase transition with a trapped ion quantum computer
June 1, Invited Speaker: Crystal Noel
2:30PM -
Many-body open quantum systems balance internal dynamics against decoherence from interactions with an environment. Here, we explore this balance via random quantum circuits implemented on
a trapped-ion quantum computer, where the system evolution is represented by unitary gates with interspersed projective measurements. As the measurement rate is varied, a purification
phase transition is predicted to emerge at a critical point akin to a fault-tolerent threshold. We probe the "pure'' phase, where the system is rapidly projected to a deterministic state
conditioned on the measurement outcomes, and the "mixed'' or "coding'' phase, where the initial state becomes partially encoded into a quantum error correcting codespace. We find evidence
of the two phases and show numerically that, with modest system scaling, critical properties of the transition emerge.
Wednesday, M11.00003: Experimental Realization of Rabi-Hubbard Model with Trapped Ions
June 1, Quanxin Mei, Bowen Li, Yukai Wu, Minglei Cai, Ye Wang, Lin Yao, Zichao Zhou, Luming Duan
3:00PM - Quantum simulation provides important tools in studying strongly correlated many-body systems with controllable parameters. As a hybrid of two fundamental models in quantum optics and in
3:12PM condensed matter physics, Rabi-Hubbard model demonstrates rich physics through the competition between local spin-boson interactions and long-range boson hopping. Here we report an
experimental realization of the Rabi-Hubbard model using up to 16 trapped ions and present a controlled study of its equilibrium properties and quantum dynamics. We observe the
ground-state quantum phase transition by slowly quenching the coupling strength, and measure the quantum dynamical evolution in various parameter regimes. With the magnetization and the
spin-spin correlation as probes, we verify the prediction of the model Hamiltonian by comparing theoretical results in small system sizes with experimental observations. For larger-size
systems of 16 ions and 16 phonon modes, the effective Hilbert space dimension exceedss 2^57, whose dynamics is intractable for classical supercomputers.
Wednesday, M11.00004: Non-equilibrium critical phenomena in a trapped-ion quantum simulator
June 1, Arinjoy De, Patrick Cook, William N Morong, Kate S Collins, Daniel A. Paz, Paraj Titum, Wen Lin Tan, Guido Pagano, Alexey V Gorshkov, Mohammad F. Maghrebi, Christopher Monroe
3:12PM - Recent work has predicted that quenched near-integrable systems can exhibit dynamics associated with thermal, quantum, or purely non-equilibrium phase transitions, depending on the initial
3:24PM state [1]. Using a trapped-ion quantum simulator with intrinsic long-range interactions, we investigate collective non-equilibrium properties of critical fluctuations after quantum
quenches. In particular, we probe the scaling behavior of fluctuations near the critical point of the ground-state disorder-to-order phase transition, after single and double quenches of
the transverse field in a long-range Ising Hamiltonian. With system sizes of up to 50 ions, we show that both the post-quench fluctuation magnitude and dynamics scale with system size with
distinct critical exponents, charaterizing the type of phase-transition. Furthermore we demonstrate that the critical exponents after a single and a double quenches are different and
correspond to effectively thermal and truly non-equilibrium behavior, respectively. Our results demonstrate the ability of quantum simulators to explore universal scaling beyond the
equilibrium context.
[1] Paraj Titum and Mohammad F. Maghrebi, Phys. Rev. Lett. 125, 040602 (2020).
Wednesday, M11.00005: Landau-Forbidden Quantum Criticality in Rydberg Atom Arrays
June 1, Jong Yeon Lee, Joshua Ramette, Wen Wei Ho, Soonwon Choi
3:24PM - A continuous transition between two distinct symmetry broken phases is generally forbidden to occur within the celebrated Landau-Ginzburg-Wilson theory of phase transitions. However, a
3:36PM quantum effect can intertwine the two symmetries, giving rise to a novel scenario called deconfined quantum criticality. In this work, we propose a model of a one-dimensional array of
strongly-interacting, individually trapped neutral atoms interacting via Rydberg states, and demonstrate through extensive numerical simulations that its ground state phase diagram
exhibits deconfined quantum criticality in certain parameter regimes. Moreover, we show how an enlarged, emergent continuous symmetry arises at these critical points, which can be directly
observed via studying the joint distribution of two competing order parameters in the natural measurement basis. Our findings highlight quantum simulators of Rydberg atoms not only as
natural platforms to experimentally realize such exotic phenomena, but also as unique ones as they allow access to physical properties not accessible in traditional condensed matter
Wednesday, M11.00006: Diagnosing dynamical phase transitions in Spin-1 Bose-Einstein condensate using classical and quantum information
June 1, Qingze Guan, Robert Lewis-Swan
3:36PM - Non-equilibrium dynamics has been used to probe quantum many-body physics and to perform state engineering which finds broad applications in various aspects of quantum technologies.
3:48PM Dynamical phase transitions (DPT), which signal different dynamical structures of a quantum system by tuning control parameters, provides a powerful tool to classify many-body dynamics in
closed quantum systems. In this work, we identify an order parameter to diagnose the DPT in quench dynamics of a Spin-1 Bose-Einstein condensate for classical initial states (coherent spin
states) which is motivated by a mean-field picture of the double-well structure in the phase space. Beyond the classical regime, a quantum probe based on the quantum Fisher information
(QFI) is shown to be able to capture such a DPT for a broader type of initial states including both coherent spin states and Fock states. The classical Fisher information (CFI), as is more
realistic to be measured in Spin-1 Bose-Einstein condensate nowadays, is also shown to mimic the role of QFI to some degrees and is useful in diagnosing such a DPT. Both the CFI and the
QFI make a smooth connection between DPTs and quantum sensing.
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Student[Statistics][OneSampleTTest] Overview
overview of the One Sample T-Test
See Also
• One Sample T Test is used to test if the sample studied follows a normal distribution [OneSampleTTest]
when its standard deviation is unknown and the mean is known. The mean is equal to the
test value based on the sample drawn.
If the standard deviation is known for the assumed normal distribution, One Sample Z Test
is used instead.
• Requirements for using One Sample T Test:
1. The sample studied is assumed to follow a normal distribution.
2. The standard deviation of the assumed normal distribution is unknown.
• The formula is:
$T=\frac{\left(\mathrm{Mean}⁡\left(X\right)-{\mathrm{\mu }}_{0}\right)
where $X$ is the sample, ${\mathrm{\mu }}_{0}$ is the test value of mean, $s$ is the
sample standard deviation, $N$ is the sample size, and $T$ follows Student's T
distribution with $N-1$ degrees of freedom.
A research team did a survey on male smokers over 20 in Ontario to figure out the age when
they started to smoke. Knowing that the starting age is normally distributed, a statistical
test was run to test whether the average starting age is 15. 1000 male smokers were
randomly selected and interviewed. The result shows that the sample standard deviation of
the 1000 smokers is 5.144 and their average age of beginning to smoke is 16.
1. Determine the null hypothesis:
Null Hypothesis: ${\mathrm{\mu }}_{0}=15$ (the actual mean)
2. Substitute the information into the formula:
3. Compute the p-value:
InvisibleTimes;6.1475\right)=p=1.136158⁢\cdot {10}^{-9}$, $T&
4. Draw the conclusion:
This statistical test provides evidence that the null hypothesis is false, so we reject
the null hypothesis.
See Also
• One Sample T Test is used to test if the sample studied follows a normal distribution [OneSampleTTest]
when its standard deviation is unknown and the mean is known. The mean is equal to the
test value based on the sample drawn.
If the standard deviation is known for the assumed normal distribution, One Sample Z Test
is used instead.
• Requirements for using One Sample T Test:
1. The sample studied is assumed to follow a normal distribution.
2. The standard deviation of the assumed normal distribution is unknown.
• The formula is:
$T=\frac{\left(\mathrm{Mean}⁡\left(X\right)-{\mathrm{\mu }}_{0}\right)
where $X$ is the sample, ${\mathrm{\mu }}_{0}$ is the test value of mean, $s$ is the
sample standard deviation, $N$ is the sample size, and $T$ follows Student's T
distribution with $N-1$ degrees of freedom.
A research team did a survey on male smokers over 20 in Ontario to figure out the age when
they started to smoke. Knowing that the starting age is normally distributed, a statistical
test was run to test whether the average starting age is 15. 1000 male smokers were
randomly selected and interviewed. The result shows that the sample standard deviation of
the 1000 smokers is 5.144 and their average age of beginning to smoke is 16.
1. Determine the null hypothesis:
Null Hypothesis: ${\mathrm{\mu }}_{0}=15$ (the actual mean)
2. Substitute the information into the formula:
3. Compute the p-value:
InvisibleTimes;6.1475\right)=p=1.136158⁢\cdot {10}^{-9}$, $T&
4. Draw the conclusion:
This statistical test provides evidence that the null hypothesis is false, so we reject
the null hypothesis.
• One Sample T Test is used to test if the sample studied follows a normal distribution when its standard deviation is unknown and the mean is known. The mean is equal to the test value based on the
sample drawn.
If the standard deviation is known for the assumed normal distribution, One Sample Z Test is used instead.
• Requirements for using One Sample T Test:
1. The sample studied is assumed to follow a normal distribution.
2. The standard deviation of the assumed normal distribution is unknown.
• The formula is:
$T=\frac{\left(\mathrm{Mean}⁡\left(X\right)-{\mathrm{\mu }}_{0}\right)⁢\sqrt{N}}{s}$
where $X$ is the sample, ${\mathrm{\mu }}_{0}$ is the test value of mean, $s$ is the sample standard deviation, $N$ is the sample size, and $T$ follows Student's T distribution with $N-1$ degrees
of freedom.
• One Sample T Test is used to test if the sample studied follows a normal distribution when its standard deviation is unknown and the mean is known. The mean is equal to the test value based on the
sample drawn.
One Sample T Test is used to test if the sample studied follows a normal distribution when its standard deviation is unknown and the mean is known. The mean is equal to the test value based on the
sample drawn.
If the standard deviation is known for the assumed normal distribution, One Sample Z Test is used instead.
1. The sample studied is assumed to follow a normal distribution.
The sample studied is assumed to follow a normal distribution.
2. The standard deviation of the assumed normal distribution is unknown.
The standard deviation of the assumed normal distribution is unknown.
where $X$ is the sample, ${\mathrm{\mu }}_{0}$ is the test value of mean, $s$ is the sample standard deviation, $N$ is the sample size, and $T$ follows Student's T distribution with $N-1$ degrees of
A research team did a survey on male smokers over 20 in Ontario to figure out the age when they started to smoke. Knowing that the starting age is normally distributed, a statistical test was run to
test whether the average starting age is 15. 1000 male smokers were randomly selected and interviewed. The result shows that the sample standard deviation of the 1000 smokers is 5.144 and their
average age of beginning to smoke is 16.
1. Determine the null hypothesis:
Null Hypothesis: ${\mathrm{\mu }}_{0}=15$ (the actual mean)
2. Substitute the information into the formula:
3. Compute the p-value:
1.136158⁢\cdot {10}^{-9}$, $T⁢˜⁢\mathrm{StudentT}\left(999\right)$
4. Draw the conclusion:
This statistical test provides evidence that the null hypothesis is false, so we reject the null hypothesis.
A research team did a survey on male smokers over 20 in Ontario to figure out the age when they started to smoke. Knowing that the starting age is normally distributed, a statistical test was run to
test whether the average starting age is 15. 1000 male smokers were randomly selected and interviewed. The result shows that the sample standard deviation of the 1000 smokers is 5.144 and their
average age of beginning to smoke is 16.
This statistical test provides evidence that the null hypothesis is false, so we reject the null hypothesis. | {"url":"https://www.maplesoft.com/support/help/view.aspx?path=Student,Statistics,OneSampleTTest,overview&L=E","timestamp":"2024-11-11T10:59:32Z","content_type":"text/html","content_length":"118231","record_id":"<urn:uuid:48a982ae-1a8f-4ed4-b982-777aff009217>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00659.warc.gz"} |
Simulation parameters
A short name for your simulation to help identify it in the queue so that you can find your results. For example, Gr-Si/NCA battery 1C discharge.
More space for additional description. For example, you might provide a citation for the parameters you are using.
For constant current, type a single value (for example, -0.015). Positive values correspond to cell charging where the anode is lithiating and the cathode is delithiating. Negative values correspond
to discharge. For time-varying currents, provide a two-column table (the first column is the time in seconds, the second column is the current in Amperes). The table can be copied and pasted into the
field directly from a spreadsheet (e.g. MS Excel). The values between the points will be linearly interpolated.
This option is recommended if you have a time-varying current; restrict the timestep to be finer than the resolution in the your current demand.
Spatial discretisation method:
Finite Volume discretisation is the most common method but it has only 1^st order of approximation. An alternative discretisation scheme uses Finite Elements in electrolyte and Control Volumes in
solid particles providing 2^nd order of approximation in the electrolyte and in the particles. Both approaches are conservative and therefore total amount of lithium is conserved exactly within the
battery cell.
Simulation stop conditions:
Simulation will stop if one of these conditions triggers.
Minimum allowed voltage (optional), V
Maximum allowed voltage (optional), V
Maximum charge/discharge time, s
Battery cell general parameters
Enter conductivity (S/m) at the reference temperature as a function of Li concentration x (mol/L) or provide a constant value. For square root use sqrt(), exponential function is exp(), x^y is pow
Enter diffusivity (cm^2/s) at the reference temperature as a function of Li concentration x (mol/L) or provide a constant value. For square root use sqrt(), exponential function is exp(), x^y is pow
Electrode cross-sectional area, cm^2
Constant absolute temperature, K
Reference temperature for the Arrhenius temperature dependence, K
Activation energy for conductivity/diffusivity in electrolyte, J·mol^-1
Initial concentration of Li ions in the electrolyte, mol·m^-3
Transference number of the electrolyte
Electrode and Separator parameters
Diffusivity in solid particles, cm^2·s^-1
Enter diffusivity (cm^2/s) at the reference temperature as a function of dimensionless Li concentration x (where x=1 corresponds to x=c[max]) or provide a constant value. For square root use sqrt(),
exponential function is exp(), x^y is pow(x,y), hyperbolic tangent is tanh().
Equilibrium potential in the electrodes, V
Enter equilibrium potential (V) as a function of dimensionless Li concentration x (where x=1 corresponds to x=c[max]). For square root use sqrt(), exponential function is exp(), x^y is pow(x,y),
hyperbolic tangent is tanh().
Constant parameters
Double particle size (optional, click to define) ▼
Output parameters
Provide a list (or a column) of the time points for the output data files and plotting (for example, 1000 2000 3000). If empty, secondary variables (e.g. concentrations, potentials) will be written
and plotted for the initial and the final time step only.
(optional, links to the parametrisation and simulation results will be unavailable on the public Simulation Queue page, only a person with the links will be able to download the results and review
the simulation parameters) | {"url":"https://simulation.dandeliion.com/legacy/simulation/e6a2be55-ea00-4bf0-963d-aaf825e942b4/","timestamp":"2024-11-06T04:49:20Z","content_type":"text/html","content_length":"24381","record_id":"<urn:uuid:f398f851-337c-4dd5-bccd-84bf2d1ee6d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00117.warc.gz"} |
Into Math Grade 3 Module 15 Lesson 1 Answer Key Compare Fractions Using Concrete and Visual Models
We included HMH Into Math Grade 3 Answer Key PDF Module 15 Lesson 1 Compare Fractions Using Concrete and Visual Models to make students experts in learning maths.
HMH Into Math Grade 3 Module 15 Lesson 1 Answer Key Compare Fractions Using Concrete and Visual Models
I Can use concrete and visual models to compare fractions.
Spark Your Learning
Rafael and Frankie each paint 4 equal parts of their doghouse walls. Each wall is the same size. Who uses more paint?
Show one way to find who uses more paint.
Rafael and Frankie each paint 4 equal parts of their doghouse walls
Each wall is of same size
So, Both Rafael and Frankie uses same amount of paint as they both painted same area of wall.
Turn and Talk Toby paints 4 of 6 sections on a small doghouse wall. Nan paints 4 of 6 sections on a large doghouse wall. Toby says that he and Nan used the same amount of paint because \(\frac{4}{6}
\) = \(\frac{4}{6}\). Does Toby’s statement make sense? Why or why not?
No, Toby’s statement is not correct
Toby paints 4 of 6 sections on a small doghouse wall
Nan paints 4 of 6 sections on a large doghouse wall
Toby says that he and Nan used the same amount of paint because \(\frac{4}{6}\) = \(\frac{4}{6}\)
As the walls are not of same size, the parts of both dog houses will be unequal.
So, Toby’s statement is not correct
Nan paints more than Toby.
Build Understanding
Question 1.
The Jasper City Parks Department marks off \(\frac{3}{4}\) of its South Rink for Family Skating. In the North Rink, \(\frac{3}{8}\) of the space is used for Family Skating. Both rinks are the same
size and shape.
Use a concrete or visual model to show and compare the parts of the rinks that are used for Family Skating.
A. Which rink has a larger area for Family Skating? How do you know?
South rink has a larger area for family skating
The Jasper City Parks Department marks off \(\frac{3}{4}\) of its South Rink for Family Skating
In the North Rink, \(\frac{3}{8}\) of the space is used for Family Skating
I shaded the fractions of north and south rinks, compared them.
Therefore, south rink has more area for family skating than the north rink as the shaded part of south rink is greater than north rink.
Connect to Vocabulary
Use these symbols to compare two fractions. means “greater than” = means “equal to”
B. Compare the fractions. \(\frac{3}{4}\)
south rink has more area for family skating than the north rink as the shaded part of south rink is greater than north rink
So, \(\frac{3}{4}\)
Question 2.
Use the visual model to compare \(\frac{4}{8}\) and \(\frac{5}{8}\).
A. Shade to show \(\frac{4}{8}\) and \(\frac{5}{8}\). Which fraction has the larger amount of the whole shaded? How do you know?
I shaded the fractions \(\frac{4}{8}\) and \(\frac{5}{8}\) in the above frames.
B. Write <, >, or =. \(\frac{4}{8}\)
As the whole part is same that fraction will be greater whose numerator that is counted parts is more.
So, \(\frac{4}{8}\)
Question 3.
Emily climbs \(\frac{2}{8}\) of the way up the climbing wall. Ryan climbs \(\frac{2}{3}\) of the way up the wall.
A. Show how you can compare the fractions on the number lines. Label the distances.
I labeled the distances both Emily and Ryan climbed on the number lines.
B. Write <, >, or =. \(\frac{2}{8}\)
I marked the fractions \(\frac{2}{8}\) and \(\frac{2}{3}\) on the number lines and Ryan climbed more than Emily.
C. Who climbs higher? How do you know?
Ryan climbed more
Ryan climbed more as shown the markings on the number lines above the distance Ryan climbed is more than the distance Emily climbed.
Turn and Talk How can the size of the equal lengths in the whole help you find which fraction is greater?
The size of equal lengths in the whole helps us to compare the fractions easily as the whole is same the number of parts counted more will be greater.
Check Understanding
Question 1.
Which spinner has a larger area shaded?
The shaded parts of the spinners are 2 of the whole.
Write the fractions. Write <, >, or =.
In the first spinner 2 parts out of 4 are shaded and in second spinner 2 parts out of 6 are shaded
So, when we compare the shaded regions we found that \(\frac{2}{4}\) > \(\frac{2}{6}\).
Shade to show each fraction. Write <, >, or =.
Question 2.
Compare \(\frac{4}{8}\) and \(\frac{4}{6}\).
I shaded the fractions, compared them
Therefore, i found that \(\frac{4}{8}\) < \(\frac{4}{6}\).
Question 3.
Compare \(\frac{4}{6}\) and \(\frac{2}{6}\).
I shaded the fractions, compared them
Therefore, i found that \(\frac{4}{6}\) > \(\frac{2}{6}\).
On Your Own
Question 4.
Social Studies Tim compared total votes for voters aged 18-24. In a 2016 election, \(\frac{2}{8}\) of people aged 18-24 voted. In 2018, \(\frac{2}{6}\) of the same age group voted. In both elections,
about the same number of votes were counted. In which election did more people aged 18-24 vote? Use a concrete or visual model to explain your answer.
Social Studies Tim compared total votes for voters aged 18-24. In a 2016 election, \(\frac{2}{8}\) of people aged 18-24 voted
In 2018, \(\frac{2}{6}\) of the same age group voted
In both elections, about the same number of votes were counted
I drew lines and shaded the fractions, when compared both the fractions, In 2018 more more people aged 18-24 voted than in the year 2016.
Question 15.
Attend to Precision Patty walks \(\frac{2}{4}\) mile on Monday and \(\frac{3}{4}\) mile on Tuesday. Show the distances on the number line. On which day does Patty walk a shorter distance? How do you
Patty walks \(\frac{2}{4}\) mile on Monday and \(\frac{3}{4}\) mile on Tuesday
I marked the distance patty walked on day 1 and day 2
When compared both the distances, the distance Patty walked on Monday is less than the distance patty walked on Tuesday
Therefore, \(\frac{2}{4}\) < \(\frac{3}{4}\).
I’m in a Learning Mindset!
How can I share the visual or concrete models I used to compare fractions to help other students with their fraction comparisons?
Leave a Comment
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