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Practice Problems 21
Problem 1: API
The Academic Performance Index (API) is computed for all California schools. It is a number, ranging from a low of 200 to a high of 1000, that reflects a school’s performance on a statewide
standardized test (http://api.cde.ca.gov). We have a SRS of 200 schools and are interested in how a school’s performance is related to the wealth of its students. The variable growth measures the
growth in API from 1999 to 2000 (API 2000 - API 1999).
(a) Categorizing wealth
Let’s define a school as “low wealth” if over 50% of its students are eligible for subsidized meals and “high wealth” otherwise. We can use an ifelse command to create a variable wealth that measures
# A tibble: 2 × 3
wealth `mean(growth)` `sd(growth)`
<chr> <dbl> <dbl>
1 high 25.2 28.8
2 low 38.8 30.0
• How many schools are “low” and “high” wealth.
• Are wealth and API growth related?
• What is the observed difference in mean API growth between high and low wealth schools. Use correct notation.
• Can we use t-inference methods to compare mean growths?
Click for answer Answer: There are \(n_h = 102\) “high” wealth and \(n_l = 98\) “low” wealth schools. The low wealth schools tend to have higher (and more variable) growth than high wealth schools.
The difference in observed mean API growth between high and low growth schools is \(\bar{x}_h - \bar{x}_l = 25.24510 - 38.82653 = -13.58\). We can use t-methods since both samples sizes (98 and 102)
can be deemed large and there isn’t severe skewness, but there are two extreme outliers that will be addressed below.
(b) SE for the sample mean difference
What is the estimated SE for the sample mean difference? Click for answer
Answer: The SE for the mean difference is 4.1544:
\[ SD_{\bar{x}_h - \bar{x}_l} = \sqrt{\dfrac{28.75380^2}{102} + \dfrac{29.95048^2}{98}} = 4.1544 \]
(c) t-test statistic
Using your SE from (b) to compute the t-test statistic that can be used to determine if mean API growth differs for low and high wealth schools. Write down your hypotheses then show how the t test
statistic is calculated. Interpret this value in context. Click for answer
Answer: The hypotheses are \(H_0: \mu_h - \mu_l = 0\) vs \(H_A: \mu_h - \mu_l \neq 0\). The test stat is
\[t = \dfrac{(25.24510 - 38.82653) - 0}{4.154404} = -3.2692\]
The observed mean difference is 3.3 SEs below the hypothesized mean difference of 0.
(d) Two-sample t-test
Is there evidence that mean API growth differs for low and high wealth schools? Give the hypotheses for this test, then run the t.test(y ~ x, data=) command below to conduct a t-test to give a
p-value and conclusion.
Welch Two Sample t-test
data: growth by wealth
t = -3.2692, df = 196.71, p-value = 0.001273
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
-21.774321 -5.388544
sample estimates:
mean in group high mean in group low
25.24510 38.82653
• What is the t test stat given in the output? Verify that it matches your answer to (c), within reasonable rounding error.
Click for answer Answer: The test stat matches, \(t = -3.2692\).
• What is the p-value for the test? Interpret this value. Click for answer
Answer: The p-value is 0.001273. If there is no difference between mean growth in the two populations, then there is just a 0.13% chance of seeing a sample mean difference that is 3.27 standard
errors or more away from 0.
• What is your test conclusion? Click for answer
Answer: We have strong evidence to suggest that the average API growth in low and high wealth schools are not the same.
(e) Consider outliers
The boxplot in (a) shows a number of outliers for the high wealth group, but two cases in particular were very high. Suppose we omitted these two (most) extreme cases when running the test in (d).
Will the p-value for this test be smaller or larger than the p-value computed in part (d)? Explain.
Click for answer Answer: Removing the two large outliers which will both reduce the mean in the high group and reduce the SD in the high group. Both actions will magnify the difference in mean growth
between the high and low groups (increasing the difference and decreasing the SE), so the test stat will increase in magnitude and the p-value will decrease.
(f) Check outlier influence
To omit these cases we have to find their row numbers, then subset them out of the data:
cds stype name sname snum
1 5.471911e+13 E Lincoln Element Lincoln Elementary 5873
2 1.975342e+13 E Washington Elem Washington Elementary 2543
dname dnum cname cnum flag pcttest api00 api99 target
1 Exeter Union Elementary 226 Tulare 53 NA 98 693 504 15
2 Redondo Beach Unified 585 Los Angeles 18 NA 100 745 615 9
growth sch.wide comp.imp both awards meals ell yr.rnd mobility acs.k3 acs.46
1 189 Yes Yes Yes Yes 50 18 <NA> 9 18 NA
2 130 Yes Yes Yes Yes 41 20 <NA> 16 19 30
acs.core pct.resp not.hsg hsg some.col col.grad grad.sch avg.ed full emer
1 NA 93 28 23 27 14 8 2.51 91 9
2 NA 81 11 26 32 16 16 2.99 100 3
enroll api.stu pw fpc wealth
1 196 177 30.97 6194 high
2 391 313 30.97 6194 high
Welch Two Sample t-test
data: growth by wealth
t = -4.395, df = 174.97, p-value = 1.916e-05
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
-23.571116 -8.961945
sample estimates:
mean in group high mean in group low
22.56000 38.82653
• How does the t-test stat change when omitting these two changes? Why does it change in this direction?
• Check your answer here with your anwer in part (e)!
Click for answer Answer: Without these outliers, the p-value decreases to 0.00001916 and we have even stronger evidence for a difference in mean API growth. Why does the p-value decrease? Omitting
the two outliers will decrease the sample SD for the high group, which in turn will (slightly) decrease the SE for the difference in means. Omitting the two outliers will also decrease the sample
mean for the high group (from 25.24510 to 22.56000), which will make the observed difference in means larger in magnitude (from -13.58 to -16.27). The test stat gets even further from 0 (drops from
-3.2692 to -4.395), meaning the observed difference with outliers omitted is further away from 0 (in terms of SEs) than it was when all data points were included. This means that the p-value will
decrease (from 0.0013 to 0.00002) since the data is deemed more ``extreme” under the null hypothesis.
(g) 95% confidence interval
Compare the two 95% CI given in the output (with and without outliers). Explain how and why the CIs change after omitting these two outliers.
Click for answer Answer: Without outliers: -23.57 to -8.96 and with outliers: -21.77 to -5.39. An mentioned above, omitting the two points makes the difference in means further away from 0. This
shifts the CI further from a difference of 0. Removing the outliers also decrease the SE of our sample difference, so the margin of error for the interval without outliers is, roughly, 7 while the
margin of error with outliers is, roughly, 8.
(h) Interpret two-sample CI
Using the results without the two outliers, interpret the 95% CI given in this output. Do not use the word ``difference’’ in your answer.
Click for answer Answer: We are 95% confident that the mean API growth between 1999 and 2000 for all low wealth schools is anywhere from 8.96 points to 23.57 points higher than the mean API growth
for all high wealth schools in California.
Problem 2: Matched Pairs
A study is conducted to determine the effect of a home meter for helping diabetics control their blood glucose levels. Researchers would like to determine if the home meter is effective in helping
patients reduce their blood glucose levels. A random sample of 36 diabetics had their blood glucose levels measured before they were taught to use the meter and again after they had utilized the
meter for 2 weeks. Researchers observed an average decrease (before - after) of blood glucose level of 2.78 mmol/liter with a standard deviation of 6.05 mmol/liter. Analysis results are shown below:
Sample mean: 2.78 ; sample standard deviation: 6.05 ; sample size: 36
Standard error: 1.0083
95 percent confidence interval for true mean: 1.0763 , Infinity
Hypothesis test H0: mu = 0 Alternative is greater
t statistic = 2.757 ; degrees of freedom = 35 ; p-value= 0.0046
(a) What conditions need to be met by this data to use \(t\) inference procedures?
Click for answer
Answer: There is a moderate sample size of \(n=36\) so we need to assume that the observed differences (before-after) are not strongly skewed and that there are no outliers. If these assumptions are
not met, then the t-inference procedures above may not be appropriate.
(b) Define the unknown parameter of interest (be very specific), then state the null and alternative hypotheses for this test. Make sure your hypotheses agree with the output!
Click for answer
Answer: Let the \(\mu\) represent the population mean decrease in glucose levels measured before and after the treatment (before - after). A positive value of \(\mu\) implies that the home meter is
effective in reducing blood glucose levels. The alternative hypothesis (the research statement) will be that \(\mu\) is greater than 0 and the null statement will be that \(\mu\) is equal to 0,
meaning there is no benefit to using the treatment. \[ H_0: \mu = 0 \textrm{ vs. } H_A: \mu > 0 \]
(c) What is the test statistic value for this test? What does this value indicate?
Click for answer
Answer: The test stat value is 2.757. The mean glucose level decrease in the sample was 2.757 SE’s above the hypothesized mean decrease of 0.
(d) Is there sufficient evidence to claim that the monitor is effective in helping patients reduce their blood glucose levels?
Click for answer
Answer: You reject \(H_0\) when the \(p\)-value is small. Since the P-value of 0.3% is quite small, we can conclude that there is strong evidence that the use of home meters lowers blood glucose
levels, on average (\(H_A\)).
(e) What type of error (1 or 2) could you have made in part (d)? If you did make this error, what are its implications for people with diabetes?
Click for answer
Answer: Since we rejected, we may have made a type 1 error of rejecting the null when it is actually true. This means we would have claimed that the home meter was useful in reducing blood glucose
levels, on average, when in fact it doesn’t reduce levels. People with diabetes would be encouraged to use these meters (at a cost to themselves or their insurance company) to help control their
glucose levels and not see any real benefit.
(f) Compute and interpret a 95% confidence interval for the true average decrease in blood glucose levels. (Note that this CI is not given above, the CI given in the output is a ``one-sided” CI.)
Click for answer
Answer: The 95% CI for the population mean decrease in glucose level is \[ \bar{x} \pm t^*_{n-1} \dfrac{s}{\sqrt{n}} = 2.78\pm 2.042 \dfrac{6.05}{\sqrt{36}} = 2.78 \pm 2.017 = (0.72, 4.84) \] where \
(t^*\) is based on 36-1=35 degrees of freedom. Using the green table, we round df down to 30 so we get \(t^*_{30} = 2.042\). Or using R command qt(.975,df=35) we get the exact value \(t^*_{35}=2.0301
\). We are 95% confident that, after learning to use a home meter, the average decrease in blood glucose in this population is between 0.72 and 4.84 mmol/liter. | {"url":"https://stat120-spring24.netlify.app/practice_problems_21","timestamp":"2024-11-10T02:03:12Z","content_type":"application/xhtml+xml","content_length":"60282","record_id":"<urn:uuid:6aec34e1-7864-48e4-879e-d1e0fef3f2f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00470.warc.gz"} |
The Power of Circular Linked Lists: Implementation in Python - Adventures in Machine Learning
Introduction to Circular Linked Lists
In the world of data structures, linked lists have always been a fundamental concept. Linked lists are a sequence of elements called nodes, which are connected through pointers, thereby creating a
chain of nodes where each node contains two fields: data and a pointer to the next node.
The data can be of any type, such as integer, character, string, etc., and the pointer points to the next node in the sequence. A common variation of the linked list is the Circular Linked List.
It is similar to a standard linked list, except that the last node of the list points to the first node, thereby creating a loop. This article will delve deeper into the concept of Circular Linked
Lists, their definition, characteristics, and implementation in Python.
Understanding of Linked Lists
Before we get into specifics, let us first understand the concept of Linked Lists. Linked lists are a dynamic data structure, meaning that nodes can be added or removed from the list without
affecting the rest of the list.
Each node holds a data field and a pointer field. The data field stores the actual data, and the pointer field points to the next node in the list.
Linked lists are an efficient data structure for dynamic data, where we do not know the size of the data beforehand. Linked lists have better insertion and deletion time complexity than arrays,
making them ideal for dynamic data.
However, arrays exhibit better performance than linked lists for accessing data since arrays use contiguous memory locations.
Circular Linked Lists in Python
Similar to standard linked lists, a circular linked list can also be implemented in Python. In a circular linked list, the last node points to the first node, creating a loop.
Consider the following example:
We have three nodes – Node 1, Node 2, and Node 3. Node 1 points to Node 2, Node 2 points to Node 3, and Node 3 points to Node 1, thereby creating a loop.
Definition and characteristics of Circular Linked Lists
As the name suggests, a circular linked list is a type of linked list where the last node points to the first node, forming a loop. This loop allows for easy traversal of the list, since we can
always start at the first node and continue until we reach the first node again, traversing the entire list.
One of the primary characteristics of circular linked lists is that they have no beginning or endpoint. Each node in the list has two fields – a data field and a pointer field.
The data field holds the actual data, and the pointer field points to the next node in the list. Circular linked lists can be used in scenarios where we need to access data in a circular manner.
For example, we could use a circular linked list to simulate a round-robin scheduling algorithm, where each process takes a turn to execute.
Visualization of Circular Linked Lists
Visualizing circular linked lists can help us understand the concept better. Consider the following example of a circular linked list of three nodes – Node 1, Node 2, and Node 3.
______ _______
| | | |
| |————>| | |
| 1 | | 2 |
|______|______________| ______|
| |
| 3 |
In the above example, Node 1 is connected to Node 2, which is connected to Node 3. Node 3 is connected back to Node 1, thereby creating a loop.
The loop represents the end of the linked list, and we can traverse through the entire list by starting at any node and following the next node until we reach the starting node.
Circular linked lists are a vital data structure in computer science, and their applications are widespread. In scenarios where we need to access data in a circular manner, circular linked lists can
be an excellent choice.
Python provides a simple and intuitive way of implementing circular linked lists. By understanding the basics of linked lists and their variation – circular linked lists, we can create efficient
solutions for a wide range of problems.
Implementing Circular Linked Lists in Python
Now that we understand how Circular Linked Lists work and their characteristics, we can move on to implementing them in Python. In this section, we will cover the implementation of the Node class and
the Circular Linked List class, along with their methods.
Implementation of Node class
The Node class represents a single node in the Circular Linked List. Each node has two fields – data and the next node.
The data field stores the actual data, and the next node field is a pointer to the next node in the list. Here is an example of a Node class implementation in Python:
class Node:
def __init__(self, data=None, next_node=None):
self.data = data
self.next_node = next_node
In the above code, we have defined the constructor for the Node class.
It takes two parameters – the data to store in the node, and the pointer to the next node. If no data or next node is specified, the default value is None.
Implementation of Circular Linked List class
The Circular Linked List class is the primary data structure we will work with. It represents the entire circular linked list and provides methods to manipulate the list.
The class contains a head node to represent the starting node of the list, and a count member to store the size of the list. Here is an example of a Circular Linked List class definition:
class CircularLinkedList:
def __init__(self):
self.head = Node()
self.count = 0
In the above code, we have defined the constructor for the Circular Linked List class.
The initializer sets the head node to a default node and sets the count of nodes in the list to zero. __init__ method
The __init__ method is responsible for initializing the Circular Linked List class and creating a head node.
It takes no input parameters and is automatically called when a new instance of the class is created. In our example implementation, the __init__ method sets the head node to a default node with no
data and no next node.
It also sets the count of nodes in the list to zero. __repr__ method
The __repr__ method is responsible for returning a string representation of the Circular Linked List object and is called when we try to print the object.
In our example implementation, we have defined the __repr__ method for the Circular Linked List class as follows:
def __repr__(self):
nodes = []
node = self.head.next_node
for i in range(self.count):
node = node.next_node
return "n".join(nodes)
In the above code, we iterate through the list and append the data of each node to a list of strings. We then join the strings with a newline character to create a string representation of the list.
append and insert method
The append method appends a new node to the end of the Circular Linked List. Here is an example implementation of the append method:
def append(self, data):
new_node = Node(data)
if self.count == 0:
self.head.next_node = new_node
new_node.next_node = new_node
node = self.head.next_node
for i in range(self.count - 1):
node = node.next_node
node.next_node = new_node
new_node.next_node = self.head.next_node
self.count += 1
In the above code, we create a new node with the specified data and add it to the end of the list.
If the list is empty, we set the head node to point to the new node, and the new node to point to itself. Otherwise, we traverse the list until we reach the last node and set its pointer to the new
We also set the new node’s pointer to point back to the head node. The insert method inserts a new node at a specified position in the list.
Here is an example implementation of the insert method:
def insert(self, data, index):
if index < 0 or index > self.count:
raise ValueError("Index out of range")
new_node = Node(data)
if index == 0:
new_node.next_node = self.head.next_node
self.head.next_node = new_node
node = self.head.next_node
for i in range(index - 1):
node = node.next_node
new_node.next_node = node.next_node
node.next_node = new_node
self.count += 1
In the above code, we first check if the index is within range. If not, we raise a ValueError.
We then create a new node with the specified data and insert it at the specified index. If the index is zero, we set the head node to point to the new node, and if not, we traverse the list until we
reach the node before the specified index and insert the new node between that node and the next node.
remove method
remove method removes the node containing the specified item from the Circular Linked List. Here is an example implementation of the
remove method:
def remove(self, item):
node = self.head.next_node
prev_node = self.head
for i in range(self.count):
if node.data == item:
prev_node.next_node = node.next_node
del node
self.count -= 1
prev_node = node
node = node.next_node
raise ValueError("Item not found")
In the above code, we traverse the list until we find the node containing the specified item and remove it.
We also handle cases where the item is not found by raising a ValueError. index, size, and display method
The index method returns the index of the item in the Circular Linked List or raises a ValueError if the item is not in the list.
Here is an example implementation of the index method:
def index(self, item):
node = self.head.next_node
for i in range(self.count):
if node.data == item:
return i
node = node.next_node
raise ValueError("Item not found")
The size method returns the size of the Circular Linked List. Here is an example implementation of the size method:
def size(self):
return self.count
The display method prints out the data of each node in the Circular Linked List.
Here is an example implementation of the display method:
def display(self):
node = self.head.next_node
for i in range(self.count):
print(node.data, end=" ")
node = node.next_node
To execute the final code, we can create an instance of the Circular Linked List class and call its methods. Here is an example of a final program that creates a Circular Linked List and appends,
inserts, removes, and displays its nodes:
clist = CircularLinkedList()
clist.insert(4, 2)
The output of the above code will be:
In conclusion, Circular Linked Lists are a useful data structure in computer science, and Python provides an intuitive and easy-to-implement way of working with them. By creating a Node class to
represent the individual nodes in the list and a Circular Linked List class to represent the entire list, we can create efficient solutions for various problems.
We have covered different methods of the Circular Linked List class, including the __init__, __repr__, append, insert, remove, index, size, and display methods. By using these methods, we can add,
remove, and modify nodes in the Circular Linked List and even search for specific items.
In this article, we have covered the basics of Circular Linked Lists and their implementation in Python. We have looked at the Node class and the Circular Linked List class, along with their methods,
including append, insert, remove, index, size, and display.
By using these methods, we can add, remove, and modify nodes in the Circular Linked List and even search for specific items. Circular Linked Lists are one of the most commonly used dynamic data
structures in computer science because they can be used in various applications such as scheduling algorithms, circular buffers, and train tracks.
With the Circular Linked List, we can represent vertices that make cycles in a graph, enabling us to perform tasks such as finding cycles or iterating through circularly connected elements
efficiently, and simulating games. The primary advantage of Circular Linked Lists over other data structures is their dynamic nature, enabling us to add, remove, and modify nodes in the list without
affecting the rest of the list.
Another benefit is the speed of traversal, since we can start at the head node and traverse through the list until we reach the head node again. Hence, Circular Linked Lists provide a powerful data
structure for scenarios that involve continuous looping or cyclic operations.
Python is an excellent language for implementing Circular Linked Lists, since it is an easy-to-learn, high-level language with built-in support for dynamic data structures and object-oriented
programming. Python provides class definitions and objects, and aided with syntactic sugar, it simplifies the process of creating data structures such as Circular Linked Lists.
To implement a Circular Linked List class using Python, we will create a Node class to represent the individual nodes of the list and then a Circular Linked List class to maintain the list of nodes.
There are various methods to manipulate the Circular Linked List, such as append, insert, remove, index, size, and display.
By using these methods, we can add, remove, and modify nodes in the Circular Linked List and even search for specific items. In conclusion, Circular Linked Lists are a powerful data structure in
computer science, and Python provides an intuitive and straightforward way of implementing them.
With their dynamic nature and cyclic operations, Circular Linked Lists are excellent for representing cyclic data structures such as circular queues or graphs that have cycles. By utilizing the Node
class and implementing the Circular Linked List class with Python, we can create efficient solutions for a wide range of problems in computer science.
Circular Linked Lists in Python are a powerful and dynamic data structure used in many applications, including scheduling algorithms, circular buffers, and train tracks. Python provides an easy and
intuitive way of implementing them by creating a Node class and a Circular Linked List class with their various methods, such as append, insert, remove, index, size, and display.
With their cyclic operations and dynamic nature, Circular Linked Lists can efficiently manage various problems in computational sciences. As such, it is crucial to incorporate this data structure for
managing cyclic data and traversing through them efficiently, which are valuable takeaways to consider in algorithm development. | {"url":"https://www.adventuresinmachinelearning.com/the-power-of-circular-linked-lists-implementation-in-python/","timestamp":"2024-11-05T23:08:22Z","content_type":"text/html","content_length":"90362","record_id":"<urn:uuid:674ba61e-a3dc-41ea-be1a-814986b0b27c>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00352.warc.gz"} |
2017-2018 University Catalogue [ARCHIVED CATALOG]
Professors Hart, Lantz, Robertson, Saracino, Schult, Strand, Valente (Chair)
Associate Professor Ay
Assistant Professors Chen, Christensen, Cipolli, Howard, Jiménez Bolaños, Seo
Visiting Assistant Professor Agbanusi
There are many good reasons to study mathematics: preparation for a career, use in another field, or the beauty of the subject itself. Students at Colgate who major in mathematics go on to careers in
medicine, law, or business administration as well as areas of industry and education having an orientation in science. Non-majors often require mathematical skills to carry on work in other
disciplines, and all students can use the study of mathematics to assist them in forming habits of precise expression, in developing their ability to reason logically, and in learning how to deal
with abstract concepts. There are also many people who view mathematics as an art form, to be studied for its own intrinsic beauty.
All mathematics courses are open to qualified students. Entering first-year students who have successfully completed at least three years of second-ary school mathematics, including trigonometry,
should be adequately prepared for MATH 161 . Students who have studied calculus in secondary school are typically ready to enter MATH 162 or MATH 163 .
Students who are planning to undertake graduate study in mathematics are advised to take MATH 485 and MATH 487 .
Course Information
The following classification scheme is used for MATH courses:
100-149: Only requires knowledge of mathematics before Calculus
150-199: Calculus-level knowledge and/or sophistication
200-249: Linear Algebra level (gentle transition-type course)
250-299: Transition to the major level
300-349: Courses with requirements at Math 150-249 level
350-399: Courses with requirements at the Math 250-299 level
400-449: Courses with requirements at the Math 300-349 level
450-474: Courses with requirements at the Math 350-399 level
475-484: Research experience seminars
485-499: Advanced material
Honors and High Honors
For a student to be considered for honors in Mathematics or in Applied Mathematics the student must achieve a 3.3 GPA in the respective major; in order for the student to be considered for high
honors, a 3.7 GPA in the major is required. For both honors and high honors, completion of a course numbered 440 or above that is not a research seminar is required.
Honors / High Honors are attained by a student’s production and defense of a thesis of distinction. A grade of A- or better is required to be considered for honors. A grade of A or better is required
to be considered for high honors. The student’s thesis advisor puts forward the thesis for honors consideration. In order for honors to be granted, the thesis must unanimously receive a grade of an
A- or better. In the event that the project is graded A or better, high honors will be granted. These are both contingent on satisfying the GPA and 440-level course requirements.
Joint theses are allowed but will not normally be considered for honors. Exceptions may be made with departmental permission.
As a reminder to the student writing theses for two different departments: In Colgate’s Honor Code, it states: Substantial portions of the same academic work may not be submitted for credit or honors
more than once without the permission of the instructor(s).
The Allen First-Year Mathematical Prize —This prize is awarded for excellence in mathematical work throughout the student’s first year.
The Edwin J. Downie ‘33 Award for Mathematics — created in memory of Edwin J. Downie ‘33, professor of mathematics emeritus, this award will be given annually to a senior majoring in mathematics who
has made outstanding contributions to the mathematics department through exemplary leadership, service, and achievement.
The Osborne Mathematics Prizes — established in honor of Professor Lucien M. Osborne, Class of 1847, to be awarded to any student who maintains a high average in mathematics courses in the junior
The Sisson Mathematics Prizes — established in honor of Eugene Pardon Sisson, a teacher of mathematics in the academy 1873–1912 and awarded to an outstanding sophomore student in mathematics.
Calculus Placement
Students should review the MATH 161 , MATH 162 , and MATH 163 course descriptions for information on topics and prerequisites, or consult with a department faculty member. In general, students are
encouraged to enroll in a higher-level course. Students may drop back from MATH 162 to MATH 161 within the first three weeks, subject to available space in an acceptable MATH 161 section.
Advanced Placement
Students who have taken the Calculus-BC, Calculus-AB, or Statistics Advanced Placement exam of the College Entrance Examination Board will be granted credit according to the following policy:
1. Students earning 4 or 5 on the Calculus-BC Advanced Placement exam will receive credit for MATH 161 and MATH 162 . Students earning 3 on the Calculus BC exam will receive credit only for MATH 161
2. Students earning 4 or 5 on the Calculus-AB Advanced Placement exam will receive credit for MATH 161 .
3. Students earning 4 or 5 on the Statistics Advanced Placement exam will receive credit for MATH 105 .
4. There are no other circumstances under which a student will receive credit at Colgate for a mathematics course taken in high school.
Transfer Credit
Transfer credit for a mathematics course taken at another college will be granted upon the pre-approval of the department chair. Mathematics and Applied Mathematics majors or minors may not receive
transfer credit for MATH 250 , MATH 260 , MATH 375 , MATH 376 , or MATH 377 , but must pass these courses at Colgate and must take them as regularly scheduled courses, not as independent studies. At
most, two transfer or independent studies courses may be counted toward a major or minor.
International Exam Transfer Credit
Transfer credit and/or placement appropriate to academic development of a student may be granted to incoming first year students who have achieved a score on an international exam (e.g., A-Levels,
International Baccalaureate) that indicates a level of competence equivalent to the completion of a specific course in the department. Requests should be directed to the department chair. Any such
credit may not be used to fulfill the university areas of inquiry requirement, but may count towards the major.
Related Majors/Minors
Teacher Certification
The Department of Educational Studies offers a teacher education program for majors in mathematics who are interested in pursuing a career in elementary or secondary school teaching. Please refer to
Educational Studies .
Study Groups
Colgate sponsors several study-abroad programs that can support continued work toward a major in mathematics. These include, but are not limited to, the Wales Study Group (U.K.), the Australia Study
Group, the Australia II Study Group, and the Manchester Study Group (U.K.). For more information about these programs, see Off-Campus Study .
Majors and Minors | {"url":"https://catalog.colgate.edu/preview_entity.php?catoid=3&ent_oid=75&returnto=50","timestamp":"2024-11-05T19:45:09Z","content_type":"text/html","content_length":"76844","record_id":"<urn:uuid:f107b98a-19ae-4c92-9992-d7c3cf152334>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00494.warc.gz"} |
Inflection Point / Turning Point: Definition & Examples
An inflection point (sometimes called a turning point, flex , inflection, or “point of diminishing returns”) is where a graph changes curvature, from concave up to concave down or vice versa.
A concave up graph is like the letter U (or, a “cup”), while a concave down graph is shaped like an upside down U, or a Cap (∩). The inflection point happens when a “cup” and a “cap” meet.
A vertical inflection point, like the one in the above image, has a vertical tangent line; It therefore has an undefined slope and a non-existent derivative.
At first glance, it might not look like there’s a vertical tangent line at the point where the two concavities meet. However, if you zoom in close enough (perhaps with a graphing calculator), you’ll
see that there’s a tiny, almost insignificant spot in the graph where the tangent line is perfectly vertical.
A horizontal point of inflection is where the tangent line is horizontal. The derivative or slope here is zero, because the tangent line is flat.
The horizontal inflection point (orange circle) has a horizontal tangent line (orange dashed line).
You can also think of an inflection point as being where the rate of change of the slope changes from increasing to decreasing, or increasing to decreasing.
When slopes of tangent lines increase (from left to right, regardless of sign), your graph is concave up:
On the other hand, concave down graphs have decreasing slopes, from left to right:
So if you know what the rates of change are at any point on your graph, you can tell where there’s an inflection point.
The second derivative test uses that information to make assumptions about inflection points. The next graph shows x^3 – 3x^2 + (x – 2) (red) and the graph of the second derivative of the graph, f” =
6(x – 1) in green. Positive x-values are to the right of the inflection point; Negative x-values are to the left.
You can find inflection points by following these general steps [1]:
Step 1: Locate all points where the second derivative equals zero or does not exist. Only include those points where the function is actually defined and there is a tangent line at that point.
Step 2: Create a sign diagram of the second derivative.
Step 3: Choose any point from in Step 1. Determine from the sign diagram in Step 2 whether or not the sign of the second derivative changes as you move across the point. If the sign changes, it is an
inflection point.
Example question: Find the inflection points of f(x) = x + (1/x^2) .
Step 1: Locate all points where the second derivative equals zero or does not exist. First we need to find the second derivative:
Then figure out where the function equals zero or DNE. The function is zero when any of the terms in the numerator equal zero: (6x^2 – 2 = 0) or (x^2 + 1 = 0).
• Solving (6x^2 – 2 = 0) for x, we get x = 1/(√3) or -1/(√3).
• The second equation, (x^2 + 1 = 0), has no solutions.
Step 2: Create a sign diagram of the second derivative.
Step 3: Determine from the sign diagram in Step 2 whether or not the sign of the second derivative changes as you move across the point. If the sign changes, it is an inflection point. For this
example, the sign changes as you move across both points, so x = 1/(√3) and -1/(√3) are both inflection points.
[1] Sec 4.2 Applications of the Second Derivative. Retrieved July 13, 2021 from: https://mathematics.ku.edu/sites/math.drupal.ku.edu/files/files/Rajaguru_CourseDocuments/class16math115.pdf
1 thought on “Inflection Point / Turning Point: Definition & Examples”
1. My favourite example is the inflection point at the origin on the curve y = cube root of x.
so simple to see and it doesn’t require d2y/dx2 to be zero or dy/dx to be defined at the point
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By What Right Do Gravitational Waves Travel at the Speed of Light?
The rationale for claiming that gravitational waves travel at the speed of light is that the "carrier" of the gravitational interaction is a massless graviton, like that of the photon for
electromagnetic interactions, and so, too, should travel at maximum speed.
1. The speed of light, c, refers to electromagnetic permeabilities, which determine how elastic the medium is, and has absolutely nothing to do with neutral matter, as that which would be the source
of gravitational waves.
2. Gravitational waves are assumed to be transverse waves, just like electromagnetic waves where the polarizations occur in the plane perpendicular to the direction of the propagation of the wave.
Whereas the polarization of light refers to 3D space, the polarization of gravitational waves refer to 4D space. This is nonsense, since polarization is a spatial phenomenon, and does not involve
the time axis. When this is realized, Eddington's claim that TL and LL waves travel at the "speed of thought" loses all meaning.
3. If "spacetime" is different than the vacuum in which electromagnetic waves propagate, then they share no common speed, c. If they are the same, electromagnetic waves do not propagate in vacuum,
and all our physical foundation of electromagnetism must be shelved.
4. Gravitation is known to affect the propagation of all types of waves, even electromagnetic ones. Then why is this not reciprocal? i.e., electromagnetic waves should affect gravitational waves
with the consequence that the LIGO instruments don't measure what they were intended to measure, i.e., the measurement of the displacement of mirrors in an interferometer using laser light.
5. Gravitational waves are said to be observed when two black holes merge. The merger of two black holes is not geodesic motion, and, therefore, lies outside the domains of general relativity. The
waves are source free solutions of the wave equation that results when the Ricci tensor is linearized. It is known that there are no periodic solutions to the nonlinear Einstein field equations.
So, the linearized solution does not refer to a weak field of some more general wave equation (e.g., solitary waves). If the merger of black holes did not occur so far away, the radiation emitted
would be much stronger. What, then, would be the equation governing the propagation of gravitational radiation?
6. In traversing a material medium the GW displaces from their equilibrium position elements of the material transversely to its direction of propagation. Consider a room in which a chandelier is
hanging from the wall. A seismic S-wave traverses the room, and the displacement of the chandelier from its equilibrium is intended to be measured by LIGO. However, the whole room, and not just
the chandelier, feels the effect of the passage of the S-wave. The measurement of the displacement of the chandelier means nothing since everything in the room has been affected, including the
electromagnetic wave that was sent out to measure the displacement.
7. The "fabric" of spacetime that supposedly supports GW requires a Poisson ratio of 1. The transverse shear velocity will thus coincide with the speed of light, while the longitudinal, pressure,
wave velocity will vanish since the bulk modulus vanishes. In general, the longitudinal wave velocity will be greater than the transverse wave velocity since the force is in the direction of
propagation in contrast to being in the plane normal to propagation. This conclusion rests upon the fact that GW are polarizable. But the polarizability in 4D is meaningless.
8. The analogy between the Cauchy stress tensor and strain in terms of Hooke's law where the spring constant is proportional to Young's modulus, and the Einstein field equations highlights the
imperfections in the latter. What would be analogous to Young's modulus, Y, is Einstein's constant k=8pi G/c^4, but the latter must be multiplied by the fourth power of the frequency, f, viz, Y=
kf^4. This would mean that the ratio of the Einstein tensor T, to the energy-stress tensor is frequency dependent. To salvage the constitutive relation relating stress to deformation, which is
what Cauchy's law is all about, and relate it to Einstein's field equation requires a bending deformation to replace straightforward stretch, contraction, or shear [Tenev & Hostemeyer, "The
Mechanics of Spacetime"] And even if it could be salvaged, it would have only physical meaning in the (3+1) Numerical relativity formulation in which time lapse is separated from spatial extent.
That is, space time is foliated into space like surfaces like the peel of an orange where the sections of the peel are related by time lapse functions. This would be catastrophic to the
interpretation of the strain, i.e. the Einstein tensor, linearized into a sourceless wave equation since the stress, i.e., the energy-stress tensor would vanish.
9. Even if a fabric of space time would exist, it must necessary explain why the speed of propagation of waves is the same as the vacuum of electromagnetism where the speed of light is related to
the "electrical elasticity" of the medium, which has been known since the time of Maxwell (1865). It will, therefore, certainly not be acceptable to state that since the graviton is massless like
the photon, and GW and EM waves are polarizable, they should have a common speed between them. | {"url":"https://www.bernardlavenda.org/post/by-what-right-do-gravitational-waves-travel-at-the-speed-of-light","timestamp":"2024-11-03T22:57:41Z","content_type":"text/html","content_length":"1050493","record_id":"<urn:uuid:48ec2752-cfac-4e5d-88ac-9256a90520f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00458.warc.gz"} |
Circumradius of Rectangle given Diagonal Calculator | Calculate Circumradius of Rectangle given Diagonal
What is a Rectangle?
A Rectangle is a two dimensional geometric shape having four sides and four corners. The four sides are in two pairs, in which each pair of lines are equal in length and parallel to each other. And
adjacent sides are perpendicular to each other. In general a 2D shape with four boundary edges are called quadrilaterals. So a rectangle is a quadrilateral in which each corner is right angle.
How to Calculate Circumradius of Rectangle given Diagonal?
Circumradius of Rectangle given Diagonal calculator uses Circumradius of Rectangle = Diagonal of Rectangle/2 to calculate the Circumradius of Rectangle, Circumradius of Rectangle given Diagonal
formula is given is defined as the radius of the circle which contains the Rectangle with all the vertices of Rectangle are lying on the circle, and calculated using diagonal of the Rectangle.
Circumradius of Rectangle is denoted by r[c] symbol.
How to calculate Circumradius of Rectangle given Diagonal using this online calculator? To use this online calculator for Circumradius of Rectangle given Diagonal, enter Diagonal of Rectangle (d) and
hit the calculate button. Here is how the Circumradius of Rectangle given Diagonal calculation can be explained with given input values -> 5 = 10/2. | {"url":"https://www.calculatoratoz.com/en/circumradius-of-rectangle-given-diagonal-calculator/Calc-630","timestamp":"2024-11-11T07:25:09Z","content_type":"application/xhtml+xml","content_length":"123024","record_id":"<urn:uuid:2591d735-f05f-459a-b827-d7abcac7a69c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00829.warc.gz"} |
A Harmonic Conundrum
This one came from user_1312 on reddit with a heading “This is a bit tricky… Enjoy!”. What else can we do but solve it?
Let $m$ and $n$ be positive numbers such that $\frac{m}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{101}$. Prove that $m-n$ is a multiple of 103.
My first approach was to invoke Wolstenholme’s theorem: the numerator of the $p-1$th harmonic number is a multiple of $p^2$ if $p$ is prime, so $H_{102} = 103^2 \frac{a}{b}$ with $b$ and $103$
(What’s a harmonic number? It’s the sum of reciprocals. $H_4$, for example, is $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$; the number we’re looking at is $H_{101}$.
What we have is $H_{102} - 1 - \frac{1}{102}$, which we could write as $H_{102} - \frac{103}{102}$. Both of those terms are multiples of 103… but (looking more closely, some time on) I don’t see that
this proves that $m-n$ is a multiple of 103.
Fortunately, I have another way
Let’s think about $\frac{m-n}{n} = \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{101}$.
If we group those, Gauss-style into 50 pairs: $\left( \frac{1}{2} + \frac{1}{101}\right) + \left( \frac{1}{3} + \frac{1}{100}\right) + \dots$, we get $\frac{103}{2 \times 101} + \frac{103}{3 \times
100} + \dots$.
We can write $\frac{m-n}{n} = 103 \br{\frac{p}{q}}$ for some integers $p$ and $q$. However, $103$ is prime, and doesn’t show up in any of the denominators, so $q$ and $103$ are coprime.
That means $m-n$ must be a multiple of 103, because it can’t cancel with anything on the bottom.
What a lovely puzzle!
* Edited 2019-11-11 to fix LaTeX.
A selection of other posts
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Spectroscopic observations of the fast X-ray transient and superluminal jet source SAX J1819.3-2525 (V4641 Sgr) reveal a best-fitting period of P-spect = 2.81678 +/- 0.00056 days and a semiamplitude
of K-2 = 211.0 +/- 3.1 km s(-1). The optical mass function is f(M) = 2.74 +/- 0.12 M-. We find a photometric period of P-photo = 2.81730 +/- 0.0001 days using a light curve measured from photographic
plates. The folded light curve resembles an ellipsoidal light curve with two maxima of roughly equal height and two minima of unequal depth per orbital cycle. The secondary star is a late B-type star
that has evolved off the main sequence. Using a moderate resolution spectrum (R = 7000) we measure T-eff = 10500 +/- 200 K, log g = 3.5 +/- 0.1, and V-rot sin i = 123 +/- 4 km s(-1) (1 sigma errors).
Assuming synchronous rotation, our measured value of the projected rotational velocity implies a mass ratio of Q equivalent to M-1/M-2 = 1.50 +/- 0.008 (1 sigma). The lack of X-ray eclipses implies
an upper limit to the inclination of i less than or equal to 70 degrees .7. On the other hand, the large amplitude of the folded light curve (approximate to0.5 mag) implies a large inclination (i
greater than or similar to 60 degrees). Using the above mass function, mass ratio, and inclination range, the mass of the compact object is in the range 8.73 less than or equal to M1 less than or
equal to 11.7 M.and the mass of the secondary star is in the range 5.49 less than or equal to M-2 less than or equal to 8.14 M. (90% confidence). The mass of the compact object is well above the
maximum mass of a stable neutron star, and we conclude that V4641 Sgr contains a black hole. The B-star secondary is by far the most massive, the hottest, and the most luminous secondary of the
dynamically confirmed black hole X-ray transients. We find that the alpha -process elements nitrogen, oxygen, calcium, magnesium, and titanium may be over-abundant in the secondary star by factors of
2-10 times with respect to the Sun. Finally, assuming E(B-V) = 0.32 +/- 0.10, we find a distance 7.4 less than or equal to d less than or equal to 12.31 kpc (90% confidence). This large distance and
the high proper motions observed for the radio counterpart make V4641 Sgr possibly the most superluminal galactic source known, with an apparent expansion velocity of greater than or similar to9.2c
and a bulk Lorentz factor of Gamma greater than or similar to 9.5, assuming that the jets were ejected during one of the bright X-ray flares observed with the Rossi X-ray Timing Explorer. | {"url":"https://iris.ucc.ie/live/!W_VA_PUBLICATION.POPUP?LAYOUT=N&OBJECT_ID=70046848","timestamp":"2024-11-03T23:31:23Z","content_type":"text/html","content_length":"7746","record_id":"<urn:uuid:341d3ec6-03ae-4003-b137-cc84bc800d1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00732.warc.gz"} |
Matrix Multiplication Inches Closer to Mythic Goal
For computer scientists and mathematicians, opinions about “exponent two” boil down to a sense of how the world should be.
“It’s hard to distinguish scientific thinking from wishful thinking,” said Chris Umans of the California Institute of Technology. “I want the exponent to be two because it’s beautiful.”
“Exponent two” refers to the ideal speed — in terms of number of steps required — of performing one of the most fundamental operations in math: matrix multiplication. If exponent two is achievable,
then it’s possible to carry out matrix multiplication as fast as physically possible. If it’s not, then we’re stuck in a world misfit to our dreams.
Matrices are arrays of numbers. When you have two matrices of compatible sizes, it’s possible to multiply them to produce a third matrix. For example, if you start with a pair of two-by-two matrices,
their product will also be a two-by-two matrix, containing four entries. More generally, the product of a pair of n-by-n matrices is another n-by-n matrix with n^2 entries.
For this reason, the fastest one could possibly hope to multiply pairs of matrices is in n^2 steps — that is, in the number of steps it takes merely to write down the answer. This is where “exponent
two” comes from.
And while no one knows for sure whether it can be reached, researchers continue to make progress in that direction.
A paper posted in October comes the closest yet, describing the fastest-ever method for multiplying two matrices together. The result, by Josh Alman, a postdoctoral researcher at Harvard University,
and Virginia Vassilevska Williams of the Massachusetts Institute of Technology, shaves about one-hundred-thousandth off the exponent of the previous best mark. It’s typical of the recent painstaking
gains in the field.
To get a sense for this process, and how it can be improved, let’s start with a pair of two-by-two matrices, A and B. When computing each entry of their product, you use the corresponding row of A
and corresponding column of B. So to get the top-right entry, multiply the first number in the first row of A by the first number in the second column of B, then multiply the second number in the
first row of A by the second number in the second column of B, and add those two products together.
Samuel Velasco/Quanta Magazine
This operation is known as taking the “inner product” of a row with a column. To compute the other entries in the product matrix, repeat the procedure with the corresponding rows and columns.
Altogether, the textbook method for multiplying two-by-two matrices requires eight multiplications, plus some additions. Generally, this way of multiplying two n-by-n matrices together requires n^3
multiplications along the way.
As matrices grow larger, the number of multiplications needed to find their product increases much faster than the number of additions. While it takes eight intermediate multiplications to find the
product of two-by-two matrices, it takes 64 to find the product of four-by-four matrices. However, the number of additions required to add those sets of matrices is much closer together. Generally,
the number of additions is equal to the number of entries in the matrix, so four for the two-by-two matrices and 16 for the four-by-four matrices. This difference between addition and multiplication
begins to get at why researchers measure the speed of matrix multiplication purely in terms of the number of multiplications required.
“Multiplications are everything,” said Umans. “The exponent on the eventual running time is fully dependent only on the number of multiplications. The additions sort of disappear.”
For centuries people thought n^3 was simply the fastest that matrix multiplication could be done. In 1969, Volker Strassen reportedly set out to prove that there was no way to multiply two-by-two
matrices using fewer than eight multiplications. Apparently he couldn’t find the proof, and after a while he realized why: There’s actually a way to do it with seven!
Strassen came up with a complicated set of relationships that made it possible to replace one of those eight multiplications with 14 extra additions. That might not seem like much of a difference,
but it pays off thanks to the importance of multiplication over addition. And by finding a way to save a single multiplication for small two-by-two matrices, Strassen found an opening that he could
exploit when multiplying larger matrices.
“This tiny improvement leads to huge improvements with big matrices,” Vassilevska Williams said.
Jared Charney; Richard TK Hawke
Say, for example, you want to multiply a pair of eight-by-eight matrices. One way to do it is to break each big matrix into four four-by-four matrices, so each one has four entries. Because a matrix
can also have entries that are themselves matrices, you can thus think of the original matrices as a pair of two-by-two matrices, each of whose four entries is itself a four-by-four matrix. Through
some manipulation, these four-by-four matrices can themselves each be broken into four two-by-two matrices.
The point of this reduction — of repeatedly breaking bigger matrices into smaller matrices — is that it’s possible to apply Strassen’s algorithm over and over again to the smaller matrices, reaping
the savings of his method at each step. Altogether, Strassen’s algorithm improved the speed of matrix multiplication from n^3 to n^2.81 multiplicative steps.
The next big improvement took place in the late 1970s, with a fundamentally new way to approach the problem. It involves translating matrix multiplication into a different computational problem in
linear algebra involving objects called tensors. The particular tensors used in this problem are three-dimensional arrays of numbers composed of many different parts, each of which looks like a small
matrix multiplication problem.
Matrix multiplication and this problem involving tensors are equivalent to each other in a sense, yet researchers already had faster procedures for solving the latter one. This left them with the
task of determining the exchange rate between the two: How big are the matrices you can multiply for the same computational cost that it takes to solve the tensor problem?
“This is a very common paradigm in theoretical computer science, of reducing between problems to show they’re as easy or as hard as each other,” Alman said.
In 1981 Arnold Schönhage used this approach to prove that it’s possible to perform matrix multiplication in n^2.522 steps. Strassen later called this approach the “laser method.”
Over the last few decades, every improvement in matrix multiplication has come from improvements in the laser method, as researchers have found increasingly efficient ways to translate between the
two problems. In their new proof, Alman and Vassilevska Williams reduce the friction between the two problems and show that it’s possible to “buy” more matrix multiplication than previously realized
for solving a tensor problem of a given size.
“Josh and Virginia have basically found a way of taking the machinery inside the laser method and tweaking it to get even better results,” said Henry Cohn of Microsoft Research.
The paper improves the theoretical speed limit on matrix multiplication to n^2.3728596.
It also allows Vassilevska Williams to regain the matrix multiplication crown, which she previously held in 2012 (n^2.372873), then lost in 2014 to François Le Gall (n^2.3728639).
Yet for all the jockeying, it’s also clear this approach is providing diminishing returns. In fact, Alman and Vassilevska Williams’s improvement may have nearly maxed out the laser method — remaining
far short of that ultimate theoretical goal.
“There isn’t likely to be a leap to exponent two using this particular family of methods,” Umans said.
Getting there will require discovering new methods and sustaining the belief that it’s even possible.
Vassilevska Williams remembers a conversation she once had with Strassen about this: “I asked him if he thinks you can get [exponent 2] for matrix multiplication and he said, ‘no, no, no, no, no.’” | {"url":"https://www.quantamagazine.org/mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/","timestamp":"2024-11-04T10:37:21Z","content_type":"text/html","content_length":"212632","record_id":"<urn:uuid:a04e57b2-fbb5-4b6b-930e-1572017d603e>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00410.warc.gz"} |
Distance-to-Slope Calculations
05 Oct 2024
Popularity: ⭐⭐⭐
Slope Distance Calculator
This calculator provides the calculation of slope distance between two points.
Calculation Example: The slope distance between two points is the distance measured along the line of sight, taking into account the vertical distance between the points. It is given by the formula S
= √(D^2 + H^2), where D is the horizontal distance between the points and H is the vertical distance between the points.
Related Questions
Q: What is the importance of slope distance in surveying?
A: Slope distance is important in surveying as it allows surveyors to determine the distance between two points, even if there are obstacles or uneven terrain between them.
Q: How is slope distance used in surveying?
A: Slope distance is used in surveying to measure distances between points that are not easily accessible or where there are obstacles in the way.
Symbol Name Unit
D Horizontal Distance m
H Vertical Distance m
Calculation Expression
Slope Distance Function: The slope distance between the two points is given by S = √(D^2 + H^2).
Calculated values
Considering these as variable values: D=100.0, H=50.0, the calculated value(s) are given in table below
Derived Variable Value
Slope Distance Function Sqrt(2500.0+D^2.0)
Similar Calculators
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Measures of Central Tendency MCQ Quiz [PDF] Questions Answers | Measures of Central Tendency MCQs App Download & e-Book
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The Measures of central tendency Multiple Choice Questions (MCQ Quiz) with Answers PDF (Measures of Central Tendency MCQ PDF e-Book) download to practice Grade 9 Math Tests. Learn Basic Statistics
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for frequency distribution, measures of central tendency test prep for online school classes.
The MCQ: The middle observation in an arranged set of data is called; "Measures of Central Tendency" App Download (Free) with answers: Mean; Mode; Median; Range; to study online certificate courses.
Solve Basic Statistics Quiz Questions, download Google eBook (Free Sample) for online learning.
Measures of Central Tendency MCQs PDF: Questions Answers Download
MCQ 1:
Value of [(n +1) ⁄ 4]^th term is the formula of
1. 1st quartile
2. 2nd quartile
3. mode
4. 3rd quartile
MCQ 2:
The middle observation in an arranged set of data is called
1. mean
2. mode
3. median
4. range
MCQ 3:
The n^th root of the product of the values x^1, x^2, x^3, ....., x^n is called
1. arithmetic mean
2. geometric mean
3. variance
4. harmonic mean
MCQ 4:
1st quartile is also known as
1. lower quartile
2. upper quartile
3. median
4. geometric mean
MCQ 5:
Value of [3(n +1) ⁄ 4]^th term is the formula of
1. variance
2. 3rd quartile
3. 2nd quartile
4. geometric mean
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Liquidation Threshold Calibration of LP Tokens | Parallel Finance
First, we calculate the liquidation risk margin implied by the ETH liquidation threshold. The liquidation risk margin denotes the margin that covers the potential loss in the case that actual
liquidation proceeds are not enough to cover the loan. Therefore, the liquidation risk margin is calculated as:
where $liq\_thld$ denotes the liquidation threshold. Therefore, we calculate the liquidation risk margin of ETH as:
\begin{aligned} LRM_{ETH}&=\frac{100\%-80\%}{80\%} \\ &=25\% \end{aligned}
This means, that when liquidation is triggered for a loan collateralized by ETH, as long as the liquidation proceeds do not fall 25% below the ETH price when liquidation is triggered, the liquidation
proceeds will be enough to cover the loan and the platform stays solvent.
Note here the numeraire is changed to the price of ETH when liquidation is triggered, which is consistent with the measure under which the health factor is calculated at the time liquidation is | {"url":"https://docs.parallel.fi/parallel-finance/staking-and-derivative-token-yield-management/borrow-against-uniswap-v3-lp-tokens/liquidation-threshold-calibration-of-lp-tokens","timestamp":"2024-11-11T08:22:43Z","content_type":"text/html","content_length":"584673","record_id":"<urn:uuid:3f609936-3b72-4ff1-805a-beb10c9647e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00713.warc.gz"} |
Forget Trading with Moving Average Indicators | Emini-Watch.com
Forget Trading with Moving Average Indicators … They’re So 1956
Barry Taylor
OK, here it is. Moving averages aren’t that great – and you can do better!
In short, here are a couple of reasons why moving averages aren’t great:
1. Moving averages assume the market is either going up or down. They are either rising or falling. But the market isn’t that simple – there aren’t just 2 states. In fact there are 3 states: the
market is either rising, falling or in congestion. A moving average can’t help you determine this third congestion phase.
2. There are literally millions of different moving averages. Simple, exponential, weighted, etc. etc. Then 9 bar, 13 bar, 21 bar, 50 bar, etc. etc. Then displaced, or not. Add all those
combinations together and you almost have an infinite number of combinations. And so which one is “right”?
But there is a better way. Let me explain. But first a little history.
Exponential Moving Average first published in 1956
1956 was the year Robert Brown and his co-creator Charles Holt published a book on the Exponential Moving Average. They both worked for the U.S. Navy, although independently, and had been using the
Exponential Moving Average to track the location of submarines for some years previously.
The beauty of the Exponential Moving Average was that you only need two numbers to make the calculation. So here’s the little formula for an Exponential Moving Average: it is yesterday’s (or the
previous bar’s ) value of the Exponential Moving Average times some factor, alpha, plus 1 minus alpha times today’s (or the current bars’) value.
// Exponential Moving Average Formula
// Example: 0.8 x 100 + (1-0.8) x 110 = 102
EMA(t) = alpha x EMA(t-1) + (1-alpha) x Price
So as an example, if alpha was 0.8, you multiply the current value of the Exponential Moving Average, say 100, by 0.8 then add 1 minus 0.8, which is 0.2, times the current data point, which might be
110. Add those two numbers together and you get 102.
If you’re using a 10 or a 25 period Simple Moving Average, you’d have to add together 10 or 25 different numbers and then divide by 25 or 10 or whatever it might be. With an Exponential Moving
Average, you only needed two numbers. And then in the days before the first pocket calculators were available, this was a godsend. Beautiful.
So 1956 was an important year. Prior to that, technical analysis has been very chart-based. It was all about trend and support resistance lines and things that you could draw on a chart. After 1956,
technical analysis was far more computationally-based, where we’re calculating moving averages and we’re using computer power to analyze charts.
The search for the BEST moving average
Now, the problem is this increased computer power we’ve had ever since 1956 has led us down the wrong path – the search for the ultimate moving average, one that smoothes out periods of
Moving Average and False Signals (Emini, 4,500 tick chart)
Here’s a 4,500 tick bar chart of the Emini continuous contract day and night session going back 7 or 8 days. I’ve just overlaid a Simple Moving Average – it doesn’t matter the length of this
particular moving average – but if it’s falling it’s colored white and if it’s rising it’s colored red.
You can see the trending moves quite nicely marked. But in between we have periods of consolidation, where the market is trying to determine which way it’s going to go. And our Simple Moving Average
is all over the place. I’ve marked these areas of consolidation and false signals with red boxes.
During these periods of uncertainty or congestion, the moving average does not do very well at signaling direction. You don’t know whether the market’s going up or going down. So, all the activity
that’s gone on in developing moving averages since 1956 has been about trying to find this “ultimate” moving average that smoothes out those periods of consolidation.
And the result is we literally have thousands of different moving averages:
• Simple Moving Average
• Exponential Moving Average
• Weighted Moving Average
• Hull Moving Average
• Multiple Moving Averages
• Moving Average Differences
• Linear regressions
• TEMA (Triple EMA)
• MAMA and FAMA
• Zero-lag Filters
• Laguerre Filter
• Median
• T3, etc.
So, as you can see, there are dozens of different types of moving averages. Then, on top of that, you’ve got to choose the length, or the smoothing factor, for each. And then you could also be
displacing these moving averages. So, put all of that together, and you’ve got a myriad of different choices of moving averages to use.
And all they’re trying to do is smooth out these periods of consolidation and limit the number of false signals you get.
The solution requires looking at market activity
Rather than trying to find this ultimate moving average, we should be looking for an indicator that tells us what kind of activity the market is going through. If the market is trending, a moving
average will be fantastic. But when a market is not trending, but is consolidating, then the moving average will be horrible.
You need an indicator that will tell you which kind of activity the market is going through right now. One that can tell is the market is trending or in consolidation. And the secret is calculating
support and resistance levels adaptively. Because once you break out of a support or resistance level, that’s when you move into a trending period. And if you keep within the support and resistance
levels you’re in consolidation.
John Ehlers to the rescue. He developed a beautiful piece of code called the Hilbert Sine Wave. I’ve improved on that algorithm, added the support and resistance levels and called it the Better Sine
Wave. I gives this alternate view of adaptive support and resistance levels.
Adaptive support and resistance levels using Better Sine Wave (Emini, 4,500 tick chart)
So going back to our 4,500 tick bar chart you can see that during periods of consolidation the Emini just bounces between support and resistance levels. But when a resistance level is broken to the
upside we break into an uptrend. And when a support level is broken to the downside we break into a downtrend.
The trending periods are also market with PB and END text. These show when a trending period is likely to stop with “Pull Back” and “End of Trend” signals. After we get an end of trend signal, we’re
going to go through some consolidation with adaptive support and resistance levels being plotted.
The Better Sine Wave indicator is showing you periods of consolidation and periods of trending. And so what you need to do is to figure out what the market is doing and to play it accordingly. If the
market is going through a consolidation phase, having been a strong trend, you need to be playing the support and resistance levels. And then waiting for the next breakout into a trend and you can
ride along.
Benefits of using an adaptive indicator
Now the beauty of using this Better Sine Wave approach is that, first of all, there are no inputs to tweak. You just load this indicator onto a chart and it dynamically calculates the support and
resistance levels. There are no inputs. Nothing to optimize.
The second thing is, because these levels are dynamically calculated, it’s an improvement on the traditional support and resistance view of the world. When traders with charting the market the
support and resistance levels were actually fixed values – which isn’t realistic.
So, that’s my pitch for the Better Sine Wave indicator and why you should be ditching your moving averages. Moving averages will always be giving you false signals through all these consolidation
periods. Great during the trending periods. Lousy during consolidation.
How to use moving averages (alternative approach)
Now, having trashed moving averages there is one instance where I think they are of tremendous value!
I have found moving averages great for smoothing incoming price data or “pre-processing” it in order to eliminate the noise. Then feeding that pre-processed or noise-reduced data into your other
indicators. But you use a very short “window” or short period for these moving averages. No more than 5 bars and something between 2 and 4 bars is ideal.
In fact, one of the simplest ways to pre-process, or smooth, input data is to use a 3-bar or 5-bar median of the price data. This will get rid of the “noise”. So that is using moving averages in a
totally different way. Very short term in order to pre-process data and reduce the noise.
So, there we go, just a quick video and article explaining my pitch for why I think you should ditch your moving averages. Forget the search for the ultimate moving average and instead understand the
psychology of the market and try and figure out when it’s trying to break support or resistance.
Now, how about checking out the Better Sine Wave indicator.
Using Average Trade Size to identify Professional and Amateur activity is totally new – and I’ve never seen it mentioned anywhere before the introduction of Better Pro Am and it’s use on Emini-Watch.
So if you want to get an edge in your trading, read on. Knowing what the Pro’s ...
Better Sine Wave Indicator: Using Price Cycles and Trends to Trade
The Better Sine Wave Indicator is unknown to the vast majority of traders. This amazes me – because if there were ONE indicator I’d recommend to traders to use, it would be the Better Sine Wave. Yes,
it really is THAT good. Here’s what some users of the Better Sine ...
Stop measuring price and start measuring demand and supply volume! Momentum indicators have been calculated based on changes in price for decades. But that doesn’t make any sense – you’re just
measuring the result, not the driving force. The driving force is demand and supply volume – the energy that ... | {"url":"https://emini-watch.com/trading-indicators/moving-average-alternatives/","timestamp":"2024-11-08T21:46:12Z","content_type":"text/html","content_length":"224296","record_id":"<urn:uuid:6fdb3544-ef5e-49a2-b9c8-aa50e66ea42e>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00479.warc.gz"} |
Garch Model: Simple Definition
The GARCH model, or Generalized Autoregressive Conditionally Heteroscedastic model, was developed by doctoral student Tim Bollerslev in 1986. The goal of GARCH is to provide volatility measures for
heteoscedastic time series data, much in the same way standard deviations are interpreted in simpler models.
The simplest GARCH model is the ARCH(1) model, which bears many similarities with AR(1) models. More complex ARCH(p) models are analogous to AR(p) models. Finally, Generalized ARCH models represent
conditional variances in the same way that ARMA models handle conditional expectation.
GARCH models have various applications for the analysis of time series data in finance and economics. They are especially useful when there are periods of fast changing variation (or volatility). For
example, they can efficiently model volatility of financial assets prices such as bonds, market indices, and stocks (Francq & Zakoian, 2011).
The utility of a GARCH model isn’t limited to financial applications. For example, Kim et al. (2014) used a GARCH model in their comparative study of different time series forecasting methods to
predict patient volumes in hospitals.
Various time series forecasting methods compared by Kim et al.
Heteroscedasticity and the GARCH model
Heteroscedasticity refers to the irregular variations of error terms seen in a time series model. Observations tend to cluster, rather than following the neat line seen in linear modeling. This makes
analysis of heteroscedastic data challenging.
In your conventional least squares model, the presence of heteroscedasticity can lead to too-narrow standard errors and confidence intervals, giving a false sense of precision. GARCH models address
this by treating heteroscedasticity as a variance which can be modeled. Predictions are then computed for each error term’s variance (Engle, 2001).
Basic Steps
A GARCH model follows three basic steps:
These steps involved (for example, finding maximum likelihood estimates of the conditionally normal model) and are usually performed with software. For example, the GARCH function in R.
Engle, R. (2001). GARCH 101: An Introduction to the Use of ARCH/GARCH models in Applied Econometrics. Retrieved May 11, 2020 from: https://www.stern.nyu.edu/rengle/GARCH101.PDF
Francq, C. & Zakoian, J. (2011). GARCH Models Structure, Statistical Inference and Financial Applications. Wiley.
GARCH models. Retrieved May 11, 2020 from: https://faculty.washington.edu/ezivot/econ589/ch18-garch.pdf
Kim et al. (2014). Predicting Patient Volumes in Hospital Medicine: A Comparative Study of Different Time Series Forecasting Methods. Retrieved May 11, 2020 from: https://www.mcs.anl.gov/~kibaekkim/ | {"url":"https://www.statisticshowto.com/garch-model-simple-definition/","timestamp":"2024-11-03T00:58:40Z","content_type":"text/html","content_length":"69435","record_id":"<urn:uuid:1ce296a0-f156-443b-b7e7-d923f4c81a90>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00876.warc.gz"} |
Introduction to centroid
Learn Mathematics through our AI based learning portal with the support of our Academic Experts!
Learn more
We know that the medians of the triangle are concurrent.
The point of concurrence of the three medians is the centroid of the triangle.
In the figure given above, \(G\) is the centroid of \(\triangle ABC\).
Based on the location of the centroid, the centroid exhibits the following properties:
[Note: Refer to the image given below for a better understanding.]
1. The centroid is always located inside the circle.
2. When a triangular plane surface is considered, the centroid acts as the centre of gravity.
3. The medians and the centroid together divide the triangle into three triangles of equal area.
In the figure we refer to, the three triangles of equal area are \(\triangle CGB\), \(\triangle GAB\), and \(\triangle AGC\).
The centroid always divides the median into two-parts to one. That is, the centroid divides the median in the ratio \(2 : 1\). The portion between the vertex and the centroid on the median occupies
two parts of the median, while the portion between the centroid and the other end of the median occupies one part of the median. | {"url":"https://www.yaclass.in/p/mathematics-state-board/class-8/geometry-3465/medians-centroid-and-altitude-of-a-triangle-17227/re-29f98914-749e-420c-a980-666eb7e64e74","timestamp":"2024-11-13T21:46:07Z","content_type":"text/html","content_length":"49671","record_id":"<urn:uuid:28465c00-23b4-4c5f-ae4e-2d140ff1309f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00182.warc.gz"} |
Can I hire a Calculus test-taker for timed exams or quizzes? | Hire Someone To Do Calculus Exam For Me
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What is the Highest Sat Score? - advisor
sat scores
Studying for the SAT is a lot of work. But it is important. The SAT is a significant part of your college application process. Most colleges either require or recommend the SAT. The SAT is a good way
for colleges to understand your academic capabilities. The tests are designed to show what you have learned in high school. And they also indicate how well-prepared you are for college coursework.
An important part of SAT preparation is having a goal. If you can identify a score you are aiming for, it will help you focus your study. Especially in terms of how much time you need to put into it.
But what is the highest SAT score?
What Is the Highest SAT Score?
The highest SAT score possible is 1600. It is incredibly difficult to score a perfect 1600 on the SAT. It is also very rare. College Board’s most recent statistics show only 7% of test takers scored
between 1400 and 1600 in 2018. And the average SAT score is 1068.
This does not mean that achieving a perfect score is impossible. It does mean that it requires a game plan and a lot of hard work.
Even if perfection is not your aim, it’s important to create an SAT scoring goal. The first place to start is with the school you plan applying to. Do a Google search to find out the average SAT
scores of students accepted into that college or university. This will help you set your ideal target score range.
From there, you will need to look at how the SAT scores work to help target your weakest areas. I will explain this below.
How Raw Scores Are Used in the SAT Score
The SAT is made up of three sections: Math, Reading, and Writing and Language. The Math section has a high score of 800. Reading, Writing, and Language are combined for a possible total 800 score.
These two sections are totaled for the highest possible SAT score—1600. This process is pretty straight forward. But figuring your section scores is a bit more complicated.
Each individual section is first given a raw score based on the number of correct answers. Then a formula is used to convert raw scores into the total section score.
The Math Calculator and Math No Calculator scores add together to get the total Math raw score. This number is then converted into a section score between 200 and 800.
Reading, and Writing and Language each use the raw scores to assign a test score between 10 and 40. Below is a sample raw score conversion table published by College Board.
These scores are then plugged into a conversion equation like this one below—also published by College Board.
Note that these are only sample conversion tables. There are many versions of the SAT taken each year to combat the possibility of cheating. There are various levels of difficulty between questions
and test versions. To make the final SAT scores fair and equal for all students, these formulas are created. This ensures that students taking different test versions around the world have fair
What is the Highest SAT Math Score
The highest score you can achieve on the Math section of the SAT is 800. There are two sections in math—calculator allowed and no calculator. These are made up of questions across various math
subtopics: Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math.
Looking at the above conversion chart, you need to get all 58 questions right for a perfect math section score. Every conversion chart is different to equate the levels of difficulty. So on a
difficult test, it may be possible to get 57 questions correct for a perfect score. But this will not always be the case.
A good way to identify your weaknesses in math for targeted study is to use the SAT practice tests or Khan Academy SAT practice. As you practice you will begin to see patterns emerge in the types of
questions you are struggling with. Those will be the areas you should begin to target more carefully to improve your scores.
What is the Highest SAT Evidence-Based Reading and Writing Score?
The SAT groups the two sections—Reading, and Language and Writing—together to get the second half of the 1600 high score. This score is the Evidence-Based Reading and Writing score—or EBRW. The
highest EBRW score is 800. Each section uses the number correct—the raw score—for a test score between 10 and 40. These are then plugged into the conversion formula to determine this total section
The Evidence-Based Reading and Writing section is made up of questions under the following subtopics: Command of Evidence, Words in Context, Expression of Ideas, and Standard English Conventions. As
you take practice tests, identify which of these subtopics you struggle most with. This is where you should focus to improve your scores and reach your goal.
How to Study for Your Highest SAT Score
Remember that there are three official sections on the SAT—Math, Reading, and Language and Writing. But there are only two section scores—Math and Evidence-Based Reading and Writing.
Reading is combined with Language and Writing to make up one half, while math makes up the other half alone. This means that math is the more important, weightier score. All three subjects are
important. But if math is a struggle for you or if you haven’t taken many upper-level math classes, you will need to make sure to target math heavily in your study.
If you use the full-length paper practice tests provided by College Board, make sure to carefully go through each corresponding “Scoring Your Practice SAT” pdf carefully. These break down each
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If you use the free Khan Academy SAT practice, much of the analysis work is done for you to help target your weaknesses. You can even send your SAT score report to Khan Academy and it will tailor
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These are the best places to begin. And for many students, this will be adequate to reach their scoring goals. But, if your goal is perfection you will need to go much more in depth in your
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Set a Goal for Your Highest SAT Score
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Numerical Analysis in JavaScript for Scientific/Mathematical Computation
Blog> Categories: Scientific
Numerical analysis is indispensable in advancing scientific understanding, technological innovation, and problem-solving capabilities across diverse fields. By leveraging numerical methods and
algorithms, researchers and practitioners can tackle complex mathematical challenges, simulate real-world phenomena, and make informed decisions based on accurate computational models. As
computational power and methodologies continue to evolve, numerical analysis remains at the forefront of driving progress in science, engineering, finance, and beyond, paving the way for new
discoveries and applications in the digital age.
What is Numerical Analysis? #
Numerical analysis is a branch of mathematics and computational science that focuses on developing algorithms and methods for solving mathematical problems by numerical approximation. It plays a
crucial role in modern scientific computing, engineering, and various fields where exact solutions are impractical or impossible to obtain. This article delves into the fundamental concepts, methods,
and applications of numerical analysis, highlighting its importance in tackling complex mathematical problems and real-world applications.
Key Concepts in Numerical Analysis #
1. Numerical Approximation: Numerical analysis involves approximating solutions to mathematical problems using computational methods rather than exact symbolic manipulations. This is often necessary
for problems involving continuous variables, complex equations, or systems with many variables.
2. Iterative Methods: Many numerical techniques rely on iterative methods, where calculations are repeated using initial estimates until a desired level of accuracy is achieved. Iterative refinement
is a common approach to improve the precision of numerical solutions.
3. Error Analysis: Central to numerical analysis is the study of errors that arise from approximations and computational limitations. Error analysis helps quantify the accuracy of numerical methods
and guides the selection of appropriate algorithms for different applications.
4. Complex Algorithms: Numerical algorithms encompass a wide range of techniques, including root-finding methods, interpolation, numerical integration, differential equation solvers, optimization
algorithms, and matrix computations. These algorithms are designed to efficiently solve specific types of problems encountered in science, engineering, finance, and other disciplines.
Importance of Numerical Analysis #
• Practical Solutions: Provides practical solutions to complex mathematical problems that cannot be solved analytically or require computationally intensive calculations.
• Accuracy and Efficiency: Enables scientists and engineers to achieve accurate results efficiently, balancing computational resources and precision.
• Innovation and Research: Facilitates innovation by enabling the development of advanced simulation techniques, optimization algorithms, and predictive models.
• Cross-Disciplinary Impact: Bridges theoretical concepts with real-world applications across disciplines, fostering interdisciplinary collaboration and problem-solving.
Challenges and Advances #
• Numerical Stability: Ensuring algorithms are robust against computational errors and numerical instability is critical for obtaining reliable results.
• High-Performance Computing: Advances in parallel computing, GPU acceleration, and cloud computing enhance the scalability and speed of numerical simulations.
• Algorithmic Complexity: Developing efficient algorithms with minimal computational complexity is essential for handling large-scale problems and big data analytics.
Why use JavaScript for Numercial Analysis #
Numerical analysis in JavaScript offers several compelling advantages. Leveraging numerical analysis in JavaScript empowers developers, researchers, and scientists to tackle complex mathematical
challenges, perform advanced data analysis, and build interactive applications that facilitate exploration and understanding of numerical models and scientific phenomena. By harnessing JavaScript’s
strengths in accessibility, performance, and integration with modern web technologies, numerical analysis becomes not only a practical tool for computational tasks but also a catalyst for innovation
and discovery across diverse domains.
1. Accessibility and Ubiquity #
JavaScript is the standard programming language for web development, supported by all major web browsers across different platforms and devices. This ubiquity allows numerical analysis to be
seamlessly integrated into web-based applications, providing accessibility to a wide audience without requiring additional software installations.
2. Interactive Data Visualization #
JavaScript libraries like D3.js, Plotly.js, and Chart.js enable interactive data visualization directly in web browsers. These libraries facilitate real-time exploration and visualization of
numerical results, enhancing the understanding and interpretation of complex data sets and computational models.
3. Asynchronous Programming Model #
JavaScript’s asynchronous programming model, supported by Promises and async/await, enables efficient handling of computational tasks that involve asynchronous data fetching, processing, and
visualization. This capability is crucial for real-time data analysis, simulations, and interactive applications where responsiveness and performance are paramount.
4. Computational Libraries and Tools #
JavaScript boasts a rich ecosystem of numerical computation libraries and tools that support a wide range of mathematical operations, statistical analysis, and scientific computing:
• Math.js: Provides comprehensive support for mathematical functions, linear algebra, and statistical operations.
• Numeric.js: Offers utilities for numerical computing, matrix operations, and solving linear systems.
• TensorFlow.js: Enables machine learning and deep learning applications directly in the browser, leveraging neural networks for predictive modeling and pattern recognition.
5. Cross-Platform Compatibility #
JavaScript’s ability to run on both client-side (browser) and server-side (Node.js) environments enhances cross-platform compatibility and facilitates seamless integration with backend systems and
data pipelines. This flexibility is advantageous for building end-to-end solutions that span from data acquisition to analysis and visualization.
6. Performance Optimization #
Advancements in JavaScript engines (such as V8 in Chrome) and optimizations like just-in-time (JIT) compilation have significantly improved JavaScript’s performance. This allows for efficient
execution of computationally intensive tasks, such as numerical simulations, algorithmic optimizations, and big data analytics.
7. Community and Collaboration #
JavaScript benefits from a large and active developer community that contributes to open-source projects, libraries, and frameworks focused on numerical computation and scientific computing. This
collaborative ecosystem fosters innovation, accelerates the development of new tools and techniques, and promotes knowledge sharing across disciplines.
8. Educational and Research Applications #
JavaScript’s accessibility and ease of use make it an ideal platform for educational purposes, allowing students, researchers, and educators to explore numerical methods, conduct experiments, and
visualize results interactively. It supports educational initiatives and facilitates hands-on learning in fields such as mathematics, physics, engineering, and data science.
Examples of Numerical Analysis in JavaScript #
Below are a few examples of numerical analysis techniques implemented in JavaScript. These examples demonstrate the implementation of the bisection method for finding the roots of equations to
solving differential equations/doing integration. You can use this notebook on Scribbler for experimentation: Numerical Analysis Recipes
These algorithms use Higher Order Functions in a functional programming paradigm.
Also check out: Numeric.js for Numerical Analysis and Linear Algebra in JavaScript and Exploring Numerical Methods for Integration Using JavaScript.
Bisection Method for Finding Solution to an Equation #
function bisectionMethod(func, a, b, tolerance) {
let c = (a + b) / 2;
while (Math.abs(func(c)) > tolerance) {
if (func(a) * func(c) < 0) {
b = c;
} else {
a = c;
c = (a + b) / 2;
return c;
// Example usage
function f(x) {
return x * x - 4; // Find the root of this function
const root = bisectionMethod(f, 1, 3, 0.0001);
console.log("Root:", root);
Newton’s Method for Finding Solution to an Equation #
function newtonsMethod(func, derivative, x0, tolerance) {
let x = x0;
while (Math.abs(func(x)) > tolerance) {
x = x - func(x) / derivative(x);
return x;
// Example usage
function f(x) {
return x * x - 4; // Find the root of this function
function fDerivative(x) {
return 2 * x; // Derivative of f(x)
const root = newtonsMethod(f, fDerivative, 2, 0.0001);
console.log("Root:", root);
Euler’s Method for Ordinary Differential Equation #
function eulerMethod(dydx, x0, y0, h, numSteps) {
let x = x0;
let y = y0;
for (let i = 0; i < numSteps; i++) {
const slope = dydx(x, y);
y = y + h * slope;
x = x + h;
return y;
// Example usage
function dydx(x, y) {
return x * x - 4 * y; // Solve the differential equation dy/dx = x^2 - 4y
const solution = eulerMethod(dydx, 0, 1, 0.1, 10);
console.log("Solution:", solution);
function simpsonsRule(func, a, b, n) {
const h = (b - a) / n;
let sum = func(a) + func(b);
for (let i = 1; i < n; i++) {
const x = a + i * h;
sum += i % 2 === 0 ? 2 * func(x) : 4 * func(x);
return (h / 3) * sum;
// Example usage
function f(x) {
return Math.sin(x); // Integrate sin(x) from 0 to π
const integral = simpsonsRule(f, 0, Math.PI, 100);
console.log("Integral:", integral);
function gaussianElimination(matrix) {
const n = matrix.length;
for (let i = 0; i < n; i++) {
let maxRow = i;
for (let j = i + 1; j < n; j++) {
if (Math.abs(matrix[j][i]) > Math.abs(matrix[maxRow][i])) {
maxRow = j;
[matrix[i], matrix[maxRow]] = [matrix[maxRow], matrix[i]];
for (let j = i + 1; j < n; j++) {
const ratio = matrix[j][i] / matrix[i][i];
for (let k = i; k < n + 1; k++) {
matrix[j][k] -= ratio * matrix[i][k];
const solution = new Array(n);
for (let i = n - 1; i >= 0; i--) {
let sum = 0;
for (let j = i + 1; j < n; j++) {
sum += matrix[i][j] * solution[j];
solution[i] = (matrix[i][n] - sum) / matrix[i][i];
return solution;
// Example usage
const augmentedMatrix = [
[2, 1, -1, 8],
[-3, -1, 2, -11],
[-2, 1, 2, -3],
const solution = gaussianElimination(augmentedMatrix);
console.log("Solution:", solution);
function rungeKuttaMethod(dydx, x0, y0, h, numSteps) {
let x = x0;
let y = y0;
for (let i = 0; i < numSteps; i++) {
const k1 = h * dydx(x, y);
const k2 = h * dydx(x + h / 2, y + k1 / 2);
const k3 = h * dydx(x + h / 2, y + k2 / 2);
const k4 = h * dydx(x + h, y + k3);
y = y + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
x = x + h;
return y;
// Example usage
function dydx(x, y) {
return x * x - 4 * y; // Solve the differential equation dy/dx = x^2 - 4y
const solution = rungeKuttaMethod(dydx,0, 1, 0.1, 10);
console.log("Solution:", solution); You can experiment with Runge-Kutta Method in this notebook on Scribbler: [Runge-Kutta Method for Differential Equations](https://app.scribbler.live/?jsnb=./examples/Runge-Kutta-for-Differential-Equations.jsnb).
Learning numerical analysis in JavaScript can empower you to perform complex numerical computations, build interactive web applications. JavaScript can provide an environment where scientific
computation can be done on the browser. This is great for interactive science. Numerical analysis tools can help solve complex scientific problems using code. This can help scientists ease out their
Applications of Numerical Analysis for Scientific Computation #
Numerical analysis, a cornerstone of computational mathematics and scientific computing, plays a pivotal role in solving complex problems across diverse fields. From engineering simulations to
financial modeling and beyond, numerical methods provide powerful tools for approximating solutions, analyzing data, and making informed decisions. This article explores the broad spectrum of
applications where numerical analysis makes a significant impact, highlighting its importance in advancing technology, research, and innovation.
Engineering and Simulation #
1. Finite Element Analysis (FEA):
□ Application: Structural engineering, aerospace engineering, and mechanical design.
□ Use: Predicts stresses, deformations, and behaviors of complex structures under various conditions, optimizing designs and ensuring safety.
2. Computational Fluid Dynamics (CFD):
□ Application: Aerospace, automotive, and environmental engineering.
□ Use: Simulates fluid flow, heat transfer, and turbulence to optimize aerodynamics, design HVAC systems, and study environmental impacts.
3. Electromagnetic Simulations:
□ Application: Electrical engineering, telecommunications, and antenna design.
□ Use: Models electromagnetic fields, wave propagation, and interactions with materials to design efficient antennas and electromagnetic devices.
Physics and Scientific Research #
1. Quantum Mechanics and Molecular Dynamics:
□ Application: Physics, chemistry, and material science.
□ Use: Simulates quantum states, molecular interactions, and chemical reactions to study atomic structures, develop new materials, and understand biological processes.
2. Astrophysical Simulations:
□ Application: Astronomy and cosmology.
□ Use: Models celestial phenomena, galaxy formation, and gravitational interactions to explore the universe’s evolution and predict astronomical events.
3. Particle Physics:
□ Application: High-energy physics experiments (e.g., Large Hadron Collider).
□ Use: Analyzes particle collisions, simulates detector responses, and validates theoretical models to discover fundamental particles and understand the universe’s fundamental laws.
Financial Modeling and Risk Analysis #
1. Derivative Pricing and Portfolio Optimization:
□ Application: Finance, investment banking, and risk management.
□ Use: Values complex financial instruments, hedges risks, and optimizes investment portfolios using Monte Carlo simulations, Black-Scholes models, and numerical methods for optimization.
2. Economic Forecasting:
□ Application: Macroeconomics and policy analysis.
□ Use: Models economic indicators, forecasts GDP growth, inflation rates, and unemployment trends to inform monetary policy decisions and assess economic impacts.
Data Analysis and Machine Learning #
1. Numerical Optimization:
□ Application: Machine learning, artificial intelligence, and data science.
□ Use: Develops optimization algorithms, trains neural networks, and performs parameter tuning to improve model accuracy, pattern recognition, and predictive analytics.
2. Big Data Analytics:
□ Application: Business intelligence and large-scale data processing.
□ Use: Analyzes massive datasets, identifies trends, and extracts actionable insights using numerical algorithms for clustering, classification, and regression analysis.
Medical and Biological Sciences #
1. Bioinformatics and Genomics:
□ Application: Genetics, personalized medicine, and pharmaceutical research.
□ Use: Analyzes genomic data, sequences DNA, predicts protein structures, and identifies genetic variations associated with diseases using numerical methods for sequence alignment and
statistical analysis.
2. Medical Imaging:
□ Application: Radiology and diagnostic imaging.
□ Use: Reconstructs 3D images from medical scans (e.g., MRI, CT) using numerical algorithms for image processing, segmentation, and visualization to aid in disease diagnosis and treatment
Environmental Modeling and Climate Science #
1. Climate Modeling:
□ Application: Meteorology and environmental science.
□ Use: Simulates climate patterns, predicts weather events, and assesses climate change impacts using numerical models for atmospheric dynamics, ocean currents, and carbon cycle interactions.
2. Environmental Fluid Dynamics:
□ Application: Hydrology, coastal engineering, and environmental impact assessments.
□ Use: Models water flow, sediment transport, and pollutant dispersion in rivers, oceans, and urban environments to manage water resources and mitigate environmental risks. | {"url":"https://scribbler.live/2023/06/07/Numerical-Analysis-in-JavaScript-for-Scientific-Computing.html","timestamp":"2024-11-07T13:59:02Z","content_type":"text/html","content_length":"55980","record_id":"<urn:uuid:ec3fc838-093a-4449-9d94-e64db60658aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00117.warc.gz"} |
Find the equation of circle passing through the intersection of ${{x}^{2}}+{{y}^{2}}=4$ and ${{x}^{2}}+{{y}^{2}}-2x-4y+4=0$ and touching the line $x+2y=0$
Hint: Equation of family of circle is${{S}_{1}}+\lambda {{S}_{2}}=0$. So, there can be infinite number of circles passing through the intersection points of ${{x}^{2}}+{{y}^{2}}=4$and ${{x}^{2}}+{{y}
^{2}}-2x-4y+4=0$. By substituting values in the Family of circle equation we get the centre of the circle. Hence, we can calculate radius using $\sqrt{{{\left( x \right)}^{2}}+{{\left( y \right)}^
{2}}-c}$ where x and y are coordinates and c is the constant in the equation.
Then find the perpendicular distance between radius and tangent to get the value of $\lambda $.
Then substitute this value in the equation of the family of circles.
Complete step by step answer:
Let us consider,
${{S}_{1}}={{x}^{2}}+{{y}^{2}}-4$ and ${{S}_{2}}={{x}^{2}}+{{y}^{2}}-2x-4y+4$
Now substitute this value in the family of circles equation.
& {{S}_{1}}+\lambda {{S}_{2}}=0 \\
& {{x}^{2}}+{{y}^{2}}-2x-4y+4+\lambda \left( {{x}^{2}}+{{y}^{2}}-4 \right)=0 \\
$\left( \lambda +1 \right){{x}^{2}}+\left( \lambda +1 \right){{y}^{2}}-2x-4y-4\lambda +4=0......(1)$
Now divide the whole equation by $\left( \lambda +1 \right)$,
$\left( \lambda +1 \right){{x}^{2}}+\left( \lambda +1 \right){{y}^{2}}-2x-4y-4\lambda +4=0$
${{x}^{2}}+{{y}^{2}}-\dfrac{2x}{\left( \lambda +1 \right)}-\dfrac{4y}{\left( \lambda +1 \right)}+\dfrac{4-4\lambda }{\left( \lambda +1 \right)}=0$
Now compare this equation with the general equation of circle $a{{x}^{2}}+b{{y}^{2}}+2gx+2fy+c=0$
Coordinates of centre are given by $\left( g,f \right)$
Therefore, coordinates of the circle will be $\left( \dfrac{1}{\lambda +1},\dfrac{2}{\lambda +1} \right)$
Radius is obtained by $\sqrt{{{\left( g \right)}^{2}}+{{\left( f \right)}^{2}}-\left( c \right)}$
Now substitute the values and find radius.
$\sqrt{{{\left( \dfrac{1}{\lambda +1} \right)}^{2}}+{{\left( \dfrac{2}{\lambda +1} \right)}^{2}}-\left( \dfrac{-4\lambda +4}{\lambda +1} \right)}$
$\sqrt{\left( \dfrac{1+4-\left( -4\lambda +4 \right)\left( \lambda +1 \right)}{{{\left( \lambda +1 \right)}^{2}}} \right)}$
$\sqrt{\left( \dfrac{5-\left( 4+4\lambda -4\lambda -4{{\lambda }^{2}} \right)}{{{\left( \lambda +1 \right)}^{2}}} \right)}$
$\sqrt{\left( \dfrac{5-4+4{{\lambda }^{2}}}{{{\left( \lambda +1 \right)}^{2}}} \right)}$
$\sqrt{\left( \dfrac{1+4{{\lambda }^{2}}}{{{\left( \lambda +1 \right)}^{2}}} \right)}$
As, our circle is touching $x+2y=0$ i.e. this line is a tangent to the circle.
We know that the tangent is perpendicular to the radius.
And the perpendicular distance is given by
Radius = $\left| \dfrac{g+2f}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right|$ where a and b are the coordinates of the perpendicular.
Now substitute the values,
$\sqrt{\left( \dfrac{1+4{{\lambda }^{2}}}{{{\left( \lambda +1 \right)}^{2}}} \right)}=\left| \dfrac{\left( \dfrac{1}{\lambda +1} \right)+\left( \dfrac{4}{\lambda +1} \right)}{\sqrt{{{\left( 1 \
right)}^{2}}+{{\left( 2 \right)}^{2}}}} \right|$
Squaring both the sides to remove root as well as modulus.
& \sqrt{\left( \dfrac{1+4{{\lambda }^{2}}}{{{\left( \lambda +1 \right)}^{2}}} \right)}=\left| \dfrac{\left( \dfrac{5}{\lambda +1} \right)}{\sqrt{{{\left( 1 \right)}^{2}}+{{\left( 2 \right)}^{2}}}} \
right| \\
& \left( \dfrac{1+4{{\lambda }^{2}}}{{{\left( \lambda +1 \right)}^{2}}} \right)=\dfrac{25}{{{\left( \lambda +1 \right)}^{2}}\times 5} \\
& 1+4{{\lambda }^{2}}=5 \\
& 4{{\lambda }^{2}}=4 \\
& \lambda =\pm 1 \\
But $\lambda $cannot be zero.
Hence, the value of $\lambda $ will be 1.
Therefore, the co-ordinates will be $\left( \dfrac{1}{2},\dfrac{2}{2} \right)$
Substitute these values in this$\left( \lambda +1 \right){{x}^{2}}+\left( \lambda +1 \right){{y}^{2}}-2x-4y+4-4\lambda =0$ equation.
Hence, the equation of our required circle will be,
& \left( 1+1 \right){{x}^{2}}+\left( 1+1 \right){{y}^{2}}-2x-4y+4-4\left( 1 \right)=0 \\
& 2{{x}^{2}}+2{{y}^{2}}-2x-4y+4-4=0 \\
& {{x}^{2}}+{{y}^{2}}-x-2y=0 \\
Hence, equation of circle is ${{x}^{2}}+{{y}^{2}}-x-2y=0$.
Note: Remember that the value of is to be taken positive. The formula of radius is $\sqrt{{{\left( g \right)}^{2}}+{{\left( f \right)}^{2}}-\left( c \right)}$ g, f being the centre of circle and c
being the constant in the equation. In the formula of the perpendicular distance applying mod is necessary. | {"url":"https://www.vedantu.com/question-answer/find-the-equation-of-circle-passing-through-the-class-11-maths-cbse-5f894cd4980bb03139dccfd6","timestamp":"2024-11-07T12:24:45Z","content_type":"text/html","content_length":"174239","record_id":"<urn:uuid:2b5824d1-dd29-4992-ab64-30ee49b4ed9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00108.warc.gz"} |
Fun Maths Puzzle for Students with AnswerFun Maths Puzzle for Students with Answer
If you are school going student, then this Fun Maths Puzzle will test your Mathematical and logical reasoning skills. In this Maths Puzzle, you challenge is to find the relationship among given
number around the Rectangle and then find the value of missing number which replaces question mark. Can you take this challenge?
Can you find the missing number?
The answer to this "Fun Maths Puzzle for Students", can be viewed by clicking on the button. Please do give your best try before looking at the answer.
No comments: | {"url":"https://www.brainsyoga.com/2018/04/fun-maths-puzzle-for-students.html","timestamp":"2024-11-13T07:50:38Z","content_type":"application/xhtml+xml","content_length":"118975","record_id":"<urn:uuid:ffb9a257-4b0d-4fb2-8e26-7a861f488644>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00131.warc.gz"} |
Polynomial equations
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• Sample Math Aptitude Test | {"url":"https://softmath.com/math-com-calculator/function-range/polynomial-equations.html","timestamp":"2024-11-14T07:42:24Z","content_type":"text/html","content_length":"180244","record_id":"<urn:uuid:0d9dd939-e0c5-4cb0-b3f5-24921e3a879c>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00710.warc.gz"} |
Why I can't constrain the objective variable with inequality？ | AIMMS Community
The objective variable is A(t) contains t = 1 to 24.
The mathematical program is to maximize the sum of all the elements in A, i. e. Max sum(A(t)), t = 1 to 24.
At the same time I hope to constrain the property of the objective variable that the standard deviation SD(A) is no larger than a constant “a”. So I defined a variable “SD(A) = standard deviation of
A”. Then a constraint “SD(A) <= a”.
When I run the program, it reminds me of “constraint programming constraints cannot be used in combination with real valued variables, only with integer valued variables, element valued variables,
and activities. The real valued objective variable is an exception. The mathematical program has both constraint programming constraints and real valued variables.”
However, when I revise the constraint from inequality to equality, i.e. SD(A) = a. Then no error arises and I get the answer.
Why dose this happen? And How can I deal with it if I want the inequality constraint? | {"url":"https://community.aimms.com/aimms-language-12/why-i-can-t-constrain-the-objective-variable-with-inequality-1448?postid=4015","timestamp":"2024-11-13T19:09:01Z","content_type":"text/html","content_length":"141843","record_id":"<urn:uuid:fcf1b0c0-e870-4ed3-8687-58c286ad74f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00240.warc.gz"} |
Hybrid Symbolic Regression with the Bison Seeker Algorithm
Hybrid Symbolic Regression with the Bison Seeker Algorithm
Keywords: genetic programming, symbolic regression, hybrid methods, local learning, bison seeker algorithm
This paper focuses on the use of the Bison Seeker Algorithm (BSA) in a hybrid genetic programming approach for the supervised machine learning method called symbolic regression. While the basic
version of symbolic regression optimizes both the model structure and its parameters, the hybrid version can use genetic programming to find the model structure. Consequently, local learning is used
to tune model parameters. Such tuning of parameters represents the lifetime adaptation of individuals. This paper aims to compare the basic version of symbolic regression and hybrid version with the
lifetime adaptation of individuals via the Bison Seeker Algorithm. Author also investigates the influence of the Bison Seeker Algorithm on the rate of evolution in the search for function, which fits
the given input-output data. The results of the current study support the fact that the local algorithm accelerates evolution, even with a few iterations of a Bison Seeker Algorithm with small
Iba H., Sato T., and de Garis, H. 1995. Recombination guidance for numerical genetic programming. In Proceedings of 1995 IEEE International Conference on Evolutionary Computation. IEEE, pp. 97. DOI:
Nikolaev, N. Y. and Iba H. 2006. Adaptive Learning of Polynomial Networks: Genetic Programming, Backpropagation and Bayesian methods. Springer, New York, USA.
Schoenauer M., Lamy, B., and Jouve, F. 1995. Identification of Mechanical Behaviour by Genetic Programming Part II: Energy formulation. Technical report, Ecole Polytechnique, France.
Sharman, K. C., Esparcia-Alcazar, A. I., and Li, Y. 1995. Evolving Signal processing Algorithms by Genetic Programming. In First International Conference on Genetic Algorithms in Engineering Systems:
Innovations and Applications, GALESIA. Volume 414, pp. 473–480, Sheffield, UK.
Koza, J. R. 1992. Genetic programming: On The Programming of Computers by Means of Natural Selection. Bradford Book, Cambridge, UK.
Poli, R., Langdon W. B., and McPhee, N. F. 2008. A Field Guide to Genetic Programming. Lulu Press, Morrisville, North Carolina, USA.
Whitley, D., Gordon, S., and Mathias, K. 1994. Lamarckian Evolution, the Baldwin Effect and Function Optimization. In Parallel Problem Solving from Nature - PPSN III. Springer, Berlin, pp. 6–15.
Le, N., Brabazon, A., and O’Neill, M. 2018. How the “Baldwin Effect” Can Guide Evolution in Dynamic Environments. Theory and Practice of Natural Computing [online]. Lecture Notes in Computer Science.
Cham: Springer International Publishing, pp. 164–175. DOI: 10.1007/978-3-030-04070-3_13
Reďko, V. G., Mosalov, O. P., and Prokhorov, D. V. 2005. A Model of Evolution and Learning. Neural Networks 18, 5–6, pp. 738–745. DOI: 10.1016/j.neunet.2005.06.005
Turney, P. D. 2002. Myths and legends of the Baldwin effect. arXiv: cs/0212036. Retrieved from https://arxiv.org/abs/cs/0212036
Hinton, G. E. and Nowlan, S. J. 1987. How learning can guide evolution. Complex Systems 1, pp. 495–502.
French, R. and Messinger, A. 1994. Genes, Phenes and the Baldwin Effect: Learning and Evolution in a Simulated Population. In Artificial Life IV. MIT Press, Cambridge, MA, USA.
Topchy, A., and Punch, W. F. 2001. Faster Genetic Programming Based on Local Gradient Search of Numeric Leaf Values. In Proceedings of the 3rd Annual Conference on Genetic and Evolutionary
Computation (GECCO'01). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, pp. 155–162.
Anderson R. W. 1995. Learning and evolution: A Quantitative Genetics Approach. Journal of Theoretical Biology 175, 1, pp. 89–101. DOI: 10.1006/jtbi.1995.0123
Hashimoto, N., Kondo, N., Hatanaka, T., and Uosaki, K. 2008. Nonlinear System Modeling by Hybrid Genetic Programming. IFAC Proceedings Volumes 41, 2, pp. 4606–4611. DOI: 10.3182/
Raidl, G. 1998. A Hybrid GP Approach for Numerically Robust Symbolic Regression. In Proc. of the 1998 Genetic Programming Conference. Madison, Wisconsin, pp. 323–328.
Brandejský, T. 2019. Dependency of GPA-ES Algorithm Efficiency on ES Parameters Optimization Strength. In AETA 2018: Recent Advances in Electrical Engineering and Related Sciences. Springer, pp.
294–302. DOI: 10.1007/978-3-030-14907-9_29
Kennedy, J., and Eberhart, R. 1995. Particle Swarm Optimization. In: Proceedings of ICNN'95 - International Conference on Neural Networks .IEEE, pp. 1942–1948. DOI: 10.1109/ICNN.1995.488968
Rosendo, M., and Pozo, A. 2010. Applying a Discrete Particle Swarm Optimization Algorithm to Combinatorial Problems. In 2010 Eleventh Brazilian Symposium on Neural Networks. IEEE, pp. 235–240. DOI:
Xing, B., and Gao, W.-J. 2014. Innovative Computational Intelligence: A Rough Guide to 134 Clever Algorithms. Springer International Publishing, New York, NY, USA.
Kazíková, A., Pluháček, M. and Šenkeřík, R. 2018. Regarding the Behavior of Bison Runners Within the Bison Algorithm. MENDEL 24, 1, pp. 63–70. DOI: 10.13164/mendel.2018.1.063
How to Cite
Merta, J. 2019. Hybrid Symbolic Regression with the Bison Seeker Algorithm. MENDEL. 25, 1 (Jun. 2019), 79-86. DOI:https://doi.org/10.13164/mendel.2019.1.079.
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The limit is the key concept that separates calculus from elementary mathematics such as arithmetic, elementary algebra or Euclidean geometry. It also arises and plays an important role in the more
general settings of topology, analysis, and other fields of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous.
Intuitive Meaning
Many people new to calculus have difficulty understanding the formal definition of a limit. Thus we begin with an informal explanation: a limit is the value to which a function grows close when its
argument is near (but not at!) a particular value. For example, $\[\lim_{x\to 2}x^2=4\]$ because whenever $x$ is close to 2, the function $f(x)=x^2$ grows close to 4.
In this case, the limit of the function happens to equal the value of the function ($\lim_{x\rightarrow c} f(x) = f(c)$). This is because the function we chose was continuous at $c$.
However, not all functions have this property. For example, consider the function $f(x)$ over the reals defined as follows: $\[f(x) = \begin{cases} 0 & \text{if } xeq 0,\\ 1 & \text{if } x=0. \end
{cases}\]$ Although the value of the function $f(x)$ at $x = 0$ is $1$, the limit $\lim_{x\rightarrow 0} f(x)$ is, in fact, zero. Intuitively, this is because the limit describes the behavior of the
function near (but not at!) the value in question: when $x$ is very close (but not equal!) to zero, $f(x)$ will always be close to (in fact equal to) zero.
Let $A$ and $B$ be metric spaces, let $A'$ be a subspace of $A$, and, let $f$ be a function from $A'$ to $B$. Let $c$ be a limit point of $A'$. (This means that in the metric space $A$, there are
elements of $A'$ arbitrarily close to $c$.) Let $L$ be an element of $B$. We say $\[\lim_{x\to c} f(x) = L,\]$ (that is, the limit of $f(x)$ as $x$ goes to $c$ equals $L$) if for every positive real
$\epsilon$ there exists a positive real $\delta$ for which $\[0 < d_A(x,c) < \delta\]$ implies $\[d_B(f(x),L) < \epsilon\]$ for all $x \in A'$. Here $d_A$ and $d_B$ are the distance functions of $A$
and $B$, respectively.
In terms of our informal definition, $\epsilon$ is a measure of "how close" we want $f(x)$ to be to its limit value. Then the formal definition says that no matter how close we want to be (for any $\
epsilon > 0$), we can make our variable close enough (within a distance $\delta$, for some $\delta$) to $c$ to achieve our goal.
In analysis and calculus, usually $A$ and $B$ are both either the set of reals $\mathbb{R}$ or complex numbers $\mathbb{C}$. In this case, the distance functions $d_A(a,b)$ and $d_B(a,b)$ are both
simply $|a-b|$. We then obtain the following definition commonly found in calculus textbooks:
Let $f$ be a function whose domain is a sub-interval of the real numbers and whose codomain is the set of reals. For a real number $L$,
$\[\lim_{x\to c} f(x) =L\]$
if for every $\epsilon >0$ there exists a $\delta>0$ such that
$\[0 < |x-c| < \delta \quad \text{implies} \quad |f(x) - L| < \epsilon .\]$
However, most theorems on real limits apply to limits in general, with identical proofs.
Existence of Limits
Limits do not always exist. For example $\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since, in fact, there exists no $\epsilon$ for which there exists $\delta$ satisfying the definition's
conditions, since $\left|\frac{1}{x}\right|$ grows arbitrarily large as $x$ approaches 0. However, it is possible for $\lim_{x\rightarrow c} f(x)$ not to exist even when $f$ is defined at $c$. For
example, consider the Dirichlet function, $D(x)$, defined to be 0 when $x$ is irrational, and 1 when $x$ is rational. Here, $\lim_{x\rightarrow c}D(x)$ does not exist for any value of $c$.
Alternatively, limits can exist where a function is not defined, as for the function $f(x)$ defined to be 1, but only for nonzero reals. Here, $\lim_{x\rightarrow 0}f(x)=1$, since for $x$ arbitrarily
close to 0, $f(x)=1$.
The notation $\lim_{x\to c}f(x) = L$ would only be justifiable if the limit $L$ were unique. Fortunately, it is always the case that if a limit exists, it is unique.
Indeed, suppose that $L'$ is also $\lim_{x\to c}f(x)$, and that $L eq L'$. Since $d_B(L,L') >0$, we can pick a positive real $\epsilon < d_B(L,L')/2$. But for any $y \in L$, $\[d_B(L,y) + d_B(L',y) \
ge d_B(L,L'),\]$ so no $y$ can simultaneously satisfy the conditions \begin{align*} d_B(L,y) &< \epsilon < \frac{d_B(L,L')}{2} \\ d_B(L',y) &< \epsilon < \frac{d_B(L,L')}{2} , \end{align*} a
contradiction. Therefore limits are unique, as we wanted.
Left and Right Hand Limits
In this section, we consider limits of functions whose domain and range are both subsets of the set of reals.
Left and right hand limits are the limits taken as a point is approached from the left and from the right, respectively. The left hand limit is denoted as $\lim_{x\to c^{-}} f(x)$, and the right hand
limit is denoted as $\lim_{x\to c^{+}} f(x)$.
If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider the step function (the greatest integer function) $f(x) = \
lfloor x \rfloor$, we have $\lim_{x\to 0^{+}} \lfloor x \rfloor = 0$, while $\lim_{x\to 0^{-}} \lfloor x \rfloor = -1$.
A limit exists if the left and right hand side limits exist, and are equal.
Sequential Criterion
Let $A\subset\mathbb{R}$ and let $c$ be a cluster point of $A$. A function $f : A \rightarrow \mathbb{R}$ has a limit $L = \lim_{x \rightarrow c} f(x)$ if for every sequence $\left\langle x_n \right\
rangle$ that converges to $c$, $\left\langle f(x_n) \right\rangle$ converges to $L$.
Other Properties
Let $f$ and $g$ be real functions. Then:
• $\lim(f+g)(x)=\lim f(x)+\lim g(x)$
• $\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
• $\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$ given that $\lim g(x)e 0$.
See also | {"url":"https://artofproblemsolving.com/wiki/index.php/Limit","timestamp":"2024-11-12T23:24:41Z","content_type":"text/html","content_length":"61845","record_id":"<urn:uuid:26092627-ee0f-4e59-8d7c-bb3088818fc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00066.warc.gz"} |
Return a Blackman window.
The Blackman window is a taper formed by using the the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only
slightly worse than a Kaiser window.
M : int
Number of points in the output window. If zero or less, an empty array is returned.
Parameters :
sym : bool, optional
When True, generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.
w : ndarray
Returns :
The window, with the maximum value normalized to 1 (though the value 1 does not appear if the number of samples is even and sym is True).
The Blackman window is defined as
Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization
(which means “removing the foot”, i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a “near optimal” tapering function, almost as
good (by some measures) as the Kaiser window.
[R95] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York.
[R96] Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fftpack import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.blackman(51)
>>> plt.plot(window)
>>> plt.title("Blackman window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Blackman window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]") | {"url":"https://docs.scipy.org/doc/scipy-0.12.0/reference/generated/scipy.signal.blackman.html","timestamp":"2024-11-05T06:25:04Z","content_type":"application/xhtml+xml","content_length":"14438","record_id":"<urn:uuid:9675cf45-eb68-4a7e-874e-b22e3a80417f>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00114.warc.gz"} |
Cube root Archives - Mathstoon
The cube root of 40 in simplified radical form is equal to 2∛5. The value of cube root of 40 is 3.42 corrected up to two decimal places. Note that a number m is called a cube root of 40 if m3 = 40.
So it is a root of the cubic equation x3=40, so … Read more
Cube Root of 81
The cube root of 81 is a number when multiplied by itself two times will be 81. The cube root of 81 is denoted by the symbol $\sqrt[3]{81}$. In this section, we will learn how to find the value of $\
sqrt[3]{81}$. The value of the cube root of 81 is $3\sqrt[3]{3}$. Important Things about Cube Root … Read more
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The cube root of 27 is a number when multiplied by itself two times will be 27. The cube root of 27 is denoted by the symbol ∛27. In this section, we will learn how to find the value of ∛27. Note
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The cube root of 125 is denoted by the symbol $\sqrt[3]{125}$. … Read more
Cube and Cube Root: Definition, Formula, How to Find, Examples
The cube root of a number is an important concept like square roots in the number system. Note that the cube root is the inverse method of finding cubes. In this section, we will discuss about the
cube root of a number. What is cube and cube root? The number obtained by multiplying a given … Read more | {"url":"https://www.mathstoon.com/category/cube-root/","timestamp":"2024-11-05T22:34:44Z","content_type":"text/html","content_length":"171078","record_id":"<urn:uuid:0a9403b9-818c-4daa-9498-89adf5633d84>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00068.warc.gz"} |
Surface evolution of elastically stressed films | EMS Magazine
An overview of recent analytical developments in the study of epitaxial growth is presented. Quasistatic equilibrium is established, regularity of solutions is addressed, and the evolution of
epitaxially strained elastic films is treated using minimizing movements.
In this paper, we give a brief overview of recent analytical developments in the study of the deposition of a crystalline film onto a substrate, with the atoms of the film occupying the substrate’s
natural lattice positions. This process is called epitaxial growth. Here we are interested in heteroepitaxy, that is, epitaxy when the film and the substrate have different crystalline structures. At
the onset of the deposition, the film’s atoms tend to align themselves with those of the substrate because the energy gain associated with the chemical bonding effect is greater than the film’s
strain due to the mismatch between the lattice parameters. As the film continues to grow, the stored strain energy per unit area of the interface increases with the film thickness, rendering the
film’s flat layer morphologically unstable or metastable after the thickness reaches a critical value. As a result, the film’s free surface becomes corrugated, and the material agglomerates into
clusters or isolated islands on the substrate. The formation of islands in systems such as In-GaAs/GaAs or SiGe/Si has essential high-end technology applications, such as modern semiconductor
electronic and optoelectronic devices (quantum dots laser). The Stranski–Krastanow (SK) growth mode occurs when the islands are separated by a thin wetting layer, while the Volmer–Weber (VW) growth
mode refers to the case when the substrate is exposed between islands.
In what follows, we adopt the variational model considered by Spencer in [41 B. J. Spencer, Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski–Krastanow
islands. Phys. Rev. B, 59, 2011 (1999) ] (see also [36 R. V. Kukta and L. B. Freund, Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate. J. Mech. Phys.
Solids45, 1835–1860 (1997) , 42 B. J. Spencer and J. Tersoff, Equilibrium shapes and properties of epitaxially strained islands. Phys. Rev. Lett.79, 4858 (1997) ], and the references contained
therein). To be precise, the free energy functional associated with the physical system is given by
Here is the function whose graph describes the profile of the film, assumed to be -periodic, with , for some , is the region occupied by the film, i.e., writing ,
is displacement of the material, is the symmetric part of . Also, the elastic energy density is a positive definite quadratic form defined on the space of symmetric matrices
with a positive definite fourth-order tensor, so that for all , is an anisotropic surface energy density evaluated at the unit normal to , and denotes the two-dimensional Hausdorff measure. We
suppose that is positively one-homogeneous and of class away from the origin, so that, in particular,
for some constant .
The substrate and the film admit different natural states corresponding to the mismatch between their respective crystalline structures. To be precise, a natural state for the substrate is given by ,
while a natural state for the film is given by for some nonzero matrix . Our models will reflect this mismatch, either by setting the elastic bulk energy as , where
or by imposing the Dirichlet boundary condition .
In the two-dimensional static case, existence of equilibrium solutions and their qualitative properties, including regularity, were studied in [3 M. Bonacini, Epitaxially strained elastic films:
The case of anisotropic surface energies. ESAIM Control Optim. Calc. Var.19, 167–189 (2013) , 4 M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions.
Adv. Calc. Var.8, 117–153 (2015) , 5 E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math.62, 1093–1121 (2002) , 15
E. Davoli and P. Piovano, Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019) , 16 E. Davoli and P. Piovano,
Derivation of a heteroepitaxial thin-film model. Interfaces Free Bound.22, 1–26 (2020) , 17 B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained
elastic films. In Topics in modern regularity theory, CRM Series 13, Ed. Norm., Pisa, 169–204 (2012) , 20 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially
strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) , 24 I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth.
Nonlinear Anal.154, 88–121 (2017) , 26 N. Fusco, Equilibrium configurations of epitaxially strained thin films. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.21, 341–348 (2010) , 29 N. Fusco and
M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012) ,
33 S. Y. Kholmatov and P. Piovano, A unified model for stress-driven rearrangement instabilities. Arch. Ration. Mech. Anal.238, 415–488 (2020) ]. The variational techniques and analytical arguments
developed in these papers have been used to treat other materials phenomena, such as voids and cavities in elastic solids [9 G. M. Capriani, V. Julin and G. Pisante, A quantitative second order
minimality criterion for cavities in elastic bodies. SIAM J. Math. Anal.45, 1952–1991 (2013) , 19 I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic
surface energies. J. Math. Pures Appl. (9)96, 591–639 (2011) ].
The scaling regimes of the minimal energy in epitaxial growth were identified in [2 P. Bella, M. Goldman and B. Zwicknagl, Study of island formation in epitaxially strained films on unbounded
domains. Arch. Ration. Mech. Anal.218, 163–217 (2015) , 30 M. Goldman and B. Zwicknagl, Scaling law and reduced models for epitaxially strained crystalline films. SIAM J. Math. Anal.46, 1–24 (2014)
] in terms of the parameters of the problem. The shape of the islands under the constraint of faceted profiles was addressed in [25 I. Fonseca, A. Pratelli and B. Zwicknagl, Shapes of epitaxially
grown quantum dots. Arch. Ration. Mech. Anal.214, 359–401 (2014) ]. A variational model that takes into account the formation of misfit dislocations was introduced in [23 I. Fonseca, N. Fusco,
G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films. J. Math. Pures Appl. (9)111, 126–160 (2018) ].
The effect of atoms freely diffusing on the surface (called adatoms) was studied in [10 M. Caroccia, R. Cristoferi and L. Dietrich, Equilibria configurations for epitaxial crystal growth with
adatoms. Arch. Ration. Mech. Anal.230, 785–838 (2018) ], where the model involves only surface energies.
A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates was undertaken in [35 L. C. Kreutz and P. Piovano, Microscopic validation of a
variational model of epitaxially strained crystalline films. SIAM J. Math. Anal.53, 453–490 (2021) , 38 P. Piovano and I. Velčić, Microscopical justification of solid-state wetting and dewetting.
arXiv:2010.08787 (2020) ].
The three-dimensional static case was studied in [6 A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications. ESAIM Control Optim.
Calc. Var.13, 717–734 (2007) , 12 A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal.39, 77–102 (2007) ] in the case in which
the symmetrized gradient is replaced by the gradient (see also [4 M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var.8, 117–153
(2015) ]). More recently, new developments in the theory of , i.e., generalized special functions of bounded deformation (see [13 V. Crismale and M. Friedrich, Equilibrium configurations for
epitaxially strained films and material voids in three-dimensional linear elasticity. Arch. Ration. Mech. Anal.237, 1041–1098 (2020) , 14 G. Dal Maso, Generalised functions of bounded deformation.
J. Eur. Math. Soc. (JEMS)15, 1943–1997 (2013) ], and the references therein) have led to considerable progress on the relaxation of the functional (1) in the three dimensional case (see [13
V. Crismale and M. Friedrich, Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity. Arch. Ration. Mech. Anal.237, 1041–1098 (2020) ]).
The regularity of equilibrium solutions remains an open problem. A local minimality sufficiency criterion, based on the strict positivity of the second variation, was established in [4 M. Bonacini,
Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var.8, 117–153 (2015) ], based on the work [29 N. Fusco and M. Morini, Equilibrium configurations
of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012) ].
To study the morphological evolution of anisotropic epitaxially strained films, we assume that the surface evolves by surface diffusion under the influence of a chemical potential . To be precise,
according to the Einstein–Nernst relation, the evolution is governed by the volume preserving equation
where , denotes the normal velocity of the evolving interface , stands for the tangential laplacian, and the chemical potential is given by the first variation of the underlying free-energy
functional. In our context, this becomes (assuming )
where stands for the tangential divergence along , and is the elastic equilibrium in , i.e., the minimizer of the elastic energy under the prescribed periodicity and boundary conditions (see (7)
If the surface energy density is highly anisotropic, there may be directions for which
fails, see for instance [18 A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math.52, 1111–1119 (1992) , 40 M. Siegel,
M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004) ]. In this case, the evolution equation (4) is backward
parabolic, and to overcome the ill-posedness of the problem we consider the following singular perturbation of the surface energy
where , stands for the sum of the principal curvatures of , and is a small positive constant (see [18 A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic
motion-by-curvature. SIAM J. Appl. Math.52, 1111–1119 (1992) , 31 M. E. Gurtin and M. E. Jabbour, Interface evolution in three dimensions with curvature-dependent energy and surface diffusion:
interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Ration. Mech. Anal.163, 171–208 (2002) , 32 C. Herring, Some theorems on the free energies of crystal
surfaces. Phys. Rev.82, 87 (1951) ]). The restriction in is motivated by the fact that the profile of the film will belong to , where , so that is continuously embedded into . This regularity is
strongly used to prove existence of solutions. In contrast, in we can assume since is embedded in .
The regularized free-energy functional becomes
and (3) is replaced by
Coupling this evolution equation on the profile of the film with the elastic equilibrium elliptic system holding in the film, and parametrizing using , we obtain the following Cauchy system of
equations with initial and natural boundary conditions:
where and is a -periodic function.
One can find in the literature sixth-order evolution equations of this type (see, e.g., [31 M. E. Gurtin and M. E. Jabbour, Interface evolution in three dimensions with curvature-dependent energy
and surface diffusion: interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Ration. Mech. Anal.163, 171–208 (2002) ] for the case without elasticity, see [40
M. Siegel, M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004) ] for the evolution of voids in elastically
stressed materials, and [7 M. Burger, F. Haußer, C. Stöcker and A. Voigt, A level set approach to anisotropic flows with curvature regularization. J. Comput. Phys.225, 183–205 (2007) , 39
A. Rätz, A. Ribalta and A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys.214, 187–208 (2006) ]).
We use the gradient flow structure of (7) with respect to a suitable -metric (see, e.g., [8 J. W. Cahn and J. E. Taylor, Overview no. 113, surface motion by surface diffusion. Acta Metall.
Mater.42, 1045–1063 (1994) ]) to solve the equation via a minimizing movement scheme (see [1 L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5)19, 191–246 (1995) ]),
i.e., we discretize the problem in time and solve suitable minimum incremental problems.
If instead of we used the gradient flow with respect to an -metric, we would obtain a fourth order evolution equation describing motion by evaporation-condensation (see [8 J. W. Cahn and J. E.
Taylor, Overview no. 113, surface motion by surface diffusion. Acta Metall. Mater.42, 1045–1063 (1994) , 31 M. E. Gurtin and M. E. Jabbour, Interface evolution in three dimensions with
curvature-dependent energy and surface diffusion: interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Ration. Mech. Anal.163, 171–208 (2002) , 37 P. Piovano,
Evolution of elastic thin films with curvature regularization via minimizing movements. Calc. Var. Partial Differ. Equ.49, 337–367 (2014) ]).
The short time existence of solutions to (7) established in [22 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with
curvature regularization. Anal. PDE8, 373–423 (2015) ] is the first such result for geometric surface diffusion equations with elasticity in three-dimensions. In the recent paper [28 N. Fusco,
V. Julin and M. Morini, The surface diffusion flow with elasticity in three dimensions. Arch. Ration. Mech. Anal.237, 1325–1382 (2020) ] (see also [27 N. Fusco, V. Julin and M. Morini, The surface
diffusion flow with elasticity in the plane. Comm. Math. Phys.362, 571–607 (2018) ] for the two-dimensional case), the authors proved short-time existence of a smooth solution without the additional
curvature regularization. They also showed asymptotic stability of strictly stable stationary sets.
The results summarized here can be found in the papers [20 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and
regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) , 21 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of elastic thin films by anisotropic surface diffusion with curvature
regularization. Arch. Ration. Mech. Anal.205, 425–466 (2012) , 22 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with
curvature regularization. Anal. PDE8, 373–423 (2015) ].
1 2D quasistatic equilibrium of epitaxially strained elastic films
In the following sections we assume self-similarity with respect to a planar axis and reduce the context to a two-dimensional framework. To be precise, we suppose that the material fills the infinite
where is a Lipschitz function representing the free profile of the film, which occupies the open set
The line corresponds to the film/substrate interface.
We assume that the mismatch strain corresponding to different natural states of the material in the substrate and in the film, respectively, is represented by
with . We will suppose that the film and the substrate share material properties, with homogeneous elasticity positive definite fourth-order tensor . Hence, bearing in mind the mismatch, the elastic
energy per unit area is given by , where
for all symmetric matrices .
In turn, the interfacial energy density has a step discontinuity at , i.e.,
where the property
will favor the SK growth mode over the VW mode. For the case , and for different crystalline materials stress tensors for the substrate and for the film, we refer to [15 E. Davoli and P. Piovano,
Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019) , 16 E. Davoli and P. Piovano, Derivation of a heteroepitaxial
thin-film model. Interfaces Free Bound.22, 1–26 (2020) ].
The total energy of the system is given by
where represents the free surface of the film, that is,
Since the functional is not lower semicontinuous, and thus, in general, does not admit minimizers, we are led to study its relaxation. Let
where stands for the pointwise variation of the function . Note that coincides with the pointwise variation of the function , and so
For define
Theorem 1 (Existence).
The following equalities hold:
We refer to [20 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186,
477–537 (2007) ] for a proof.
Next we study regularity properties of minimizers of in . As customary in constrained variational problems, in order to have more flexibility in the choice of test functions, we prove that the volume
constraint can be replaced by a volume penalization.
Theorem 2 (Volume penalization).
Let be a minimizer of the functional defined in (17) with . Then there exists such that for every integer , is a minimizer of the penalized functional
over all .
Proof. An argument similar to that of the proof of Theorem 1 guarantees that for every there exists a minimizer of . If for all sufficiently large, then
and so is a minimizer of .
Assume now that there is a subsequence, not relabeled, such that for all . If
for countably many , define
where has been chosen so that . Note that . Indeed, for every partition , we have that
for all . Hence,
which is a contradiction. Therefore, for all sufficiently large
it follows from (18) and (20) that as and that . In turn, by (16), for some constant independent of .
Let be so large that for all . Then
and the function , , satisfies
Consider a partition . Then
where we used the fact that . Hence,
and so, by (20),
We deduce that
For define
By a change of variables and (10), we have
where is the matrix whose entries are
Observe that
Since is a positive definite quadratic form over the symmetric matrices (see (11)), we have that
for all symmetric matrices and . Hence by (1), (10) and (LABEL:601)
where depends only on the ellipticity constants of and . By (20), (21), and (24), we have that
Thus, if
we get a contradiction, and this completes the proof. ∎
To prove the regularity of the free boundary we use the following internal sphere condition.
Theorem 3 (Internal Sphere’s Condition).
Let be a minimizer of the functional defined in (17). Then there exists with the property that for every there exists an open ball , with , such that
This result was first proved in a slightly different context by Chambolle and Larsen [11 A. Chambolle and C. J. Larsen, C∞ regularity of the free boundary for a two-dimensional optimal compliance
problem. Calc. Var. Partial Differential Equations18, 77–94 (2003) ] (see also [9 G. M. Capriani, V. Julin and G. Pisante, A quantitative second order minimality criterion for cavities in elastic
bodies. SIAM J. Math. Anal.45, 1952–1991 (2013) , 20 I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity
results. Arch. Ration. Mech. Anal.186, 477–537 (2007) ]). The argument is entirely two-dimensional and its extension to three dimensions is open.
Remark 4. Note that if is the outward unit normal to at , then . Thus, the set
is nonempty.
In the next theorem we prove that admits a left and right derivative at all but countably many points.
Theorem 5 (Left and Right Derivatives of ).
Let be a minimizer of the functional defined in (17). Then admits a left and a right tangent at every point not of the form with , where
where is the set defined in (25) and is the set defined in (26).
Theorem 6 (Cusps and Cuts).
Let be a minimizer of the functional defined in (17). Then the sets and contain at most finitely many vertical segments.
Remark 7. If , then since and is lower semicontinuous, for all sufficiently close to , we have that | {"url":"https://euromathsoc.org/magazine/articles/6","timestamp":"2024-11-04T04:31:55Z","content_type":"text/html","content_length":"1049075","record_id":"<urn:uuid:b47da484-57e6-4551-ab51-bf6836454048>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00264.warc.gz"} |
Proportional Parts In Triangles And Parallel Lines Worksheet Answers - TraingleWorksheets.com
Parallel Lines And Triangles Worksheet Answers – Triangles are one of the most fundamental patterns in geometry. Understanding triangles is crucial for learning more advanced geometric concepts. In
this blog post, we will cover the various types of triangles triangular angles, the best way to calculate the areas and perimeters of a triangle, and present details of the various. Types of
Triangles There are three kinds of triangulars: Equilateral isosceles, as well as scalene. Equilateral … Read more | {"url":"https://www.traingleworksheets.com/tag/proportional-parts-in-triangles-and-parallel-lines-worksheet-answers/","timestamp":"2024-11-13T10:39:01Z","content_type":"text/html","content_length":"48231","record_id":"<urn:uuid:02b9fad6-e983-4445-9ced-3575c87079d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00009.warc.gz"} |
- C
An inch (″ for short) is 2.54 centimeters. Inches are used as a measure of length for screen diagonals or car tires, for example. Converted, this means that 1 cm is 0.39 inches. To calculate this,
the number of inches is multiplied by 2.54 or the number of centimeters by 0.39.
Table - Customs in cm
Customs (″ / in) Centimeter (cm)
1 in 2,54 cm
5 in 12,70 cm
10 in 25,40 cm
25 in 63,50 cm
50 in 127,00 cm
Table - cm in Customs
Centimeter (cm) Customs (″ / in)
1 cm 0,39 in
10 cm 3,90 in
25 cm 9,75 in
50 cm 19,50 in
100 cm 39,00 in
All information is without guarantee
Use calculator:
• Select what you want to convert.
• Enter the number of inches or centimeters into the calculator.
• Click on "generate" to get the result.
Converting centimeters to inches is a common task, especially in situations where metric and imperial units are used. Although the metric system is predominant in most parts of the world, the
imperial system, which uses inches, is still used in some countries such as the USA, UK and Canada.
Centimeters (cm) and inches (in) are units of measurement for length or distance. The centimeter is a metric unit, while the inch belongs to the imperial system. 1 centimeter corresponds to 0.3937
The conversion formula for cm to inches is: inch = centimeter / 2.54
This formula is based on the fact that 1 inch equals exactly 2.54 centimeters.
Example 1: We have an object with a length of 50 centimeters and want to convert it to inches. Inch = 50 / 2.54 = 19.68 inches (rounded to two decimal places)
Example 2: Suppose we have a length of 30 centimeters that we want to convert to inches. Inch = 30 / 2.54 = 11.81 inches (rounded to two decimal places)
Conclusion: Converting centimeters to inches is a useful skill, especially when working with different systems of units. The conversion formula inches = centimeters / 2.54 provides a simple and
accurate conversion. However, it is also helpful to memorize the conversion factor of 2.54 to make quick estimates.
Calculator | Table
On this website, calculations, formulas and sample calculations with simple explanations are provided online free of charge by the author. | {"url":"https://www.fastcalc.net/en/length/inch-cm/","timestamp":"2024-11-03T04:06:46Z","content_type":"text/html","content_length":"178415","record_id":"<urn:uuid:fded2a1b-fa59-4b94-ba38-836d1a7f042b>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00389.warc.gz"} |
Diagrams and graph are useful to convey informations. Right now I have to jump through hoops to get these from other sources.
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• Yeah, the diagrams and graphs are the most useful way to convey the information. We can visualize the complete information and data via graphs and diagrams. We can make quick decisions after
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most useful to convey information. | {"url":"https://help.slides.com/forums/175819-general/suggestions/19734529-diagrams-and-graph-are-useful-to-convey-informatio","timestamp":"2024-11-09T09:27:26Z","content_type":"text/html","content_length":"66685","record_id":"<urn:uuid:ca38129a-69da-41ec-b3ed-5a42cba288b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00403.warc.gz"} |
Is it possible to hire someone for assistance with quantum algorithms for solving problems in quantum computing for optimization in my assignment? Computer Science Assignment and Homework Help By CS Experts
Is it possible to hire someone for assistance with quantum algorithms for solving problems in quantum computing for optimization in my assignment? I’m highly on a project where I’m working on quantum
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Taylor expansion. It isn’t all the same. And this is not going to be easy either. I prefer at least a 2nd order Taylor-expansion and it has a good answerIs it possible to hire someone for assistance
with quantum algorithms for solving problems in quantum computing for optimization in my assignment? My problem is extremely simple, the problem is stated some way, and you just need to perform the
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you can see, the QuantumEngine is about Quantum algebra. Suppose we have $.
People In My Class
~+5*2×1 + 4×2 and (1+(x 0)+(4)(x 1)+(x 2)+(4)(x 3)), where p1,p2 may be a bitfield or bitfield-tree. Therefore, $$\frac{a_1^{(1)}}{x^2} + \frac{a_2^{(4)}}{x^3} + \ldots + \frac{a_n^{(4)}}{x^n} = 0.$$
Now we can have a slightly simplified ansatz here. Does this problem happen in your particular problem? A: Quantum math is easy to describe in terms of sets. When given an array of numbers (not just
numbers) and its standard multiplication, quantum computation has simple arithmetic which is called quantum algebra (remember that we use the index pattern to write a list of strings). For example,
if you go to the quantum computer simulation language in its most elementary form: The program $$\frac{x_1^3}{x^2}+\frac{y_1y_2^2}{x^3} – \dfrac{x_3y_1G_2}{x^2}=0,$$ so let $$y_3=y_1^
2+y_2y_3+y_1y_1+y_2y_2+y_3y_1=0.$$ It can take an integer $n$ and sum values of all characters of the mod set of characters of all numbers. Then this hyperlink with $$x_1=-2n, \qquad x_1^3=-x_1^3, \
qquad x_2^3=-3n, \qquad |x_1|=1/2, \qquad |x_2|=2/3$$ The fact $$T_1=n \qquad T_2=n^2 \qquad T_3=n ^3$$ tells us that to find the solution $$\begin{split}\begin{aligned} T_1=n-\sqrt{2}n & \qquad + n
\\+n & \qquad +n\Is it possible to hire someone for assistance with quantum algorithms for solving problems in quantum computing for optimization in my assignment? Thanks! A: As DOUB @DeVries pointed
out, for this question you should not call the designer of your Q.In this more information she would be trying to apply an optimization technique to this particular instance (based on the
implementation of quantum algorithms) into this specific instance; however given the circumstances, your query, given in your question, may well be correct. However, in my experience, since you’re
about to compare a small number of algorithms that are both significantly different helpful site complexity, you should be careful to treat your query as though it had to be simple, and simply have a
pointer look up any relevant information about the algorithm. This Q is not all that different from any other. For instance if performance is really important and you want to optimize fast ancillary
query, this line of C computes a normalised regression term for each query (which, given correct computation time, is a special case of his reduced-sum approach, and should be automatically optimized
for the whole algorithm unless you really really like your query in general): Cmatrix1*Smatrix1* As you move upwards, down slightly and then next line to the next line provides more approximate
version of the regression term, which can usually be guaranteed to be called “fractional”. To note, The regression term can definitely be adjusted very drastically, for instance by putting weighting
the factor A on the time variance from the next linked here When the regression term goes back down to its initial value: This time variance is shifted down even more, as the shift is not used to
compute the second term. | {"url":"https://csmonsters.com/is-it-possible-to-hire-someone-for-assistance-with-quantum-algorithms-for-solving-problems-in-quantum-computing-for-optimization-in-my-assignment","timestamp":"2024-11-10T02:18:12Z","content_type":"text/html","content_length":"85249","record_id":"<urn:uuid:7f81d693-d702-4c27-82f9-2957b7810121>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00246.warc.gz"} |
Square Roots Definition, Methods and Cube Roots List
Square Roots Definition, Methods and Cube Roots List
Read the article to get detailed information about square roots, methods to find the square roots of a number, square roots and cube roots, and a list of square roots.
Square Roots: The square root of a number is multiplying factor of a number which when multiplied by itself hives the actual number. The square root is just the opposite method of squaring a number.
In squaring of a number same number is multiplied by itself giving the square of a number while in square root the original number is obtained on squaring the number. The square of any number is
always a positive number while the square root of a number has both positive and negative values. We can understand the square and square root easily by taking a simple example. Suppose if ‘x’ is a
square of the number ‘y’ means x × x = y but if ‘x’ is a square root of ‘y’ means x2= y.
Square Roots
A Square root is a number that when multiplying gives the original number. It is like a factor, both square and square roots are a type of exponent. In other square roots can be defined as the number
whose value has the power of 1/2 of a number. The symbol to represent the square root is √ which is called radical in mathematics and the number inside or called the radicand. For example, suppose a
number 16 which can be obtained by multiplying 4 by 4 i.e. 4 × 4 = 16 me, and 16 is a square of 4. And suppose a number √9 on squaring the number √32 = 3 so the square root of √9 is 3. Here we will
discuss the square root in detail. So refer to the below content to understand the square root in a good manner.
How to find Square Root?
The square root of a number is the square of a number that gives the original number. When finding a square root of a number firstly we need to find whether the number is a perfect square or
imperfect. A perfect square is a number that can be expressed in the form of power 2. The square root of the perfect square can be found easily by factoring it into the prime factors. If a number is
an imperfect square then the long division method needs to be performed on it which is discussed below in the article.
To find the square root of a number following four methods are used:
• Square Root by Prime Factorization
• Square Root by Repeated Subtraction Method
• Square Root by Long Division Method
• Square Root by Estimation Method
We will discuss all the above four methods in detail with a suitable example.
1. Square Root by Prime Factorisation Method
The square root of a perfect square number can be found easily by Prime Factorisation Method. In this method, some steps need to be followed to find the square root of a number listed below:
• Divide the given number into its prime factors.
• Form pairs of similar factors such that both factors in each pair are equal.
• Take one factor from the pair.
• Find the product of the factors obtained by taking one factor from each pair.
• The product gives the square root of the given number.
64 = 2×2×2×2×2×2×2
On pairing, we get 2×2×2 = 8
So the square root of 64 is 8.
2. Square Root by Repeated Subtraction Method
In the Repeated Subtraction Method, the square root of a number can be found by following the below steps if a given number is a perfect square.
• Repeatedly subtract the consecutive odd numbers from the given number
• Subtract till the difference remains zero
• The number of times we subtract will be the required square root
For example, let us find the square root of the number 36
1. 36 – 1 = 35
2. 35 – 3 = 32
3. 32 – 5 = 27
4. 27 – 7 = 20
5. 20 – 9 = 11
6. 11 – 11 = 0
Since, the subtraction is done 6 times, hence the square root of 36 is 6
3. Square Root by Long Division Method
Using the long division method the square roots of imperfect squares can be found easily. The method can be understood by the below steps.
• Place a bar over every pair of digits of the number starting from the units’ place (right-side).
• Then we divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.
Calculate the square root of 17.64
The square root of 17.64 by the long division method is found to be 4.2
4. Square Root by Estimation Method
This method used approximation to find the square root of a number. The square root can be found by guessing the approximate value.
For example, we know that the square root of 4 is 2 and the square root of 9 is 3, thus we can guess, that the square root of 5 will lie between 2 and 3.
But, we need to check the value of √5 is nearer to 2 or 3. Let us find the squares 2.2 and 2.8.
2.22 = 4.84
2.82 = 7.84
Since the square of 2.2 gives 5 approximately, thus we can estimate the square root of 5 is equal to 2.2 approximately.
Square Root 4
The square root of 4 is denoted by √4. We know that 4 is a perfect square whose square root can be found easily by the prime factorization method or repeated subtraction method. The square root of 4
is 2. But the square roots have two values always positive and negative as we already discussed. So √4 has two values +2 and -2.
Square roots 1 to 30
As we discussed the square riots can be found easily using the four methods mentioned above. The list of square roots from 1 to 30 is mentioned here.
Square Roots 1 to 30
= 1
√2 = 1.4142
√3 = 1.732
√4 = 2
√5 = 2.236
√6 = 2.4494
√7 = 2.6457
√8 = 2.8284
√9 = 3
√10 = 3.1622
√11 = 3.3166
√12 = 3.4641
√13 = 3.6055
√14 = 3.7416
√15 = 3.8729
√16 = 4
√17 = 4.1231
√18 = 4.2426
√19 = 4.3588
√20 = 4.4721
√21 = 4.5825
√22 = 4.6904
√23 = 4.7958
√24 = 4.8989
√25 = 5
√26 = 5.099
√27 = 5.1961
√28 = 5.2915
√29 = 5.3851
√30 = 5.4772
Square Roots and Squares
The square root is a factor multiplying with a given number that gives the original number while the square is a number that is multiplied by itself. Both roots and squares are considered exponents.
They are both opposite to each other. The examples of both square and square roots are discussed above already.
Square Roots and Cube Roots
To find the square root of any number, we need to find a number that when multiplied twice by itself gives the original number. Similarly, to find the cube root of any number we need to find a number
that when multiplied three times by itself gives the original number.
Notation: The square root is denoted by the symbol ‘√’, whereas the cube root is denoted by ‘∛’.
√4 = √(2 × 2) = 2
∛27 = ∛(3 × 3 × 3) = 3
Square Roots: FAQs
Que.1 What are squares?
Ans – The squares are a twice multiplication of the number itself. For example, 25 is a square of 5 i.e. 5×5 = 25
Que.2 What are square roots?
Ans – Square root is just opposite to square. Is the 1/2 power of a number. For example, the square root of 9 is 3 i.e. √9 = 3 | {"url":"https://sscegy.testegy.com/2023/11/square-roots-and-cube-roots.html","timestamp":"2024-11-12T02:15:58Z","content_type":"text/html","content_length":"1049064","record_id":"<urn:uuid:7db71e10-6695-44e8-a394-8d124ab2907a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00722.warc.gz"} |
How to measure the peak power of a pulsed laser - Ophir Photonics
Here’s a peak power calculator, if that’s what you wanted.
Read on if you’re interested in how to measure the real pulse shape of your laser – and use it to calculate the peak power.
When working with pulsed laser sources, laser developers and scientists are often interested in knowing the peak power, the highest power output from the laser. However, most pulsed laser power
meters display the total energy of a pulse or alternatively the average power, not the peak power. How can a user measure the peak power of a pulsed laser beam using Ophir laser measurement equipment
Ophir offers a nanosecond response time photodetector which is designed to measure the temporal behavior of pulsed lasers, the FPS-1 photodetector. The FPS-1 is easily connected to an oscilloscope
which displays a temporal trace of the power output during the pulse. Since the oscilloscope does not display a trace of absolute power output over time, but rather the relative pulse behavior and
shape, it cannot be used directly to find the peak power. However, with a simple calculation the trace may be used to derive the peak power of a pulse.
In order to measure the peak power of a laser pulse given the oscilloscope trace and the total energy of the laser pulse:
1) Construct a rectangle to represent a pulse with height equal to the peak power in arbitrary units of the oscilloscope trace, and with area equal to the area of the oscilloscope trace. Note the
constructed time duration of the constructed rectangular pulse. (This construction can be most easily done by manipulation of the numerical data log of the oscilloscope trace).
2) Since the total energy of the pulse is known, the peak power and consequently the non-arbitrary vertical scale of the oscilloscope trace may be determined by dividing the known total energy of the
laser pulse by the constructed time duration of the constructed rectangular pulse.
For example in the trace below let us say the total energy measured is 10mJ.
a) The trace has a peak reading of 55 A.U. (arbitrary units)
b) Sum the reading values in A.U. = total 268
c) Find the approximate area under the trace by multiplying the sum by the step size, 268 * 0.1 ms = 26.8 ms
d) Find the time duration that when multiplied by the peak power in A.U. will give the same area as the preceding.
The equation is:
(time duration) = (area under trace) / ( peak power A.U.) i.e.
(time duration) = (26.8 ms)/55 = 0.49 ms.
Thus, for instance if the total pulse energy is 10 mJ, the peak power is 10 mJ divided by 0.49 ms, approximately 20.5 W.
This video also explains how to measure peak power, and includes a lot more about the differences between peak and average power:
You might also like to read:
Laser Peak Power and Average Power: What’s the Difference?
A shortcut for calculating Power Density of a laser beam | {"url":"https://blog.ophiropt.com/how-to-measure-the-peak-power-of-a-pulsed-laser/","timestamp":"2024-11-09T19:23:39Z","content_type":"text/html","content_length":"93999","record_id":"<urn:uuid:538897f4-f3f6-424b-a8b0-78a6e9151edb>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00855.warc.gz"} |
How would i make an rts camera for mouse?
How would I make an RTS camera similar to warcraft 3? Where your mouse moving in any direction would then move your camera once you reach the edge of the screen?
Im not sure where to start with this, is it possible to detect mouse movement and direction like this? Thank you.
You could get the position of mouse , and then compare it by using crocodiles like this >= <= > < from centre
How would I get the direction of the mouse and convert that to a vector 3 movement for my camera?
Why do you need to convert that into vector3, sorry just asking because I don’t play warcraft3
I need to pan the camera in the direction of the mouse. So id need to get the direction somehow. Comparing center of screen and mouse only gets me distance.
I quickly came up with this, I think it is what you want, but I have never played warcraft 3, so I just pulled up a vid to see how it works. Comments should explain what it does.
local uis = game:GetService("UserInputService") -- used to get the mouse position
local camera = workspace.CurrentCamera -- the current camera
local maxDist = 100 -- 100 studs away from center
local camSpeed = 1 * 2 -- the max speed of the cam, multiplied by two for reason seen later
local angle = math.rad(-75) -- the angle of the camera pointing down
local x, z = 0, 0 -- the curent positions of the x and z
camera.CameraType = Enum.CameraType.Scriptable
local viewport = camera.ViewportSize -- the size of the cam viewport for percentages for the mouse
local mousePos = uis:GetMouseLocation()
local perX = (mousePos.X / viewport.X - 0.5) * camSpeed -- get the percentage that the mouse is away from the center, that is why there is a -0.5 and * 2 for cam speed
local perZ = (mousePos.Y / viewport.Y - 0.5) * camSpeed -- without the -0.5, it would only move positively
x = math.clamp(x + perX, -maxDist, maxDist) -- clamp the x value between the max distance
z = math.clamp(z + perZ, -maxDist, maxDist) -- clamp the z value between the max distance
local currentPos = CFrame.new(x, 50, z) -- set the current position
camera.CFrame = currentPos * CFrame.Angles(angle, 0, 0) -- apply and angle
Yes, using Roblox’s built-in UserInputService and Workspace services, you can make an RTS camera that functions like the camera in Warcraft 3. An illustration of how you could achieve this is as
You must first write a script that will control how the camera is moved. To operate the camera, you can either build a separate object or attach this script to the camera itself.
The UserInputService must be used in the script to determine when the mouse is getting close to the screen’s edge. The ViewSizeX and ViewSizeY parameters of the Workspace can be used to calculate
how far the mouse is from the screen’s edge after using the GetMouseLocation function to obtain the position of the mouse right now.
Once you’ve established that the mouse is close to the edge of the screen, you can move the camera in that direction by using the Move function. To find out how quickly the mouse is moving,
multiply the unit vector you obtain from the GetNormalizedVector function of the Vector2 class by the speed value.
This is perfect! But one issue, in WC3 at least. The camera only pans if your mouse reaches the edges of the screen. I’m not sure how would I go about adding this feature? I thought to only jump
the camera if it’s in those regions, but if I do that wouldn’t the camera all of a sudden jump and it not be smooth?
Any idea?
Yeah, it is possible. Here is what I would do to make this work. First, you need some value to represent how far from the center the mouse has to be to activate the camera. Then you need to check if
the mouse is actually that far in the positive or negative direction. After that, there is some complicated math IDK if it is actually complicated, but I just got off of work and I don’t really
want to think it through right now to figure out how far from that edge it is. Then you can just use that in the other code I provided. Here is what I did. It works perfectly if you have 1/2 as both
the mins for X and Z, but if you change some values around, it can be pretty smooth for others also.
If someone else wants to do the correct math, then feel free.
local uis = game:GetService("UserInputService") -- used to get the mouse position
local camera = workspace.CurrentCamera -- the current camera
local maxDist = 100 -- 100 studs away from center
local camSpeed = 1 -- the max speed of the cam
local angle = math.rad(-75) -- the angle of the camera pointing down
local minPerX, minPerZ = 1 / 2, 1 / 2 -- the minimum percent that the mouse must be at to activate
local x, z = 0, 0 -- the curent positions of the x and z
-- start of stuff
camSpeed = camSpeed / minPerX
camera.CameraType = Enum.CameraType.Scriptable
local viewport = camera.ViewportSize -- the size of the cam viewport for percentages for the mouse
local mousePos = uis:GetMouseLocation()
local perX = 2 * (mousePos.X / viewport.X - 0.5) -- get the percentage that the mouse is away from the center, that is why there is a -0.5 and * 2 for cam speed
local perZ = 2 * (mousePos.Y / viewport.Y - 0.5) -- without the -0.5, it would only move positively
if perX >= minPerX or perX <= - minPerX then -- if it is in the activation zone
perX = (perX - (math.sign(perX) * minPerX)) / minPerX -- set the percentage from the screen. works perfect for 1/2 as minPerX
x = math.clamp(x + perX, -maxDist, maxDist) -- clamp it
if perZ >= minPerZ or perZ <= -minPerZ then -- same as above
perZ = (perZ - (math.sign(perZ) * minPerZ)) / minPerZ
z = math.clamp(z + perZ, -maxDist, maxDist)
local currentPos = CFrame.new(x, 50, z) -- set the current position
camera.CFrame = currentPos * CFrame.Angles(angle, 0, 0) -- apply and angle
2 Likes
Thank you so much, now just to wrap my head around this math and how it works haha
Appreciate it
1 Like
Hey, just trying to understand this. How does the camera not just jump to the next position.
Like if you are moving your mouse around the center and that doesnt activate the camera cframe adjustments. But then all of a sudden you reach the corners of the screen in your code its not just
teleporting the camera?
Like when i did this before you posted. I had a similar approach that detected when you reached the corners and then would assign the camera CFrame but youd see a sharp jump where the camera
teleports to the new mouse location.
Im having trouble understanding your code how this accounts for that jump and is smooth? Thank you.
Likely what you were doing is using the position of the mouse and directly converting it to a point in world space. What I did was use how close the mouse was to the edge of the screen to decide how
much to move the camera each frame. As for the math, here is my detailed explanation of each part, and it now works with any percentage as the margin of activation.
First, you need some sort of value to represent the minimum percentage of the screen away the mouse must be to activate the camera. It’ll look something like this:
local minPerX, minPerZ = 1 / 3, 1 / 4
This will be used later to decide when to move the camera based on the position of the mouse.
(Shows what a few places of activation are based on your minPerX. Just flip vertically for Y)
You also need the reciprocal of those two numbers minus 1 for later:
local minXActivation = 1 / minPerX - 1
local minZActivation = 1 / minPerZ - 1
From here on out, this will be in a render stepped loop.
You need the viewportSize and the mousePosition:
local viewport = camera.ViewportSize
local mousePos = uis:GetMouseLocation()
You need to get the ratio of the viewport size and the minPerX and minPerZ by doing something like this:
local viewX, viewZ = viewport.X * minPerX, viewport.Y * minPerZ
This gives the viewport as if it was only 1/3 or 1/4 (or whatever ratio you chose) the size. That means that if the mouse is at the far edge of the screen and you divide the mouse location by it as
is done later, you will get a number near the reciprocal of the ratio.
Now is where the real math comes in.
You need to know what percent of the way the mouse is based on the percent of the viewport. If it is less than one, it is on the left side ready to activate, but if it is more than the reciprocal of
the ratio-1 then it is on the right side.
The code for this would look like this:
local perX = mousePos.X / viewX
Now there are just some checks to see which side of the screen it is on.
if perX < 1 then
-- on the left side
elseif perX > minXActivation then
-- on the right side
Now to find the actual percent away from there, it depends on which side of the screen it is at. For the left side it will just be equal to 1-perX because perX is one when it is exactly on the viewX
ratio. 1-perX flips it and gives the distance from the boundary not the screen.
If it is on the left side, there is a different step. It will be perX - minXActivation. This means that we take the perX, which is going to be more than the minXActivation, but less than the full
reciprocal, and subtract minXActivation from it to get a number between 0 and 1, where 1 is at the far-right edge, and 0 is on the boundary line.
Now just multiply the perX by the speed value, clamp the value, and apply!
(Then do the same thing for the Horizontal axis.)
local uis = game:GetService("UserInputService")
local camera = workspace.CurrentCamera
local maxDist = 500
local camSpeed = 100 -- in studs/sec
local angle = math.rad(-75)
local minPerX, minPerZ = 1 / 3, 1 / 4
local x, z = 0, 0
local minXActivation = 1 / minPerX - 1
local minZActivation = 1 / minPerZ - 1
camera.CameraType = Enum.CameraType.Scriptable
local viewport = camera.ViewportSize
local mousePos = uis:GetMouseLocation()
local viewX, viewZ = viewport.X * minPerX, viewport.Y * minPerZ
local perX = mousePos.X / viewX
if perX < 1 then
-- it is on the left side of the margin
perX = (1 - perX) * (camSpeed * dt)
x = math.clamp(x - perX, -maxDist, maxDist)
elseif perX > minXActivation then
-- right side
perX = (perX - minXActivation) * (camSpeed * dt)
x = math.clamp(x + perX, -maxDist, maxDist)
local perZ = mousePos.Y / viewZ
if perZ < 1 then
perZ = (1 - perZ) * (camSpeed * dt)
z = math.clamp(z - perZ, -maxDist, maxDist)
elseif perZ > minZActivation then
perZ = (perZ - minZActivation) * (camSpeed * dt)
z = math.clamp(z + perZ, -maxDist, maxDist)
local currentPos = CFrame.new(x, 50, z)
camera.CFrame = currentPos * CFrame.Angles(angle, 0, 0)
3 Likes
Thank you for explaining that! <3
I was never great with math, so it’s a bit hard to comprehend all of it. But I get the idea though, thanks again!
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Kalman Filter
Learn the intuition of one of the world's most applied algorithm
It's an iterative algorithm to reduce noise (or uncertainty) and arrive at a really good approximate value efficiently.
Kalman filter has these high-level components:-
• PREDICTION - Predict the next state of the object.
• MEASUREMENT - Measure the current state of the object.
• CONFIDENCE - How good (or bad) the PREDICTED or MEASUREMENT values
Kalman filters (or variants) have the following intuition.
• Suppose the current time is 3:00 pm
PREDICTION over MEASUREMENT
• After a few mins, we look at the watch again and we PREDICT it was 3:04 pm, but showed 3:55 pm!!
• We know the MEASUREMENT is wrong and have every reason to believe the PREDICTION is more accurate and set the watch to be 3:04 pm. Our CONFIDENCE in the MEASUREMENT is shaken, we believe it is so
noisy to show 3:55 pm
MEASUREMENT over PREDICTION
On the other hand, if the measurement shows 3:05 pm, we have greater confidence that the MEASUREMENT is correct, since it's more or less in line with our expectations and since it's measured, that
should be more accurate. Here we trust or more CONFIDENCE in the MEASUREMENT.
3 Main Calculations
There are 3 main calculations involved in the calculation of Kalman Filter
• Calculate the Kalman Gain
• Calculate the current estimate
• Calculate the error in the current estimate
Before we show the calculations, here are all the definitions used below
• \(EST_{t-1}\) = Previous Estimate
• \(EST_{t}\) = Current Estimate
• MEA = Measurement
• \(E_{mea}\) = Error in measurement
• \(E_{est}\) = Error in the estimate
• KG = Kalman Gain
Calculate the Kalman gain
KG = \({{E_{est}} / {(E_{est} + E_{mea})}}\)
• Measurements are more accurate if KG is close to 1
• Estimates are more correct if KG is close to 0
Calculate the current estimate
• \(EST_{t} = EST_{t-1} \) + KG * \( [MEA - EST_{t-1}] \)
Calculate the error in the estimate
• \(E_{est_{t}}\) = \({E_{mea} * E_{est_{t-1}}} / {E_{mea} + E_{est_{t-1}}}\)
• \(E_{est_{t}}\) = \([1-KG] E_{est_{t-1}}\)
What does this mean?
If measurement comes with very little error, then we are closing in on the true value. \(E_{mea}\) is low, then KG is high (close to 1). Then (1-KG) is very low. That means its drops the value of \
(E_{est_{t-1}}\) significantly, hence moving towards the "true" value. Intuitively it makes sense, measurements are good, and the system should converge to a good estimate. Note that, in the real
world, no matter how good the measurement is there is always noise, it will never be perfect.
The opposite is also true. Measurements are unstable, then \(E_{mea}\) is large, KG is low, and 1-KG is high (or close to 1). Then we can trust the measurements coming, so we do not move the needle
as much and try to keep the value close to the current error in estimate value as much as possible. Here is a fantastic video explanation here
If you are interested to tinker with the code, here is the link to the Google Colab.
Did you find this article valuable?
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Distinguished University Professor Ed Ott Retires
Distinguished University Professor Ed Ott retired in December, having served on the UMD faculty for a remarkable and stellar 43 years. Ott is globally known for his pioneering contributions in
nonlinear dynamics and chaos theory.
"Ed has had a magnificient career, exploring and explaining chaos and helping researchers to understand its impact across disciplines," said Physics chair Steve Rolston.
In recent years, Ott was instrumental in sparking intense activity in applying machine learning to nonlinear dynamics, giving keynote lectures and invited talks in several countries. For the AIP
journal Chaos he was asked to co-edit a special 2020 issue: When machine learning meets complex systems: Networks, chaos, and nonlinear dynamics.
Ott, a member of the National Academy of Sciences, is a University of Maryland Distinguished University Professor and holder of the Yuen Sang and Yu Yuen Kit So Endowed Professorship in nonlinear
dynamics. He received the 2014 Julius Edgar Lilienfeld Prize of the American Physical Society, and in 2016, with Celso Grebogi and James A. Yorke, was named a Thomson Reuters Citation Laureate in
physics for "...development of a control theory of chaotic systems."
In 2017, Ott received the Lewis Fry Richardson Medal of the European Geosciences Union for pioneering contributions in the theory of chaos. Also in 2017, he was selected for the Jürgen Moser Lecture
and Award, of the Society for Industrial and Applied Mathematics "... for his extensive and influential contributions to nonlinear dynamics, including seminal work on chaos theory and on the dynamics
of physical systems." He was elected a foreign member of the Academia Europaea for his outstanding achievements and international scholarship as a researcher.
Ott is a Fellow of the Society for Industrial and Applied Mathematics, the American Physical Society and the Institute of Electrical and Electronics Engineers. He has served as an editor or editorial
board member for most renowned journals in his field, including Physica D, Physical Review Letters, Physics of Fluids, Physical Review, Chaos and Dynamics and Stability of Systems.
Ott received his B.S. in Electrical Engineering at The Cooper Union and his M.S. and Ph.D. in Electrophysics from the Brooklyn Polytechnic Institute, then enjoyed a postdoctoral fellowship at the
Department of Applied Mathematics and Theoretical Physics of Cambridge University. Upon his return to the U.S., he joined the Electrical Engineering faculty at Cornell. He left Ithaca in 1979 to join
the Department of Physics and Department of Electrical Engineering on this campus. He is a member of the Institute for Research in Electronics and Applied Physics (IREAP), and has held appointments
at the Naval Research Lab and what is now the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara.
In addition to more than 500 papers, Ott has written the book "Chaos in Dynamical Systems", and edited "Coping with Chaos," a collection of reprints that focuses on how scientists observe, quantify,
and control chaos. He has advised more than 50 doctoral students, starting with Distinguished University Professor Tom Antonsen at Cornell University (1977) and most recently including Amitava
Banerjee (2022). | {"url":"https://umdphysics.umd.edu/about-us/news/department-news/1855ott-retires.html","timestamp":"2024-11-13T21:56:51Z","content_type":"text/html","content_length":"165717","record_id":"<urn:uuid:6d054fe5-18ce-45c5-a535-5fe3676344b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00808.warc.gz"} |
Explain the effect of correcting the error has
Explain the effect of correcting the error has on the IQR and the standard deviation. The professor calculated the IQR and the standard deviation of the test scores in question 6 before realizing her
mistake that is she entered the top score as 46 but it was actually 56.
Efan Halliday
Answered question
Explain the effect of correcting the error has on the IQR and the standard deviation.
The professor calculated the IQR and the standard deviation of the test scores in question 6 before realizing her mistake that is she entered the top score as 46 but it was actually 56.
Answer & Explanation
Inter quartile range:
In contrast to the range, which measures only differences between the extremes, the inter quartile range (also called mid spread) is the difference between the third quartile and the first quartile.
Thus, it measures the variation in the middle 50 percent of the data, and, unlike the range, is not affected by extreme values. $IQR={Q}_{3},-{Q}_{1}$,
Here, the IQR of the test scores is based on 50% of the observations only hence it is not affected but the standard deviation of the test scores would be affected because the mean changes which will
affect the deviations and increase the standard deviation of the test scores. | {"url":"https://plainmath.org/college-statistics/951-correcting-standard-deviation-professor-calculated-deviation-realizing","timestamp":"2024-11-04T12:15:23Z","content_type":"text/html","content_length":"155314","record_id":"<urn:uuid:1f1139b6-0967-448b-8579-ebca3b2a47de>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00096.warc.gz"} |
Pressure Conversion Calculator
Q: What is air pressure?
A: Air pressure, also known as atmospheric pressure, is the force exerted per unit area by the weight of the air surrounding an object or location. The atmosphere is composed of air molecules, which
have a certain weight. As these molecules stack up in layers, they create a substantial weight that presses down on objects below.
Q: What are the units of pressure?
A: Pressure can be expressed in various units, including:
- Pascal (Pa): The SI unit, equal to one newton per square meter.
- Bar: Part of the SI and metric system, equal to 100,000 Pascals.
- Pounds per square inch (psi): Approximately 6.895 Pascals.
- Hectopascal (hPa): Mainly used by meteorologists to express atmospheric air pressure, equivalent to 100 Pascals.
- Atmosphere (atm): Also called Standard Pressure, it is the air pressure at sea level with a fixed value of 101,325 Pascals. Additionally, other pressure units such as millimeters of mercury (mmHg)
or torr (1 atm = 760 Torr) may be encountered.
Q: How much is 2.24 atm in Pa and bar?
A: If the air pressure in a tire is 2.24 atm, it would be equivalent to 226,968 Pascals (Pa) and approximately 2.27 bar. To calculate it, you can follow these steps:
- Convert 2.24 atm to Pascals: 2.24 atm × 101,325 Pa = 226,968 Pa.
- Divide by 100,000 to find pressure in bars: 226,968 Pa ÷ 100,000 = 2.26968 bar ≈ 2.27 bar.
Q: How can I convert pressure to atmospheres?
A: To convert pressure from different units to atmospheres (atm), you can use the following methods:
- Divide the pressure in Pascals by 101,325 Pa/atm. For example, if the pressure is 97,560 Pa, the conversion would be 97,560 Pa ÷ 101,325 Pa/atm ≈ 0.963 atm.
- For psi to atm conversion, divide the pressure in psi by 14.7, as 1 atm is approximately equal to 14.7 psi.
- To convert pressure in bars to atm, multiply the pressure in bars by 0.987. For instance, if the pressure is 0.98 bars, the conversion would be 0.98 bar × 0.987 ≈ 0.967 atm.
Q: Which instrument is used to measure pressure?
A: The measurement of atmospheric pressure is typically done using a barometer. For gases, a manometer is commonly used to measure pressure. Interestingly, many modern smartphones, including the
iPhone, now come equipped with built-in barometers, enabling users to measure air pressure. This feature is useful for various purposes, from detecting weather changes to determining altitude. | {"url":"https://www.digicalculators.com/conversion/pressure_converter","timestamp":"2024-11-06T13:55:04Z","content_type":"text/html","content_length":"44639","record_id":"<urn:uuid:8602d65b-ec7f-4881-867b-c081961cd9d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00518.warc.gz"} |
Dashboard Report - Spotlight Dashboard
The Spotlight Dashboard is available for you to see you Actual and goals for the previous quarter, current quarter and next quarter. Use the Edit Goals option to create your quarterly goals.
The rows include the following:
Revenue -- Actual and Goal. Compare how much Actual revenue you've made vs. the Goals you've set. Compare previous quarter to current to next quarter.
Efficiency Factor -- Offers a clickable blue link for an Actual percentage. The Employee Time Records can be viewed By Employee By Date & Employee or By Date. You can also Search from this tab, and
export to Excel. Efficiency is calculated by taking Job Hours divided by Clock Hours.
Rev/Job Hr -- Actual and Goal numbers are your total revenue divided by the total job hours.
Recurring Service Sets -- Actual and Goal total Service Sets. (Every Week, Every Two Weeks, Every Three Weeks and Every Four Weeks recurring frequencies)
Scorecard -- Is your Average Score. EX. Scorecards are ranked on the following scale: 4 = 100% 3 = 75% 2 = 50% 1 = 25% and 0 = 0%. If I have 4 4s and 3 3s I have 7 total scorecards returned. My AVG %
would then be calculated by taking 4 * 1 = 4 and 3 * .75 = 2.25. Add 4 + 2.25 and my total is 6.25 divided by 7 total scorecards returned gives me a Scorecard % of 89.29%.
Rev/Job -- Actual and Goal Total Revenue divided by Total Jobs.
Customer Attrition -- Is measured for a given period (EX. Previous Quarter Actual and Goal) by dividing the number of customers your company had at the beginning of the period by the number of
customers at the end of the period. Below is how this is calculated.
For the given time frame (Period) we do the following calculation:
Recurring Lost Customer / (Recurring Customer Count Start of Period + Recurring Customer Count at End of Period) / 2 * (Average Days Per Month / Total Days in Period)
*Average Days Per Month = 365 / 12
Recurring Customers: More than 6 Pending future jobs and Included in the Kiosk Dashboard
0 comments
Please sign in to leave a comment. | {"url":"https://support.maidcentral.com/hc/en-us/articles/4402072978708-Dashboard-Report-Spotlight-Dashboard","timestamp":"2024-11-08T17:57:07Z","content_type":"text/html","content_length":"58678","record_id":"<urn:uuid:f923d167-6d04-4951-a4df-d01c07ebef2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00484.warc.gz"} |
TR19-082 | 2nd June 2019 05:58
Approximate degree, secret sharing, and concentration phenomena
The $\epsilon$-approximate degree $\widetilde{\text{deg}}_\epsilon(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $\epsilon$.
The approximate degree of $f$ is at least $k$ iff there exists a pair of probability distributions, also known as a dual polynomial, that are perfectly $k$-wise indistinguishable, but are
distinguishable by $f$ with advantage $1 - \epsilon$. Our contributions are:
We give a simple new construction of a dual polynomial for the AND function, certifying that $\widetilde{\text{deg}}_\epsilon(f) \geq \Omega(\sqrt{n \log 1/\epsilon})$. This construction is the first
to extend to the notion of weighted degree, and yields the first explicit certificate that the $1/3$-approximate degree of any read-once DNF is $\Omega(\sqrt{n})$.
We show that any pair of symmetric distributions on $n$-bit strings that are perfectly $k$-wise indistinguishable are also statistically $K$-wise indistinguishable with error at most $K^{3/2} \cdot \
exp(-\Omega(k^2/K))$ for all $k \leq K \leq n/64$.
This implies that any symmetric function $f$ is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-$K$ coalitions with statistical error $K
^{3/2} \exp(-\Omega(\widetilde{\text{deg}}_{1/3}(f)^2/K))$ for all values of $K$ up to $n/64$ simultaneously.
Previous secret sharing schemes required that $K$ be determined in advance, and only worked for $f=$ AND.
Our analyses draw new connections between approximate degree and concentration phenomena.
As a corollary, we show that for any $d \leq n/64$, any degree $d$ polynomial approximating a symmetric function $f$ to error $1/3$
must have $\ell_1$-norm at least $K^{-3/2} \exp({\Omega(\widetilde{\text{deg}}_{1/3}(f)^2/d)})$, which we also show to be tight for any $d > \widetilde{\text{deg}}_{1/3}(f)$.
These upper and lower bounds were also previously only known in the case $f=$ AND. | {"url":"https://eccc.weizmann.ac.il/report/2019/082/","timestamp":"2024-11-07T02:36:27Z","content_type":"application/xhtml+xml","content_length":"22437","record_id":"<urn:uuid:1b74a85e-df6a-47c1-bab1-68958fd01ed1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00846.warc.gz"} |
Induction, recursion, replacement and the ordinals
Paul Taylor
1990s and 2019–23
My papers — slides — MathOverflow — the Pataraia fixed point theorem.
The purpose of Categorical Set Theory is to study well-foundedness and extensionality as a mathematical structure, using the tools of category theory, but stripped of its ontological pretensions.
A coalgebra α:A → T A for a functor T:C→C is well founded if, for every mono i:U ↪ A such that the pullbackH factors throughU in the diagram on the left, the map i must be an isomorphism.
Under conditions that are discussed in the papers below, any well founded coalgebra satisfies the recursion scheme that, for any algebra θ:T θ→Θ, there is a unique map f:A→Θ making the square on the
right commute.
This notion is in principle powerful enough to study quantifier complexity, although I haven’t exploited that yet.
A coalgebra is extensional if its structure mapα is mono. Category theory has already contributed here, identifying exactly what is required of the underlying category and endofunctor. Principally,
it has replaced “monos” with factorisation systems, to study different systems of constructive ordinals.
In set theory, extensional well founded relations provide partial models for the non-existent free algebra for the covariant powerset functor. Our techniques generalise this to other endofunctors,
with or without free algebras.
The case without free algebras suggests a categorical way to approach the axiom-scheme of replacement from ZFC set theory. Transfinite iteration of functors can be expressed using the same techniques
as our version of Mostowski’s extensional quotient. The aim is to give a replacement for replacement in the native language of category theory, using adjointness in foundations.
Recent work (since 2019):
Well founded coalgebras and recursion: currently with a journal referee; abstract below.
Ordinals as Coalgebras: coherent draft; abstract below.
Transfinite iteration of functors: work in progress, abstract below.
Older work (1990s):
Intuitionistic Sets and Ordinals: Journal of Symbolic Logic, 61 (1996) 705–744. Abstract
Towards a Unified Treatment of Induction: An obsolete incomplete manuscript that is only included here for the historical record; all of the material in it is superseded by the recent papers above.
Abstract etc
The Fixed Point Property in Synthetic Domain Theory: Presented at Logic in Computer Science 6, Amsterdam, July 1991. Abstract
Seminar Slides
A Fixed Point Theorem for Categories at the British Logic Colloquium in Birmingham, 7 September 2024 (abstract). See also the version of the proof on MathOverflow.
Ordinals as Coalgebras, 27 June 2024, Category Theory 2024, Santiago de Compostela.
Ordinals as Coalgebras, 23 August 2023, Symposium in honour of Andy Pitts, Cambridge.
A novel fixed point theorem, towards a replacement for replacement, 6July 2023, Category Theory 2023, abstract and YouTube video of the lecture.
Well Founded Coalgebras, slides for a seminar given on 2March 2023 in person in Birmingham and online at the invitation of the Logic and Semantics and Applied Category Theory research groups in
Tallinn University of Technology.
Order-theoretic fixed point theorems, slides for a seminar given on 8December 2022 in Birmingham, which was is an extended version of one given on 29September 2022 in Ljubljana.
Free algebras for functors, without an ordinal in sight, slides for a seminar given onn 26April 2024 in Birmingham. This was superseded by the Fixed Point Theorem for Categories above, which has a
much much simpler proof.
Miscellaneous slides from several lectures given in the 1990s. NB: This is a 21Mb scanned image of overhead projector slides in no particular order, but some of them are still interesting.
MathOverflow questions and answers
MathOverflow is the only inter-disciplinary site that we have for Mathematics. So it is — or should be — the appropriate place to ask about history and the inter-relationship of ideas. However,
whenever I have tried to use it in this way or to advertise this research programme I have received abuse and at least down-votes.
So I ask my colleagues to visit the following pages, up-vote them and make positive comments. Better still, answer the historical questions.
Constructive ordinals and Mostowski extensional reflection:
Mostowski collapses and universal extensional relational classes, asked by Martin Brandenburg in 2010 and answered by me in September 2023.
Ordinals in constructive mathematics, asked by Simon Henry in 2015 and answered by me in September 2023.
A categorical characterization of ordinal numbers, asked by Fosco Loregian in May 2014 and then answered by me.
Pataraia and Bourbaki–Witt fixed point theorems:
Bourbaki-Witt in a textbook, other than in logic, asked by me in December 2022.
A new and subtle order-theoretic fixed point theorem, ie my version of Pataraia’s theorem, asked by me in March 2023, together with my proof of the categorical version as an Answer (Seotember 2024).
That was a re-written version of Uses of Zorn’s Lemma when the thing is actually unique, asked by me in June 2020. This gives several non-constructive proofs as well as Pataraia’s constructive one. I
was trying to ask whether similar ideas (especially my fifth condition) had arisen elsewhere. Unfortunately, all I got was a lecture on transfinite recursion, but the author of that was eventually
persuaded to delete it.
Other history of induction and recursion:
History of well founded relations, asked by me in April 2020.
Well founded induction attributed to Noether, asked by me in 2013.
Foundations without Set Theory:
Categorical foundations without set theory, an anonymous question with two answers by me in 2010.
Pataraia’s fixed point theorem
Slides of my seminar on 9 December 2022
For the central induction results in Well founded coalgebras and recursion I needed to use the following novel principle:
Let s:X→ X be an endofunction of a poset such that
• X has a least element ⊥;
• X has joins of directed subsets (or chains, classically),
• s is monotone: ∀ x y.x≤ y⇒ s x≤ s y;
• s is inflationary: ∀ x.x≤ s x;
• the special condition, ∀ x y.x=s x≤ y=s y⇒ x=y.
• X has a greatest element ⊤;
• ⊤ is the unique fixed point of s;
• if ⊥ satisfies some predicate and it is preserved by s and directed joins then it holds for ⊤.
The constructive proof for the related fixed point theorem was found by Dito Pataraia in 1996 but he never published it and died in 2011. His proof was dramatically simplified by Alex Simpson and the
above is a further simplification of that. The induction principle was apparently first identified by Martín Escardó.
The special condition above seems to be new with me; it is what is needed to force X to have ⊤.
The categorical version explains where the “special condition” comes from: see the BLC24 slides and the proof on MathOverflow.
History and translations
To come, but see my new translations page.
Well founded coalgebras and recursion
2019–23 paper currently with a journal referee (PDF).
Sections 2 and 8 have been re-written since the November 2022 version.
We define well founded coalgebras and prove the recursion theorem for them: that there is a unique coalgebra-to-algebra homomorphism to any algebra for the same functor. The functor must preserve
monos, whereas earlier work also required it to preserve their pullbacks. The argument is based on von Neumann’s recursion theorem for ordinals. Extensional well founded coalgebras are seen as
initial segments of the free algebra, even when that does not exist.
The assumptions about the underlying category, originally sets, are examined thoroughly, with a view to ambitious generalisation. In particular, the “monos” used for predicates and extensionality are
replaced by a factorisation system. Future work will obtain much more powerful results by using this in a categorical form of Mostowski’s theorem that imposes extensionality.
These proofs exploit Pataraia’s fixed point theorem for dcpos, which Section2 advocates (independently of the rest of the paper) for much wider deployment as a much prettier (as well as
constructive) replacement for the use of the ordinals, the Bourbaki–Witt theorem and Zorn’s Lemma.
See below for my 1990s work that this supersedes.
Ordinals as Coalgebras
Slides, 27 June 2024, Category Theory 2024, Santiago de Compostela.
Slides, 23 August 2023, Symposium in honour of Andy Pitts, Cambridge.
This works out the theory from Well Founded Coalgebras and Recursion in the category of posets using both general subsets with the restricted order and lower subsets as the class of “monos”.
This yields the “thin” and “plump” ordinals from Intuitionistic Sets and Ordinals and also another system that we call “slim”, although this is much less well behaved.
The first few plump ordinals are calculated in the topos of presheaves on a single arrow over classical sets, showing that this requires Replacement for ω· 2.
However, the paper is still work in progress, so please don’t distribute or file it. As you can see from the number of declared counterexamples, this is a particularly fiddly subject and likely to be
riddled with more than the usual quota of errors.
Seminar slides for Ordinals as Coalgebras, Symposium in honour of Andy Pitts, Cambridge, 23 August 2023.
Transfinite iteration of functors
This is the most common use of Replacement in mainstream mathematics.
The paper will demonstrate how “extensional quotient” àla Mostowski can be used to formulate a characterisation of transfinite iteration in the native language of category theory,
So it is in accordance with Bill Lawvere’s dictum of using Adjointness in Foundations.
Please contact me if you know of other uses of Replacement that are not instances of transfinite iteration of functors.
Things like “the (2-)category of all categories” are just uses of Universes and so are not interesting.
Iwant to hear about areas of Mathematics that can be expressed in its lingua franca, namely Category Theory. I don’t care about problems invented by set theorists in their obscure notation and
ontology in order to try to claim that category theory is incapable of serving all of the foundational needs of mathematics.
What interests me about Replacement is that it builds skyscrapers out of plans on the ground. Using Universes to do this is like building them from orbiting satellites.
Based on Section 9.5 of my book Practical Foundations of Mathematics, published by Cambridge University Press in 1999.
Practical Foundations of Mathematics
My work on this topic in the 1990s is described briefly in my book Practical Foundations of Mathematics, published by Cambridge University Press in 1999:
Intuitionistic Sets and Ordinals
Journal of Symbolic Logic, 61 (1996) 705–744
Presented at Category Theory and Computer Science 5, Amsterdam, September 1993.
Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is
poorly behaved, leading toproblems of functoriality.
We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifies the traditional development ofsuccessors and unions, making it intuitionistic;
even the (classical) proof oftrichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly.
Directedness must be defined hereditarily. Itis orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it.
We treat ordinals as order-types, and develop a corresponding set theory similar to Osius’ transitive set objects. This presents Mostowski’s theorem as a reflection of categories, and set-theoretic
union is a corollary of the adjoint functor theorem. Mostowski’s theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to
be unavoidable for the plump rank.
The comparison between sets and toposes is developed as far as the identification of replacement with completeness and there are some suggestions for further work in this area.
Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions
such as x≤ s x, monotonicity or s(x∨ y)≤ s x∨ s y.
Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We
explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.
Towards a Unified Treatment of Induction
slides (scanned PDF) NB: This file is a 21Mb scanned image of a merged collection of overhead projector slides from several lectures.
This manuscript is obsolete and only included here for the historical record. All of the material in it is superseded by Well founded coalgebras and recursion above.
The Fixed Point Property in Synthetic Domain Theory
Presented at Logic in Computer Science 6, Amsterdam, July 1991.
We present an elementary axiomatisation of synthetic domain theory and show that it is sufficient to deduce the fixed point property and solve domain equations. Models of these axioms based on
partial equivalence relations have received much attention, but there are also very simple sheaf models based on classical domain theory. In any case the aim of this paper is to show that an
important theorem can be derived from an abstract axiomatisation, rather than from a particular model. Also, by providing a common framework in which both PER and classical models can be expressed,
this work builds a bridge between the two.
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vs whole-test measures
Combining Part-test (subtest) measures vs whole-test measures
From a person's raw score on a whole-test, perhaps several hundred items long, that person's overall measure can be estimated. But the whole-test often contains several content-homogeneous subsets of
items. From the person's raw score on each subset that person's measure on each subset can also estimated. How do the subset scores and measures compare with whole-test scores and measures?
For complete data, subset raw scores have the convenient numerical property that however the right answers are exchanged among subsets, the sum of part-test scores for each person still adds up to
the same whole-test score. This forces the plot of mean part-test percent corrects against whole-test percent corrects into an identity line.
This simple additivity of subset raw scores, however, is somewhat illusory. It is an addition of instances, but not of meanings. To obtain meaning from raw scores, the raw scores must be converted
into measures. But raw scores are necessarily non-linear (because they are bound by zero "right" and zero "wrong"). To convert them into linear measures, a linearizing transformation is required.
This transformation must be ogival, as in [log[e]("rights"/"wrongs")].
Once the raw scores have been transformed non-linearly, the sum of parts can no longer correspond exactly to the whole. In fact, when mean measures for parts are plotted against the measure for the
whole, the mean measures on part-tests are necessarily more extreme than the measures on the whole-test. Low part-test mean measures must be lower than their whole-test counterparts. High part-test
mean measures are higher than their whole-test counterparts. Thus, when mean part-test measures are compared with whole-test measures, the mean part-test measures spread out more than their
whole-test counterparts. In short, the variance of mean subtest measures is greater than the variance of whole test measures.
│ │
│ Part-test vs. Whole-test Measures │
│ blue line, parallel to x-axis: full-test measure │
│ red line: average of part-test measures │
│ green line: composite test measure │
│ = average of standard-error-weighted part-test measures │
│ │
│ │
│ S.E.(Composite measure) = 1 / √ ( Σ ( 1 / S.E.(Measure(subtest))² )) │
Why does this happen? There are two causes:
1. The effect of asymmetric non-linearity on score exchanges among part-tests. The Figure shows what happens when a person takes a whole-test with 200 dichotomous items, each of difficulty 0 logits
and achieves a whole-test score of 150 correct. Look at the red line in Figure 1. The whole-test measure is log[e](150/50) = 1.1 logits. But what happens when this performance is partitioned into two
subtests of 100 items? When the score on both subtests is 75, then the mean subtest measure is log[e](75/25) = 1.1 logits. But when one subset score is 90, then the other must be 60. Now the mean of
the two subset measures becomes [log[e](90/10) + log[e](60/40)]/2 = [log[e](9) + log[e](1.5)]/2 = 1.3, i.e., higher! So, as the scores on the two subtests diverge, while maintaining the same total
score, the mean subtest measure becomes higher. The mean measure becomes infinite when the score on one subtest is a perfect 100, and on the other test a middling 50.
The effect of asymmetric non-linearity on score exchanges results from the difference between any original raw (once only, specific, now a matter of history) data and its (inferential, general, to be
useful in the future) meaning. Only when all part-test percent corrects are identical to their whole-test percent corrects, and all the part-test measures they imply are identical to the whole-test
measures they imply, do mean part-test measures approach equality to whole-test measures.
As we investigate this, at first surprising but later persuasive, realization we discover that as the (surely to be expected) within person variation among part-test measures increases, the
asymmetric effect of score exchanges on measures increases the variance of part-test measures and their means, making the distribution of part-test measures and means wider than the distribution of
their corresponding whole-test measures. This happens no matter how carefully all item difficulties are anchored in the same well-constructed item bank and even when the part-tests exhaust the items
in the whole test.
2. The increase in measurement error due to fewer items in the part-tests. Measurement error inflates measure variance. The observed sample measure variance on any test has two components: the
"true", unobservably exact, variance of the sample and the error inevitably inherent in estimating each measure.
The measurement error component is dominated by the number of useful (targeted on, with difficulties near, the person measures) items. This factor is the usual reciprocal square root of replications.
When the number of useful items decreases to a fourth, measurement error doubles. Thus, even for tests with identical targeting and content, the shorter the test, the wider the variance of the
estimated measures.
A second factor contributing to "real" (as compared to theoretical, modelled) measurement error is response pattern misfit. The more improbable a person's response pattern, the more uncertain their
measure. Misfit increases "real" measurement error beyond the modelled error specified by the stochastic model used to govern measure estimation. The more subsets comprise a whole-test, the more
likely it is that local misfit will pile up unevenly in particular subsets and so perturb a person's measure for those subsets. Thus further increasing overall subset measure variance.
Scores vs. measures: The ogival shape of the linearizing transformation from scores to measures shows that one more "right" answer implies a greater increase in inferred measure at the extremes of a
test (near 100 or zero percent) than in the middle (near 50 percent). So we must choose between scores and measures. But raw scores cannot remain our ultimate goal. What we need for the construction
of knowledge is what raw scores imply about a person's typical, expected to recur behavior. When we take the step from data (raw experience) to inference (knowledge) by transforming an exactly
observed raw score into an uncertain (qualified by model error and misfit) estimate of an inferred linear measure, the concrete one-for-one raw score exchanges between parts and wholes are left
Implication: How, in everyday practice, are we to keep part-test measures quantitatively comparable to whole-test measures? First, we must construct all measures in the same frame of reference.
Estimate item calibrations on the whole-test, then anchor all items at these calibrations when estimating part-test measures. Then, when the set of part-tests is complete and stable, one might
consider simplifying scale management by formulating an empirically-based conversion between mean part-test and whole-test measures. For instance, look at the green line in Figure 1. Weighting
part-test measures by their standard errors, Mean = (M[1]/SE[1] + M[2]/SE[2])/(1/SE[1] + 1/SE[2]), brings them into closer conformity with the whole-test measures. This conversion will make things
look more "right" to audiences who are still uncomfortable with the transition from concrete non-linear raw scores to the abstract linear measures they imply.
The utility of the mean of any set of parts, however, depends on the homogeneity of the part-measures. When the part-measures are statistically equivalent so that it is reasonable to think of them as
replications on a common line of inquiry, then their mean is a reasonable summary and will be close to the whole-test measure. But then, if statistical equivalence among part-measures is so evident,
what is the motivation for the partitions? In that case the most efficient and simplest interpretation of test performance is the single whole-test measure.
When, however, the differences among part-measures are significant enough (statistically or substantively) to become interesting, what then is the logic for either a part-measure mean or even a
whole-test measure? Surely then one must deal with each part-test measure on its own, or else admit that the whole test measure is a compromise. The whole-test measure may be useful for coordinating
representations of related, if now discovered to be usefully distinct, variables. But this "for-the-time-being" representationally convenient "common" metric loses its inferential significance as an
unambiguous representation of a single variable.
Ben Wright
Combining Part-test vs whole-test measures. Wright BD. Rasch Measurement Transactions, 1994, 8:3 p.376
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Methods To Get Geometry Homework Answers Online
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The best methods are the ones which work for you. Find that method or methods and reap the rewards. | {"url":"https://www.takebackyourschools.com/methods-to-get-geometry-homework-answers-online","timestamp":"2024-11-11T00:49:06Z","content_type":"text/html","content_length":"16571","record_id":"<urn:uuid:bd73ded1-ebe8-48ac-abac-a1504e593f13>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00154.warc.gz"} |
Green Bridge Learning Resource For The Cell And Its Environment: Physical And Biophysical Processes
Living Things and Their Environment:
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Living organisms exhibit varying levels of organization, ranging from cells to tissues, organs, organ systems, and finally, the whole organism. This hierarchical organization allows for
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cell membranes. | {"url":"https://www.supergb.com/cbt/users/learning/resource/dd107606-d9e2-42d4-993d-df1ebef037c8","timestamp":"2024-11-12T06:19:04Z","content_type":"text/html","content_length":"85329","record_id":"<urn:uuid:00d2451f-772c-46b2-a2ff-f3ab21270cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00133.warc.gz"} |
re: st: Checking reliability of a measurement device
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re: st: Checking reliability of a measurement device
From David Airey <[email protected]>
To [email protected]
Subject re: st: Checking reliability of a measurement device
Date Fri, 26 Nov 2004 11:55:28 -0600
This package seems to have an aspect of what you want:
------------------------------------------------------------------------ -------
help for concord (STB-43: sg84; STB-45: sg84.1)
------------------------------------------------------------------------ -------
Concordance correlation coefficient
concord var1 var2 [weight] [if exp] [in range]
[ , summary graph(ccc|loa) snd(sndvar[, replace])
noref reg by(byvar) level(level) graph_options ]
concord computes Lin's (1989) concordance correlation coefficient
for agreement on a continuous measure obtained by two persons
or methods. The Lin coefficient combines measures of both precision
and accuracy to determine whether the observed data significantly
deviate from the line of perfect concordance (i.e., the line at 45
degrees). Lin's coefficient increases in value as a function of the
nearness of the data's reduced major axis to the line of perfect
concordance (the accuracy of the data) and of the tightness of the
data about its reduced major axis (the precision of the data). The
Pearson correlation coefficient, r, the bias-correction factor, C_b,
and the equation of the reduced major axis are reported to show these
components. Note that the concordance correlation coefficient, rho_c,
can be expressed as the product of r, the measure of precision,
and C_b, the measure of accuracy. The optional concordance graph
plots the observed data, the reduced major axis of the data, and the
line of perfect concordance as a graphical display of the observed
concordance of the measures.
concord also provides statistics and optional graphics for Bland and
Altman's limits-of-agreement, "loa", procedure (1986). The loa, a
data-scale assessment of the degree of agreement, is a complementary
approach to the relationship-scale approach of Lin.
The user provides the pairs of measurements for a single property as
observations in variables var1 and var2. Frequency weights may
be specified and used. Missing values (if any) are deleted in a
casewise manner.
graph(ccc) requests a graphical display of the data, the line of
perfect concordance and the reduced major axis of the data. The
reduced major axis or SD line goes through the intersection of the
means and has slope given by the sign of Pearson's r and the ratio
of the standard deviations. The SD line serves as a summary of
the center of the data.
graph(loa) requests a graphical display of the loa, the mean
difference, and the data presented as paired differences plotted
against pair-wise means. A Normal plot for the differences is
also shown.
snd(sndvar[, replace]) saves the standard normal deviates
produced for the Normal plot generated by graph(loa). The values
are saved in variable sndvar. If sndvar does not exist, it is
created. If sndvar exists, an error will occur unless replace
is also specified. This option is ignored if graph(loa) is not
noref suppresses the reference line at y=0 in the loa plot.
This option is ignored if graph(loa) is not requested.
reg adds a regression line to the loa plot fitting the paired
differences to the pair-wise means. This option is ignored if
graph(loa) is not requested.
summary requests summary statistics.
by(byvar) produces separate results for groups of observations
defined by byvar.
level sets the confidence level % for the CI; default is 95%.
graph_options are those allowed with graph, twoway. Setting
t1title(.) blanks out the default t1title. The default
graph_options for graph(ccc) are connect(.l) symbol(o.) pen(22)
for the data points and SD line, respectively, along with default
titles and labels. The default graph_options for graph(loa)
include connect(lll.l) symbol(...o.) pen(35324) for the lower
confidence interval limit, the mean difference, the upper confidence
interval limit, the data points, and the regression line (if
requested) respectively, along with default titles and labels.
(The user is not allowed to modify the graph options for the Normal
probability plot.)
Saved values (if by option not used)
S_1 number of observations compared
S_2 concordance correlation coefficient rho_c
S_3 standard error of rho_c
S_4 lower CI limit (asymptotic)
S_5 upper CI limit (asymptotic)
S_6 lower CI limit (z-transform)
S_7 upper CI limit (z-transform)
S_8 bias-correction factor C_b
S_9 mean difference
S_10 standard deviation of mean difference
S_11 lower loa CI limit
S_12 upper loa CI limit
. concord rater1 rater2
. concord rater1 rater2 [fw=freq]
. concord rater1 rater2, s g(c)
. concord rater1 rater2, level(90) by(grp)
. concord rater1 rater2, g(l)
Thomas J. Steichen
[email protected]
Nicholas J. Cox
University of Durham, UK
[email protected]
Bland, J. M., Altman, D. G. 1986. Statistical methods for
assessing agreement between two methods of clinical measurement.
Lancet I, 307-310.
Lin, L. I-K. 1989. A concordance correlation coefficient to
evaluate reproducibility. Biometrics 45: 255-268.
Also see
STB: sg84.1 (STB-45), sg84 (STB-43)
(remote file ends)
------------------------------------------------------------------------ -------
(click here to return to the previous screen)
Hi Statalisters,
a Dentistry PhD student did some measurements on 12
teeth with varying conditions and he asked me how
could he show that the device used for the
measurements is reliable. More specifically each one
of the 12 teeth has been measured by this device by 2
raters (a and b) X 2 time points (week 1 and week 2) X
6 relative positions = 24 measurements. The goal is
to show that the discrepancies among measurements are
not statistically significant.
You might look at Stata's "kappa" and "alpha" commands...
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
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Hilbert series and Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which
measure the growth of the dimension of the homogeneous components of the algebra.
These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.
The typical situations where these notions are used are the following:
• The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree.
• The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
• The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.
The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety
defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties because a flat family ${\displaystyle \pi :X\to S}$ has the same Hilbert
polynomial over any closed point ${\displaystyle s\in S}$. This is used in the construction of the Hilbert scheme and Quot scheme.
Definitions and main properties
Consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that
${\displaystyle S=\bigoplus _{i\geq 0}S_{i}}$
and that ${\displaystyle S_{0}=K}$.
The Hilbert function
${\displaystyle HF_{S}:n\longmapsto \dim _{K}S_{n}}$
maps the integer n to the dimension of the K-vector space S[n]. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series
${\displaystyle HS_{S}(t)=\sum _{n=0}^{\infty }HF_{S}(n)t^{n}.}$
If S is generated by h homogeneous elements of positive degrees ${\displaystyle d_{1},\ldots ,d_{h}}$, then the sum of the Hilbert series is a rational fraction
${\displaystyle HS_{S}(t)={\frac {Q(t)}{\prod _{i=1}^{h}\left(1-t^{d_{i}}\right)}},}$
where Q is a polynomial with integer coefficients.
If S is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as
${\displaystyle HS_{S}(t)={\frac {P(t)}{(1-t)^{\delta }}},}$
where P is a polynomial with integer coefficients, and ${\displaystyle \delta }$ is the Krull dimension of S.
In this case the series expansion of this rational fraction is
${\displaystyle HS_{S}(t)=P(t)\left(1+\delta t+\cdots +{\binom {n+\delta -1}{\delta -1}}t^{n}+\cdots \right)}$
${\displaystyle {\binom {n+\delta -1}{\delta -1}}={\frac {(n+\delta -1)(n+\delta -2)\cdots (n+1)}{(\delta -1)!}}}$
is the binomial coefficient for ${\displaystyle n>-\delta ,}$ and is 0 otherwise.
${\displaystyle P(t)=\sum _{i=0}^{d}a_{i}t^{i},}$
the coefficient of ${\displaystyle t^{n}}$ in ${\displaystyle HS_{S}(t)}$ is thus
${\displaystyle HF_{S}(n)=\sum _{i=0}^{d}a_{i}{\binom {n-i+\delta -1}{\delta -1}}.}$
For ${\displaystyle n\geq i-\delta +1,}$ the term of index i in this sum is a polynomial in n of degree ${\displaystyle \delta -1}$ with leading coefficient ${\displaystyle a_{i}/(\delta -1)!.}$ This
shows that there exists a unique polynomial ${\displaystyle HP_{S}(n)}$ with rational coefficients which is equal to ${\displaystyle HF_{S}(n)}$ for n large enough. This polynomial is the Hilbert
polynomial, and has the form
${\displaystyle HP_{S}(n)={\frac {P(1)}{(\delta -1)!}}n^{\delta -1}+{\text{ terms of lower degree in }}n.}$
The least n[0] such that ${\displaystyle HP_{S}(n)=HF_{S}(n)}$ for n ≥ n[0] is called the Hilbert regularity. It may be lower than ${\displaystyle \deg P-\delta +1}$.
The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients (Schenck 2003, pp. 41).
All these definitions may be extended to finitely generated graded modules over S, with the only difference that a factor t^m appears in the Hilbert series, where m is the minimal degree of the
generators of the module, which may be negative.
The Hilbert function, the Hilbert series and the Hilbert polynomial of a filtered algebra are those of the associated graded algebra.
The Hilbert polynomial of a projective variety V in P^n is defined as the Hilbert polynomial of the homogeneous coordinate ring of V.
Graded algebra and polynomial rings
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g[1], ..., g[n] of
degree 1, then the map which sends X[i] onto g[i] defines an homomorphism of graded rings from ${\displaystyle R_{n}=K[X_{1},\ldots ,X_{n}]}$ onto S. Its kernel is a homogeneous ideal I and this
defines an isomorphism of graded algebra between ${\displaystyle R_{n}/I}$ and S.
Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will
be restricted to the quotients of polynomial rings by ideals.
Properties of Hilbert series
Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if
${\displaystyle 0\;\rightarrow \;A\;\rightarrow \;B\;\rightarrow \;C\;\rightarrow \;0}$
is an exact sequence of graded or filtered modules, then we have
${\displaystyle HS_{B}=HS_{A}+HS_{C}}$
${\displaystyle HP_{B}=HP_{A}+HP_{C}.}$
This follows immediately from the same property for the dimension of vector spaces.
Quotient by a non-zero divisor
Let A be a graded algebra and f a homogeneous element of degree d in A which is not a zero divisor. Then we have
${\displaystyle HS_{A/(f)}(t)=(1-t^{d})\,HS_{A}(t)\,.}$
It follows from the additivity on the exact sequence
${\displaystyle 0\;\rightarrow \;A^{[d]}\;{\xrightarrow {f}}\;A\;\rightarrow \;A/f\rightarrow \;0\,,}$
where the arrow labeled f is the multiplication by f, and ${\displaystyle A^{[d]}}$ is the graded module which is obtained from A by shifting the degrees by d, in order that the multiplication by f
has degree 0. This implies that ${\displaystyle HS_{A^{[d]}}(t)=t^{d}\,HS_{A}(t)\,.}$
Hilbert series and Hilbert polynomial of a polynomial ring
The Hilbert series of the polynomial ring ${\displaystyle R_{n}=K[x_{1},\ldots ,x_{n}]}$ in ${\displaystyle n}$ indeterminates is
${\displaystyle HS_{R_{n}}(t)={\frac {1}{(1-t)^{n}}}\,.}$
It follows that the Hilbert polynomial is
${\displaystyle HP_{R_{n}}(k)={{k+n-1} \choose {n-1}}={\frac {(k+1)\cdots (k+n-1)}{(n-1)!}}\,.}$
The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here ${\displaystyle x_{n}}$) and remarking
that ${\displaystyle HS_{K}(t)=1\,.}$
Shape of the Hilbert series and dimension
A graded algebra A generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is
nilpotent. This implies that the dimension of A as a K-vector space is finite and the Hilbert series of A is a polynomial P(t) such that P(1) is equal to the dimension of A as a K-vector space.
If the Krull dimension of A is positive, there is a homogeneous element f of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension
of A/(f) is the Krull dimension of A minus one.
The additivity of Hilbert series shows that ${\displaystyle HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)}$. Iterating this a number of times equal to the Krull dimension of A, we get eventually an algebra of
dimension 0 whose Hilbert series is a polynomial P(t). This show that the Hilbert series of A is
${\displaystyle HS_{A}(t)={\frac {P(t)}{(1-t)^{d}}}}$
where the polynomial P(t) is such that P(1) ≠ 0 and d is the Krull dimension of A.
This formula for the Hilbert series implies that the degree of the Hilbert polynomial is d, and that its leading coefficient is ${\displaystyle {\frac {P(1)}{d!}}}$.
Degree of a projective variety and Bézout's theorem
The Hilbert series allows us to compute the degree of an algebraic variety as the value at 1 of the numerator of the Hilbert series. This provides also a rather simple proof of Bézout's theorem.
For showing the relationship between the degree of a projective algebraic set and the Hilbert series, consider an projective algebraic set V, defined as the set of the zeros of a homogeneous ideal $
{\displaystyle I\subset k[x_{0},x_{1},\ldots ,x_{n}]}$, where k is a field, and let ${\displaystyle R=k[x_{0},\ldots ,x_{n}]/I}$ be the ring of the regular functions on the algebraic set.
In this section, one does not need irreducibility of algebraic sets nor primality of ideals. Also, as Hilbert series are not changed by extending the field of coefficients, the field k is supposed,
without loss of generality, to be algebraically closed.
The dimension d of V is equal to the Krull dimension minus one of R, and the degree of V is the number of points of intersection, counted with multiplicities, of V with the intersection of ${\
displaystyle d}$ hyperplanes in general position. This implies the existence, in R, of a regular sequence ${\displaystyle h_{0},\ldots ,h_{d}}$ of d + 1 homogeneous polynomials of degree one. The
definition of a regular sequence implies the existence of exact sequences
${\displaystyle 0\longrightarrow \left(R/\langle h_{0},\ldots ,h_{k-1}\rangle \right)^{[1]}{\stackrel {h_{k}}{\longrightarrow }}R/\langle h_{1},\ldots ,h_{k-1}\rangle \longrightarrow R/\langle h_
{1},\ldots ,h_{k}\rangle \longrightarrow 0,}$
for ${\displaystyle k=0,\ldots ,d.}$ This implies that
${\displaystyle HS_{R/\langle h_{0},\ldots ,h_{d-1}\rangle }(t)=(1-t)^{d}\,HS_{R}(t)={\frac {P(t)}{1-t}},}$
where ${\displaystyle P(t)}$ is the numerator of the Hilbert series of R.
The ring ${\displaystyle R_{1}=R/\langle h_{0},\ldots ,h_{d-1}\rangle }$ has Krull dimension one, and is the ring of regular functions of a projective algebraic set ${\displaystyle V_{0}}$ of
dimension 0 consisting of a finite number of points, which may be multiple points. As ${\displaystyle h_{d}}$ belongs to a regular sequence, none of these points belong to the hyperplane of equation
${\displaystyle h_{d}=0.}$ The complement of this hyperplane is an affine space that contains ${\displaystyle V_{0}.}$ This makes ${\displaystyle V_{0}}$ an affine algebraic set, which has ${\
displaystyle R_{0}=R_{1}/\langle h_{d}-1\rangle }$ as its ring of regular functions. The linear polynomial ${\displaystyle h_{d}-1}$ is not a zero divisor in ${\displaystyle R_{1},}$ and one has thus
an exact sequence
${\displaystyle 0\longrightarrow R_{1}{\stackrel {h_{d}-1}{\longrightarrow }}R_{1}\longrightarrow R_{0}\longrightarrow 0,}$
which implies that
${\displaystyle HS_{R_{0}}(t)=(1-t)HS_{R_{1}}(t)=P(t).}$
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.
Thus ${\displaystyle R_{0}}$ is an Artinian ring, which is a k-vector space of dimension P(1), and Jordan–Hölder theorem may be used for proving that P(1) is the degree of the algebraic set V. In
fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.
For proving Bézout's theorem, one may proceed similarly. If ${\displaystyle f}$ is a homogeneous polynomial of degree ${\displaystyle \delta }$, which is not a zero divisor in R, the exact sequence
${\displaystyle 0\longrightarrow R^{[\delta ]}{\stackrel {f}{\longrightarrow }}R\longrightarrow R/\langle f\rangle \longrightarrow 0,}$
shows that
${\displaystyle HS_{R/\langle f\rangle }(t)=\left(1-t^{\delta }\right)HS_{R}(t).}$
Looking on the numerators this proves the following generalization of Bézout's theorem:
Theorem - If f is a homogeneous polynomial of degree ${\displaystyle \delta }$, which is not a zero divisor in R, then the degree of the intersection of V with the hypersurface defined by ${\
displaystyle f}$ is the product of the degree of V by ${\displaystyle \delta .}$
In a more geometrical form, this may restated as:
Theorem - If a projective hypersurface of degree d does not contain any irreducible component of an algebraic set of degree δ, then the degree of their intersection is dδ.
The usual Bézout's theorem is easily deduced by starting from a hypersurface, and intersecting it with n − 1 other hypersurfaces, one after the other.
Complete intersection
A projective algebraic set is a complete intersection if its defining ideal is generated by a regular sequence. In this case, there is a simple explicit formula for the Hilbert series.
Let ${\displaystyle f_{1},\ldots ,f_{k}}$ be k homogeneous polynomials in ${\displaystyle R=K[x_{1},\ldots ,x_{n}]}$, of respective degrees ${\displaystyle \delta _{1},\ldots ,\delta _{k}.}$ Setting
${\displaystyle R_{i}=R/\langle f_{1},\ldots ,f_{i}\rangle ,}$ one has the following exact sequences
${\displaystyle 0\;\rightarrow \;R_{i-1}^{[\delta _{i}]}\;{\xrightarrow {f_{i}}}\;R_{i-1}\;\rightarrow \;R_{i}\;\rightarrow \;0\,.}$
The additivity of Hilbert series implies thus
${\displaystyle HS_{R_{i}}(t)=(1-t^{\delta _{i}})HS_{R_{i-1}}(t)\,.}$
A simple recursion gives
${\displaystyle HS_{R_{k}}(t)={\frac {(1-t^{\delta _{1}})\cdots (1-t^{\delta _{k}})}{(1-t)^{n}}}={\frac {(1+t+\cdots +t^{\delta _{1}})\cdots (1+t+\cdots +t^{\delta _{k}})}{(1-t)^{n-k}}}\,.}$
This shows that the complete intersection defined by a regular sequence of k polynomials has a codimension of k, and that its degree is the product of the degrees of the polynomials in the sequence.
Relation with free resolutions
Every graded module M over a graded regular ring R has a graded free resolution, meaning there exists an exact sequence
${\displaystyle 0\to L_{k}\to \cdots \to L_{1}\to M\to 0,}$
where the ${\displaystyle L_{i}}$ are graded free modules, and the arrows are graded linear maps of degree zero.
The additivity of Hilbert series implies that
${\displaystyle HS_{M}(t)=\sum _{i=1}^{k}(-1)^{i-1}HS_{L_{i}}(t).}$
If ${\displaystyle R=k[x_{1},\ldots ,x_{n}]}$ is a polynomial ring, and if one knows the degrees of the basis elements of the ${\displaystyle L_{i},}$ then the formulas of the preceding sections
allow deducing ${\displaystyle HS_{M}(t)}$ from ${\displaystyle HS_{R}(t)=1/(1-t)^{n}.}$ In fact, these formulas imply that, if a graded free module L has a basis of h homogeneous elements of degrees
${\displaystyle \delta _{1},\ldots ,\delta _{h},}$ then its Hilbert series is
${\displaystyle HS_{L}(t)={\frac {t^{\delta _{1}}+\cdots +t^{\delta _{h}}}{(1-t)^{n}}}.}$
These formulas may be viewed as a way for computing Hilbert series. This is rarely the case, as, with the known algorithms, the computation of the Hilbert series and the computation of a free
resolution start from the same Gröbner basis, from which the Hilbert series may be directly computed with a computational complexity which is not higher than that the complexity of the computation of
the free resolution.
Computation of Hilbert series and Hilbert polynomial
The Hilbert polynomial is easily deducible from the Hilbert series (see above). This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered
or graded by the total degree.
Thus let K a field, ${\displaystyle R=K[x_{1},\ldots ,x_{n}]}$ be a polynomial ring and I be an ideal in R. Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the
elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the
leading monomials of the elements of B.
The computation of the Hilbert series is based on the fact that the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series.
Thus the computation of the Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the
computation of the Gröbner basis. The computational complexity of the whole computation depends mainly on the regularity, which is the degree of the numerator of the Hilbert series. In fact the
Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
The computation of Hilbert series and Hilbert polynomials are available in most computer algebra systems. For example in both Maple and Magma these functions are named HilbertSeries and
Generalization to coherent sheaves
In algebraic geometry, graded rings generated by elements of degree 1 produce projective schemes by Proj construction while finitely generated graded modules correspond to coherent sheaves. If ${\
displaystyle {\mathcal {F}}}$ is a coherent sheaf over a projective scheme X, we define the Hilbert polynomial of ${\displaystyle {\mathcal {F}}}$ as a function ${\displaystyle p_{\mathcal {F}}(m)=\
chi (X,{\mathcal {F}}(m))}$, where χ is the Euler characteristic of coherent sheaf, and ${\displaystyle {\mathcal {F}}(m)}$ a Serre twist. The Euler characteristic in this case is a well-defined
number by Grothendieck's finiteness theorem.
This function is indeed a polynomial.^[1] For large m it agrees with dim ${\displaystyle H^{0}(X,{\mathcal {F}}(m))}$ by Serre's vanishing theorem. If M is a finitely generated graded module and ${\
displaystyle {\tilde {M}}}$ the associated coherent sheaf the two definitions of Hilbert polynomial agree.
Graded free resolutions
Since the category of coherent sheaves on a projective variety ${\displaystyle X}$ is equivalent to the category of graded-modules modulo a finite number of graded-pieces, we can use the results in
the previous section to construct Hilbert polynomials of coherent sheaves. For example, a complete intersection ${\displaystyle X}$ of multi-degree ${\displaystyle (d_{1},d_{2})}$ has the resolution
${\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} ^{n}}(-d_{1}-d_{2}){\xrightarrow {\begin{bmatrix}f_{2}\\-f_{1}\end{bmatrix}}}{\mathcal {O}}_{\mathbb {P} ^{n}}(-d_{1})\oplus {\mathcal {O}}_{\
mathbb {P} ^{n}}(-d_{2}){\xrightarrow {\begin{bmatrix}f_{1}&f_{2}\end{bmatrix}}}{\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}_{X}\to 0}$
See also | {"url":"https://wiki.wikirank.net/en/Hilbert_series_and_Hilbert_polynomial","timestamp":"2024-11-09T00:44:10Z","content_type":"text/html","content_length":"187820","record_id":"<urn:uuid:bdcca4d1-b0a5-4ed0-87df-3c2407aa167a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00052.warc.gz"} |
Seeing Blue: Exoplanet With Water-Rich Atmosphere Observed - NASA Science
4 min read
Seeing Blue: Exoplanet With Water-Rich Atmosphere Observed
A Japanese research team of astronomers and planetary scientists has used Subaru Telescope's two optical cameras, Suprime-Cam and the Faint Object Camera and Spectrograph (FOCAS), with a blue
transmission filter to observe planetary transits of super-Earth GJ 1214 b (Gilese 1214 b). The team investigated whether this planet has an atmosphere rich in water or hydrogen. The Subaru
observations show that the sky of this planet does not show a strong Rayleigh scattering feature, which a cloudless hydrogen-dominated atmosphere would predict. When combined with the findings of
previous observations in other colors, this new observational result implies that GJ 1214 b is likely to have a water-rich atmosphere.
Super-Earths are emerging as a new type of exoplanet (i.e., a planet orbiting a star outside of our Solar System) with a mass and radius larger than the Earth's but less than those of ice giants in
our Solar System, such as Uranus or Neptune. Whether super-Earths are more like a "large Earth" or a "small Uranus" is unknown, since scientists have yet to determine their detailed properties. The
current Japanese research team of astronomers and planetary scientists focused their efforts on investigating the atmospheric features of one super-Earth, GJ 1214 b, which is located 40 light years
from Earth in the constellation Ophiuchus, northwest of the center of our Milky Way galaxy. This planet is one of the well-known super-Earths discovered by Charbonneau et. al. (2009) in the MEarth
Project, which focuses on finding habitable planets around nearby small stars. The current team's research examined features of light scattering of GJ 1214 b's transit around its star.
Current theory posits that a planet develops in a disk of dense gas surrounding a newly formed star (i.e., a protoplanetary disk). The element hydrogen is a major component of a protoplanetary disk,
and water ice is abundant in an outer region beyond a so-called "snow line." Findings about where super-Earths have formed and how they have migrated to their current orbits point to the prediction
that hydrogen or water vapor is a major atmospheric component of a super-Earth. If scientists can determine the major atmospheric component of a super-Earth, they can then infer the planet's
birthplace and formation history.
Planetary transits enable scientists to investigate changes in the wavelength in the brightness of the star (i.e., transit depth), which indicate the planet's atmospheric composition. Strong Rayleigh
scattering in the optical wavelength is powerful evidence for a hydrogen-dominated atmosphere. Rayleigh scattering occurs when light particles scatter in a medium without a change in wavelength. Such
scattering strongly depends on wavelength and enhances short wavelengths; it causes greater transit depth in the blue rather than in the red wavelength.
The current team used the two optical cameras Suprime-Cam and FOCAS on the Subaru Telescope fitted with a blue transmission filter to search for the Rayleigh scattering feature of GJ 1214 b's
atmosphere. This planetary system's very faint host star in blue light poses a challenge for researchers seeking to determine whether or not the planet's atmosphere has strong Rayleigh scattering.
The large, powerful light-collecting 8.2 m mirror of the Subaru Telescope allowed the team to achieve the highest-ever sensitivity in the bluest region.
The team's observations showed that GJ 1214 b's atmosphere does not display strong Rayleigh scattering. This finding implies that the planet has a water-rich or a hydrogen-dominated atmosphere with
extensive clouds.
Although the team did not completely discount the possibility of a hydrogen-dominated atmosphere, the new observational result combined with findings from previous research in other colors suggests
that GJ 1214 b is likely to have a water-rich atmosphere. The team plans to conduct follow-up observations in the near future to reinforce their conclusion.
Although there are only a small number of super-Earths that scientists can observe in the sky now, this situation will dramatically change when the Transiting Exoplanet Survey Satellite (TESS) begins
its whole sky survey of small transiting exoplanets in our solar neighborhood. When new targets become available, scientists can study the atmospheres of many super-Earths with the Subaru Telescope
and next generation, large telescopes such as the Thirty Meter Telescope (TMT). Such observations will allow scientists to learn even more about the nature of various super-Earths. | {"url":"https://science.nasa.gov/universe/exoplanets/seeing-blue-exoplanet-with-water-rich-atmosphere-observed/","timestamp":"2024-11-07T13:23:27Z","content_type":"text/html","content_length":"316440","record_id":"<urn:uuid:599294af-9d2b-4f67-a307-9d2d784aa791>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00016.warc.gz"} |
Ideal Gas Law Calculator - calculator
What is the Ideal Gas Law Calculator?
The Ideal Gas Law Calculator helps users determine the number of moles of gas in a system given the pressure, volume, and temperature. It employs the ideal gas equation to establish relationships
among these variables, facilitating quick calculations that are essential in chemistry and physics.
The Ideal Gas Law formula is: pV = nRT
• p = Pressure (Pa)
• V = Volume (m³)
• n = Moles of gas
• R = Ideal gas constant (8.3144598 J/(mol·K))
• T = Temperature (K)
How to Use the Calculator
To use the Ideal Gas Law Calculator, fill in the fields for Temperature (K), Pressure (Pa), and Volume (m³). As you enter values for Pressure, Volume, and Temperature, the calculator automatically
computes the number of moles of gas. The results are displayed in a table format for easy interpretation. You can also clear the inputs to start fresh.
Temperature (K) Pressure (Pa) Volume (m³)
Moles of Gas Result
What is the Ideal Gas Law?
The Ideal Gas Law describes the relationship between pressure, volume, temperature, and moles of a gas. It's used in chemistry and physics to model gas behavior under ideal conditions.
How accurate is the Ideal Gas Law?
The Ideal Gas Law is a good approximation for many gases under standard conditions. However, it may not hold true for gases at high pressure or low temperature, where intermolecular forces become
What is R in the Ideal Gas Law?
R is the ideal gas constant, which is approximately 8.3144598 J/(mol·K). It relates the energy scale to the temperature scale in the equation.
What units should I use for pressure?
In the Ideal Gas Law, pressure should be measured in Pascals (Pa) for consistent results. Ensure all units are compatible to avoid calculation errors.
Can I use this calculator for real gases?
This calculator is designed for ideal gases. Real gases may deviate from ideal behavior, especially under high pressure or low temperature.
How do I convert temperature to Kelvin?
To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. For example, 25°C is 298.15 K.
What if I enter zero for pressure or volume?
Entering zero for pressure or volume will lead to undefined results in the calculation. Ensure all values are greater than zero.
Is this calculator suitable for educational purposes?
Yes, this calculator is an excellent educational tool for students learning about gas laws and chemistry principles.
Can I use different units for volume?
While this calculator accepts volume in cubic meters (m³), you can convert from other units as long as you maintain consistent units for pressure and temperature.
Where can I find more information about gas laws?
Additional resources on gas laws can be found in chemistry textbooks, online educational platforms, and scientific websites dedicated to chemistry.
Method of Solving | {"url":"https://calculatordna.com/ideal-gas-law-calculator/","timestamp":"2024-11-10T06:26:48Z","content_type":"text/html","content_length":"84275","record_id":"<urn:uuid:84efd5c7-6473-4586-bc40-a547688fb8d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00287.warc.gz"} |
Dynamic Wind Analysis of a Billboard - Part 1
Wind Analysis is very common, but the dynamic effects are often over estimated or even worse overlooked. In this case study, we walk through the approach taken to simulate the stress due to
aerodynamic loading through a billboard. This case study walks through the Method, the Physics, and the Results of a case study that was originally presented at Autodesk University 2019.
Billboards come in a wide variety of layouts and structural configurations. The goal here is to provide a billboard with which to highlight the associativity between Autodesk CFD and Autodesk Nastran
In-CAD, as well as to leverage a structural configuration that may not be ideal. Thus would provide us great foundation for learning how to best apply these tools.
After some discussion, the design chosen for this investigation was the "Back-to-back" Cantilevered Monopole with bolt cage. This allows us to analyze "head-on" wind loading, which is predicted to be
worst case. The overall design is relatively simple and so it can be compared to hand calcs of a vertical flat plate. The overall approach here is to start simple, gain confidence, and then add
complexity as you build competence. This is works for students as well as designers.
CREATING THE RIGHT "LEVEL OF DETAIL" FOR CFD
One of the "classic" problems we have in Computer Aided Engineering (whether for structural or fluids simulation), is the need to defeature the CAD file. In this case, where the Inventor model has
the fully detailed weldment (needed for structure analysis) that detail is not needed in CFD. It's not the geometry detail that is our focus (we want to keep that!) but rather the computational
"weight" of the model.
In other words, trying to bring 150+ parts from CAD to CFD, is going to be a waste of time and most likely a huge headache no matter what software you are using. Instead, our strategy here is to
strip the model (not the detail) down to one part, so we can then use it to create a cavity in our air volume. Remember, in CFD the focus is on analyzing the air around our structure, not the
structural assembly itself.
This is quite easily done in Inventor using a Derived Part and the sculpt feature. Although there are other ways to do this, this method allows us to easily create and modify on-the-fly additional,
parametric air volumes around our structure to manage our mesh density as needed. As can be seen below, creating this geometry parametrically is really easy in Inventor, and it will allow us to fine
tune our mesh later - a step that is often overlooked and can result in a lot of overall time savings.
With the CAD model ready, the next step is to set up the CFD model. To do that, accurate velocity boundary conditions are needed. In architectural aerodynamics, the wind is constantly changing - both
in magnitude and direction. To distill this case study down to a manageable investigation, the wind velocity was chosen at just two states, Average and Gust, where each would be analyzed in Steady
State (for now!).
Simple internet research pointed us to the Water and Atmospheric Resources Monitoring (WARM) Program, which offers a lot of detailed data, including a really neat wind map animation of the United
To be timely and relevant, we decided to use the November 2019 Chicago Area Wind Speeds (the month we originally presented this topic at Autodesk University). , and on the right you can see the
detailed data we extracted our wind velocities from.
With everything needed to start simulating, we jumped right into Autodesk CFD, and built out our model using the the proper velocity, pressure, and slip boundary conditions. At Fastway, we are always
striving for a mesh independent solution, and every analyst should do this. Therefore, early simulations start relatively coarse ,and tighten up as we review the results and how they look. For
example, we know for our average wind speed steady state analysis, that our free stream velocity is 10 mph. If the walls of your analysis are too close to your flow impedance, then it will start to
"squeeze" the flow, and create additional error into the analysis (yes, this is true even if you use the slip/symmetry condition). If the analysis volume is too large, then you just end up waiting
too long for results.
To optimize our air volume, we used an iso-volume set to 9.5mph and watched it's shape as we changed the size of the "virtual wind tunnel" that we had created. After just a handful of iterations were
able to cut down the size of the volume quite a bit, and then use the extra computing power to add finer mesh details at the billboard. This ensured that we could get the proper boundary layer mesh
height associated to our turbulence model. This increases the numerical accuracy of our force predictions. Recall that for K-epsilon, you should have a y+ value between 30 and 300, and for K-omega,
y+ should be equal to 1.
Before we start looking at CFD results (and especially before we start extracting force values), it is critical to do some simple hand calcs. If you recall, we strategically chose a "simple"
billboard design (the cantilevered Monopole), which is basically just a vertical plate. Assuming the airflow is perfectly normal to the billboard, the force acting on it is equivalent to the change
in momentum of the air. Assuming the air hits the billboard and comes to a complete stop (velocity = zero), that results in the equation below.
When we enter the wind velocities and compare to the theoretical values, we get correlation within 15%, which is not bad considering the actual CAD is more complex than a theoretical vertical plate,
and we are using a steady-state, Reynolds Averaged Navier Stokes approach - there are always several sources of potential error (hopefully they cancel each other out!).
With the results of the CFD analysis in our hands, now we can turn our attention to the structural aspects of our investigation. First, we need to prepare our CAD model for import into Nastran
In-CAD. Since our area of interest has shifted to the mechanical structure, and not the airflow around it, we must ensure the we have the proper level of detail.
MODELING 2D BEAMS VS 3D SOLIDS
Full disclosure: in order to properly map the CFD results to FEA, we need to have the same 3D volumes in both models. Simplifying weldments to 2D Beam elements is an accepted practice, but in our
case won't work for mapping the direct CFD pressures. We include it here for an academic exercise, however if you are more interested in directly mapping the CFD results, skip to the next section.
DISCUSSION AND NEXT STEPS... | {"url":"https://blog.fastwayengineering.com/dynamic-wind-analysis-billboard-autodesk-cfd-nastran-fea","timestamp":"2024-11-09T22:09:04Z","content_type":"text/html","content_length":"67897","record_id":"<urn:uuid:ceb9c81a-a04a-4735-afed-8facb878af4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00555.warc.gz"} |
School of Mathematical and Data Sciences | Lori Ogden
Dr. Lori Ogden is a Teaching Associate Professor of Mathematics and an Associate Director of the School of Mathematical and Data Sciences at West Virginia University. She is the course coordinator
for college algebra and has taught a variety of undergraduate mathematics courses including precalculus, trigonometry, and calculus. Her role as Associate Director includes running the Institute for
Mathematics Learning where she leads initiatives that promote student success within all undergraduate mathematics courses through Calculus 3. Recently she led the redesign of several entry-level
courses by fostering the use of undergraduate learning assistants and the integration of assignments that teach students how to study, where to get help, and how to manage their time. In addition,
she has developed and coordinated training opportunities for undergraduate learning assistants and professional development opportunities for faculty, instructors, and graduate students.
Ph.D., Education, West Virginia University
M.S., Mathematics, West Virginia University
B.S., Secondary Education (Mathematics, grades 5-12), West Virginia University
Research Interests Include:
• Online and Blended Learning Environments
• Course Design and Development
• Teacher Education and Professional Development
Courses offered at WVU:
• Math 126 College Algebra
• Math 128 Trigonometry
• Math 129 Precalculus
• Math 150 Applied Calculus
• Math 155 Calculus 1
• Math 318 Perspectives on Mathematics and Science
Recent Publications:
Ogden, L., Kearns, J., & Bowman, N. (2018). Prerequisite knowledge of mathematics and success in calculus 1. Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics
Education, San Diego, CA
Ogden, L., & Shambaugh, N. (2017). Professional Development for Teaching College Mathematics Using an Integrated Flipped Classroom. In J. Keengwe (Ed.) Handbook of Research on Pedagogical Models for
Next-Generation Teaching and Learning (pp. 154-176). Hershey, PA: IGI Global.
Ogden, L. (2016). Students perceptions of learning college algebra online using adaptive learning technology.
Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education,
Pittsburgh, PA
Ogden, L. & Shambaugh, N. (2016). Best Teaching and Technology Practices for the Hybrid Flipped College Classroom. In P. Vu, S. Fredrickson, & C. Moore (Eds.),
Handbook of Research on Innovative Pedagogies and Technologies for Online Learning in Higher Education (pp.286-308).
Hershey, PA: IGI Global.
Ogden, L. (2015). Student Perceptions of the Flipped Classroom in College Algebra.
PRIMUS: Problems, Resources, Issues in Mathematics Undergraduate Studies, 25
(9-10), 782-791. | {"url":"https://mathanddata.wvu.edu/directory/leadership/lori-ogden","timestamp":"2024-11-03T03:06:03Z","content_type":"text/html","content_length":"69586","record_id":"<urn:uuid:59d0acfa-0185-4e43-944c-6f07d59063e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00632.warc.gz"} |
Your number was...
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
This problem follows on from Your Number Is... Press the button in the interactivity below to investigate what this machine does:
Try starting with other numbers - maybe decimals or negative numbers.
How is the machine working out your starting numbers?
Can you devise your own set of instructions so you can work out someone else's starting numbers in the same way?
Getting Started
One way to begin thinking about this is to collect together some outputs and their corresponding inputs:
│ Finishing Number │ Starting Number │
│ │ │
│ │ │
│ │ │
│ │ │
│ │ │
Can you write each step of the machine's instructions as a function machine?
What happens if you work backwards?
Student Solutions
Students from Westridge School for Girls in the USA, Cedric from Australian International School Malaysia, Poppy and Gracie from Walton High School in England, students from Humanitree in Mexico,
Florence from Walthamstow Hall Junior School in the UK, Eric and Lucia from Willowbank School in New Zealand, Abdullah from Poland, students from Frederick Irwin Anglican School in Australia, John
from Poolesville ES in the USA, Bracha and Antoine from St Philip's CofE Primary School Cambridge in the UK, Park from Renaissance International School Sigon in Korea, Diya from the UK, Deniz from
North Leamington School in the UK, Ira from Coldspring3 and Adi from the New Beacon in the UK all described how the computer works out your starting number.
Kristin from Westridge School for Girls sent this explanation:
All you have to do is do the problem backwards.
For example, if your number is 1 then you add 4, double that, subtract 7, and you will get 3.
All the computer has to do is do the problem backwards.
3$+$7 (opposite of $-$7) is 10.
10$\div$2 (opposite of $\times$2) is 5.
$-$4 (opposite of $+$4) is 1, which was my original number.
Luigi from Humanitree and Karrthic from Australian International School Malaysia found an algebraic expression for the final number. This is Karrthic's work:
The operation is
$(x +4) \times 2 -7= y$
For example
$(5 + 4) = 9\\
9 \times2 = 18\\
When you reverse the operation, you find $x$
Furkan from TED Ankara Koleji in Turkey, Bernd and Qianwei from Humanitree and Shriya from International School Frankfurt in Germany showed what the computer does algebraically. This is Qianwei's
$(x+4) \times 2 - 7=y$
If we want to get $x$, we need to do the opposite of this equation.
Start with $y.$ In the equation $7$ [was subtracted], so we need to add $7$ to $y$
In the equation, $(x+4)$ was doubled, we need to divide it by two to get the opposite.
In the equation, $x$ was added [to] $4$, to get the opposite, we need to take away $4$ from $x.$
Finally the equation we used to find $x$ will be:
Furkan and Shriya simplified the equation before 'reversing' it. This is Furkan's work:
$(x+4) \times2 - 7$ is equal to $2x + 1,$ so we subtract $1$ from the result and divide by $2$ to get the starting number.
Alex from Humanitree and Fernando from Academia Britanica Cuscatleca in El Salvador noticed a different pattern in the numbers from the computer. This is Fernando's work:
When following the instructions given in the problem, if [you give the computer random integers as the result of your calculation], you will get random numbers.
However, when we start with 1 and try until five, you will notice the sequence:
│Number you tell the computer was your answer │Number the computer says you started from│
│1 │0 │
│2 │0.5 │
│3 │1 │
│4 │1.5 │
│5 │2 │
As you can see, the sequence is progressing by 0.5, thus is linear. It is adding 0.5 so, and we can see that when 0.5$\times n$ is done, you the sequence now looks like this:
│Number you tell the computer was your answer ($n$)│Number the computer says you started from│ 0.5$\times n$ │
│1 │0 │0.5 │
│2 │0.5 │1 │
│3 │1 │1.5 │
│4 │1.5 │2 │
│5 │2 │2.5 │
The answer to the subsitution is always 0.5 greater than the actual answer,
therefore, we need to always substract 0.5
Therefore, the instructions are directly 0.5$n-$0.5 !
Can you see how this result is related to Furkan's equation?
We received lots of sets of instructions for similar activities.
These instructions are from Tamara from Humanitree:
.Think of a number
.multiply it by 4
.subtract 2
.add 1
Your number was”¦
So if the final number was 19 first you would need to subtract 1. That would equal to 18. Then you would need to add 2, which would equal to 20. And [finally] you would need to divide it by 4. This
equals to 5 so the first number would be 5.
Federico from Humanitree used algebra:
Subtract 10
Divide by 4
Add 5
If the number was $n$
You would do
$(n-5) \times4 +10 = $ end number ($x$)
Then the computer would do this operation to get the original value of $n:$
$(x-10)\div4 +5 =$ starting number
This operation would give you the original results.
Abdullah from Poland wrote a Python program to guess your number:
print("YOUR NUMBER WAS...")
print("Press [ENTER] after you made each move.")
input("Think of a number.")
input("Add 9")
input("Divide it by 4")
input("Subtract 6")
input("Triple it")
input("Subtract 23")
number = float(input("Now, type in number you finished with and then press enter."))
answer = (((number + 23) / 3) + 6) * 4 - 9
print("You chose the number %s" % answer)
Leila C and Scarlett K from Westridge School for Girls and KSc from Cheadle Academy sent in interesting sets of instructions which will always give the same result, like in the problem Your Number
If you follow these instructions, will it ever be possible for the computer to work out what number you started from?
These are KSc's instructions:
1. Think of a number: $x$ $\lt\lt\lt$ [Input] number
2. Add $3$: $x+3$
3. Double it: $2x+6$ $\lt\lt\lt$ Begin to balance out dividable equation
4. Add $4$: $2x+10$ $\lt\lt\lt$ Equation can now be easily divided
5. Divide by $2$: $x+5$ $\lt\lt\lt$ Simplify equation
6. Subtract [original] number from [current] number: $x+5\hspace{2mm}-x$ $\lt\lt\lt$ Balace equation [balancing $x$ with $-x$]
7. $x=$ always $5$ $\lt\lt\lt x$ is always fixed at $5$
Teachers' Resources
Why do this problem?
This problem provides a great opportunity to introduce the concept of representing operations on unknown numbers algebraically. It leads to work on inverse operations and solving simple linear
equations. When first shown the interactivity, students may be amazed and astonished that it can always work out the starting number. The intention is to provoke their curiosity so they become
determined to explain the mystery.
Possible approach
Either work through a few examples using the interactivity, or read out the instructions:
Think of a number
Add 4
Subtract 7
Ask a few students what they finished with and then surprise them by revealing their starting number!
As we don't want students to end up thinking maths is a strange magical mystery, it's time to explain what's going on!
One way to begin thinking about this is to collect together some outputs and their corresponding inputs, for example:
│Finishing Number │Starting Number │
│11 │5 │
│17 │8 │
│3 │1 │
│23 │11 │
│14 │6.5 │
This is not merely an exercise in pattern spotting - once learners have figured out HOW the machine is working out the starting numbers, they need to understand WHY this works.
It would be good to introduce a variety of methods for explaining what's going on:
• Represent the instructions as a function machine, and explore what happens when the functions are inverted.
• Represent the chosen number as $x$ and write an expression for the final number $y$ in terms of $x$. Then rearrange to get $x$ in terms of $y$.
• Discuss simplification to make the functions easier to invert (so representing it as $y=2x+1$ rather than $y=2(x+4)-7$.
Now challenge the class to come up with their own examples of "Think of a number" instructions, which they can invert to deduce someone's starting number from their final number. Encourage them to
record their working using function machines and algebra. Set challenges such as having to include particular operations, at least 6 different operations, and a set of instructions that's really
quick to invert.
At the end of the lesson, a selection of students can read out their instructions for the class to try, and then work out starting numbers from the final numbers.
In the "Settings" menu for the interactivity, you can edit the functions used by the machine. Click below to learn more.
To access the settings menu, click on the purple cog in the top right corner of the interactivity. The screenshot below shows the menu:
You can enter your own sequence, separated by hyphens, using the code "a" for add, "s" for subtract, "m" for multiply and "d" for divide. The sequence a4-m2-s7 in the screenshot represents "Add 4,
Multiply by 2, Subtract 7".
Once you have entered your chosen sequence, click to load it in the current window or a new resizable window.
Key questions
How does the computer "know" what your starting number was?
If you think of a number and add four, what do you need to do to get back to where you started?
If you think of a number, add four, and then double, what do you need to do to get back to where you started?
Is there a way of representing the "Think of a number" trick that helps you to get back to the starting number?
Possible support
In the problem
Your Number Is...
, students can explore visual and symbolic ways of representing number tricks.
Possible extension | {"url":"https://nrich.maths.org/problems/your-number-was","timestamp":"2024-11-07T17:24:32Z","content_type":"text/html","content_length":"51575","record_id":"<urn:uuid:09044b1f-1bab-4ada-a3e7-dfd0c627d27f>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00782.warc.gz"} |
Understanding the Impact of Lens Design on Focal Distance in context of focal distance
31 Aug 2024
Understanding the Impact of Lens Design on Focal Distance
The focal length of a lens is a critical parameter that determines its ability to capture images or project light onto a surface. In this article, we explore the impact of lens design on focal
distance, including the effects of refractive index, curvature, and aperture size.
A lens is an optical device that focuses light by bending it through refraction. The focal length of a lens is the distance between its principal point and the image formed by the lens when an object
is placed at infinity. In this article, we will examine how different design parameters affect the focal length of a lens.
Refractive Index
The refractive index (n) of a lens material determines its ability to bend light. The refractive index is defined as the ratio of the speed of light in vacuum (c) to the speed of light in the medium
n = c / v
A higher refractive index means that the lens will have a shorter focal length, since it can bend light more efficiently.
The curvature of a lens affects its ability to focus light. A convex lens has a positive curvature, while a concave lens has a negative curvature. The focal length (f) of a thin lens is related to
its curvature (C) by the following formula:
1/f = 1/do + 1/di
where do and di are the object and image distances, respectively.
Aperture Size
The aperture size of a lens affects its ability to collect light. A larger aperture allows more light to enter the lens, but it also increases the risk of aberrations. The focal length of a lens is
affected by the aperture size through the following formula:
f = f0 / (1 + (A/2))
where f0 is the focal length without an aperture and A is the aperture size.
In conclusion, the design parameters of a lens have a significant impact on its focal distance. The refractive index, curvature, and aperture size all play important roles in determining the focal
length of a lens. Understanding these relationships is critical for designing high-quality lenses that meet specific optical requirements.
• Hecht, E., & Zajac, A. (2007). Optics. Addison-Wesley.
• Smith, W. J. (1990). Modern Optical Engineering. McGraw-Hill.
• Jenkins, F. A., & White, H. E. (1957). Fundamentals of Optics. McGraw-Hill.
Related articles for ‘focal distance’ :
• Reading: Understanding the Impact of Lens Design on Focal Distance in context of focal distance
Calculators for ‘focal distance’ | {"url":"https://blog.truegeometry.com/tutorials/education/6aed788bf7561382c35f904408659171/JSON_TO_ARTCL_Understanding_the_Impact_of_Lens_Design_on_Focal_Distance_in_conte.html","timestamp":"2024-11-06T19:53:51Z","content_type":"text/html","content_length":"16092","record_id":"<urn:uuid:e0590b13-b4e4-4198-b2c0-bcf3f37efd89>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00126.warc.gz"} |
Information Theory #5 - City | Ivan Carvalho
Information Theory #5 - City
I am writing a series of posts about Information Theory problems. You may find the previous one here.
The problem we will analyze is City from the Japanese Olympiad in Informatics Spring Camp (JOI SC 2017). You may find the problem statement and a place to submit here .
A synthesis of the problem statement is : you are given a rooted tree of N vertices (N <= 250000) that has a depth that is at most 18. You must write an integer on each node so that given only the
integers written on X and Y, without knowing the structure of the tree, you can answer the question : is X on the path from Y to the root, Y on the path from X to the root or neither of them?
The scoring of the problem is based on the maximum integer you write on each node. There is also a subtask for N <= 10.
Subtask 1 : N <= 10 (8 points)
In this subtask, the graph is small and the maximum integer does not matter as long it is smaller than 2^60 - 1.
Because of this, we may choose a suboptimal strategy to solve the problem. The simplest one is to encode the vertex and the whole adjacency matrix on the integer we send.
We use the binary encoding to send the current node (this uses 4 bits) and use another 45 bits to send the matrix. Then, we run a DFS to see which case happens
L <= 2^36 - 1 : 22 points
With very big N, it is impossible to send the adjacency matrix. So we will focus on another technique to solve the problem.
This technique is commonly called “Euler Tour on a Tree” , even tough it is not exactly an Euler Tour. You can interpret it as noticing a special property of the initial and final time when you do a
DFS on the tree.
The image above represents the “Euler Tour”. Every time you enter a vertex v, you increase the DFS counter and says this is number is the start time of v . At the end of the DFS on v, we also save
the DFS counter and says it is the end time of v. The vector in the image represents this DFS ordering.
The cool thing about this is that it allows us to check if a vertex u is on the subtree of a vertex v easily. Because the vertices of the subtree form a contiguous subarray in the DFS order array, we
can simply check if the start of u is contained in the interval [s,e] where s is the start time of v and e the end time of v.
If you think a little bit about it, checking if a vertex v is on a subtree of another is exactly the same as checking if that another vertex is on the path from v to the root. So we found a solution
to the problem we want to solve!
For each vertex, we send an integer that represents the start and end time of the DFS. To do that, we send 18 bits for the start and another 18 bits for the end. In the decoding part, we simply
recover the starts and ends and run the interval check described above.
The maximum number if 2^36 - 1 , thus we score 22 points.
An unused property
So far we did not the property that the depth of the tree is at most 18. The solution described above works for any tree, and therefore does not exploit the special property of the depth.
Because the maximum depth is 18, every node has at most 18 ancestors and is therefore contained in at most 18 intervals. Thus, the sum of the lengths of all intervals is at most 18*N. So on average
each interval is at most 18. That is not much.
It seems to be a waste to send 18 bits of the end when we could send the size, which is on average 5 bits. So we will do exactly that : instead of sending the end, we send the size of the interval.
This optimization alone does not affect our punctuation, because there are causes in which we will still send integers up to 2**36 - 1. However, this idea is a crucial step to solve the problem.
An efficient way of sending the size : 100 points solution
We need to optimize the way we send the size. In a perfect environment, we could create a smaller dictionary that represented the sizes and send the position in that dictionary.
However, it is difficult to adjust a dictionary to the sizes of the interval. We will try a different approach : adjust the interval size to the dictionary. But how do we do that ?
Suppose that the interval size is a. How do we fit that size to the dictionary , if it is not in it? Let b be the smallest integer of the dictionary greater than our equal to a. We can always add
some extra leaves as dummy vertices in order to reach the size b. The image above exemplifies that, by adding two leaves to reach the size 5 from the original size 3.
In practice, we are adding empty spaces to our DFS array. The challenge now is to build that dictionary of sizes.
Building the dictionary : 100 points solution
The challenge of finding a dictionary of sizes is that it should not add many extra vertices, because in that case the start of the vertices would be big and thus the dictionary would need to be
If the total number of vertices (original and dummy) of the final solution is close to 2^K, then we must construct a dictionary of size 2^(28 - K) .
Additionally , this dictionary should vary gradually. A function that varies gradually is a power function, so we can describe our dictionary as a set of {r^0, r^1, r^2…} for some number r. The
optimal r is the solution for r^(2^(28 - K)) = 2^K
If you experiment with K, you may find that K = 20 is a good choice. Therefore, the maximum number of vertices if 2^20 and the dictionary size is 2^8 = 256.
The value of r is 2^(20/256) ≈ 1.055645 (we must truncate because we cannot represent an irrational number in an exact decimal form).
Because r is not an integer, our dictionary is not exactly described by a power function, but by an approximation. Let A be our dictionary and a0 = 0 the first element. We say that ai = max{ a(i-1) +
1, ai*r } , where ai*r is rounded down.
In the final code, I opted to send the difference between the start and the end instead of the size. The result is identical, tough. | {"url":"https://ivaniscoding.github.io/posts/informationtheory5/","timestamp":"2024-11-04T21:22:06Z","content_type":"text/html","content_length":"19647","record_id":"<urn:uuid:3601394e-4251-4c12-97be-0d8b03bffd33>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00473.warc.gz"} |
Feedback Linearizing Control Strategies for Chemical Engineering Applications.
Document Type
Degree Name
Doctor of Philosophy (PhD)
Chemical Engineering
First Advisor
Michael A. Henson
Two widely studied control techniques which compensate for process nonlinearities are feedback linearization (FBL) and nonlinear model predictive control (NMPC). Feedback linearization has a low
computational requirement but provides no means to explicitly handle constraints which are important in the chemical process industry. Nonlinear model predictive control provides explicit constraint
compensation but only at the expense of high computational requirements. Both techniques suffer from the need for full-state feedback and may have high sensitivities to disturbances. The main work of
this dissertation is to eliminate some of the disadvantages associated with FBL techniques. The computation time associated with solving a nonlinear programming problem at each time step restricts
the use of NMPC to low-dimensional systems. By using linear model predictive control on top of a FBL controller, it is found that explicit constraint compensation can be provided without large
computational requirements. The main difficulty is the required constraint mapping. This strategy is applied to a polymerization reactor, and stability results for discrete-time nonlinear systems are
established. To alleviate the need for full-state feedback in FBL techniques it is necessary to construct an observer, which is very difficult for general nonlinear systems. A class of nonlinear
systems is studied for which the observer construction is quite easy in that the design mimics the linear case. The class of systems referred to are those in which the unmeasured variables appear in
an affine manner. The same observer construction can be used to estimate unmeasured disturbances, thereby providing a reduction in the controller sensitivity to those disturbances. Another
contribution of this work is the application of feedback linearization techniques to two novel biotechnological processes. The first is a mixed-culture bioreactor in which coexistence steady states
of the two cell populations must be stabilized. These steady states are unstable in the open-loop system since each population competes for the same substrate, and each has a different growth rate.
The requirement of a pulsatile manipulated input complicates the controller design. The second process is a bioreactor described by a distributed parameter model in which undesired oscillations must
be damped without the use of distributed control.
Recommended Citation
Kurtz, Michael James, "Feedback Linearizing Control Strategies for Chemical Engineering Applications." (1997). LSU Historical Dissertations and Theses. 6576. | {"url":"https://repository.lsu.edu/gradschool_disstheses/6576/","timestamp":"2024-11-12T00:44:58Z","content_type":"text/html","content_length":"40622","record_id":"<urn:uuid:d4cebe3f-85c9-4d08-ace2-adec9b3314ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00687.warc.gz"} |
9 class circle
Ex 7 circle 8:23 PM
Thursday, December 3, 2020
1a) A 16 cm long chord is drawn in a circle l having radius 10 cm .What is the distance of the
chord from the Centre?
Length of chord AB= 16
AC= perpendicular drawn from the centre
of circle to chord bisect the chord
Radius(OC)= 10 cm
using pythagorous theorem
b) A 24 cm long chord is drawn in a circle having diameter r 26 cm . What is the distance of the
chord from the center of the circle?
Length of chord PQ= perpendicular drawn from the centre
AC= of circle to chord bisect the chord
Radius(OR)= [ diameter = 2 radius]
using pythagorous theorem
c) Find the length of a chord , which is at a distance of 6 cm from the center of circle of radius
10 cm
Radius(OC)= 10 cm
using pythagorous theorem
AC= perpendicular drawn from the centre
of circle to chord bisect the chord
5 Geometry Page 1
d) the radius of a circle is 13 cm and the length of chord is 24cm find the distance of the chord
form the center
Length of chord AB= 24
AC= perpendicular drawn from the centre
of circle to chord bisect the chord
Radius(OC)= 13 cm
using pythagorous theorem
2a) in the given figure , o is the center of a circle > if
Solution: perpendicular drawn from the centre
Length of chord MN= of circle to chord bisect the chord
using pythagorous theorem
diameter =
b) In the given figure O is the center of a circle . if Find
the length of diameter
Length of chord AB= 16
AM= perpendicular drawn from the centre
of circle to chord bisect the chord
using pythagorous theorem
diameter =
5 Geometry Page 2
3a) in the given figure QR is the diameter of a circle with cente O . If OP=13 cm and PR=
10 find the length of PQ
Given :
To find :
statements Reasons
using pythagorous
b) in the given figure ,MN is the diameter of a circle with center O .If QM =24 c m and QN=
32 cm find the length of OQ
solution Reasons
Given :
To find :
Or 2x+2y=
using pythagorous
5 Geometry Page 3
4) a) In the figure along side O is the center of the circle of radius 13 cm .A chord MN is at a
distance of 5 cm from the center . Find the area of the
in right triangle OPM
Using Pythagoras theorem
MP perpendicular drawn from the centre
of circle to chord bisect the chord
Area of OM=5 cm and AB=8 cm . Find the area of
4b) in the given figure, O is the centre
in right triangle OBD
Using Pythagoras theorem
BD perpendicular drawn from the centre
of circle to chord bisect the chord
Area of
5a) In the adjoining circle radius is 10 cm if chord AB= chord CD=16 cm
and What is the meaure of OP?
Given :
To prove :
AX= the
5 Geometry Page 4
To prove :
AX= the
in right angle triangle AOX
OX=OY equal chord are equidistance from the
OXPY is a square
OX=XD= 6 being the sides of the
using pythagorus theorem
Q6 same as example 5 and 6
7a) in the given figure ,O is the cente of the circle. If AC=BD then proce that
Given : Reason s
to prove : supplement of equal angle
construction : given
1. OC=OD
4 in
corresponding sides of
Q 8 is similar to example 9
9) in the given figure PQ and RS are two chords of a circle with center O where PQ =16 cm RS=
30 cm and PQ//RS . If the lie MON of length 23 cm is perpendicular to both PA an RS , fin the
radius of the circle .
Join OP and OR
let OM=x then OR= 23-x
use Pythagoras theorem in
5 Geometry Page 5
5 Geometry Page 6 | {"url":"https://anyflip.com/hcjpe/zuqc/basic","timestamp":"2024-11-14T11:59:33Z","content_type":"text/html","content_length":"34042","record_id":"<urn:uuid:ff03921a-ff88-4caa-bd7a-581691a014c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00202.warc.gz"} |
What is Weight? Definition and Examples
What is weight? Definition and examples
What is weight? Weight, also called force of gravity, is how heavy an object is taking into account the location of the object.
For example, the weight of an object on the moon is less than the weight of an object on the earth.
Even here on earth, the weight of an object may vary. You will weigh slightly less at the top of Mount Everest than you would at sea level.
In reality, when you step onto a scale, the scale will return your mass, not your weight. However, people usually refer to the mass as weight.
How to find the weight of an object
One way to find the weight or the force of gravity (F[g]) is to multiply the mass of object by the gravity.
If weight = F[g] = w, then, Fg = w = mg
m is measured in kg
F[g] is measured in Newtons or N
Suppose you step onto the scale and you see 75 kg. What is your weight?
Since you are measuring your weight on the earth, g = 9,807 m/s²
Then, your weight is 75(9.807) = 735.525 N
If you are measuring your weight on the Moon instead,
g = 1,62 m/s²
Then, your weight is 75(1.62) = 121.5 N | {"url":"https://www.math-dictionary.com/what-is-weight.html","timestamp":"2024-11-14T01:07:33Z","content_type":"text/html","content_length":"24434","record_id":"<urn:uuid:3a613dea-0232-4013-8425-04ba07a2d34b>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00779.warc.gz"} |
?(Term structure of interest rates?) You want to invest your
savings of ?$26,000 in government securities...
?(Term structure of interest rates?) You want to invest your savings of ?$26,000 in government securities...
?(Term structure of interest rates?) You want to invest your savings of ?$26,000 in government securities for the next 2 years.? Currently, you can invest either in a security that pays interest of
8.1 percent per year for the next 2 years or in a security that matures in 1 year but pays only 6.1 percent interest. If you make the latter? choice, you would then reinvest your savings at the end
of the first year for another year.
a. Why might you choose to make the investment in the 1?-year security that pays an interest rate of only 6.1 ?percent, as opposed to investing in the 2-year security paying 8.1 ?percent? Provide
numerical support for your answer. Which theory of term structure have you supported in your? answer?
b. Assume your required rate of return on the? second-year investment is 11.1 ?percent; otherwise, you will choose to go with the 2?-year security. What rationale could you offer for your?
The 2 year rate is given as 8.1% and 1 year rate is 6.1%. As per the pure expectations theory, the two year rate should essentially be 1 year rate today plus expected rate from Year 1 to 2, as below:
(1+r[2])^2 = (1+r[1]) * (1+r[1,2]). Using this equation, we find the expected Year 1 one year rate :
(1+r[1,2]) = (1+8.1%)^2/(1+6.1%) = 10.14%
Thus an investor might choose to invest into 1 year rate in the expectation for the Year 1 to 2 rate to be higher. However if the requires rate for second year is 11.1% then rationally investor
should lock into 2 year rate today unless the investor believes that the rates 1 year hence should be higher than as predicted / expected by the above equation . | {"url":"https://justaaa.com/finance/13628-term-structure-of-interest-rates-you-want-to","timestamp":"2024-11-05T03:57:06Z","content_type":"text/html","content_length":"42096","record_id":"<urn:uuid:c85cb1b7-d8e2-4de7-bff5-7827634cb45b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00611.warc.gz"} |
All right. Now let's discuss another ratio, the inventory turnover ratio. So, the amount of cost of goods sold to average inventory levels. Right? We'll discuss this in a little more detail, but the
inventory turnover ratio, this is a really common efficiency ratio that you study in this class. Okay? So, we're trying to see how efficiently we use our inventory. Remember, when we have inventory,
it costs us money to store whatever boxes. If we sell t-shirts, we've got boxes of t-shirts sitting in a warehouse. Well, that warehouse costs us money, right? So we want to manage that as well as
So, let's look at how we calculate this ratio right here. Inventory turnover ratio, well, it equals cost of goods sold divided by average inventory. Okay? In a lot of ratios, we use this average
balance idea. Okay? In the average balance, we're always going to calculate it the same way. We're going to take the beginning balance in that account, plus the ending balance in that account. So
we're going to do that first. We're going to add those together and then we're going to divide by 2. Just like you see up here, right? The average inventory, that's the beginning inventory plus the
ending inventory divided by 2, right? So that's how we're always going to calculate the average.
Now, sometimes when they give you the inventory turnover ratio, they might just give you one number for inventory. They may not say beginning inventory and ending inventory. Well, if they just give
you one number, well, that's going to be your average. That's just going to be what you use for inventory.
In this case, cost of goods sold over average inventory. That's our inventory turnover. COGSavg inventory Okay? So what does this really tell us? It tells us how many times we turned our inventory
into COGS. So if you imagine, we're going to have some average level of inventory. Say there's $10,000 worth of t-shirts that sit in our warehouse at any given time. Right? But that $10,000 of
t-shirts, well, it doesn't just sit there—that's always there. Right? We're selling t-shirts, we're buying more t-shirts. So, it's kind of a cycle. Right? If you think about it, we're going to have
some number for cost of goods sold and it's related to inventory because every time we make a sale, right, it comes out of inventory. The cost of goods sold come out of inventory.
We're going to have this average balance that we keep in our inventory, but we're going to be selling it. So, we would expect that we would turn that inventory into COGS. Hopefully, a few times,
right? It'd be more efficient the lower amount of inventory we can keep and just sell COGS and COGS and just be making lots of sales, right? So the idea here is how efficiently are we using those
inventory levels?
Think about it. Wouldn't it be most efficient if we didn't have to hold any t-shirts at all? If our inventory was 0, but we were able to just make tons and tons of sales. So, every time we needed a
t-shirt, instead of going to our warehouse and having all these costs to maintain a warehouse, well that t-shirt just went straight to the customer from the factory that prints them or whatever it
is. Well, that would be very efficient because we don't have any warehouse cost at that point. But it's still efficient to keep a very manageable amount of inventory. Maybe we can keep just the right
amount of inventory, just a small amount of inventory that we can keep turning without ever running out, right? If we keep too little inventory, maybe we run into the problem that we get orders from
customers and we can't fill them because we don't have enough inventory. So, we want to be able to manage that inventory properly so that we were able to keep up with our sales, but also maintain our
warehouse cost at a reasonable level.
So that's what the inventory turnover is trying to tell us here. Okay? So how do we compare our inventory turnover? Like a lot of ratios, we use benchmarking. We want to be able to compare our
inventory turnover to other companies in our industry. How well are we managing our inventory compared to our competitors, right? If it takes us huge warehouses to manage our t-shirt business and
we've got to have thousands of boxes of t-shirts just sitting there costing us money and there's another company that, hey, they just print it and send it to the customer right away. Well, they're
going to have a lot more efficient management of their inventory, right? So we want to benchmark against our competitors and see if we're turning our inventory faster, right? So what we want is high
inventory turnovers. Higher inventory turnover ratios means we're being more efficient with our inventory, right? So that's basically how we analyze inventory turnover.
Why don't we go ahead and move into this example and see how we calculate an inventory turnover ratio? Let's do it right now. So, this first question right here, XYZ company had net sales of 500,000
and COGS of 320,000. If the beginning balance of inventory was 60,000 and the ending balance of inventory was 100,000, what is the inventory turnover ratio?
Notice they talk to us about net sales in this question and this is a classic way that they try and trip you up with inventory turnover as they talk about net sales. But remember when we're doing
inventory turnover, when we sell inventory, well, our sales revenue is related to the cash we receive from customers, right? The selling price of the good. But inventory and COGS, that relates to
what we paid for the goods. So inventory is related to COGS where sales revenue is more related to the cash we receive from customer, the selling price of the good.
So they're generally going to try and trick you like this and throw in a net sales number. But remember, inventory turnover, we want to use COGS here. Okay? So that's our good number.
COGS,320,000avg inventory, that's going to be our numerator. What about our denominator, right? We have to calculate that average inventory balance. So, in this case, they gave us two numbers, right?
They gave us a beginning and ending balance, so we have to calculate an average.
Let's do our average inventory first, right. So, it gave us a beginning balance of inventory was 60,000. Our ending balance of inventory was 100,000. Well, the average inventory 60,000+100,0002 is
going to give us 80,000 as our average inventory balance. So, we're not done yet. Now, we've got to actually calculate our inventory turnover.
So inventory turnover, remember that's going to be COGSavg inventory. We've got those numbers ready. We've got our COGS of 320,000. We've got our average inventory of 80,000. So, we've got an
inventory turnover of 4 in this case, right? Four times. So that means during the year, we were able to turn our average inventory balance, this 80,000 that's usually sitting in our warehouse, we
turned it into $320,000 worth of sales. We were able to turn that average inventory into 4 times the amount of sales in COGS. Right?
Hopefully, that makes sense to you right there and that is our answer here. So, 4.0, that is our inventory turnover. Turnover ratios are usually a little more complicated to understand the logic, but
if you think about it like that, right, how many times we turn that average inventory, what we have sitting in our warehouse, how many times we're able to flip that and turn it into COGS, right? Make
sales and turn that inventory, get it out of the warehouse. So, basically everything that was sitting in the warehouse was able to move out of the warehouse 4 times during the year. Cool? Let's go
ahead and move on to this practice problem where you guys get to try and solve inventory turnover. | {"url":"https://www.pearson.com/channels/financial-accounting/learn/brian/ch-14-financial-statement-analysis/ratios-inventory-turnover?chapterId=3c880bdc","timestamp":"2024-11-15T03:40:19Z","content_type":"text/html","content_length":"340676","record_id":"<urn:uuid:4fa63232-9b29-4051-ac7f-77d921091c91>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00015.warc.gz"} |
Lana earned $49on friday $42 on saturday,and $21 on sunday selling bracelets she sold each bracelet for the same amount what is the most she could have charged for eac bracelet?
1. Home
2. General
3. Lana earned $49on friday $42 on saturday,and $21 on sunday selling bracelets she sold each bracelet... | {"url":"http://thibaultlanxade.com/general/lana-earned-49on-friday-42-on-saturday-and-21-on-sunday-selling-bracelets-she-sold-each-bracelet-for-the-same-amount-what-is-the-most-she-could-have-charged-for-eac-bracelet","timestamp":"2024-11-08T11:34:39Z","content_type":"text/html","content_length":"29912","record_id":"<urn:uuid:15eb3b71-4e26-44c1-a862-263c7101eab8>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00865.warc.gz"} |
Borrowing Limits | GLIF Docs
Before jumping into the limits of the Pool, it's important to understand three metrics: loan-to-value (LTV), debt-to-equity (DTE) and debt-to-income (DTI). The DTE and DTI ratios are used to limit
the amount SPs can borrow from the Pool.
Loan-to-value ratio
The loan-to-value ratio - measures the amount of borrowed funds against the SP's liquidation value.
$LTV = borrowedFunds/liquidationValue$
For example:
Borrowed Funds Liquidation Value LTV
Debt-to-equity ratio
The debt-to-equity ratio - otherwise known as the "skin in the game" ratio - measures the amount of borrowed funds against the SP's equity value.
$DTE = borrowedFunds / equityValue$
For example:
Borrowed Funds Equity Value Debt-to-equity
Debt-to-income ratio
The debt-to-income ratio measures the expected weekly payment against the expected weekly earnings of the SP.
For example:
Borrow limits
The Pool enforces two borrowing limits:
LTV < 80% - the SP can only borrow an amount less than or equal to 80% of their liquidation value.
DTE < 200% - the SP can only borrow an amount less than or equal to 2x the amount of equity they bring to the protocol. As the SP earns more equity through block rewards, it can continuously
borrow more.
DTI < 25% - the SP can only borrow an amount such that the weekly expected payment can be covered by 25% of the SP's weekly expected earnings.
Some examples of accepted/rejected borrow requests:
The GLIF Website will be updated with an SP calculator, which will show each SP how much they can borrow.
The Borrow Cycle
The Pool is meant to grow with the SP. As the SP borrows funds and pledges them to the Filecoin network, their expected earnings increase, allowing them to borrow more. As the SP earns block rewards,
its equity value increases, increasing the SP's borrow cap. | {"url":"https://docs.glif.io/v1-english/storage-provider-economics/borrowing-limits","timestamp":"2024-11-10T12:56:47Z","content_type":"text/html","content_length":"376632","record_id":"<urn:uuid:fe7b0593-0aeb-46ed-9d99-65f24100368a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00698.warc.gz"} |
Two upper bounds on the A_α-spectral radius of a connected graph
[1] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer, New York, 2012.
[2] Y. Chen, D. Li, Z. Wang, and J. Meng, Aα-spectral radius of the second power of a graph, Appl. Math. Comput. 359 (2019), 418–425.
[3] D. Li, Y. Chen, and J. Meng, The Aα-spectral radius of trees and unicyclic graphs with given degree sequence, Appl. Math. Comput. 363 (2019), Article ID: 124622.
[4] H. Lin, X. Huang, and J. Xue, A note on the Aα-spectral radius of graphs, Linear Algebra Appl. 557 (2018), 430–437.
[5] T.S. Motzkin and E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán, Canad. J. Math. 17 (1965), 533–540.
[6] V. Nikiforov, Merging the A and Q spectral theories, Appl. Analysis Discrete Math. 11 (2017), no. 1, 81–107.
[7] S. Pirzada, An Introduction to Graph Theory, Universities Press Orient BlackSwan, Hyderabad, 2012.
[8] S. Wang, D. Wong, and F. Tian, Bounds for the largest and the smallest Aαeigenvalues of a graph in terms of vertex degrees, Linear Algebra Appl. 590 (2020), 210–223. | {"url":"https://comb-opt.azaruniv.ac.ir/article_14178.html","timestamp":"2024-11-03T16:25:16Z","content_type":"text/html","content_length":"45263","record_id":"<urn:uuid:214a953b-76e8-4d2a-a870-c3bf824395c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00068.warc.gz"} |
Metric Measures | Metric System of Measurement | The Metric System
Metric Measures
We will discuss here about the metric measures of length, mass and capacity.
Measurements of length, mass and capacity:
We know about the different units for measuring length, mass and capacity. We also know about the relationship between the units.
Let us revise them.
Length is measured in kilometre, metre and centimetre.
Mass is measured in kilogram and gram.
Capacity is measured in litre and millilitre.
You know that
1 metre = 100 centimetres
1 kilometre = 1000 metres
1 kilogram = 1000 grams
1 litre = 1000 millilitres
The basic unit of length is metre, the basic unit of mass is kilogram and the basic unit of capacity is litre. This system of measurement of length, mass and capacity is known as Metric system.
In metric system, we represent thousands by kilo, hundreds by hector, tens by deca, tenths by deci, hundredths by centi and thousandths by milli. This system is decimal since each unit is 10 times
the next smaller unit.
As we know, the standard units of length, weight and capacity are metre (m), gram (g) and litre (l) respectively. Kilometre (km) kilogram (kg) and kilolitre (kl) are the bigger units of length,
weight and capacity respectively. The three-basic unit systems of measurement of length, mass and capacity is known as metric system.
This is decimal system based on the multiples of 10.
Measures of Length
10 millimetres (mm) = 1 centimetre (cm)
10 centimetres = 1 decimetre (dm)
10 decimetres = 1 metre (m)
10 metres = 1 decametre (dam)
10 decametres = 1 hectometre (hm)
10 hectometres = 1 kilometre (km)
Measures of Weight
10 milligrams (mg) = 1 centigram (cg)
10 centigrams = 1 decigram (dg)
10 decigrams = 1 gram (g)
10 grams = 1 decagram (dag)
10 decagrams = 1 hectogram (hg)
10 hectograms = 1 kilogram (kg)
100 kg = 1 quintal
10 quintals = 1 tonne
1000 kg = 1 tonne
Measures of Capacity
10 millilitres (ml) = 1 centilitre (cl)
10 centilitres = 1 decilitre (dl)
10 decilitres = 1 litre (l)
10 litres = 1 decalitre (dal)
10 decalitres = 1 hectolitre (hl)
10 hectolitres = 1 kilolitre (kl)
We can understand the metric system more clearly through the following table of metric measures.
Thousand Hundred Ten One Tenth Hundredth Thousandth
1000 100 10 1 1/10 1/100 1/1000
Metric Kilo Hecto Deca Basic Deci Centi Milli
Unit (x1000) (x100) (x10) Unit (÷ 10) (÷ 100) (÷ 100)
Kilo- Hecto- Deca- Deci- Centi- Milli-
Mass Gram
gram gram gram gram gram gram
Kilo- Hecto- Deca- Deci- Centi- Milli-
Length Metre
metre metre metre metre metre metre
Kilo- Hecto- Deca- Deci- Centi- Milli-
Capacity Litre
litre litre litre litre litre litre
Measurement of length, mass and capacity tables can be represented in the form of a place-value chart as shown below.
From the above table of metric system we come to know that:
Deca means 10 times
Hecto means 100 times
Kilo means 1000 times
These are the bigger units of metric system
Deci means \(\frac{1}{10}\) times
Centi means \(\frac{1}{100}\) times
Millimeans \(\frac{1}{1000}\) times
These are the smaller units of metric system
Questions and Answers on Metric Measures:
I. Convert 1629 grams into
(i) kilograms
(ii) hectograms
(iii) decagrams
I. (i) 1.629 kilograms
(ii) 16.29 hectograms
(iii) 162.9 decagrams
II. Convert 4525 millilitres into
(i) litres
(ii) decilitres
(iii) centilitres
II. (i) 4.525 litres
(ii) 45.25 decilitres
(iii) 452.5 centilitres
III. Convert 4020 metres into
(i) kilometres
(ii) hectometres
(iii) decametres
III. (i) 4.02 kilometres
(ii) 40.2 hectometres
(iii) 402 decametres
IV. Convert 1756 kilograms into
(i) quintals
(ii) tonnes
IV. Convert 1756 kilograms into
(i) quintals
(ii) tonnes
V. Convert 726 quintals into tonnes
VI. Convert 125 kilometres into
(i) hectometres
(ii) decametres
(iii) metres
VII. Convert 75 kilograms into
(i) decigrams
(ii) centigrams
(iii) milligrams
VIII. Convert 120 litres into
(i) kilolitres
(ii) decalitres
(iii) decilitres
(iv) millilitres
IX. Convert 7 tonnes into
(i) kilograms
(ii) quintals
X. Convert 825 quintals into kilograms.
XI. Fill in the blanks:
(i) 5 kg 600 g = 4 kg ……… g
(ii) 3 hm 20 m = ……… m
(iii) 6 hl 4 dal = ……… l
(iv) 5 m 6 dm = ……… cm
(v) 1 km 2 hm 3 dam 2 m = ……… m
From Metric Measures to HOME PAGE
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FindRoot help with precision and accuracy
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11 Replies
1 Total Likes
FindRoot help with precision and accuracy
I'm solving a non-linear system of equations with FindRoot. The system is very sparse, if I can say that about a non-linear system. After a few seconds of computation, Mathematica outputs the answer
and gives me the error
FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal
and PrecisionGoal but was unable to find a sufficient decrease in the merit function.
You may need more than MachinePrecision digits of working precision to meet these tolerances. >>
I have played around with DampingFactor, WorkingPrecision and AccuracyGoal to no avail: When I verify the solutions that Mathematica gives me, at best about 1/3 of them are not satisfied. Can you
help me figure out what should I be looking at?
11 Replies
I did not do a very complete search, as the DifficultyFactor, which determines the number of function evaluations, was set to a low value. I was trying to illustrate that minimizing the sum of
squares of the equations is a viable approach. Do you know what the bounds on the variables are? I based the bounds I used (0 and 200) on the starting values you set up for FindRoot.
Thanks Frank, what does that mean? The system has no solution (at least with the restriction that the values must lie in [0,200])? If you think about it, it must have a solution given that it can be
viewed as a system of recursive equations with terminal conditions. Using backwards substitution, one should be able to reach to the solution.
Using MathOptimizer Professional http://www.wolfram.com/products/applications/mathoptpro/ to do a global search, I was able to reduce the sum of squares error from 1.88 * 10^7 to 1.99 * 10^6. I
assumed the variables have lower bounds of 0 and upper bounds of 200.
In[31]:= Dimensions[
varrule = init /. ( {var_, initval_} -> var -> initval)]
Out[31]= {1368}
In[34]:= equations.equations /. varrule // N
Out[34]= 1.88301*10^7
In[35]:= Needs["MathOptimizerProLF`callLGO`"]
In[36]:= initbnds = init /. {var_, val_} -> {var, 0, val, 200};
In[41]:= AbsoluteTiming[
mosln = callLGO[equations.equations, {}, initbnds,
DifficultyFactor -> 0.001];]
Out[41]= {22.9501, Null}
In[40]:= First @mosln
Out[40]= 1.98812*10^6
I don't think this is a precision or accuracy issue. I think that FindRoot, starting at the initial values, cannot improve the solution in its first step, so it is returning the starting values.
Given the complexity and number of equations, very good starting values are probably a necessity.
I previously tried using FindMinimum on the sum of squares like you suggested. But it takes way longer with a simplified version of the model, I have not tried it with the full version but presumably
would take even longer. I am wondering if compiling the system of equations would be faster. I thought Mathematica did that automatically for functions like FindMinimum.
Also, if I use machine-precision numbers with FindMinimum, do you think I would get the same precision error as before? As you know, machine-precision numbers allow the computation to run faster.
Like I said, it's probably better to form a sum of squares of differences and use FindMinimum. Getting a good approximate root with FindRoot might be too hard, given the complexity of the system of
Hi guys,
I avoided using inexact numbers an still get the same error. Can you please take a look at the attached notebook?
Looking at the last part of the notebook, I suggest that you calculate the constraint violations to test the result, rather than checking if they are within your defined tolerance. That may give a
clue as to what is going on.
You might do better to form a sum of squares and use FindMinimum or NMinimize. Also read what I wrote about using higher WorkingPrecision: it requires higher precision input (easy in this case; make
every constant exact instead of machine number).
Thanks Daniel. I have attached a notebook with the system. I increased the precision to 30 with the same results. Hopefully my code is not too unreadable for you.
If the input has machine doubles, maybe change them to high precision using SetPrecision. Other than that, without the actual system it's difficult to guess what might be at issue.
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/488913","timestamp":"2024-11-07T04:35:33Z","content_type":"text/html","content_length":"147785","record_id":"<urn:uuid:c8d37699-ac49-40e8-ba1a-7d136c03165d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00424.warc.gz"} |
Private Inexpensive Norm Enforcement (PINE) VDAF
This document is an Internet-Draft (I-D). Anyone may submit an I-D to the IETF. This I-D is
not endorsed by the IETF
and has
no formal standing
in the
IETF standards process
Document Type Active Internet-Draft (individual)
Authors Junye Chen , Christopher Patton
Last updated 2024-09-27
RFC stream (None)
Intended RFC status (None)
Stream Stream state (No stream defined)
Consensus boilerplate Unknown
RFC Editor Note (None)
IESG IESG state I-D Exists
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Send notices to (None)
Crypto Forum J. Chen
Internet-Draft Apple Inc.
Intended status: Informational C. Patton
Expires: 29 March 2025 Cloudflare
25 September 2024
Private Inexpensive Norm Enforcement (PINE) VDAF
This document describes PINE, a Verifiable Distributed Aggregation
Function (VDAF) for secure aggregation of high-dimensional, real-
valued vectors with bounded L2-norm. PINE is intended to facilitate
private and robust federated machine learning.
About This Document
This note is to be removed before publishing as an RFC.
The latest revision of this draft can be found at
cfrg-vdaf-pine.html. Status information for this document may be
found at https://datatracker.ietf.org/doc/draft-chen-cfrg-vdaf-pine/.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (mailto:cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg. Subscribe
at https://www.ietf.org/mailman/listinfo/cfrg/.
Source for this draft and an issue tracker can be found at
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
Chen & Patton Expires 29 March 2025 [Page 1]
Internet-Draft Private Inexpensive Norm Enforcement (PI September 2024
This Internet-Draft will expire on 29 March 2025.
Copyright Notice
Copyright (c) 2024 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights
and restrictions with respect to this document.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 5
3. PINE Overview . . . . . . . . . . . . . . . . . . . . . . . . 6
4. The PINE Proof System . . . . . . . . . . . . . . . . . . . . 7
4.1. Measurement Encoding . . . . . . . . . . . . . . . . . . 9
4.1.1. Encoding Range-Checked Results . . . . . . . . . . . 9
4.1.2. Encoding Gradient and L2-Norm Check . . . . . . . . . 9
4.1.3. Running the Wraparound Checks . . . . . . . . . . . . 10
4.1.4. Encoding the Range-Checked, Wraparound Check
Results . . . . . . . . . . . . . . . . . . . . . . . 10
4.2. The FLP Circuit . . . . . . . . . . . . . . . . . . . . . 11
4.2.1. Range Check . . . . . . . . . . . . . . . . . . . . . 12
4.2.2. Bit Check . . . . . . . . . . . . . . . . . . . . . . 12
4.2.3. L2 Norm Check . . . . . . . . . . . . . . . . . . . . 12
4.2.4. Wraparound Check . . . . . . . . . . . . . . . . . . 13
4.2.5. Putting All Checks Together . . . . . . . . . . . . . 13
5. The PINE VDAF . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 15
5.3. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 15
5.4. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 16
6. Variants . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7. PINE Auxiliary Functions . . . . . . . . . . . . . . . . . . 16
8. Security Considerations . . . . . . . . . . . . . . . . . . . 16
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 16
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 16
10.1. Normative References . . . . . . . . . . . . . . . . . . 16
10.2. Informative References . . . . . . . . . . . . . . . . . 16
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 17
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
Chen & Patton Expires 29 March 2025 [Page 2]
Internet-Draft Private Inexpensive Norm Enforcement (PI September 2024
1. Introduction
The goal of federated machine learning [MR17] is to enable training
of machine learning models from data stored on users' devices. The
bulk of the computation is carried out on-device: each user trains
the model on its data locally, then sends a model update to a central
server. These model updates are commonly referred to as "gradients"
[Lem12]. The server aggregates the gradients, applies them to the
central model, and sends the updated model to the users to repeat the
+---------+ gradients +--------+
| Clients |-+ -----------------------------------> | Server |
+---------+ |-+ +--------+
+---------+ | |
+---------+ |
^ |
| updated model |
Figure 1: Federated learning
Federated learning improves user privacy by ensuring the training
data never leaves users' devices. However, it requires computing an
aggregate of the gradients sent from devices, which may still reveal
a significant amount of information about the underlying data. One
way to mitigate this risk is to distribute the aggregation step
across multiple servers such that no server sees any gradient in the
With a Verifiable Distributed Aggregation Function [VDAF], this is
achieved by having each user shard their gradient into a number of
secret shares, one for each aggregation server. Each server
aggregates their shares locally, then combines their share of the
aggregate with the other servers to get the aggregate result.
Chen & Patton Expires 29 March 2025 [Page 3]
Internet-Draft Private Inexpensive Norm Enforcement (PI September 2024
v gradient aggregate
+---------+ shares +-------------+ shares +-----------+
| Clients |-+ ----> | Aggregators |-+ -----> | Collector |
+---------+ |-+ +-------------+ | +-----------+
+---------+ | +-------------+ |
+---------+ |
^ |
| updated model |
Figure 2: Federated learning with a VDAF
Along with keeping the gradients private, it is desirable to ensure
robustness of the overall computation by preventing clients from
"poisoning" the aggregate and corrupting the trained machine learning
model. A client's gradient is typically expressed as a vector of
real numbers. A common goal is to ensure each gradient has a bounded
"L2-norm" (sometimes called Euclidean norm): the square root of the
sum of the squares of each entry of the input vector. Bounding the
L2 norm is used in federated learning to limit the contribution of
each client to the aggregate, without over constraining the
distribution of inputs. [CP: Add a relevant reference.]
In theory, Prio3 (Section 7 of [VDAF]) could be adapted to support
this functionality, but for high-dimensional data, the concrete cost
in terms of runtime and communication would be prohibitively high.
The basic idea is simple. An FLP ("Fully Linear Proof", see
Section 7.3 of [VDAF]) could be used to compute the L2 norm of the
secret shared gradient and check that the result is in the desired
range, all without learning the gradient or its norm. This
computation, on its own, can be done efficiently: the challenge lies
in ensuring that the computation itself was carried out correctly,
while properly accounting for the relevant mathematical details of
the proof system and the range of possible inputs.
This document describes PINE ("Private Inexpensive Norm
Enforcement"), a VDAF for secure aggregation of gradients with
bounded L2-norm [ROCT23]. Its design is based largely on Prio3 in
that the norm is computed and verified using an FLP. However, PINE
uses a new technique for verifying the correctness of the norm
computation that is incompatible with Prio3.
We give an overview of this technique in Section 3. In Section 4 we
specify an FLP circuit and accompanying encoding scheme for computing
and verifying the L2 norm of each gradient. Finally, in Section 5 we
specify the complete multi-party, 1-round VDAF.
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NOTE As of this draft, the algorithms are not yet fully specified.
We are still working out some of the minor details. In the
meantime, please refer to the reference code on which the spec
will be based: https://github.com/junyechen1996/draft-chen-cfrg-
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
This document uses the same parameters and conventions specified for:
* Clients, Aggregators, and Collectors from Section 5 of [VDAF].
* Finite fields from Section 6.1 of [VDAF]. All fields in this
document have prime order.
* XOFs ("eXtendable Output Functions") from Section 6.2 of [VDAF].
A floating point number, denoted float, is a IEEE-754 compatible
float64 value [IEEE754-2019].
A "gradient" is a vector of floating point numbers. Each coordinate
of this vector is called an "entry". The "L2 norm", or simply
"norm", of a gradient is the square root of the sum of the squares of
its entries.
The "dot product" of two vectors is to compute the sum of element-
wise multiplications of the two vectors.
The user-specified parameters to initialize PINE are defined in
Table 1.
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| Parameter | Type | Description |
| l2_norm_bound | float | The L2 norm upper bound |
| | | (inclusive). |
| dimension | int | Dimension of each gradient. |
| num_frac_bits | int | The number of bits of precision |
| | | to use when encoding each |
| | | gradient entry into the field. |
Table 1: User parameters for PINE.
3. PINE Overview
This section provides an overview of the main technical contribution
of [ROCT23] that forms the basis of PINE. To motivate their idea,
let us first say how Prio3 from Section 7 of [VDAF] would be used to
aggregate vectors with bounded L2 norm.
Prio3 uses an FLP ("Fully Linear Proof"; see Section 7.3 of [VDAF])
to verify properties of a secret shared measurement without revealing
the measurement to the Aggregators. The property to be verified is
expressed as an arithmetic circuit over a finite field (Section 7.3.2
of [VDAF]). Let q denote the field modulus.
In our case, the circuit would take (a share of) the gradient as
input, compute the squared L2-norm (the sum of the squares of the
entries of the gradient), and check that the result is in the desired
range. Note that we do not compute the exact norm: it is
mathematically equivalent to compute the squared norm and check that
it is smaller than the square of the bound.
Crucially, arithmetic in this computation is modulo q. This means
that, for a given gradient, the norm may have a different result when
computed in our finite field than in the ring of integers. For
example, suppose our bound is 10: the gradient [99, 0, 7] has squared
L2-norm of 9850 over the integers (out of range), but only 6 modulo q
= 23 (in range). This circuit would therefore deem the gradient
valid, when in fact it is invalid.
Thus the central challenge of adapting FLPs to this problem is to
prevent the norm computation from "wrapping around" the field
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One way to achieve this is to ensure that each gradient entry is in a
range that ensures the norm is sufficiently small. However, this
approach has high communication cost (roughly num_frac_bits *
dimension field elements per entry), which becomes prohibitive for
high-dimensional data.
PINE uses a different strategy: rather than prevent wraparounds, we
can try to detect whether a wraparound has occurred.
[ROCT23] devises a probabilistic test for this purpose. A random
vector over the field is generated (via a procedure described in
Section 4.1.3) where each entry is sampled independently from a
particular probability distribution. To test for wraparound, compute
the dot product of this vector and the gradient, and check if the
result is in a specific range determined by parameters in Table 1.
If the norm wraps around the field modulus, then the dot product is
likely to be large. In fact, [ROCT23] show that this test correctly
detects wraparounds with probability 1/2. To decrease the false
negative probability (that is, the probability of misclassifying an
invalid gradient as valid), we simply repeat this test a number of
times, each time with a vector sampled from the same distribution.
However, [ROCT23] also show that each wraparound test has a non-zero
false positive probability (the probability of misclassifying a valid
gradient as invalid). We refer to this probability as the "zero-
knowledge error", or in short, "ZK error". This creates a problem
for privacy, as the Aggregators learn information about a valid
gradient they were not meant to learn: whether its dot product with a
known vector is in a particular range. [CP: We need a more intuitive
explanation of the information that's leaked.] The parameters of
PINE are chosen carefully in order to ensure this leakage is
4. The PINE Proof System
This section specifies a randomized encoding of gradients and FLP
circuit (Section 7.3 of [VDAF]) for checking that (1) the gradient's
squared L2-norm falls in the desired range and (2) the squared
L2-norm does not wrap around the field modulus. We specify the
encoding and validity circuit in a class PineValid.
The encoding algorithm takes as input the gradient and an XOF seed
used to derive the random vectors for the wraparound tests. The seed
must be known both to the Client and the Aggregators: Section 5
describes how the seed is derived from shares of the gradient.
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Operational parameters for the proof system are summarized below in
Table 2.
| Parameter | Type | Description |
| alpha | float | Parameter in wraparound |
| | | check that determines the |
| | | ZK error. The higher alpha |
| | | is, the lower ZK error is. |
| num_wr_checks | int | Number of wraparound checks |
| | | to run. |
| num_wr_successes | int | Minimum number of |
| | | wraparound checks that a |
| | | Client must pass. |
| sq_norm_bound | Field | The square of l2_norm_bound |
| | | encoded into a field |
| | | element. |
| wr_check_bound | Field | The bound of the range |
| | | check for each wraparound |
| | | check. |
| num_bits_for_sq_norm | int | Number of bits to encode |
| | | the squared L2-norm. |
| num_bits_for_wr_check | int | Number of bits to encode |
| | | the range check in each |
| | | wraparound check. |
| bit_checked_len | int | Number of field elements in |
| | | the encoded measurement |
| | | that are expected to be |
| | | bits. |
| chunk_length | int | Parameter of the FLP. |
Table 2: Operational parameters of the PINE FLP.
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4.1. Measurement Encoding
The measurement encoding is done in two stages: * Section 4.1.2
involves encoding floating point numbers in the Client gradient into
field elements Section 4.1.2.1, and encoding the results for L2-norm
check Section 4.1.2.2, by computing the bit representation of the
squared L2-norm, modulo q, of the encoded gradient. The result of
this step allows Aggregators to check the squared L2-norm of the
Client's gradient, modulo q, falls in the desired range of [0,
sq_norm_bound]. * Section 4.1.4 involves encoding the results of
running wraparound checks Section 4.1.3, based on the encoded
gradient from the previous step, and the random vectors derived from
a short, random seed using an XOF. The result of this step, along
with the encoded gradient and the random vector that the Aggregators
derive on their own, allow the Aggregators to run wraparound checks
on their own.
4.1.1. Encoding Range-Checked Results
Encoding range-checked results is a common subroutine during
measurement encoding. The goal is to allow the Client to prove a
value is in the desired range of [B1, B2], over the field modulus q
(see Figure 1 in [ROCT23]). The Client computes the "v bits", the
bit representation of value - B1 (modulo q), and the "u bits", the
bit representation of B2 - value (modulo q). The number of bits for
the v and u bits is ceil(log2(B2 - B1 + 1)).
As an optimization for communication cost per Remark 3.2 in [ROCT23],
the Client can skip sending the u bits if B2 - B1 + 1 (modulo q) is a
power of 2. This is because the available v bits can naturally bound
value - B1 to be B2 - B1.
4.1.2. Encoding Gradient and L2-Norm Check
We define a function PineValid.encode_gradient_and_norm(self,
measurement: list[float]) -> list[Field] that implements this
encoding step.
4.1.2.1. Encoding of Floating Point Numbers into Field Elements
TODO Specify how floating point numbers are represented as field
4.1.2.2. Encoding the Range-Checked, Squared Norm
TODO Specify how the Client encodes the norm such that the
Aggregators can check that it is in the desired range.
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TODO Put full implementation of encode_gradient_and_norm() here.
4.1.3. Running the Wraparound Checks
Given the encoded gradient from Section 4.1.2 and the XOF to generate
the random vectors, the Client needs to run the wraparound check
num_wr_checks times. Each wraparound check works as follows.
The Client generates a random vector with the same dimension as the
gradient's dimension. Each entry of the random vector is a field
element of 1 with probability 1/4, or 0 with probability 1/2, or q-1
with probability 1/4, over the field modulus q. The Client samples
each entry by sampling from the XOF output stream two bits at a time:
* If the bits are 00, set the entry to be q-1. * If the bits are 01
or 10, set the entry to be 0. * If the bits are 11, set the entry to
be 1.
Finally, the Client computes the dot product of the encoded gradient
and the random vector, modulo q.
Note the Client does not send this dot product to the Aggregators.
The Aggregators will compute the dot product themselves, based on the
encoded gradient and the random vector derived on their own.
4.1.4. Encoding the Range-Checked, Wraparound Check Results
We define a function PineValid.encode_wr_checks(self,
encoded_gradient: list[Field], wr_joint_rand_xof: Xof) ->
tuple[list[Field], list[Field]] that implements this encoding step.
It returns the tuple of range-checked, wraparound check results that
will be sent to the Aggregators, and the wraparound check results
(i.e., the dot products from Section 4.1.3) that will be passed as
inputs to the FLP circuit.
The Client obtains the wraparound check results, as described in
Section 4.1.3. For each check, the Client runs the range check on
the result to see if it is in the range of [-wr_check_bound + 1,
wr_check_bound]. Note we choose wr_check_bound, such that
wr_check_bound is a power of 2, so the Client does not have to send
the u bits in range check. The Client also keeps track of a success
bit wr_check_g, which is a 1 if the wraparound check result is in
range, and 0 otherwise.
The Client counts how many wraparound checks it has passed. If it
has passed fewer than num_wr_successes of them, it should retry, by
using a new XOF seed to re-generate the random vectors and re-run
wraparound checks Section 4.1.3.
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The range-checked results and the success bits are sent to the
Aggregators, and the wraparound check results are passed to the FLP
4.2. The FLP Circuit
Evaluation of the validity circuit begins by unpacking the encoded
measurement into the following components:
* The first dimension entries are the encoded_gradient, the field
elements encoded from the floating point numbers.
* The next bit_checked_len entries are expected to be bits, and
should contain the bits for the range-checked L2-norm, the bits
for the range-checked wraparound check results, and the success
bits in wraparound checks.
* The last num_wr_checks are the wraparound check results, i.e., the
dot products of the encoded gradient and the random vectors.
It also unpacks the "joint randomness" that is shared between the
Client and Aggregators, to compute random linear combinations of all
the checks:
* The first joint randomness field element is to reduce over the bit
checks at all bit entries.
* The second joint randomness field element is to reduce over all
the quadratic checks in wraparound check.
* The last joint randomness field element is to reduce over all the
checks, which include the reduced bit check result, the L2 norm
equality check, the L2 norm range check, the reduced quadratic
checks in wraparound check, and the success count check for
wraparound check.
In the subsections below, we outline the various checks computed by
the validity circuit, which includes the bit check on all the bit
entries Section 4.2.2, the L2 norm check Section 4.2.3, and the
wraparound check Section 4.2.4. Some of the auxiliary functions in
these checks are defined in Section 7.
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4.2.1. Range Check
In order to verify the range-checked results reported by the Client
as described in Section 4.1.1, the Aggregators will verify (1) v bits
and u bits are indeed composed of bits, as described in
Section 4.2.2, and (2) the verify the decoded value from the v bits,
and the decoded value from the u bits sum up to B2 - B1 (modulo q).
If the Client skips sending the u bits as an optimization mentioned
in Section 4.1.4, then the Aggregators only need to verify the
decoded value from the v bits is equal to B2 - B1 (modulo q).
4.2.2. Bit Check
The purpose of bit check on a field element is to prevent any
computation involving the field element from going out of range. For
example, if we were to compute the squared L2-norm from the bit
representation claimed by the Client, bit check ensures the decoded
value from the bit representation is at most 2^(num_bits_for_norm) -
To run bit check on an array of field elements bit_checked, we use a
similar approach as Section 7.3.1.1 of [VDAF], by constructing a
polynomial from a random linear combination of the polynomial x(x-1)
evaluated at each element of bit_checked. We then evaluate the
polynomial at a random point r_bit_check, i.e., the joint randomness
for bit check:
f(r_bit_check) = bit_checked[0] * (bit_checked[0] - 1) + \
r_bit_check * bit_checked[1] * (bit_checked[1] - 1) + \
r_bit_check^2 * bit_checked[2] * (bit_checked[2] - 1) + ...
If one of the entries in bit_checked is not a bit, then
f(r_bit_check) is non-zero with high probability.
TODO Put PineValid.eval_bit_check() implementation here.
4.2.3. L2 Norm Check
The purpose of L2 norm check is to check the squared L2-norm of the
encoded gradient is in the range of [0, sq_norm_bound].
The validity circuit verifies two properties of the L2 norm reported
by the Client:
* Equality check: The squared norm computed from the encoded
gradient is equal to the bit representation reported by the
Client. For this, the Aggregators compute their shares of the
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squared norm from their shares of the encoded gradient, and also
decode their shares of the bit representation of the squared norm
(as defined above in Section 4.2.2), and check that the values are
* Range check: The squared norm reported by the Client is in the
desired range [0, sq_norm_bound]. For this, the Aggregators run
the range check described in Section 4.2.1.
TODO Put PineValid.eval_norm_check() implementation here.
4.2.4. Wraparound Check
The purpose of wraparound check is to check the squared L2-norm of
the encoded Client gradient hasn't overflown the field modulus q.
The validity circuit verifies two properties for wraparound checks:
* Quadratic check (See bullet point 3 in Figure 2 of [ROCT23]):
Recall in Section 4.1.4, the Client keeps track of a success bit
for each wraparound check, i.e., whether it has passed that check.
For each check, the Aggregators then verify a quadratic constraint
that, either the success bit is a 0 (i.e., the Client has failed
that check), or the success bit is a 1, and the range-checked
result reported by the Client is correct, based on the wraparound
check result (i.e., the dot product) computed by the Aggregators
from the encoded gradient and the random vector. For this, the
Aggregators multiply their shares of the success bit, and the
difference of the range-checked result reported by the Client, and
that computed by the Aggregators. We then construct a polynomial
from a random linear combination of the quadratic check at each
wraparound check, and evaluate it at a random point r_wr_check,
the joint randomness.
* Success count check: The number of successful wraparound checks,
by summing the success bits, is equal to the constant
num_wr_successes. For this, the Aggregators sum their shares of
the success bits for all wraparound checks.
TODO Put PineValid.eval_wr_check() implementation here.
4.2.5. Putting All Checks Together
Finally, we will construct a polynomial from a random linear
combination of all the checks from PineValid.eval_bit_checks(),
PineValid.eval_norm_check(), and PineValid.eval_wr_check(), and
evaluate it at the final joint randomness r_final. The full
implementation of PineValid.eval() is as follows:
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TODO Specify the implementation of Valid from Section 7.3.2 of
5. The PINE VDAF
This section describes PINE VDAF for [ROCT23], a one-round VDAF with
no aggregation parameter. It takes a set of Client gradients
expressed as vectors of floating point values, and computes an
element-wise summation of valid gradients with bounded L2-norm
configured by the user parameters in Table 1. The VDAF largely uses
the encoding and validation schemes in Section 4, and also specifies
how the joint randomness shared between the Client and Aggregators is
derived. There are two kinds of joint randomness used:
* "Verification joint randomness": These are the field elements used
by the Client and Aggregators to evaluate the FLP circuit. The
verification joint randomness is derived similar to the joint
randomness in Prio3 Section 7.2.1.2 of [VDAF]: the XOF is applied
to each secret share of the encoded measurement to derive the
"part"; and the parts are hashed together, using the XOF once
more, to get the seed for deriving the joint randomness itself.
* "Wraparound joint randomness": This is used to generate the random
vectors in the wraparound checks that both the Clients and
Aggregators need to derive on their own. It is generated in much
the same way as the verification joint randomness, except that
only the gradient and the range-checked norm are used to derive
the parts.
In order for the Client to shard its gradient into input shares for
the Aggregators, the Client first encodes its gradient into field
elements, and encodes the range-checked L2-norm, according to
Section 4.1.2. Next, it derives the wraparound joint randomness for
the wraparound checks as described above, and uses that to encode the
range-checked, wraparound check results as described in
Section 4.1.4}. The encoded gradient, range-checked norm, and range-
checked wraparound check results will be secret-shared to (1) be sent
as input shares for the Aggregators, and (2) derive the verification
joint randomness as described above. The Client then generates the
proof with the FLP and secret shares it. The secret-shared proof,
along with the input shares, and the joint randomness parts for both
wraparound and verification joint randomness, are sent to the
Then the Aggregators carry out a multi-party computation to obtain
the output shares (the secret shares of the encoded Client gradient),
and also reject Client gradients that have invalid L2-norm. Each
Aggregator first needs to derive wraparound and verification joint
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randomness. Similar to Prio3 preparation Section 7.2.2 of [VDAF],
the Aggregator does not derive every joint randomness part like the
Client does. It only derives the joint randomness part from its
secret share via the XOF, and applies its part and and other
Aggregators' parts sent by the Client to the XOF to obtain the joint
randomness seed. Then each Aggregator runs the wraparound checks
Section 4.1.3 with its share of encoded gradient and the wraparound
joint randomness, and queries the FLP with its input share, proof
share, the wraparound check results, and the verification joint
randomness. All Aggregators then exchange the results from the FLP
and decide whether to accept that Client gradient.
Next, each Aggregator sums up their shares of the encoded gradients
and sends the aggregate share to the Collector. Finally, the
Collector sums up the aggregate shares to obtain the aggregate
result, and decodes it into an array of floating point values.
Like Prio3 Section 7.1.2 of [VDAF], PINE supports generation and
verification of multiple FLPs. The goal is to improve robustness of
PINE (Corollary 3.13 in [ROCT23]) by generating multiple unique
proofs from the Client, and only accepting the Client gradient if all
proofs have been verified by the Aggregators. The benefit is that
one can improve the communication cost between Clients and
Aggregators, by instantiating PINE FLP with a smaller field, but
repeating the proof generation (Flp.prove) and validation (Flp.query)
multiple times.
The remainder of this section is structured as follows. We will
specify the exact algorithms for Client sharding Section 5.1,
Aggregator preparation Section 5.2 and aggregation Section 5.3, and
Collector unsharding Section 5.4.
5.1. Sharding
TODO Specify the implementation of Vdaf.shard().
5.2. Preparation
TODO Specify the implementations of Vdaf.prep_init(),
.prep_shares_to_prep(), and .prep_next().
5.3. Aggregation
TODO Specify the implementation of Vdaf.aggregate().
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5.4. Unsharding
TODO Specify the implementation of Vdaf.unshard().
6. Variants
TODO Specify concrete parameterizations of VDAFs, including the
choice of field, number of proofs, and valid ranges for the
parameters in Table 1.
7. PINE Auxiliary Functions
TODO Put all auxiliary functions here, including range_check(),
8. Security Considerations
Our security considerations for PINE are the same as those for Prio3
described in Section 9 of [VDAF].
9. IANA Considerations
TODO Ask IANA to allocate an algorithm ID from the VDAF algorithm
ID registry.
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[VDAF] Barnes, R., Cook, D., Patton, C., and P. Schoppmann,
"Verifiable Distributed Aggregation Functions", Work in
Progress, Internet-Draft, draft-irtf-cfrg-vdaf-11, 22
August 2024, <https://datatracker.ietf.org/doc/html/draft-
10.2. Informative References
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[BBCGGI19] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and
Y. Ishai, "Zero-Knowledge Proofs on Secret-Shared Data via
Fully Linear PCPs", CRYPTO 2019 , 2019,
"IEEE Standard for Floating-Point Arithmetic", 2019,
[Lem12] Lemaréchal, C., "Cauchy and the gradient method", 2012,
[MR17] McMahan, B. and D. Ramage, "Federated Learning:
Collaborative Machine Learning without Centralized
Training Data", 2017, <https://ai.googleblog.com/2017/04/
[ROCT23] Rothblum, G. N., Omri, E., Chen, J., and K. Talwar, "PINE:
Efficient Norm-Bound Verification for Secret-Shared
Vectors", 2023, <https://arxiv.org/abs/2311.10237>.
[Tal22] Talwar, K., "Differential Secrecy for Distributed Data and
Applications to Robust Differentially Secure Vector
Summation", 2022, <https://arxiv.org/abs/2202.10618>.
Guy Rothblum Apple Inc. gn_rothblum@apple.com
Kunal Talwar Apple Inc. ktalwar@apple.com
Authors' Addresses
Junye Chen
Apple Inc.
Email: junyec@apple.com
Christopher Patton
Email: chrispatton+ietf@gmail.com
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Basics and types of Polynomial Equations
Basics of Polynomial Equations
1. Different types of Polynomial Equations
We already know that, for any non–negative integer n , a polynomial of degree n in one variable
x is an expression given by
P ≡ P(x)= a[n] xn + a[n][-1] xn-1 +...+ a[1] x + a[0] ……….(1)
where a[r] ∈ C are constants, r = 0,1, 2,K, n with a[n] ≠ 0 . The variable x is real or complex.
When all the coefficients of a polynomial P are real, we say “P is a polynomial over R ”. Similarly we use terminologies like “P is a polynomial over C ”, “P is a polynomial over Q”, and P is a
polynomial over Z ”
The function P defined by P ( x) = a[n] x^n + a[n][−1] x^n^−1 +...+ a[1]x + a[0] is called a polynomial function.
The equation
an xn + a[n][-1] x^n^-1 +...+ a[1] x + a[0] = 0 ……….(2)
is called a polynomial equation.
If a[n]c^n + a[n][−1]c^n^−1 +...+ a[1]c + a[0] = 0 for some c ∈ C , then c is called a zero of the polynomial (1) and root or solution of the polynomial equation (2).
If c is a root of an equation in one variable x, we write it as“ x = c is a root”. The constants a[r] are called coefficients. The coefficient a[n] is called the leading coefficient and the term a[n]
x^n is called the leading term. The coefficients may be any number, real or complex. The only restriction we made is that the leading coefficient a[n] is nonzero. A polynomial with the leading
coefficient 1 is called a monic polynomial.
We note the following:
· Polynomial functions are defined for all values of x .
· Every nonzero constant is a polynomial of degree 0 .
· The constant 0 is also a polynomial called the zero polynomial; its degree is not defined.
· The degree of a polynomial is a nonnegative integer.
· The zero polynomial is the only polynomial with leading coefficient 0 .
· Polynomials of degree two are called quadratic polynomials.
· Polynomials of degree three are called cubic polynomials.
· Polynomial of degree four are called quartic polynomials.
It is customary to write polynomials in descending powers of x . That is, we write polynomials having the term of highest power (leading term) as the first term and the constant term as the last
For instance, 2x +3y +4z= 5 and 6x^2 + 7x^2 y^3 + 8z =9 are equations in three variables x , y , z ; x^2- 4x + 5 =0 is an equation in one variable x. In the earlier classes we have solved
trigonometric equations, system of linear equations, and some polynomial equations.
We know that 3 is a zero of the polynomial x^2 − 5x + 6 and 3 is a root or solution of the equation x^2 − 5x + 6 = 0 . We note that cos x = sin x and cos x + sin x = 1 are also equations in one
variable x.
However, cos x − sin x and cos x + sin x −1 are not polynomials and hence cos x = sin x and cos x + sin x = 1 are not “polynomial equations”. We are going to consider only “polynomial equations” and
equations which can be solved using polynomial equations in one variable.
We recall that sin^2 x + cos^2 x = 1 is an identity on R , while sin x + cos x = 1 and sin^2 x + cos^2 x = 1 are equations.
It is important to note that the coefficients of a polynomial can be real or complex numbers, but the exponents must be nonnegative integers. For instance, the expressions 3x^−^2 +1 and 5x^1/2 +1 are
not polynomials. We already learnt about polynomials and polynomial equations, particularly about quadratic equations. In this section let us quickly recall them and see some more concepts.
2. Quadratic Equations
For the quadratic equation ax^2 + bx + c =0, b^2 - 4ac is called the discriminant and it is usually denoted by Δ. We know that
are roots of the ax^2 + bx + c = 0 . The two roots together are usually written as
It is unnecessary to emphasize that a ≠ 0 , since by saying that ax^2 + bx + c is a quadratic polynomial, it is implied that a ≠ 0.
• ∆ > 0 if, and only if, the roots are real and distinct
• ∆ < 0 if, and only if, the quadratic equation has no real roots. | {"url":"https://www.brainkart.com/article/Basics-and-types-of-Polynomial-Equations_39114/","timestamp":"2024-11-13T20:55:02Z","content_type":"text/html","content_length":"93092","record_id":"<urn:uuid:bd11a3bf-e29b-445d-946a-e0dda73eb6f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00356.warc.gz"} |
Extended Dyalog APL
Extended Dyalog APL features extended domains of existing primitives and quad names and adds a few new ones to Dyalog APL. It was an experimental project that is no longer maintained. Quoting from
project README.md:
This project serves as a breeding ground for ideas. While some have been adopted into Dyalog APL proper, it is unlikely that many will be. Furthermore, Dyalog 18.0 gave a different meaning to
monadic ≠ than proposed here, leaving Extended Dyalog APL as a deadend.
The role of Extended Dyalog APL as breeding ground for ideas was followed Dyalog APL Vision, also by Adám Brudzewsky.
The following extensions were made:
Name Glyph Type* Extension
Back Slash \ 🔶 ∘.fwhen dyadic, allows short and/or multiple left args
Back Slash Bar ⍀ 🔶 ⊢∘fwhen dyadic, allows short and/or multiple left args
Bullet ∙ 🔺 Inner product and Alternant
Circle Diaeresis ⍥ 🔺 Over and Depth
Circle Jot ⌾ 🔺 Complex/Imaginary
Del Diaeresis ⍢ 🔺 Under (a.k.a. Dual)
Del Tilde ⍫ 🔺 Obverse;⍺⍺but with inverse⍵⍵
Diaeresis ¨ 🔵 allows constant operand
Divide ÷ 🔵 monadic converts letters to title case when possible
Dollar Sign $ 🔺 string enhancement ${1}:1⊃⍺, ${expr}:⍎expr,\n:JSON
Down Arrow ↓ 🔵 allows long⍺
Down Shoe ∪ 🔵 allows rank>1
Downstile ⌊ 🔵 monadic lowercases letters
Down Tack ⊤ 🔶 2s as default left argument
Ellipsis … 🔺 fill sequence gaps (dfns workspace'sto⍤1
Epsilon Underbar ⍷ 🔶 monadic is Type∊with⎕ML←0
Equals = 🔶 with TAO; monad: is-type
Greater Than > 🔶 with TAO; monad: is-strictly-negative/is-visible
Greater Than Or Equal To ≥ 🔶 with TAO; monad: is non-positive/is-not-control-character
house ⌂ 🔺 prefix for contents of dfns workspace
infinity ∞ 🔺 largest integer (for use with⍤and⍣)
Iota ⍳ 🔵 Unicode version of dfns workspace's iotag
Iota Underbar ⍸ 🔵 allows duplicates/non-Booleans
Iota Underbar Inverse ⍸⍣¯1 🔵 givenr, findsnso thatr≡⍸n
Jot Diaeresis ⍤ 🔵 allows constant left operand, Atop with function right operand
Jot Underbar ⍛ 🔺 reverse compositionX f⍛g Yis(f X) g Y
Left Shoe ⊂ 🔵 allows partitioning along multiple trailing axes, with short ⍺s, and inserting/appending empty partitions
Left Shoe Stile ⍧ 🔺 monad: nub-sieve; dyad: count-in
Left Shoe With Axis ⊂[k] 🔵 as⊂, but called with left operand
Less Than < 🔶 with TAO; monad: is-strictly-positive/is-control-character
Less Than Or Equal To ≤ 🔶 with TAO ; monad: is-non-negative/is-invisible
Minus - 🔵 monadic flips letter case
macron ¯ 🔵 as prefix to name or primitive means its inverse
negative Infinity ¯∞ 🔺 smallest integer (for use with⍣)
Nand ⍲ 🔶 monad: not all equal to type
Nor ⍱ 🔶 monad: not any equal to type
Not Equal To ≠ 🔶 with TAO; monad: is-non-type
Percent % 🔺 f%andA%: probability-logical function (mapping arrays)
Quad Diamond ⌺ 🔶 auto-extended⍵⍵, allows small⍵, optional edge spec(s) (0:Zero; 1:Repl; 2:Rev; 3:Mirror; 4:Wrap; -:Twist) with masks as operand's⍺
Question Mark ? 🔵 ⍺?¯⍵as norm dist stddev ⍵and optional mean⍺←0
Rho ⍴ 🔵 allows omitting one dimension length with¯1
Right Shoe Underbar ⊇ 🔺 monadic discloses if scalar, dyadic indexes sanely
Right Shoe Underbar With Axis ⊇[k] 🔺 as above, but called with left operand
Root √ 🔺 (Square) Root
Semicolon Underbar ⍮ 🔺 (Half) Pair; use↑⍤⍮to add axis
Slash / 🔵 allows short and/or multiple left args
Slash Bar ⌿ 🔵 allows short and/or multiple left args
Star Diaeresis ⍣ 🔵 allows non-scalar right operand incl.∞and¯∞and array left operand
Stile | 🔵 monadic normalises letters to lowercase (upper then lower)
Stile Tilde ⍭ 🔺 monadic is factors; dyadic depends on⍺: 0=non-prime?, 1=prime?, ¯1=primes less than⍵, ¯2=⍵th prime, 4=next prime, ¯4=prev prime
Tilde ~ 🔵 monadic allows probabilities, dyadic allows rank>1
Tilde Diaeresis ⍨ 🔵 allows constant operand
Times × 🔵 set/query letter case (lower:¯1, title:0, upper:1)
Up Arrow ↑ 🔵 allows long⍺
Up Shoe ∩ 🔶 monadic is self-classify; dyadic allows rank>1
Upstile ⌈ 🔵 monadic uppercases letters
Up Tack ⊥ 🔶 2 as default left argument
Vel ∨ 🔶 monadic is Descending Sort
Wedge ∧ 🔶 monadic is Ascending Sort
Case Convert ⎕C 🔺 fn ⎕Capplies case-insensitively,array ⎕Ccase-folds
Error Message ⎕EM 🔺 Self-inverse⎕EM
Namespace ⎕NS 🔵 allows⎕NS names values(tries to resolve⎕ORs)
Namespace inverse ⎕NS⍣¯1 🔺 allows(names values)←⎕NS⍣¯1⊢ns(returns⎕ORs for ns/fns)
Unicode Convert ⎕UCS 🔵 scalar when monadic
*🔺 means new feature🔶 means added valence🔵 means expanded domain | {"url":"https://aplwiki.com/index.php?title=Extended_Dyalog_APL&mobileaction=toggle_view_mobile","timestamp":"2024-11-12T13:33:51Z","content_type":"text/html","content_length":"47886","record_id":"<urn:uuid:62d1299b-84d1-4a25-a930-4d7700485d75>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00621.warc.gz"} |
LAMINATED SPRINGS CLASS NOTES FOR MECHANICAL ENGINEERING - ME Subjects - Concepts Simplified
This type of spring is very common in use.
There is an advantage over the coiled
spring. It acts as a structural member. In
addition it absorbs a shock. Used in cars,
trucks, buses and trains. This type of spring
has number of plates of constant
width and thickness. However, the length
of each leaf is different. These plates are
placed over one another and then
connected by a U-bolt in the center. The
plates are lamination’s. Thus, it is
called a laminated spring. There are two
types of laminated springs.
(i) Semi-elliptic type
Fig. Semi-elliptical Spring
Semi-elliptic type spring is simply supported at the ends. Bolts connects two eye ends to the vehicle body. In this case, load acts at the center. It is considered in simple bending.
(ii) Quarter elliptic type
Fig. Quarter Elliptical Spring
It is fixed at one. It acta as a cantilever. Load acts at the free end of the cantilever. These lamination’s have initial curvature. The load which can straighten these lamination’s is proof load. It
is the maximum load, this type of spring can sustain. It absorbs the shocks due to unevenness of the road. Load acting on this spring tends to decrease the depth or try to straighten the initially
curved plates. Hence design this type of spring in pure bending. Since maximum bending moment acts at the center, the other lamination’s decrease in length. Where a quarter elliptic acts as a
cantilever, maximum bending moment acts at the fixed end. Thus, the maximum number of lamination’s are at the fixed end.
MATERIALS USED IN LEAF SPRINGS
Yield stress Endurance Limit
Sr. No. Material
N/mm^2 N/mm^2
Spring carbon steel
1. 1170 670
Chrome Vanadium Steels
2. SAE -6140 1350 700
Silicon Manganese Steels
3. SAE-9250
Commonly used plain carbon steels are with 0.9 to 1 % of carbon properly heat treated. Use Chrome-Vanadium and Silicon-Manganese steels for better performance. These alloy steels do not have strength
more than that of carbon steels. But these have greater toughness and a higher endurance limit. These materials suit to springs subjected to rapidly fluctuating loads.
Semi-Elliptical Type
This spring is a beam of uniform strength. Load acts at the center. It is supported at the ends. Spring has number of lamination’s. One or two are of full length. Remaining plates reduce in length
successively. Reduced length plates are truncated leaves. These are of constant width and constant thickness. Bending moment is different at different locations. Therefore, depth of the spring
varies. Hence the section modulus varies.
Calculate the followings:
(a) Uniform stress
AT the center, bending moment is M = WL/4
AT the center, section modulus is n bt^2/6
Uniform stress is = M/Z= (WL/4)/ n bt^2/6= 6WL/4nbt^2
σ =3WL/2nbt^2
(b) Overlap
OVERLAP= a = (L/2) / n L/2n
(c) Number of plates
n = (L/2)/a= L/2a
(d) Central deflection
For an initially curved beam, deflection is
δ = ML^2 / 8EI = (WL/4)L^2/[8E(1/12) nbt^3]
δ = 3WL^3/8Enbt^3
(e) Radius of curvature
σ/y = E/R
R = (Ex t/2)/σ= Et/2σ also = L^2/8δ’
Where δ’ = R –(R^2—L^2/4)^0.5
Where R is the initial radius of curvature
Δ’ is corresponding of proof load
Proof load is that load which can straighten the initially curved plate
(f) Strain energy and resilience
u = U/V= σ^2/6E
We know U = (1/2) W δ= (W/2)( 3WL^3/8Enbt^3)
U = [(3/2) WL/nbt^2 ]^2 (n b t L/12E) = (σ^2/6E)x( volume of the spring)
Therefore resilience u = σ^2/6E
(g) Length of leaves
Full length = L
Firstly length of second leaf = L – 2a
Secondly length of third leaf= L-4a
Thirdly length of fourth leaf = L-6a
And so on
In the above equations, assume originally straight spring. Plates are relatively thin and free to take lateral strains. Spring plates have initial curvature. Load acting at the center tries to
straighten the curved plates.
https://www.mesubjects.net/wp-admin/post.php?post=7602&action=edit Salient features of springs | {"url":"https://mesubjects.net/laminated-springs/","timestamp":"2024-11-11T07:49:45Z","content_type":"text/html","content_length":"104053","record_id":"<urn:uuid:9d61f14e-24c5-4647-a6e4-07625d773c52>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00488.warc.gz"} |
Coupled allpass IIR filter
The dsp.CoupledAllpassFilter object implements a coupled allpass filter structure composed of two allpass filters connected in parallel. Each allpass branch can contain multiple sections. The overall
filter output is computed by adding the output of the two respective branches. An optional second output can also be returned, which is power complementary to the first. For example, from the
frequency domain perspective, if the first output implements a lowpass filter, the second output implements the power complementary highpass filter. For real signals, the power complementary output
is computed by subtracting the output of the second branch from the first. dsp.CoupledAllpassFilter supports double- and single-precision floating point and allows you to choose between different
realization structures. This System object™ also supports complex coefficients, multichannel variable length input, and tunable filter coefficient values.
To filter each channel of the input:
1. Create the dsp.CoupledAllpassFilter object and set its properties.
2. Call the object with arguments, as if it were a function.
To learn more about how System objects work, see What Are System Objects?
caf = dsp.CoupledAllpassFilter returns a coupled allpass filter System object, caf, that filters each channel of the input signal independently. The coupled allpass filter uses the default inner
structures and coefficients.
caf = dsp.CoupledAllpassFilter(AllpassCoeffs1,AllpassCoeffs2) returns a coupled allpass filter System object, caf, with Structure set to 'Minimum multiplier', AllpassCoefficients1 set to
AllpassCoeffs1, and AllpassCoefficients2 set to AllpassCoeffs2.
caf = dsp.CoupledAllpassFilter(struc,AllpassCoeffs1,AllpassCoeffs2) returns a coupled allpass filter System object, caf, with Structure set to struc and the relevant coefficients set to
AllpassCoeffs1 and AllpassCoeffs2. struc can be 'Minimum multiplier' | 'Wave Digital Filter' | 'Lattice'.
caf = dsp.CoupledAllpassFilter(Name,Value) returns a Coupled allpass filter System object, caf, with each property set to the specified value.
Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
Structure — Internal structure of allpass branches
'Minimum multiplier' (default) | 'Wave Digital Filter' | 'Lattice'
Specify the internal structure of allpass branches as one of 'Minimum multiplier', 'Wave Digital Filter', or 'Lattice'. Each structure uses a different pair of coefficient values, independently
stored in the relevant object property.
AllpassCoefficients1 — Allpass polynomial coefficients of branch 1
[0 0.5] (default) | row vector | cell array
Specify the polynomial filter coefficients for the first allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as many
cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Minimum multiplier'.
Data Types: single | double
WDFCoefficients1 — Wave Digital Filter coefficients of branch 1
[0.5 0] (default) | row vector | cell array
Specify the Wave Digital Filter coefficients for the first allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as
many cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Wave Digital Filter'.
Data Types: single | double
LatticeCoefficients1 — Lattice coefficients of branch 1
[0.5 0] (default) | row vector | cell array
Specify the allpass lattice coefficients for the first allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as many
cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Lattice'.
Data Types: single | double
Delay — Length in samples for branch 1
0 (default) | positive integer scalar
Integer number of the delay taps in the top branch, specified as a positive integer scalar.
Tunable: Yes
This property is applicable only if you set the PureDelayBranch property to true.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Gain1 — Independent Branch 1 Phase Gain
1 (default) | '–1' | '0+i' | '0–i'
Gain1 is the individual branch phase gain. This property can accept only values equal to '1', '–1', '0+i', or '0–i'. This property is nontunable.
Data Types: char
AllpassCoefficients2 — Allpass polynomial coefficients of branch 2
{[ ]} (default) | row vector | cell array
Specify the polynomial filter coefficients for the second allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as
many cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Minimum multiplier'.
Data Types: single | double
WDFCoefficients2 — Wave Digital Filter coefficients of branch 2
{[ ]} (default) | row vector | cell array
Specify the Wave Digital Filter coefficients for the second allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as
many cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Wave Digital Filter'.
Data Types: single | double
LatticeCoefficients2 — Lattice coefficients of branch 2
{[ ]} (default) | row vector | cell array
Specify the allpass lattice coefficients for the second allpass branch. This property can accept values either in the form of a row vector (single-section configuration) or a cell array with as many
cells as filter sections.
Tunable: Yes
This property is applicable only if you set the Structure property to 'Lattice'.
Data Types: single | double
Gain2 — Independent Branch 2 Phase Gain
'1' (default) | '–1' | '0+1i' | '0–1i'
Specify the value of the independent phase gain applied to branch 2. This property can accept only values equal to '1', '–1', '0+i', or '0–i'. This property is nontunable.
Data Types: char
Beta — Coupled phase gain
1 (default) | complex value with magnitude equal to 1
Specify the value of the phasor gain in complex conjugate form, in each of the two branches, and in complex coefficient configuration. The absolute value of this property should be 1 and its default
value is 1.
Tunable: Yes
This property is applicable only when the selected Structure property supports complex coefficients.
Data Types: single | double
PureDelayBranch — Replace allpass filter in first branch with pure delay
false (default) | true
If you set PureDelayBranch to true, the property holding the coefficients for the first allpass branch is disabled and Delay becomes enabled. You can use this property to improve performance, when
one of the two allpass branches is known to be a pure delay (e.g. for halfband filter designs)
Data Types: logical
ComplexConjugateCoefficients — Allow inferring coefficients of second allpass branch as complex conjugate of first
false (default) | true
When the input signal is real, this property triggers the use of an optimized structural realization. This property enables providing complex coefficients for only the first branch. The coefficients
for the second branch are automatically inferred as complex conjugate values of the first branch coefficients
This property is only enabled if the currently selected structure supports complex coefficients. Use it only if the filter coefficients are actually complex.
Data Types: logical
y = caf(x) filters the input signal x to produce the output y. When x is a matrix, each column is filtered independently as a separate channel over time.
[y,ypc] = caf(x) also returns ypc, the power complementary signal to the primary output y.
Input Arguments
x — Data input
vector | matrix
Data input, specified as a vector or a matrix. This object also accepts variable-size inputs. Once the object is locked, you can change the size of each input channel, but you cannot change the
number of channels.
Data Types: single | double
Complex Number Support: Yes
Output Arguments
y — Lowpass filtered output
vector | matrix
Lowpass filtered output, returned as a vector or a matrix. The size and data type of the output signal matches that of the input signal.
Data Types: double | single
Complex Number Support: Yes
ypc — Highpass filtered output
vector | matrix
Power complimentary highpass filtered output, returned as a vector or a matrix. The size and data type of the output signal matches that of the input signal.
Data Types: double | single
Complex Number Support: Yes
Object Functions
To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:
Specific to dsp.CoupledAllpassFilter
getBranches Return internal allpass branches
freqz Frequency response of discrete-time filter System object
impz Impulse response of discrete-time filter System object
info Information about filter System object
coeffs Returns the filter System object coefficients in a structure
cost Estimate cost of implementing filter System object
grpdelay Group delay response of discrete-time filter System object
Common to All System Objects
step Run System object algorithm
release Release resources and allow changes to System object property values and input characteristics
reset Reset internal states of System object
Allpass Realization of a Butterworth Lowpass Filter
Realize a Butterworth lowpass filter of order 3. Use a coupled allpass structure with inner minimum multiplier structure.
Fs = 48000; % in Hz
Fc = 12000; % in Hz
frameLength = 1024;
[b, a] = butter(3,2*Fc/Fs);
AExp = [freqz(b,a,frameLength/2); NaN];
[c1, c2] = tf2ca(b,a);
caf = dsp.CoupledAllpassFilter(c1(2:end),c2(2:end));
tfe = dsp.TransferFunctionEstimator('FrequencyRange', 'onesided',...
'SpectralAverages', 2);
aplot = dsp.ArrayPlot('PlotType', 'Line',...
'YLimits', [-40 5],...
'YLabel', 'Magnitude (dB)',...
'SampleIncrement', Fs/frameLength,...
'XLabel', 'Frequency (Hz)',...
'Title', 'Magnitude Response',...
'ShowLegend', true,'ChannelNames',{'Actual','Expected'});
Niter = 200;
for k = 1:Niter
in = randn(frameLength,1);
out = caf(in);
A = tfe(in,out);
Allpass Realization of an Elliptic Highpass Filter
Remove a low-frequency sinusoid using an elliptic highpass filter design implemented through a coupled allpass structure.
Fs = 1000;
f1 = 50; f2 = 100;
Fpass = 70; Apass = 1;
Fstop = 60; Astop = 80;
filtSpecs = fdesign.highpass(Fstop,Fpass,Astop,Apass,Fs);
hpSpec = design(filtSpecs,'ellip','FilterStructure','cascadeallpass',...
frameLength = 1000;
nFrames = 100;
sine = dsp.SineWave('Frequency',[f1,f2],'SampleRate',Fs,...
'SamplesPerFrame',frameLength); % Input composed of two sinusoids.
sa = spectrumAnalyzer('SampleRate',Fs,...
'Title','Original (Channel 1) Filtered (Channel 2)',...
for k = 1:1000
original = sum(sine(),2); % Add the two sinusoids together
filtered = hpSpec(original);
View Power Complementary Output of Coupled Allpass Filter
Design a Butterworth lowpass filter of order 3. Use a coupled allpass structure with inner minimum multiplier structure.
Fs = 48000; % in Hz
Fc = 12000; % in Hz
frameLength = 1024;
[b,a] = butter(3,2*Fc/Fs);
AExp = [freqz(b,a,frameLength/2); NaN];
[c1,c2] = tf2ca(b,a);
caf = dsp.CoupledAllpassFilter(c1(2:end),c2(2:end));
Using the 'SubbandView' option of the dsp.CoupledAllpassFilter, you can visualize the lowpass filter output, the power complementary highpass filter output, or both using the fvtool.
To view the lowpass filter output, set 'SubbandView' to 1.
To view the highpass filter output, set 'SubbandView' to 2.
To view both the outputs, set 'SubbandView' to 'all', [1 2] or [1;2].
The following three figures summarize the main structures supported by dsp.CoupledAllpassFilter.
• Minimum Multiplier and WDF
• Lattice
• Lattice with Complex Conjugate Coefficients
[1] Regalia, Philip A., Mitra, Sanjit K., and P.P Vaidyanathan “ The Digital All-Pass Filter: A Versatile Signal Processing Building Block.” Proceedings of the IEEE 1988, Vol. 76, No. 1, pp. 19–37.
[2] Mitra, Sanjit K., and James F. Kaiser, "Handbook for Digital Signal Processing" New York: John Wiley & Sons, 1993.
Version History
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558 Sign/Square Day to Turn/Square Nanosecond
Sign/Square Day [sign/day2] Output
558 sign/square day in degree/square second is equal to 0.0000022424768518519
558 sign/square day in degree/square millisecond is equal to 2.2424768518519e-12
558 sign/square day in degree/square microsecond is equal to 2.2424768518519e-18
558 sign/square day in degree/square nanosecond is equal to 2.2424768518519e-24
558 sign/square day in degree/square minute is equal to 0.0080729166666667
558 sign/square day in degree/square hour is equal to 29.06
558 sign/square day in degree/square day is equal to 16740
558 sign/square day in degree/square week is equal to 820260
558 sign/square day in degree/square month is equal to 15508629.14
558 sign/square day in degree/square year is equal to 2233242596.25
558 sign/square day in radian/square second is equal to 3.9138604464572e-8
558 sign/square day in radian/square millisecond is equal to 3.9138604464572e-14
558 sign/square day in radian/square microsecond is equal to 3.9138604464572e-20
558 sign/square day in radian/square nanosecond is equal to 3.9138604464572e-26
558 sign/square day in radian/square minute is equal to 0.00014089897607246
558 sign/square day in radian/square hour is equal to 0.50723631386085
558 sign/square day in radian/square day is equal to 292.17
558 sign/square day in radian/square week is equal to 14316.24
558 sign/square day in radian/square month is equal to 270676.64
558 sign/square day in radian/square year is equal to 38977436.3
558 sign/square day in gradian/square second is equal to 0.0000024916409465021
558 sign/square day in gradian/square millisecond is equal to 2.4916409465021e-12
558 sign/square day in gradian/square microsecond is equal to 2.4916409465021e-18
558 sign/square day in gradian/square nanosecond is equal to 2.4916409465021e-24
558 sign/square day in gradian/square minute is equal to 0.0089699074074074
558 sign/square day in gradian/square hour is equal to 32.29
558 sign/square day in gradian/square day is equal to 18600
558 sign/square day in gradian/square week is equal to 911400
558 sign/square day in gradian/square month is equal to 17231810.16
558 sign/square day in gradian/square year is equal to 2481380662.5
558 sign/square day in arcmin/square second is equal to 0.00013454861111111
558 sign/square day in arcmin/square millisecond is equal to 1.3454861111111e-10
558 sign/square day in arcmin/square microsecond is equal to 1.3454861111111e-16
558 sign/square day in arcmin/square nanosecond is equal to 1.3454861111111e-22
558 sign/square day in arcmin/square minute is equal to 0.484375
558 sign/square day in arcmin/square hour is equal to 1743.75
558 sign/square day in arcmin/square day is equal to 1004400
558 sign/square day in arcmin/square week is equal to 49215600
558 sign/square day in arcmin/square month is equal to 930517748.44
558 sign/square day in arcmin/square year is equal to 133994555775
558 sign/square day in arcsec/square second is equal to 0.0080729166666667
558 sign/square day in arcsec/square millisecond is equal to 8.0729166666667e-9
558 sign/square day in arcsec/square microsecond is equal to 8.0729166666667e-15
558 sign/square day in arcsec/square nanosecond is equal to 8.0729166666667e-21
558 sign/square day in arcsec/square minute is equal to 29.06
558 sign/square day in arcsec/square hour is equal to 104625
558 sign/square day in arcsec/square day is equal to 60264000
558 sign/square day in arcsec/square week is equal to 2952936000
558 sign/square day in arcsec/square month is equal to 55831064906.25
558 sign/square day in arcsec/square year is equal to 8039673346500
558 sign/square day in sign/square second is equal to 7.4749228395062e-8
558 sign/square day in sign/square millisecond is equal to 7.4749228395062e-14
558 sign/square day in sign/square microsecond is equal to 7.4749228395062e-20
558 sign/square day in sign/square nanosecond is equal to 7.4749228395062e-26
558 sign/square day in sign/square minute is equal to 0.00026909722222222
558 sign/square day in sign/square hour is equal to 0.96875
558 sign/square day in sign/square week is equal to 27342
558 sign/square day in sign/square month is equal to 516954.3
558 sign/square day in sign/square year is equal to 74441419.88
558 sign/square day in turn/square second is equal to 6.2291023662551e-9
558 sign/square day in turn/square millisecond is equal to 6.2291023662551e-15
558 sign/square day in turn/square microsecond is equal to 6.2291023662551e-21
558 sign/square day in turn/square nanosecond is equal to 6.2291023662551e-27
558 sign/square day in turn/square minute is equal to 0.000022424768518519
558 sign/square day in turn/square hour is equal to 0.080729166666667
558 sign/square day in turn/square day is equal to 46.5
558 sign/square day in turn/square week is equal to 2278.5
558 sign/square day in turn/square month is equal to 43079.53
558 sign/square day in turn/square year is equal to 6203451.66
558 sign/square day in circle/square second is equal to 6.2291023662551e-9
558 sign/square day in circle/square millisecond is equal to 6.2291023662551e-15
558 sign/square day in circle/square microsecond is equal to 6.2291023662551e-21
558 sign/square day in circle/square nanosecond is equal to 6.2291023662551e-27
558 sign/square day in circle/square minute is equal to 0.000022424768518519
558 sign/square day in circle/square hour is equal to 0.080729166666667
558 sign/square day in circle/square day is equal to 46.5
558 sign/square day in circle/square week is equal to 2278.5
558 sign/square day in circle/square month is equal to 43079.53
558 sign/square day in circle/square year is equal to 6203451.66
558 sign/square day in mil/square second is equal to 0.000039866255144033
558 sign/square day in mil/square millisecond is equal to 3.9866255144033e-11
558 sign/square day in mil/square microsecond is equal to 3.9866255144033e-17
558 sign/square day in mil/square nanosecond is equal to 3.9866255144033e-23
558 sign/square day in mil/square minute is equal to 0.14351851851852
558 sign/square day in mil/square hour is equal to 516.67
558 sign/square day in mil/square day is equal to 297600
558 sign/square day in mil/square week is equal to 14582400
558 sign/square day in mil/square month is equal to 275708962.5
558 sign/square day in mil/square year is equal to 39702090600
558 sign/square day in revolution/square second is equal to 6.2291023662551e-9
558 sign/square day in revolution/square millisecond is equal to 6.2291023662551e-15
558 sign/square day in revolution/square microsecond is equal to 6.2291023662551e-21
558 sign/square day in revolution/square nanosecond is equal to 6.2291023662551e-27
558 sign/square day in revolution/square minute is equal to 0.000022424768518519
558 sign/square day in revolution/square hour is equal to 0.080729166666667
558 sign/square day in revolution/square day is equal to 46.5
558 sign/square day in revolution/square week is equal to 2278.5
558 sign/square day in revolution/square month is equal to 43079.53
558 sign/square day in revolution/square year is equal to 6203451.66 | {"url":"https://hextobinary.com/unit/angularacc/from/signpd2/to/turnpns2/558","timestamp":"2024-11-11T08:16:18Z","content_type":"text/html","content_length":"113101","record_id":"<urn:uuid:bb63fbb1-a6b5-4b6e-aa6f-3dc9dc0da698>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00826.warc.gz"} |
Increasing x - (AP Calculus AB/BC) - Vocab, Definition, Explanations | Fiveable
Increasing x
from class:
AP Calculus AB/BC
"Increasing x" refers to when the x-coordinate component (horizontal component) of a vector is getting larger as time progresses. In other words, the object is moving to the right on the x-axis.
"Increasing x" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/ap-calc/increasing-x","timestamp":"2024-11-04T16:44:09Z","content_type":"text/html","content_length":"241460","record_id":"<urn:uuid:d99ed81e-ab95-4034-910a-4b300288ad8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00776.warc.gz"} |
Problems on Train
In a kilometer race, A beats B by 40 meters or by 5 seconds. What is the time taken by A over the course?
(a) 1.3 mins
5 (b) 2 mins
(c) 1.5 mins
(d) 2.5 mins
Option (b)
A train travels from City X to City Y at a speed of 44 kmph, while on its return journey the train was travelling at speed of 77 kmph. Find the average speed of the train.
(a) 56
4 (b) 58.5
(c) 60
(d) 60.5
Option (a)
A train travelling at 60 kmph crosses a man in 6 seconds. What is the length of the train?
(a) 100
3 (b) 150
(c) 200
(d) 250
Option (a)
A train covers a distance of 12 km in 10 minutes. If it takes 6 seconds to pass a telegraph post, then the length of the train is:
(a) 90 m
2 (b) 120 m
(c) 100 m
(d) 140 m
Option (c)
A train 150 m long is running with a speed of 68 kmph. In what time will it pass a man who is running at 8 kmph in the same direction in which the train is going?
(a) 6 sec
1 (b) 9 sec
(c) 12 sec
(d) 10 sec
Option (b)
Pratik travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Then, find his average speed for the entire distance travelled?
(a) 18 kmph
(b) 13 kmph
(c) 14.25 kmph
(d) 16.75 kmph
Option (b)
A boat can travel with a speed of 13 km/hr in still water. If the speed of the stream is4 km/hr, find the time taken by the boat to go 68 km downstream.
(a) 2 hrs
7 (b) 4 hrs
(c) 3 hrs
(d) 5 hrs
Option (b)
A boatman goes 2 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water?
(a) 40 mins
8 (b) 1 hr
(c) 1 hr 15 mins
(d) 1 hr 30 mins
Option (c)
A train 108 meters long is moving at a speed of 50 km/hr. It crosses a train 112 meters long coming from opposite direction in 6 seconds. What is the speed of the second train?
(a) 82 KMPH
9 (b) 58 KMPH
(c) 44 KMPH
(d) 76 KMPH
Option (a)
A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:
(a) 45 KMPH
10 (b) 50 KMPH
(c) 54 KMPH
(d) 55 KMPH
Option (b) | {"url":"https://placementprep.in/QuantitativeAptitude/Problems-on-Train/MCQs/PN=1","timestamp":"2024-11-12T08:48:08Z","content_type":"text/html","content_length":"55189","record_id":"<urn:uuid:ec31db72-3a68-477a-b610-1d7f4f768578>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00253.warc.gz"} |
Root Locus
The root locus provides an incredibly elegant and insightful way to investigate feedback control systems. It provides a graphical depiction of how a system responds to a whole family of feedback
laws. It allows one to see what is possible with different types of feedback.
Table of Links
Root Locus Rules
I do not have Rules 0 through 3 yet, but here are the two parts of Rule 4. This rule deals with the branches as the gain K approaches positive inifinity. Part A of the rule is provided in the
following video.
Part B of Rule 4 is provided in the video below.
Example Problem A
Here’s a simple insightful example problem to get you started. You should try it out before watching the video.
Example Problem B
Here’s another one. Again, try sketching the plot before watching the video.
Root Locus for PD & PID Controller, and rltool
In this video, I consider P, PD, and PID controllers for a single system. We sketch root locus plots for these cases by hand. Then we discuss how to use Matlab’s rltool and root locus to determine
controller gains. [Note. In the video, I use something in Matlab called sisotool. Since then, I have discovered that you can use the Matlab command ‘rltool’ to activate only the part of sisotool that
we care about here.] | {"url":"http://www.spumone.org/courses/control-notes/root-locus/","timestamp":"2024-11-07T19:29:01Z","content_type":"text/html","content_length":"49089","record_id":"<urn:uuid:31827f85-faca-4b15-8266-6178cc2b07da>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00323.warc.gz"} |
Can't find potential of vector field
• Thread starter Addez123
• Start date
In summary: If a potential ##\phi## exists, then$$\phi = \int^{\vec x} \vec A\cdot d\vec x.$$This is coordinate independent. In polar coordinates however$$d\vec x = \vec e_r dr + \vec e_v r \,
dv$$and so$$\phi = \int^{\vec x} (A_r dr + rA_v dv).$$Integrate along any curve to find the result.
Homework Statement
There's two exercises where I'm suppose to find if there's a potential and if so determine it.
$$1. A = 3r^2sin v e_r + r^3cosv e_v$$
$$2. A = 3r^2 sin v e_r + r^2 cos v e_v$$
Relevant Equations
1. To find the solution simply integrate the e_r section by dr.
$$\nabla g = A$$
$$g = \int 3r^2sin v dr = r^3sinv + f(v)$$
Then integrate the e_v section similarly:
$$g = \int r^3cosv dv = r^3sinv + f(r)$$
From these we can see that ##g = r^3sinv + C##
But the answer is apparently that there is no solution since ∇ × A does not equal zero.
Wtf? Take the derivative of ##r^3sinv## with respect to r and v and you'll literally get A!
So how does it "not exist"?
Same problem with number 2.
$$g = \int 3r^2 sin v dr = r^3 sin v + f(v)$$
$$g = \int r^2 cos v dv = r^2 sin v + f(r)$$
From this you can CLEARLY see there is NO solution available.
Yet this exercise has the audacity to claim that
$$g = r^3 cos v$$
would be a solution!
Have they even tried taking the derivative of that with respect to literally ANYTHING?!
##d/dr (r^3 cos v) = 3r^2 COS v##, which is not ##A_r##.
Are they trying to make me go mad??
Orodruin said:
That's not how the gradient works in curvilinear coordinates.
My textbook does not cover potentials except in cartesian coordinates.
What's wrong with my approach?
I need to add a jacobian somewhere or something?
Science Advisor
Homework Helper
In plane polars, [tex]
\nabla g = \frac{\partial g}{\partial r} e_r + \frac 1r \frac{\partial g}{\partial v} e_v.[/tex]
Addez123 said:
My textbook does not cover potentials except in cartesian coordinates.
What's wrong with my approach?
I need to add a jacobian somewhere or something?
You would need to convert r, ##e_r##, v, and ##e_v## to Cartesian coordinates. It's technically possible but it's going to be a real mess.
The concepts are identical if you use polar coordinates and it will give you some good experience to work it that way. Coordinate transformations are a good technique to learn when solving these
types of problems, and many others.
Ahh ok I get it!
So when solving for ##e_v## in 1.
I should have done
$$g = \int r^3cosv * 1/r dv = r^2sinv + f(r)$$
which then has no solution.
Is that correct?
Addez123 said:
My textbook does not cover potentials except in cartesian coordinates.
But does it cover the derivative operators in curvilinear coordinates? That should be enough since the potential concept in itself is not coordinate dependent.
Edit: … and if it doesn’t, then consider another textbook that does (nudge, nudge, wink, wink)
Science Advisor
Homework Helper
Addez123 said:
Ahh ok I get it!
So when solving for ##e_v## in 1.
I should have done
$$g = \int r^3cosv * 1/r dv = r^2sinv + f(r)$$
which then has no solution.
Is that correct?
No, you have [tex]
\frac{\partial g}{\partial r} &= A_r \\
\frac 1r \frac{\partial g}{\partial v} &= A_v \end{split}[/tex] so if [itex]g[/itex] exists then [tex]
\int A_r\,dr + B(v) = \int rA_v\,dv + C(r).[/tex]
pasmith said:
No, you have [tex]
\frac{\partial g}{\partial r} &= A_r \\
\frac 1r \frac{\partial g}{\partial v} &= A_v \end{split}[/tex] so if [itex]g[/itex] exists then [tex]
\int A_r\,dr + B(v) = \int rA_v\,dv + C(r).[/tex]
I would write it slightly differently. If a potential ##\phi## exists, then
\phi = \int^{\vec x} \vec A\cdot d\vec x.
This is coordinate independent. In polar coordinates however
d\vec x = \vec e_r dr + \vec e_v r \, dv
and so
\phi = \int^{\vec x} (A_r dr + rA_v dv).
Integrate along any curve to find the result.
The result is of course the same.
Science Advisor
Gold Member
Don't we need the condition ## U_x=V_y## , in Cartesian , as a necessary condition?
Science Advisor
Gold Member
2023 Award
First calculate the curl (I'd extend it to 3D to use the standard formulae for cylinder coordinates ;-)). If the curl doesn't vanish there's no scalar potential. Otherwise, I'd use the hint in #8.
FAQ: Can't find potential of vector field
1. What is the potential of a vector field?
The potential of a vector field is a scalar function that describes the energy associated with the movement of a particle in that field. It is a mathematical concept used in physics and engineering
to analyze and understand the behavior of vector fields.
2. How is the potential of a vector field calculated?
The potential of a vector field can be calculated by taking the negative gradient of the vector field. This involves finding the partial derivatives of each component of the vector field with respect
to the coordinates and then combining them using the gradient operator. The resulting function is the potential of the vector field.
3. Why can't I find the potential of a vector field?
There are a few reasons why you may not be able to find the potential of a vector field. One possibility is that the vector field is not conservative, meaning that the work done by the field on a
particle depends on the path taken and not just the endpoints. In this case, the potential does not exist. Another reason could be that the vector field is defined on a non-simply connected domain,
which can also make it non-conservative and therefore not have a potential.
4. Can a vector field have multiple potentials?
No, a vector field can only have one potential. This is because the potential is a unique function that describes the energy of the field at any given point. If a vector field had multiple
potentials, it would mean that the energy at a single point can have different values, which is not possible.
5. What are some real-world applications of finding the potential of a vector field?
The concept of potential in vector fields has many practical applications. Some examples include analyzing the flow of fluids in engineering, understanding the behavior of electric and magnetic
fields, and studying the motion of celestial bodies in astronomy. It is also used in various areas of physics, such as quantum mechanics and thermodynamics, to solve complex problems and make
predictions about the behavior of particles and systems. | {"url":"https://www.physicsforums.com/threads/cant-find-potential-of-vector-field.1047795/","timestamp":"2024-11-09T17:38:23Z","content_type":"text/html","content_length":"135338","record_id":"<urn:uuid:e521aa2a-b79e-491c-a78d-2b5aa168129e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00864.warc.gz"} |
Internet of things
1. The use of IPsec (IP security) in IPv6 is
Evaluate your answer
2. The number of bits used by IPv4 (Internet protocol version 4 ) for the address is
Evaluate your answer
3. The architecture of IPv6 is based on
Evaluate your answer
4. First applications of IoT were
Evaluate your answer
5. What part of the world leads the number of M2M connections?
Evaluate your answer
6. The application service layer of IoT provides
Evaluate your answer
7. IoT was born
Evaluate your answer
8. What are tags?
Evaluate your answer
9. CGA (Cryptographically Generated Address) is used in IPv6
Evaluate your answer
10. The number of bits used by IPv6 (Internet protocol version 6 ) for the address is
Evaluate your answer
11. What will the IoT change considerably?
Evaluate your answer
12. What is one of the main barriers for the IoT?
Evaluate your answer
13. Which one is not a future trend of IoT?
Evaluate your answer
14. Which of the following is not a barrier of IoT?
Evaluate your answer
15. What is not about the IoT?
Evaluate your answer
16. One of the following is not an enabling factor?
Evaluate your answer
17. Which is not a newer form of battery technology?
Evaluate your answer
18. How many devices can be connected directly to the internet if the IPv6 protocol is used?
Evaluate your answer
19. _______ memory is not the required type of storage by devices.
Evaluate your answer
20. Integration of smart devices into packaging _____________________
Evaluate your answer | {"url":"https://techpedia.fel.cvut.cz/html/frame.php?oid=120&pid=LM08_F_EN_TE&finf=html-test&fp=","timestamp":"2024-11-02T01:33:56Z","content_type":"text/html","content_length":"29332","record_id":"<urn:uuid:ea32dcfc-7ee3-4ebd-a6da-af3de8c64db6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00471.warc.gz"} |
A Direction in Space
Space is isotropic – the same in all directions – and homogeneous – the same at each point in a certain direction. This implies that we cannot determine our position in some absolute space, because
absolute space does not exist.
We can however find our speed through space, or at least our speed relative to things in space, by measuring the doppler shift of the radiation they give off. For example, we might assume the most
distant stars are fixed in space so if the stars in one direction have a larger redshift than the stars in another direction, then we must be moving in the direction of the stars with the smaller
redshift. It is hard though to disentangle the redshift from the distance. Much better to use something that pervades all space and is thought to be truly isotropic- the cosmic microwave background
radiation, the echo of the big bang. The cosmic microwave background radiation has the spectrum of a black body with a characteristic temperature of 2.73 K and a characteristic wavelength given by
Wien's Law of
If we are not moving through space, hence not moving relative to this background radiation it will have the same wavelength over the whole sky. If we are moving through space, it will display a
shorter wavelength in the direction in which we are moving and a longer wavelength in the direction from which we are moving.
We can estimate our speed through space using the formula for the redshift:
This is illustrated below. We are moving in the direction of the blue part, which illustrates the blueshifting of light. | {"url":"https://mail.astarmathsandphysics.com/a-level-physics-notes/cosmology/2499-a-direction-in-space.html","timestamp":"2024-11-10T05:04:02Z","content_type":"text/html","content_length":"27951","record_id":"<urn:uuid:0434623c-0d88-44d6-b0e3-d9b4c8cb2860>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00777.warc.gz"} |
Bit shifting ( layer mask)
So I’m learning to code through programming games , and it’s kinda hard understanding things like bit shifting , masking , accessing files ect … anyway I used this like 20 times and whenever I look
at it ,it feels kinda weird using stuff that I dont really understand it’s functionality .
an int number is represented as a 32 bit .
So 1<<8 means something like this right ? : 00000000000000000000000010000000
it positions the 1 in the 8th position and applies a logical && operation with the layers so only the layer with the ID 8 which actually got the same binary representation will return true .
So only if an object is in the layer with the ID 8 will be considered as a collider , please correct me if I’m wrong , I hate using things That I dont understand .
Thanks in advance .
And one more question , if I use this << it means I start counting from left to right , so if Use this >> does it mean the opposite ?
Thanks again .
When you are using bit operations, you always start from the right.
1 in decimal is represented by 1 in binary (I removed the leftmost 0s since they are irrelevant, and pollute the reading)
a << n means you “shift to the left all the bits of a n times”. So 1 << 8 is 100000000 (binary) (not 10000000 as you indicated) = 256 (decimal)
In the opposite, a >> n means you “shift to the right all the bits of a n times”. So 8 >> 2 is 1000 >> 2 = 10 (binary) = 2 (decimal)
When using NameToLayer(string layerName), you get n (the layer index as indicated in the documentation), so if you want to use it in Physics.Raycast, don’t forget to use bitshifting :
int layerIndex = NameToLayer( "MyLayer" ) ;
Physics.Raycast(origin, direction, Mathf.Infinity, 1 << layerIndex ) ;
I know a lot of people are learning to write in C# from the perspective of Unity, but really the subject is so large you can earn PhD’s in the science.
What you’re asking about is part of computer science. You’ll need to consider studying programming in C# (and/or other languages) as part of your own growth, because if you insist on learning
programming from a Unity origin, you are going to be deluged with concepts far more advanced than students of programming would have to endure.
It’s like you’re trying to perform on a guitar in a professional band by having played on the “Garage Band” app.
That said, the subject you’ve inquired about is similar to what people learn to do when multiplying and dividing by 10 - remember about moving the decimal point to the right one notch for every
mutliplication by 10, or to the left one notch when dividing by 10? That’s what shifting does, but in powers of 2.
What is happening, more fundamentally, is that you’re starting with a 1 (which is your example, but you could start with any number). It has a value of 1 just like in decimal math, but it is
represented as a single bit. Then, if you shift it to the left 1, as in 1<<1, you’re moving that bit to the left to become 10, just as you’ve indicated. This is a multiplication, of 1 x 2. If you
shift 1<<3, the result is 1000, which is 8 in the binary numbering system. You’re correct that 1>>4 is division by a power of 2, but in this case that 1 is shifted right and out of scope - it falls
away, and you get 0 (because 1 / 2^4 is less than 1, it is a fraction, and integers just truncate fractions).
However, if you start with something like 8, and write 8>>2, then 8 is divided by 2^2 (which is 4), so the result will be 2 decimal, or 10 binary.
This has nothing to do with layers specifically. The layers respond to which bits are “on” or “off” using a mask. What you’re doing is fashioning the math.
I should point out that you are incorrect about one point. The <> do not perform any kind of logical operation like “and” or “xor”, or any other form like them. The shift operators <> only shift bits
left or right. I think you intended to say that this mask is used in a logical operation to identify layers, but your phrase suggested the <> did that, which it doesn’t.
I should note, too, that this is such a fundamental operation that the CPU does this natively. There are shift instructions, and special circuitry (called barrel shifters) designed to perform this
work very quickly, and it comes in more forms than the operators provide. Shifting with <> perform the kind of shift where bits that exceed either end of the “word” (32 bit word in your example, or
64 bit word in others) are simply dropped. However, there are “rolls”, where the bits that are tossed out of one end are brought back end from the other. A “roll right”, or ROR instruction, of
00000001 (an 8 bit value in an 8 bit register), would push the 1 bit out of the right, then back in from the left, which becomes 10000000. It has special use cases beyond the scope of our discussion,
but it is useful to understand how large the subject is, and just where it applies.
I know a lot of people like to use the form 1<<8 to put a bit where the want it, but us old timers (I’ve been at this for decades) often prefer HEX representation to do the same. With practice we see
hex values for their binary counterparts automatically, so if I needed the pattern 10100101, I’d merely use hex 0xA5. To me, 0xA5 looks like 10100101, and visa versa. Instead of 1<<8, I’d just write
To me, after practice, each “place” of the HEX number is 4 bits. 0x10 (a hex value) means, to me, the right most 4 bits are all zero, in the next group, going left, it’s a 1, so the binary is 10000.
After a little practice, this is so natural that we just don’t bother with shifts to place bits. | {"url":"https://discussions.unity.com/t/bit-shifting-layer-mask/211874","timestamp":"2024-11-07T03:49:14Z","content_type":"text/html","content_length":"34538","record_id":"<urn:uuid:770c486f-f8cf-45e9-9a23-2a467cf7abed>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00014.warc.gz"} |
Related Queries: PyTorch MCQ | PyTorch Quiz | PyTorch Questions and Answers
How does torch.nn.functional.kl_div work in PyTorch?
It divides KL values
For computing Kullback-Leibler divergence loss
To optimize distribution comparisons
To implement custom loss functions
What are some PyTorch best practices mentioned in the text?
Using GPUs, employing data loaders, selecting appropriate loss functions and optimizers
Using only CPU computations
Avoiding the use of pre-trained models
Maximizing model complexity
What are some ways to initialize weights in PyTorch?
Using methods like uniform, normal, xavier, and kaiming initialization
Weights are always initialized randomly
Weights must be set manually
PyTorch doesn't support weight initialization
What is a hyperparameter in deep learning?
A learnable parameter
A configuration setting not learned from data
A type of activation function
A loss function
Which PyTorch function is used to compute the softmax of a tensor?
All of the above
Which PyTorch function is used to compute the binary cross-entropy loss?
Both a and c
Which PyTorch library is commonly used for image augmentation?
How do you perform element-wise addition of two tensors?
torch.add(a, b)
a + b
torch.sum(a, b)
torch.elementwise_add(a, b)
How does torch.autograd.grad work in PyTorch?
It computes gradients automatically
For computing gradients of outputs with respect to inputs
To optimize gradient calculations
To implement custom gradient functions
Which of the following is NOT a valid initialization method in PyTorch?
Score: 0/10
Which of the following is NOT a valid PyTorch tensor creation method?
What is the purpose of the torch.nn.Embedding layer in PyTorch?
To convert indices into dense vectors, often used for word embeddings
To embed images into lower-dimensional spaces
To create embeddings for graph data
To compress model weights
How can you use torch.compile in PyTorch 2.0?
To compile models to C++
For ahead-of-time compilation of models
To optimize Python code
To create standalone executables
Which PyTorch function is used to compute the cosine similarity between tensors?
Both b and c
How can you use torch.package in PyTorch?
To create installation packages
For bundling models and code
To optimize package imports
To implement custom packaging logic
How can you use torch.nn.utils.vector_to_parameters in PyTorch?
To vectorize parameters
For converting a vector back to model parameters
To implement vector operations
To optimize parameter updates
How do you compute the dot product of two tensors?
torch.dot(a, b)
torch.matmul(a, b)
a @ b
What type of computation graph does PyTorch use?
What is the main purpose of attention mechanisms?
To reduce model size
To focus on relevant input parts
To speed up training
To normalize inputs
In PyTorch, what does the backward() method do?
Computes gradients
Reverses tensor operations
Undoes the last operation
Moves data to CPU
Score: 0/10
Which parameter in SGD controls the momentum?
What is the purpose of torch.cuda.is_available() in PyTorch?
To check if CUDA is available for GPU computations
To enable CUDA for the current model
To move the model to GPU
To optimize CUDA performance
What is the purpose of torch.nn.LayerNorm in PyTorch?
To apply layer normalization
To create normalized layers
To initialize layer weights
To optimize layer computations
Which method is used to get predictions from a PyTorch model?
How does torch.nn.functional.kl_div work in PyTorch?
It divides KL values
For computing Kullback-Leibler divergence loss
To optimize distribution comparisons
To implement custom loss functions
What is the main advantage of PyTorch's dynamic computational graphs?
They allow for more flexible and intuitive model building
They are faster than static graphs
They use less memory
They are easier to deploy
How can you use torch.nn.init.normal_ in PyTorch?
To normalize tensors
For initializing tensor values from a normal distribution
To implement Gaussian noise
To optimize weight distributions
Which method is used to compute the gradient of a PyTorch tensor?
What is the purpose of torch.stack() in PyTorch?
To stack tensors along a new dimension
To stack tensors along an existing dimension
To unstack tensors
To restack tensors
Which is NOT a type of generative model in PyTorch?
Variational Autoencoder (VAE)
Generative Adversarial Network (GAN)
Long Short-Term Memory (LSTM)
Normalizing Flow
Score: 0/10
How can you train a PyTorch model on multiple GPUs?
Using torch.nn.DataParallel or DistributedDataParallel
PyTorch doesn't support multi-GPU training
By manually copying the model to each GPU
By training separate models on each GPU
Which PyTorch function is used to compute the exponential of a tensor?
What method is used to compute the element-wise maximum of two tensors?
torch.maximum(a, b)
torch.max(a, b)
torch.elementwise_max(a, b)
What company actively uses PyTorch for its deep learning requirements?
What is the purpose of torch.nn.init module in PyTorch?
To initialize tensors
To initialize model parameters
To initialize optimizers
To initialize datasets
What is an example of unsupervised learning in PyTorch?
Image classification
Sentiment analysis
K-means clustering
Named entity recognition
What does LSTM stand for in neural networks?
Long Short-Term Memory
Large Scale Training Method
Linear Sequence Training Model
Layered System for Text Mining
Which PyTorch function computes the inverse of a matrix?
Both a and b
What is the primary function of the squeeze() method in PyTorch?
To add dimensions
To remove dimensions of size 1
To reshape tensor
To flatten tensor
What is the main purpose of batch normalization in deep learning?
To increase batch size
To normalize layer inputs
To reduce model complexity
To speed up convergence
Score: 0/10
Related Reads: | {"url":"https://coolgenerativeai.com/pytorch-mcq/","timestamp":"2024-11-12T09:39:02Z","content_type":"text/html","content_length":"190559","record_id":"<urn:uuid:ea7de421-ef18-4515-aa44-420ea2d77164>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00081.warc.gz"} |
Sizes of Objects and Stuff - Page 2 of 13 -
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Homework In Schools - WriteWork
In school today homework is a key tool of the learning process. Homework teaches students to gather information, solve problems, and learn material. My question is this. Are teachers abusing their
right to give homework? Are the students getting to much homework? Being a student myself, I believe so.
Teachers abuse the use of homework in several ways. One way in which this happens is that teachers use homework instead of actually teaching. In this case the teacher will do an example or two and
assign homework. This forces students teach themselves everything, and defeats the whole purpose of having teachers. Another way teachers abuse this is by giving too much homework. Most of the time a
teacher will give a fairly reasonable amount of homework, but when you add up seven periods of "reasonable"� homework, it no longer is "reasonable."� Both of these will hurt the student's
grade whether he/she rushes through homework, or if he/she does not do it at all.
I would like to say that this does not apply to all teachers. I have many teachers that are exceptional educators. They understand that teaching is not about how much homework they can give, but how
much they can help students to understand.
Time after school is very limited because of homework. Personal time is very is scarce, especially when attempting to juggle sports and academics. Many times the solution for this is to either quit
the sport, or drop the harder classes, but I don't believe either of these is the answer.
The way to solve this problem would be to cut down homework with repetitious problems, and to actually teach the material the homework is over before assigning it! In other words, stop giving so | {"url":"https://www.writework.com/essay/homework-schools","timestamp":"2024-11-11T01:41:41Z","content_type":"application/xhtml+xml","content_length":"27624","record_id":"<urn:uuid:a495175a-8d90-4631-b1df-bfc206c13477>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00003.warc.gz"} |
Recursively enumerable language
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In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it
is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language.
Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively
The class of all recursively enumerable languages is called RE.
There are three equivalent definitions of a recursively enumerable language:
1. A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language.
2. A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) which will enumerate all valid strings of the language. Note that if
the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is "already" produced for a
number which is less than n. If it already is produced, use the output for input n+1 instead (recursively), but again, test whether it is "new".
3. A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any string in the
language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this to recursive languages, which require that the Turing machine
halts in all cases.
All regular, context-free, context-sensitive and recursive languages are recursively enumerable.
Post's theorem shows that RE, together with its complement co-RE, correspond to the first level of the arithmetical hierarchy.
The set of halting turing machines is recursively enumerable but not recursive. Indeed one can run the Turing Machine and accept if the machine halts, hence it is recursively enumerable. On the other
hand the problem is undecidable.
Some other recursively enumerable languages that are not recursive include:
Closure properties[edit]
Recursively enumerable languages (REL) are closed under the following operations. That is, if L and P are two recursively enumerable languages, then the following languages are recursively enumerable
as well:
Recursively enumerable languages are not closed under set difference or complementation. The set difference L − P may or may not be recursively enumerable. If L is recursively enumerable, then the
complement of L is recursively enumerable if and only if L is also recursive.
• Sipser, M. (1996), Introduction to the Theory of Computation, PWS Publishing Co.
• Kozen, D.C. (1997), Automata and Computability, Springer.
External links[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/v%C3%A9ges_%C3%A1llapot%C3%BA_transzducereket_(FST)/en.wikipedia.org/wiki/Recursively_enumerable_language.html","timestamp":"2024-11-13T04:48:38Z","content_type":"text/html","content_length":"54004","record_id":"<urn:uuid:6d335b0c-7472-4fc5-ac04-d362be14cffe>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00525.warc.gz"} |
Recursion is an important programming technique. It is used to have a function call itself from within itself. One example is the calculation of factorials. The factorial of 0 is defined specifically
to be 1. The factorials of larger numbers are calculated by multiplying 1 * 2 * ..., incrementing by 1 until you reach the number for which you are calculating the factorial.
The following paragraph is a function, defined in words, that calculates a factorial.
"If the number is less than zero, reject it. If it is not an integer, round it down to the next integer. If the number is zero, its factorial is one. If the number is larger than zero, multiply it by
the factorial of the next lesser number."
To calculate the factorial of any number that is larger than zero, you need to calculate the factorial of at least one other number. The function you use to do that is the function you're in the
middle of already; the function must call itself for the next smaller number, before it can execute on the current number. This is an example of recursion.
Recursion and iteration (looping) are strongly related - anything that can be done with recursion can be done with iteration, and vice-versa. Usually a particular computation will lend itself to one
technique or the other, and you simply need to choose the most natural approach, or the one you feel most comfortable with.
Clearly, there is a way to get in trouble here. You can easily create a recursive function that does not ever get to a definite result, and cannot reach an endpoint. Such a recursion causes the
computer to execute a so-called "infinite" loop. Here's an example: omit the first rule (the one about negative numbers) from the verbal description of calculating a factorial, and try to calculate
the factorial of any negative number. This fails, because in order to calculate the factorial of, say, -24 you first have to calculate the factorial of -25; but in order to do that you first have to
calculate the factorial of -26; and so on. Obviously, this never reaches a stopping place.
Thus, it is extremely important to design recursive functions with great care. If you even suspect that there is any chance of an infinite recursion, you can have the function count the number of
times it calls itself. If the function calls itself too many times (whatever number you decide that should be) it automatically quits.
Here is the factorial function again, this time written in JScript code.
// Function to calculate factorials. If an invalid
// number is passed in (ie, one less than zero), -1
// is returned to signify an error. Otherwise, the
// number is converted to the nearest integer, and its
// factorial is returned.
function factorial(aNumber) {
aNumber = Math.floor(aNumber); // If the number is not an integer, round it down.
if (aNumber < 0) { // If the number is less than zero, reject it.
return -1;
if (aNumber == 0) { // If the number is 0, its factorial is 1.
return 1;
else return (aNumber * factorial(aNumber - 1)); // Otherwise, recurse until done. | {"url":"http://jsdoc.inflectra.com/HelpReadingPane.ashx?href=js56jsconrecurse.htm","timestamp":"2024-11-06T05:58:18Z","content_type":"text/html","content_length":"4851","record_id":"<urn:uuid:8d3bfdfb-4f8f-4258-bff6-b23a57451c3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00545.warc.gz"} |
Small and Big Business
Full text: Small and Big Business
developed at all, and diminishing returns to capital-intensifi
cation exist only theoretically beyond the point at which
actually developed methods end. This point is, however, shifted
further on in the course of technical progress. This avoidance
of the range of increasing capital-output ratios may result
from the above stated influence of high profit margins.
What can be the relevance of this analysis to the question
of scale ? In Chapter III I used it to explain why the highest
size groups of corporations show signs of diminishing profit
rate, for which I could find no technical reasons (dis
economies or scale I assumed to be unimportant). But if the
capital coefficient does not rise with Increasing scale, the
whole argument falls to tiie ground. There is an even more
direct objection against it: Why, one might ask, do the big
concerns apply such methods of production, if they yield them
a smaller profit rate than less capital-using methods would ?
In fact, the analysis of Chapter III is much more suitable
for explaining why the large firms do not increase the capital
I thought for some time that Chapter III is altogether out of
place in this book. But this is not true. In feet the analysis
is relevant for the whole relation of capital intensity,
technical progress, economies of scale and the distribution
of income | {"url":"https://viewer.wu.ac.at/viewer/fulltext/AC14446393/7/","timestamp":"2024-11-12T13:38:00Z","content_type":"application/xhtml+xml","content_length":"67919","record_id":"<urn:uuid:62cf7435-ab9b-46a9-b06f-2b1166d9dda8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00761.warc.gz"} |
eulerr 7.0.2
Bug Fixes
• Fix order and layout of strips when grouping (#108, by @altairwei)
eulerr 7.0.1
Minor Changes
• Fixed some incorrect documentation of internal functions.
• Corrected an url for eulerAPE.
eulerr 7.0.0
New Features
• It is now possible to set the loss function to be used when trying to optimize the Euler diagram layout via loss and loss_aggregator. There is a new vignette that showcases this new feature.
Minor Changes
• C++14 is now required for the package.
Bug Fixes
• Label repelling via adjust_labels in plot.euler() has been deprecated and removed to fix sanitizer warnings.
eulerr 6.1.1
Minor changes
• citation to conference paper added to inst/CITATION
• error messages for erroneous input have been improved in several places
• switched PI to M_PI to support STRICT_R_HEADERS in C++ code (#82, thanks @eddelbuettel)
Bug fixes
• error in documentation for list method has been fixed, thanks @gprezza (##77)
eulerr 6.1.0
Minor changes
• Label repelling (activated by calling euler() with adjust_labels = TRUE) no longer repels text labels away from the edges of the shapes in the diagram.
Bug fixes
• Rectify sanitizer error from clang-ASAN test environment.
eulerr 6.0.2
Bug fixes
• Set stringsAsFactors = TRUE inside all relevant functions in euler() to avoid errors in upcoming R version.
• Fix broken link in eulerr under the hood vignette.
eulerr 6.0.1
Minor changes
• Throw an error message when the number of sets in venn() exceeds 5 (##65)
• Performance improved when large lists are used as input to euler() and venn() (##64, @privefl)
Bug fixes
• Correctly handle data.frame inputs to euler() when categorical variables are character vectors and not factors.
eulerr 6.0.0
New features
• In plot.euler(), percentages can be added to the plot in addition to or instead of counts by providing a list to the quantities argument with an item type that can take any combination of counts
and percent. This change also comes with a redesign of the grid graphics implementation for labels.
• eulerr_options() gains a new argument padding which controls the amount of padding between labels and quantities. (##48)
• plot.euler() now uses code from the ggrepel package to prevent labels from overlapping or escaping the plot area if adjust_labels is set to TRUE.
• A new vignette featuring a gallery of plots from the package has been added.
Minor changes
• The default cex for quantity labels has changed from 1.0 to 0.9.
• Labels for sets that overlap are now merged (partly fixes ##45)
• The fill colors for sets which are completely contained within another set are now once again composed of a mix of the color of the subset and the superset.
• Plotting data has been exposed in a data slot in the object created by calling to plot.euler() (##57)
Bug fixes
• An error in layout normalization that occurred sometimes with ellipses has been fixed.
eulerr 5.1.0
New features
• venn() is a new function that produces Venn diagrams for up to 5 sets. The interface is almost identical to euler() except that a single integer can also be provided. A new vignette, Venn
diagrams with eulerr, exemplifies its use.
Minor changes
• Calculations for the strips in plot.euler() when a list of Euler diagrams is given has been improved. Setting fontsize or cex now results in appropriately sized strips as one would expect.
• Tiny overlaps (where the fraction of the area is less than one thousandth of the largest overlap) in the final diagram are no longer plotted.
• eulergram() objects from plot.euler() now have a proper grob name for the canvas grob, so that extracting information from them is easier.
Bug fixes
• Return value documentation for euler() now correctly says “ellipses” and not “coefficients”.
• data.frame or matrix inputs now work properly when values are numerical. (##42)
• Fixed some spelling errors in news and vignettes.
eulerr 5.0.0
New features
• error_plot() is a new function that offers diagnostic plots of fits from euler(), letting the user visualize the error in the resulting Euler diagram.
Major changes
• euler() once again uses the residual sums of squares, rather than the stress metric, as optimization objective, which means that output is always scaled appropriately to input (##28).
• plot.euler() now uses the polylabelr package to position labels for the overlaps of the ellipses, which has improved performance in plotting complicated diagrams considerably and reduced the
amount of code in this package greatly.
• The c++ internals have been rewritten using more memory-efficient, performant and expressive code.
Minor changes
• The euler.data.frame() method (and by proxy the euler.matrix() method) can now take matrices with factors in addition to the previously supported logical and integer (binary) input. The function
will dummy code the variables for the user.
• A few performance fixes.
• Additional unit tests.
• Previously deprecated arguments to plot.euler() have been made defunct.
• Added a data set, plants, to exemplify the list method for euler().
• Added a data set, fruits, to exemplify the data.frame method for euler().
• euler.data.frame() gains an argument sep, which is a character vector used to separate dummy-coded factors if there are factors or characters in the input.
• Added a data set, organisms, to exemplify the matrix method for euler().
• Added a data set, pain, to exemplify the table method for euler().
• euler.table() gains an argument, factor_names, for specifying whether the factor names should be included when generating dummy-coded variables in case the input is a data.frame with character or
factor vectors or if the input is a table with more than two columns or rows.
• Parts of the eulerr under the hood vignette has been branched off into a new vignette regarding visualization.
Bug fixes
• Empty combinations can now be provided and will be plotted (generating completely blank plots).
• euler.list() now passes its ellipsis argument along properly. (##33, thanks, @banfai)
• Several spelling and grammar mistakes were corrected in vignettes and documentation.
eulerr 4.1.0
Minor changes
• plot.euler() now returns a gTree object. All of the plotting mechanisms are now also found in this function and plot.eulergram() and print.eulergram() basically just call grid::grid.draw() on the
result of plot.euler(). This change means that functions such as gridExtra::grid.arrange() now work as one would intuit on the objects produced by plot.euler().
• Fitting and plotting Euler diagrams with empty sets is now allowed (##23). Empty sets in the input will be returned as NA in the resulting data.frame of ellipses.
• The last-ditch optimizer has been switched back to GenSA::GenSA() from RcppDE::DEoptim().
Bug fixes
• The grid parameters available for edges are now correctly specified in the manual for plot.euler().
• euler.data.frame() now works as expected for tibbles (from the tibble package) when argument by is used.
eulerr 4.0.0
Major changes
• plot.euler() has been rewritten completely from scratch, now using a custom grid-based implementation rather than lattice. As a result, all panel.*() functions and label() have been deprecated as
well as arguments fill_alpha, auto.key, fontface, par.settings, default.prepanel, default.scales, and panel. The method for plotting diagrams has also changed—rather than overlaying shapes on top
of each other, the diagram is now split into separate polygons using the polyclip package. Instead of relying on semi-transparent fills, the colors of the fills are now blended in the CIELab
color space (##16).
• The default color palette has been redesigned from scratch to suit the new plot method.
• A new function eulerr_options() have been provided in order to set default graphical parameters for the diagrams.
Minor changes
• Arguments counts and outer_strips to plot.euler() are now defunct.
• euler() now always returns ellipse-based parameters with columns h, k, a, b, and phi, regardless of which shape is used. This item was previously named “coefficients”, but it now called
“ellipses” instead and a custom coef.euler() method has been added to make cure that coef() still works.
• Layouts are now partially normalized so that diagrams will look approximately the same even with different random seeds.
Bug fixes
• Providing custom labels to quantities and labels arguments of plot.euler() now works correctly (##20).
eulerr 3.1.0
Major changes
• The last-ditch optimizer switched from GenSA::GenSA() to RcppDE::DEoptim().
• The optimizer used in all the remaining cases, including all circular diagrams and initial layouts, was switched back to stats::nlm() again.
• In final optimization, we now use stress instead of residual sums of squares as a target for our optimizer.
Minor changes
• label is now a proper generic with an appropriate method (label.euler()).
• The eulerr under the hood vignette has received a substantial update.
Bug fixes
• Fixed warnings resulting from the deprecated counts argument in one of the vignettes.
• Fixed memcheck errors in the final optimizer.
• Corrected erroneous labeling when auto.key = TRUE and labels were not in alphabetic order. (##15)
eulerr 3.0.1
Bug fixes
• Added the missing %\VignetteEngine{knitr::knitr} to both vignettes. It had mistakenly been left out, which had mangled the resulting vignettes.
eulerr 3.0.0
Major changes
• Ellipses are now supported by setting the new argument shape = "ellipse" in euler(). This functionality accompanies an overhaul of the innards of the function.
• Initial optimization function and gradient have been ported to C++.
• The initial optimizer has been switched from stats::optim(..., method = "L-BFGS-B") to stats::nlminb().
• The final optimizer now falls back to GenSA::GenSA() when the fit from nlminb() isn’t good enough, by default for 3 sets and ellipses, but this behavior can be controlled via a new argument
• A packing algorithm has been introduced to arrange disjoint clusters of ellipses/circles.
• The label placement algorithm has been rewritten to handle ellipses and been ported to C++. It now uses numerical optimization, which should provide slightly more accurate locations.
• The initial optimizer now uses an analytical Hessian in addition to gradient.
Minor changes
• The initial optimizer now restarts up to 10 times and picks the best fit (unless it is perfect somewhere along the way).
• The default palette has been changed to a fixed palette, still adapted to color deficiency, but with some manual adjustments to, among other things, avoid unnecessary use of color.
• The names of the diagError and regionError metrics have been changed from diag_error and region_error to reflect the original names.
• The coordinates for the centers are now called h and k instead of x and y, respectively.
• A new label() function has been added to extract locations for the overlaps for third party plotting (##10).
• The counts argument to plot.euler() and panel.euler.labels() have been deprecated in favor of the more appropriate quantities.
• Argument fill_opacity in plot.euler() that was deprecated in v2.0.0 has been made defunct.
eulerr 2.0.0
Major changes
• eulerr() has been replaced with euler() (see update 1.1.0) and made defunct.
• There are two new methods for euler:
□ euler.list() produces diagrams from a list of sample spaces.
□ euler.table() produces diagrams from a table object, as long as there are no dimensions with values greater than 2.
• plot.euler() has been rewritten (again) from the ground up to better match other high-level functions from lattice. This change is intended to be as smooth as possible and should not make much of
a difference to most users.
• Arguments polygon_args, mar, and text_args to plot.euler() have been made defunct.
Minor changes
• plot.euler() handles conflicting arguments better.
• c++ routines in eulerr now use registration.
• euler() now allows single sets (##9).
• Labels in plot.euler() now use a bold font face by default in order to distinguish them from the typeface used for counts.
• Argument key in plot.euler() has been deprecated and replaced with auto.key. Notice that using key does not throw a warning since the argument is used in lattice::xyplot() (which plot.euler()
relies on).
• Argument fill_opacity is softly deprecated and has been replaced with fill_alpha for consistency with other lattice functions.
Bug fixes
• border argument in plot.euler() works again (##7).
eulerr 1.1.0
Major changes
• eulerr() and its related methods been deprecated and are being replaced by euler(), which takes slightly different input. Notably, the default is now to provide input in the form of disjoint
class combinations, rather than unions. This is partly to make the function a drop-in replacement for venneuler::venneuler.
• plot.euler() has been completely revamped, now interfacing xyplot() from lattice. As a result, arguments polygon_args, mar, and text_args have been deprecated.
Minor changes
• Added a counts argument to plot.eulerr, which intersections and complements with counts from the original set specification (##6).
• Added a key argument to plot.eulerr that prints a legend next to the diagram.
• Switched to atan2() from RcppArmadillo.
• Added version requirement for RcppArmadillo.
• Dropped dependency on MASS for computing label placement, replacing it with a faster, geometric algorithm.
• Dropped the cost function argument cost and now forces the function to use sums of squares, which is more or less equivalent to the cost function from venneuler.
• Color palettes in plot.euler() now chooses colors adapted to color vision deficiency (deuteranopia). With increasingly large numbers of sets, this adaptation is relaxed to make sure that colors
are kept visually distinct.
• euler() now uses nlm() instead of optim(method = "Nelder-Mead") for its final optimization.
Bug fixes
• The previous algorithm incorrectly computed loss from unions of sets. It now computes loss from disjoint class combinations.
• Added missing row breaks in print.eulerr.
eulerr 1.0.0
New features
• Final optimization routines have been completely rewritten in C++ using Rcpp and RcppArmadillo.
• Switched to the cost function from EulerAPE for the default optimization target but added the possibility to choose cost function via a cost argument (currently eulerAPE or venneuler).
• Added the option to produce conditional eulerr plots via a by argument to eulerr. The result is a list of Euler diagrams that can be plotted in a grid arrangement via a new plot method.
• Improved label placement by using a two-dimensional kernel density estimation instead of means to calculate label centers.
Bug fixes and minor improvements
• Cleaned up typos and grammar errors in the Introduction to eulerr vignette.
• Added mar argument to plot.eulerr with a default that produces symmetric margins.
• Corrected the implementation of the stress statistic from venneuler.
• Switched to Vogel sampling to generate points to choose label positions from.
• Minor clean up and performance fixes all around.
• Added a print.eulerr method.
• Updated vignette to cover new features and changes.
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3 Digit By 1 Digit Multiplication Worksheets
Mathematics, specifically multiplication, creates the cornerstone of numerous scholastic techniques and real-world applications. Yet, for numerous students, mastering multiplication can posture a
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Introduction to 3 Digit By 1 Digit Multiplication Worksheets
3 Digit By 1 Digit Multiplication Worksheets
3 Digit By 1 Digit Multiplication Worksheets -
Country Bahamas School subject Math 1061955 Main content Multiplication 2013181 Multiply 3 digits by 1 digit
Multiplication Worksheets 3 Digits by 1 Digit This printable worksheets and activities on this page feature 3 digit times 1 digit multiplication problems Download task cards a Scoot game word problem
practice pages and math riddle worksheets Most files are aligned with the Common Core standards 3 Digit Times 1 Digit Printables
Importance of Multiplication Practice Recognizing multiplication is essential, laying a solid structure for sophisticated mathematical ideas. 3 Digit By 1 Digit Multiplication Worksheets supply
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Development of 3 Digit By 1 Digit Multiplication Worksheets
1 Digit X 2 Digit Multiplication Worksheets
1 Digit X 2 Digit Multiplication Worksheets
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher 3 digit by 1 digit multiplication
David Lopes Member for 2 years 10 months Age 9 13 Level 4 Language English en ID 990248 11 05 2021 Country code US
From traditional pen-and-paper exercises to digitized interactive styles, 3 Digit By 1 Digit Multiplication Worksheets have actually developed, dealing with varied understanding styles and choices.
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Word Trouble Worksheets
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Multiplying 3 Digit By 3 Digit Numbers A
Multiplying 3 Digit By 3 Digit Numbers A
Utilize our free printable 3 digit by 1 digit multiplication word problems worksheets and get your child s multiplication skills a much deserved upgrade Students can use techniques such as vertical
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The 3 digit by 1 digit multiplication worksheets are easy to use interactive and have several visual simulations to support the learning process Download 3 Digit by 1 Digit Multiplication Worksheet
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reinforces multiplication skills, advertising retention and fluency. Balancing Rep and Variety A mix of repetitive exercises and varied problem layouts keeps rate of interest and comprehension.
Giving Useful Responses Feedback aids in recognizing locations of improvement, motivating ongoing development. Challenges in Multiplication Method and Solutions Inspiration and Interaction
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Final thought
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Free 1 Digit Multiplication Worksheet By 0s And 1s Free4Classrooms
3 digit By 1 digit Multiplication Worksheets
Check more of 3 Digit By 1 Digit Multiplication Worksheets below
3 digit By 2 digit multiplication Games And worksheets
Multiplying 2 And 3 Digit Numbers By 1 Digit
17 Best Images Of Three Digit Addition Worksheets Three Digit Addition And Subtraction
Multiplication 3x3 Digit Worksheet Have Fun Teaching
Math Multiplication Worksheet
Multiplication Worksheets 3 Digits by 1 Digit Super Teacher Worksheets
Multiplication Worksheets 3 Digits by 1 Digit This printable worksheets and activities on this page feature 3 digit times 1 digit multiplication problems Download task cards a Scoot game word problem
practice pages and math riddle worksheets Most files are aligned with the Common Core standards 3 Digit Times 1 Digit Printables
Multiply 3 x 1 digits worksheets K5 Learning
Students multiply numbers up to 1 000 by single digit numbers in these multiplication worksheets Regrouping carrying will by required for most questions Worksheet 1 Worksheet 2 Worksheet 3 Worksheet
4 Worksheet 5 10 More Similar Multiply 4 x 1 digits Multiply 2 x 2 digits What is K5
Multiplication Worksheets 3 Digits by 1 Digit This printable worksheets and activities on this page feature 3 digit times 1 digit multiplication problems Download task cards a Scoot game word problem
practice pages and math riddle worksheets Most files are aligned with the Common Core standards 3 Digit Times 1 Digit Printables
Students multiply numbers up to 1 000 by single digit numbers in these multiplication worksheets Regrouping carrying will by required for most questions Worksheet 1 Worksheet 2 Worksheet 3 Worksheet
4 Worksheet 5 10 More Similar Multiply 4 x 1 digits Multiply 2 x 2 digits What is K5
Multiplying 2 And 3 Digit Numbers By 1 Digit
Multiplication 3x3 Digit Worksheet Have Fun Teaching
Math Multiplication Worksheet
3 Digit By 1 Digit Multiplication Worksheets Printable Printable Worksheets
Free Printable 3 Digit By 3 Digit Multiplication Worksheets 2 digit multiplication worksheets
Free Printable 3 Digit By 3 Digit Multiplication Worksheets 2 digit multiplication worksheets
Printable 3 Digit Multiplication Worksheets 2023 Template Printable
Frequently Asked Questions (Frequently Asked Questions).
Are 3 Digit By 1 Digit Multiplication Worksheets ideal for any age groups?
Yes, worksheets can be customized to various age and ability degrees, making them adaptable for various learners.
Exactly how commonly should pupils exercise making use of 3 Digit By 1 Digit Multiplication Worksheets?
Regular method is crucial. Regular sessions, preferably a couple of times a week, can produce substantial enhancement.
Can worksheets alone enhance mathematics skills?
Worksheets are a beneficial tool yet ought to be supplemented with different understanding methods for extensive skill growth.
Exist on the internet systems providing complimentary 3 Digit By 1 Digit Multiplication Worksheets?
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Just how can moms and dads support their children's multiplication technique in the house?
Motivating consistent method, supplying assistance, and creating a favorable learning environment are advantageous actions. | {"url":"https://crown-darts.com/en/3-digit-by-1-digit-multiplication-worksheets.html","timestamp":"2024-11-05T07:19:29Z","content_type":"text/html","content_length":"28922","record_id":"<urn:uuid:8bc79956-cead-4807-a1f7-641f51da0fb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00589.warc.gz"} |
EOCT Geometry Test Prep Course - 1 Time Fee / Unlimited Access / No Expiration!! (Best Value) or 1 Month Plan
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15-399 Constructive Logic
Fall 2000
Frank Pfenning
TuTh 10:30-11:50
BH A51
Recitation Sec A, Wed 10:30-11:20, DH A317, Steven Awodey
Recitation Sec B, Wed 11:30-12:20, DH A317, Andreas Abel
9 units
This multidisciplinary junior/senior-level course is designed to provide a thorough introduction to modern constructive logic, its roots in philosophy, its numerous applications in computer science,
and its mathematical properties. Some of the topics to be covered are intuitionistic logic, inductive definitions, functional programming, type theory, realizability, connections between classical
and constructive logic, decidable classes, temporal logic, model checking.
What's New?
• (Dec 22) The final has been gradesd and course grades assigned. A model solution is available, and you may view your final in the instructors office (WeH 8117)
Class Material
Course Information
Lectures TuTh 10:30-11:15, BH A51, Frank Pfenning
Recitations Section A, Wed 10:30-11:20, DH A317, Steven Awodey
Section B, Wed 11:30-12:20, DH A317, Andreas Abel
Prerequisites CS Majors: 15-151 or equivalent and 15-212
Philosophy Majors: one programming course and either 80-210 or 80-211
Mathematics Majors: 21-127 and one of 21-228, 21-484, 21-373, 21-132
Textbook There is no textbook.
Notes will be handed out throughout the class.
Credit 9 units
Grading 40% Homework and Tests, 15% Midterm I, 15% Midterm II, 30% Final
Homework Weekly homework is assigned each Thursday and due the following Thursday.
Late homework will be accepted only under exceptional circumstances.
Pre-Test Wednesday, Aug 20, in recitation.
You are required to take this test.
Every answer receives full credit.
Midterm I Thursday, Oct 5, in class.
Closed book, one two-sided sheet of notes permitted.
Exam, Model Solution
Midterm II Thursday, Nov 9, in class.
Closed book, one two-sided sheet of notes permitted.
Exam, Model Solution
Post-Test Tuesday, Dec 12, in class.
You are required to take this test.
Every answer receives full credit.
Final Tuesday, Dec 19, 5:30-8:30, BH A51
Open book.
Final, Model Solution
Topics Intuitionistic Logic, Inductive Definitions,
Functional Programming, Type Theory,
Realizability, Classical Logic, Decidable Classes
Temporal Logic, Model Checking
Home http://www.cs.cmu.edu/~fp/courses/logic/
Newsgroup academic.cs.15-399
Email to bb+academic.cs.15-399@andrew.cmu.edu.
Directory /afs/andrew.cmu.edu/scs/cs/15-399/
Teaching Staff
Office Office Hours Phone Email
Lecturer Frank Pfenning WeH 8117 W 2:30-3:30 x8-6343 fp@cs
Section A Steven Awodey BH 152 M 1:00-2:00 x8-8947 awodey@cmu.edu
Section B Andreas Abel WeH 8104 T 1:00-2:00 x8-2582 abel@cs.cmu.edu
Exec. Asst. Maury Burgwin WeH 8124 x8-4740 mburgwin@cs.cmu.edu
[ Home | Schedule | Assignments | Handouts | Software | Overview ]
Frank Pfenning | {"url":"https://www.cs.cmu.edu/~fp/courses/15317-f00/","timestamp":"2024-11-08T21:26:02Z","content_type":"text/html","content_length":"18872","record_id":"<urn:uuid:ae971639-10f8-4df7-9553-119d07280df8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00330.warc.gz"} |
If, as is usually the case, an input series is autocorrelated, the direct cross-correlation function between the input and response series gives a misleading indication of the relation between the
input and response series.
One solution to this problem is called prewhitening. You first fit an ARIMA model for the input series sufficient to reduce the residuals to white noise; then, filter the input series with this model
to get the white noise residual series. You then filter the response series with the same model and cross-correlate the filtered response with the filtered input series.
The ARIMA procedure performs this prewhitening process automatically when you precede the IDENTIFY statement for the response series with IDENTIFY and ESTIMATE statements to fit a model for the input
series. If a model with no inputs was previously fit to a variable specified by the CROSSCORR= option, then that model is used to prewhiten both the input series and the response series before the
cross-correlations are computed for the input series.
For example,
proc arima data=in;
identify var=x;
estimate p=1 q=1;
identify var=y crosscorr=x;
Both X and Y are filtered by the ARMA(1,1) model fit to X before the cross-correlations are computed.
Note that prewhitening is done to estimate the cross-correlation function; the unfiltered series are used in any subsequent ESTIMATE or FORECAST statements, and the correlation functions of Y with
its own lags are computed from the unfiltered Y series. But initial values in the ESTIMATE statement are obtained with prewhitened data; therefore, the result with prewhitening can be different from
the result without prewhitening.
To suppress prewhitening for all input variables, use the CLEAR option in the IDENTIFY statement to make PROC ARIMA disregard all previous models.
Prewhitening and Differencing
If the VAR= and CROSSCORR= options specify differencing, the series are differenced before the prewhitening filter is applied. When the differencing lists specified in the VAR= option for an input
and in the CROSSCORR= option for that input are not the same, PROC ARIMA combines the two lists so that the differencing operators used for prewhitening include all differences in either list (in the
least common multiple sense). | {"url":"http://support.sas.com/documentation/cdl/en/etsug/63939/HTML/default/etsug_arima_sect033.htm","timestamp":"2024-11-04T03:00:06Z","content_type":"application/xhtml+xml","content_length":"16170","record_id":"<urn:uuid:7973ea80-2890-4f7b-86f1-cb0b3c6b7e95>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00402.warc.gz"} |
Plus One Maths Notes Chapter 13 Limits and Derivatives - A Plus Topper
Plus One Maths Notes Chapter 13 Limits and Derivatives is part of Plus One Maths Notes. Here we have given Kerala Plus One Maths Notes Chapter 13 Limits and Derivatives.
Board SCERT, Kerala
Text Book NCERT Based
Class Plus One
Subject Maths Notes
Chapter Chapter 13
Chapter Name Limits and Derivatives
Category Plus One Kerala
Kerala Plus One Maths Notes Chapter 13 Limits and Derivatives
Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain changes.
I. Limit
Limit of a function f(x) at x = a is the behaviors of f(x) at x = a.
x → a^–: Means that ‘x’ takes values less than ‘a’ but not ‘a’.
x → a^+: Means that ‘x’ takes values greater than ‘a’ but not ‘a’.
x → a: Read as ‘x’ tends to ‘a’, means that ‘x’ takes values very close to ‘a’ but not ‘a’.
\(\lim _{x \rightarrow a^{-}} f(x)=A\): Read as left limit of f(x) is ‘A’, means that f(x) → A as x → a^–. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{-}} f
(x)=\lim _{h \rightarrow 0} f(a-h)\)
\(\lim _{x \rightarrow a^{+}} f(x)=B\): Read as right limit of f(x) is ‘B’, means that f(x) → B as x → a^+. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{+}}
f(x)=\lim _{h \rightarrow 0} f(a+h)\).
If left limit and right limit of f(x) at x = a are equal, then we say that the limit of the function f(x) exists at x = a and is denoted
by lim \(\lim _{x \rightarrow a} f(x)\). Otherwise we say that \(\lim _{x \rightarrow a} f(x)\) does not exist.
II. Evaluation Methods
1. Direct substitution method
2. Factorisation method
3. Rationalisation method
4. Using standard results.
III. Algebra of Limits:
For functions f and g the following holds;
IV. Standard Results
\(\lim _{x \rightarrow a} k=k\), where k is constant.
\(\lim _{x \rightarrow a} f(x)=f(a)\), if f(x) is a polynomial function.
1. \(=\frac{0}{0}\), if possible we can factorise the numerator and denominator and then, cancel the common factors and again put x = a. This factorization method is not possible in all cases so we
are studying some standard limits.
V. Derivatives
A derivative of f at a: Suppose f is a real-valued function and a is a point in its domain of definition. The derivative of f at a is defined by \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)
Provided this limit exists. A derivative of f (x) at a is denoted by f'(a).
Derivative of f at x. Suppose f is a real valued function, the function defined by \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
Wherever this limit exists is defined as the derivative of f at x and is denoted by f”(x) |\(\frac{d y}{d x}\)| |y[1]| y’. This definition of derivative is also called the first principle of the
VI. Algebra of Derivatives
For functions f and g are differentiable following holds;
VII. Standard Results
We hope the Plus One Maths Notes Chapter 13 Limits and Derivatives help you. If you have any query regarding Kerala Plus One Maths Notes Chapter 13 Limits and Derivatives, drop a comment below and we
will get back to you at th earliest. | {"url":"https://www.aplustopper.com/plus-one-maths-notes-chapter-13/","timestamp":"2024-11-06T03:02:43Z","content_type":"text/html","content_length":"46402","record_id":"<urn:uuid:a285f1b4-f0d9-4237-9497-19e8785406d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00509.warc.gz"} |
Find Element in Rotated Sorted Array
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83. Find Element in Rotated Sorted Array
Objective: Find the given element in the rotated sorted array.
What is a Rotated Sorted Array?
A sorted array is rotated around some pivot element. See the Example Below, the array is rotated after 6.
Naive Approach: Do the linear scan and find the element in O(n) time
Better Approach: Binary Search
• Since the array is still sorted we can use binary search to find the element in O(logn) but we cannot apply the normal binary search technique since the array is rotated.
• Say the array is arrA[] and the element that needs to be searched is 'x'.
• Normally when arrA[mid]>arrA[low] we check the left half of the array, and make low = mid-1 but here is the twist, the array might be rotated.
• So first we will check arrA[mid]>arrA[low], if true that means the first half of the array is in strictly increasing order. so next check if arrA[mid] > x && arrA[low] <= x, if true then we can
confirm that element x will exist in the first half of the array (so high = mid-1) else array is rotated in the right half and element might be in the right half so (low = mid+1).
• If arrA[mid]>arrA[low], if false that means there is rotation in the first half of the array. so next check if arrA[mid] < x && arrA[high] >= x, if true then we can confirm that element x will
exist in the right half of the array (so low = mid+1) else element might be in the first half so (high = mid-1).
Index of element 5 is 7 | {"url":"https://tips.tutorialhorizon.com/algorithms/find-element-in-rotated-sorted-array/","timestamp":"2024-11-04T20:43:07Z","content_type":"text/html","content_length":"87118","record_id":"<urn:uuid:3377ab85-2df1-4f1d-95ab-ac17d99e3d29>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00505.warc.gz"} |
Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of mathematics which deals with the study of random events. One of the essential theories in probability theory is the geometric distribution. The
geometric distribution is a distinct probability distribution which models the number of tests required to get the first success in a sequence of Bernoulli trials. In this article, we will explain
the geometric distribution, derive its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the number of experiments needed to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial
is an experiment which has two viable results, typically indicated to as success and failure. Such as flipping a coin is a Bernoulli trial since it can likewise come up heads (success) or tails
The geometric distribution is used when the trials are independent, meaning that the consequence of one test does not impact the result of the next trial. In addition, the chances of success remains
unchanged across all the trials. We could denote the probability of success as p, where 0 < p < 1. The probability of failure is then 1-p.
Formula for Geometric Distribution
The probability mass function (PMF) of the geometric distribution is specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the amount of trials needed to achieve the first success, k is the count of trials required to achieve the initial success, p is the probability of success
in a single Bernoulli trial, and 1-p is the probability of failure.
Mean of Geometric Distribution:
The mean of the geometric distribution is described as the likely value of the number of experiments required to achieve the first success. The mean is given by the formula:
μ = 1/p
Where μ is the mean and p is the probability of success in an individual Bernoulli trial.
The mean is the anticipated count of trials needed to get the first success. For example, if the probability of success is 0.5, therefore we expect to obtain the first success following two trials on
Examples of Geometric Distribution
Here are few primary examples of geometric distribution
Example 1: Flipping a fair coin up until the first head appears.
Suppose we flip an honest coin till the initial head appears. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random
variable that portrays the count of coin flips required to obtain the initial head. The PMF of X is given by:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of obtaining the first head on the first flip is:
P(X = 1) = 0.5^(1-1) * 0.5 = 0.5
For k = 2, the probability of getting the first head on the second flip is:
P(X = 2) = 0.5^(2-1) * 0.5 = 0.25
For k = 3, the probability of achieving the first head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling an honest die up until the initial six appears.
Suppose we roll a fair die till the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular
variable which portrays the number of die rolls required to obtain the first six. The PMF of X is given by:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of getting the first six on the first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of getting the initial six on the second roll is:
P(X = 2) = (5/6)^(2-1) * 1/6 = (5/6) * 1/6
For k = 3, the probability of obtaining the first six on the third roll is:
P(X = 3) = (5/6)^(3-1) * 1/6 = (5/6)^2 * 1/6
And so on.
Get the Tutoring You Need from Grade Potential
The geometric distribution is an essential concept in probability theory. It is applied to model a wide range of practical phenomena, such as the count of trials needed to achieve the first success
in different scenarios.
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offer customized and productive tutoring services to help you be successful. Contact us right now to schedule a tutoring session and take your math abilities to the next stage. | {"url":"https://www.irvineinhometutors.com/blog/geometric-distribution-definition-formula-mean-examples","timestamp":"2024-11-07T03:57:13Z","content_type":"text/html","content_length":"74827","record_id":"<urn:uuid:4b37796e-a790-4162-9689-9a4e10900704>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00433.warc.gz"} |
Sum Values Less Than a Particular Value (SUMIF Less Than)
- Written by Puneet
To sum values below a particular value, you need to use the SUMIF function in Excel.
You need to specify the range of cells to check for the criteria, a value to use as criteria, and then the range from where you want to sum values. But you can also use the same range to sum values
and check for criteria.
As I said, SUMIF is great for adding up values that are less than a specific value because it’s easy to use and very efficient.
It allows you to quickly specify a condition and sum only those numbers that meet the condition, all within a single, simple formula. And in this tutorial, we will learn to write a formula to sum
values less than a particular value.
SUMIF Less Than (Sum Less than Values)
SUMIF is used to add up values in a range that meets a single criteria. When you want to sum values that are less than a specific value, you can specify this in the criteria of the SUMIF.
You can use the following steps:
1. First, in a cell, enter the SUMIF function.
2. After that, in the first argument, refer to the range where you need to check for the criteria. In our example, you need to refer to the range of lot sizes.
3. Now, you need to specify the criteria here. So, we will use a 40 with a lower than operator to sum all the less than values.
4. Next, we will refer to the range where we have values to sum. In our example, you need to refer to the range of quantity. In the end, hit enter to get the result.
As you can see, we have got 685 in the result. That means the sum of all the values from the quantity column where the value in the lot size column is less than 40 is 685.
When you use a lower than sign with the value, it doesn’t include that value while sum less than values. That means with <40, you don’t have the value 40 included. For this, you need to use the
equals sign as well, just like the following.
Using the Less Than Value from a Cell
Let’s say instead of specifying the value into the function, you want to use for a cell. In that case, you need to write your formula like the following example:
Here you need to use the lower than sign using double quotation marks, and then by using an ampersand combine it with the cell in which you have the less than value.
• A2:A10: This is the range of cells that the function will check according to the criterion entered. The function will look at the set of cells to determine whether to include the corresponding
value from the third argument (C2:C10) in the sum.
• “<“&E5: This is the function’s criterion to decide whether to sum the corresponding value from the third argument. In this case, the formula checks whether the value in the range A2:A10 is less
than in cell E5. The “<” means less than, and the “&” concatenates this operator with the value in E5 to create the entire criterion.
• C2:C10: The function will sum up the actual range of cells based on the criterion. For each cell in the range A2:A10 that meets the criterion (i.e., is less than the value in E5), the function
will include the corresponding cell from the range C2:C10 in the sum.
• Ensure no extra spaces are in the concatenation; ” < ” will not work correctly, it should be “<“.
• The cell reference in the criteria allows you to change the value in C1 without altering the formula.
SUMIF Less than When You have a Same Range to check for Condition
Now let’s think about another situation where you have the same range to sum the values and to check the less than criteria.
In the above example, we have referred to the quantity column in the criteria_range, and in the criteria have specified <150, but we have specified no range in the sum_range argument. In SUMIF,
sum_range is optional. And when you skip it, the function will use the criteria range to sum values.
In the above formula, SUMIF sums values less than 150 from the quantity column.
Leave a Comment | {"url":"https://excelchamps.com/formulas/sumif-less-than/","timestamp":"2024-11-02T11:02:15Z","content_type":"text/html","content_length":"402749","record_id":"<urn:uuid:b60ed866-a299-4d4c-9d94-73abc675a834>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00060.warc.gz"} |
CUNY Probability Seminars, Fall 2021
The CUNY Probability Seminar will be held by videoconference for the entire semester. Its usual time will be Tuesdays from 4:30 to 5:30 pm EST. The exact dates, times, and seminar links are mentioned
below. If you are interested in speaking at the seminar or would like to be added or to be removed from the seminar mailing list, then please contact either of the Seminar Coordinators
Videoconference Link (via Zoom
Time: August 31, 4:30 – 5:30 pm EDT
Title: Dynamical First-Passage Percolation
Abstract: In first-passage percolation (FPP), we place i.i.d. nonnegative weights on the edges of the cubic lattice Z^d and study the induced weighted graph metric T = T(x,y). Letting F be the common
distribution function of the weights, it is known that if F(0) is less than the threshold p_c for Bernoulli percolation, then T(x,y) grows like a linear function of the distance |x-y|. In 2015,
Ahlberg introduced a dynamical model of first-passage percolation, in which the weights are resampled according to Poisson clocks, and considered the growth of T(x,y) as time varies. He showed that
when F(0) < p_c, the model has no “exceptional times” at which the order of the growth is anomalously large or small. I will discuss recent work with J. Hanson, D. Harper, and W.-K. Lam, in which we
study this question in two dimensions in the critical regime, where F(0) = p_c, and T(x,y) typically grows sublinearly. We find that the existence of exceptional times depends on the behavior of F(x)
for small positive x, and we characterize the dimension of the exceptional sets for all but a small class of such F.
Time: September 14, 4:30 – 5:30 pm EDT
Title: Pandemic REUs
Abstract: The pandemic changed many things, REU Programs included. I will discuss challenges and advantages of mentoring undergraduates in math research from afar. Some results about interacting
particle systems—namely, the frog model and ballistic annihilation—from this summer will also be presented.
Time: October 05, 4:30 – 5:30 pm EDT
Speaker: David Aldous,
Professor, Emeritus and Professor of the Graduate School, UC Berkeley
Title: Two processes on compact spaces
Abstract: It can be found here.
Time: October 12, 4:30 – 5:30 pm EDT
Title: Chemical distance for 2d critical percolation and random cluster model
Abstract: In 2-dimensional critical percolation, with positive probability, there is a path that connects the left and right side of a square box. The chemical distance is the expected length of the
shortest such path conditional on its existence. In this talk, I will introduce the best known estimates for chemical distance. I will then discuss analogous estimates for the radial chemical
distance (the expected length of the shortest path from the origin to distance n), as well as recent extensions of these estimates to the random cluster model. A portion of this talk is based on
joint work with Philippe Sosoe.
Time: October 19, 3:30 – 4:30 pm EDT
Title: Non-equilibrium multi-scale analysis and coexistence in competing
first-passage percolation
We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes FPP_1 and FPP_\
lambda that compete for the occupancy of sites. Initially FPP_1 occupies the origin and spreads through the edges of Z^d at rate 1, while FPP_\lambda is initialised at sites called seeds that are
distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP_1 or FPP_\lambda attempts to occupy it after which it spreads through the edges of Z^d at
rate \lambda. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on Z^d. We remark that, though
counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_\lambda could favor FPP_1. A central contribution of our work is the development of a novel multi-scale
analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of
Based on a joint work with Tom Finn (Univ. of Bath).
Time: October 26, 3:30 – 4:30 pm EDT
Title: Branching Brownian motion with self repulsion
Abstract: We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an \epsilon-neighbourhood
of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers
and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion
equation with a time dependent reaction term. This is joint work with Anton Bovier.
Time: November 02, 4:30 – 5:30 pm EDT
Title: An unexpected phase-transition for percolation on networks
Abstract: The talk concerns the critical behavior for percolation on inhomogeneous random networks on n vertices, where the weights of the vertices follow a power-law distribution with exponent τ∈
(2,3). Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero, as n→ ∞. We show that the critical window for percolation
phase transition is given by πc(λ)=λn^−^(3^−^τ)/2, for λ∈(0,λc), where λc>0 is an explicit constant. Rather surprisingly, it turns out that a giant component of size √n emerges for λ > λc. Thus, the
critical window turns out to be of finite length, which is in sharp contrast with the previously studied critical behaviors for τ∈(3,4) and τ >4 regimes. The rescaled vector of maximum component
sizes are shown to converge in distribution to an infinite vector of non-degenerate random variables that can be described in terms of components of a one-dimensional inhomogeneous percolation model
on Z[+] studied in a seminal work by Durrett and Kesten (1990). Based on joint work with Shankar Bhamidi, Remco van der Hofstad.
Time: November 09, 4:30 – 5:30 pm EDT
Title: Cooperative motion random walks
Abstract: Cooperative motion random walks form a family of random walks where each step is dependent upon the distribution of the walk itself. Movement is promoted at locations of high probability
and dampened in locations of low probability. These processes are a generalization of the hipster random walk introduced by Addario-Berry et. al. in 2020. We study the process through a recursive
equation satisfied by its CDF, allowing the evolution of the walk to be related to a finite difference scheme. I will discuss this relationship and how PDEs can be used to describe the distributional
convergence of asymmetric and symmetric cooperative motion. This talk is based on joint work with Louigi Addario-Berry and Jessica Lin.
Time: November 23, 4:30 – 5:30 pm EDT
Title: Busemann functions and semi-infinite geodesics in a semi-discrete space
Abstract: In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the
semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization
of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new
phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that
direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and
never come back together. We derive these results by studying variational problems for Brownian motion with drift.
Time: November 30, 4:30 – 5:30 pm EST
Title: Exact sampling and fast mixing of Activated Random Walk
Abstract: Activated Random Walk (ARW) is an interacting particle system on the d-dimensional lattice Z^d: Random walkers fall asleep at a fixed rate and wake up any sleeping particles they encounter.
On a finite subset V of Z^d it defines a Markov chain on {0,1}^V. We prove that when V is a Euclidean ball intersected with Z^d, the mixing time of the ARW Markov chain is at most 1+o(1) times the
volume of the ball. The proof uses an exact sampling algorithm for the stationary state, a coupling with internal DLA, and an upper bound on the time when internal DLA fills the entire ball. We
conjecture cutoff at time z times the volume of the ball, where z<1 is the limiting density of the stationary state. Joint work with Feng Liang.
Time: December 07, 4:30 – 5:30 pm EST
Title: Limit Profiles of Reversible Markov Chains
Abstract: The limit profile captures the exact shape of the distance of a Markov chain from stationarity. Lately, there have been new techniques developed with the aim of studying limit profiles.
These techniques have been particularly helpful in studying the limit profile of certain interchange processes, such as the random-transpositions and star transpositions. In this talk, I will give an
overview of these new results and discuss a few open questions. | {"url":"https://probability.commons.gc.cuny.edu/cuny-probability-seminars-fall-2021/","timestamp":"2024-11-02T15:29:36Z","content_type":"text/html","content_length":"85151","record_id":"<urn:uuid:744d1784-5e63-49ce-9cc4-d36d7ce30ded>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00330.warc.gz"} |
Contribution Margin Ratio: Definition, Calculation, and Example - Angola Transparency
Contribution Margin Ratio: Definition, Calculation, and Example
Contribution Margin Ratio: Definition, Calculation, and Example
The contribution margin ratio (CM ratio) is a financial ratio that measures the proportion of revenue that exceeds variable costs. It is calculated by dividing the contribution margin by the revenue
and multiplying the result by 100 to express it as a percentage.
Key Facts
1. Calculate the Contribution Margin: The Contribution Margin is the difference between the revenue and the variable costs. It represents the amount of money available to cover fixed costs and
contribute to profit. The formula to calculate the Contribution Margin is Revenue – Variable Costs.
2. Calculate the Contribution Margin Ratio: The Contribution Margin Ratio is the Contribution Margin expressed as a percentage of the revenue. It helps to understand the specific costs of a
particular product and guide pricing decisions. The formula to calculate the Contribution Margin Ratio is (Contribution Margin / Revenue) * 100.
Here is an example to illustrate the calculation:
Suppose a company has a product that generated $1 million in revenue, with variable costs of $600,000. To find the CM ratio:
1. Calculate the Contribution Margin: Contribution Margin = Revenue – Variable Costs = $1,000,000 – $600,000 = $400,000.
2. Calculate the Contribution Margin Ratio: Contribution Margin Ratio = (Contribution Margin / Revenue) * 100 = ($400,000 / $1,000,000) * 100 = 40%.
Calculating the Contribution Margin Ratio
The contribution margin is the difference between the revenue and the variable costs. It represents the amount of money available to cover fixed costs and contribute to profit. The formula to
calculate the Contribution Margin is:
Contribution Margin = Revenue – Variable Costs
The contribution margin ratio is calculated by dividing the contribution margin by the revenue and multiplying the result by 100. The formula to calculate the Contribution Margin Ratio is:
Contribution Margin Ratio = (Contribution Margin / Revenue) * 100
Example of Contribution Margin Ratio Calculation
Suppose a company has a product that generated $1 million in revenue, with variable costs of $600,000. To find the CM ratio:
1. Calculate the Contribution Margin:
Contribution Margin = Revenue – Variable Costs
= $1,000,000 – $600,000
= $400,000
2. Calculate the Contribution Margin Ratio:
Contribution Margin Ratio = (Contribution Margin / Revenue) * 100
= ($400,000 / $1,000,000) * 100
= 40%
Therefore, the contribution margin ratio for this product is 40%. This means that for every $1 of revenue generated, $0.40 is available to cover fixed costs and contribute to profit.
The contribution margin ratio is a useful tool for analyzing a company’s profitability and pricing strategy. It can also be used to compare the profitability of different products or services.
What is the contribution margin ratio?
The contribution margin ratio is a financial ratio that measures the proportion of revenue that exceeds variable costs. It is expressed as a percentage.
How do you calculate the contribution margin ratio?
The contribution margin ratio is calculated by dividing the contribution margin by the revenue and multiplying the result by 100.
What does the contribution margin ratio tell you?
The contribution margin ratio tells you how much of each dollar of revenue is available to cover fixed costs and contribute to profit.
What is a good contribution margin ratio?
A good contribution margin ratio varies by industry, but generally speaking, a ratio of 30% or higher is considered to be good.
How can I improve my contribution margin ratio?
There are a few ways to improve your contribution margin ratio, such as increasing revenue, reducing variable costs, or both.
What is the difference between the contribution margin and the contribution margin ratio?
The contribution margin is the difference between revenue and variable costs, while the contribution margin ratio is the contribution margin expressed as a percentage of revenue.
How is the contribution margin ratio used in pricing decisions?
The contribution margin ratio can be used to help set prices that cover variable costs and contribute to profit.
How is the contribution margin ratio used in profitability analysis?
The contribution margin ratio can be used to analyze a company’s profitability and compare the profitability of different products or services. | {"url":"https://angolatransparency.blog/en/how-do-you-find-the-cm-ratio/","timestamp":"2024-11-09T12:53:44Z","content_type":"text/html","content_length":"61595","record_id":"<urn:uuid:51511c34-9c32-48e7-988f-63228d3ec0fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00773.warc.gz"} |
bool operator!= (const QgsMargins &lhs, const QgsMargins &rhs)
Returns true if lhs and rhs are different; otherwise returns false.
QgsMargins operator* (const QgsMargins &margins, double factor)
Returns a QgsMargins object that is formed by multiplying each component of the given margins by factor.
QgsMargins operator* (double factor, const QgsMargins &margins)
Returns a QgsMargins object that is formed by multiplying each component of the given margins by factor.
QgsMargins operator+ (const QgsMargins &lhs, double rhs)
Returns a QgsMargins object that is formed by adding rhs to lhs.
QgsMargins operator+ (const QgsMargins &m1, const QgsMargins &m2)
Returns a QgsMargins object that is the sum of the given margins, m1 and m2; each component is added separately.
QgsMargins operator+ (const QgsMargins &margins)
Returns a QgsMargins object that is formed from all components of margins.
QgsMargins operator+ (double lhs, const QgsMargins &rhs)
Returns a QgsMargins object that is formed by adding lhs to rhs.
QgsMargins operator- (const QgsMargins &lhs, double rhs)
Returns a QgsMargins object that is formed by subtracting rhs from lhs.
QgsMargins operator- (const QgsMargins &m1, const QgsMargins &m2)
Returns a QgsMargins object that is formed by subtracting m2 from m1; each component is subtracted separately.
QgsMargins operator- (const QgsMargins &margins)
Returns a QgsMargins object that is formed by negating all components of margins.
QgsMargins operator/ (const QgsMargins &margins, double divisor)
Returns a QgsMargins object that is formed by dividing the components of the given margins by the given divisor.
bool operator== (const QgsMargins &lhs, const QgsMargins &rhs)
Returns true if lhs and rhs are equal; otherwise returns false.
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markov decision process inventory example
Other state transitions occur with 100% probability when selecting the corresponding actions such as taking the Action Advance2 from Stage2 will take us to Win. The probability of being in state-1
plus the probability of being in state-2 add to one (0.67 + 0.33 = 1) since there are only two possible states in this example. Markov processes 23 2.1. The states are independent over time. The
Markov property 23 2.2. 8.1Markov Decision Process (MDP) Toolbox The MDP toolbox provides classes and functions for the resolution of descrete-time Markov Decision Processes. Markov Decision Theory
In practice, decision are often made without a precise knowledge of their impact on future behaviour of systems under consideration. Markov Decision Processes Andrey Kolobov and Mausam Computer
Science and Engineering University of Washington, Seattle 1 TexPoint fonts used in EMF. Disclaimer 8. Applications. The first and most simplest MDP is a Markov process. The process is represented in
Fig. Inventory Problem – Certain demand You sell souvenirs in a cottage town over the summer (June-August). The probability of going to each of the states depends only on the present state and is
independent of how we arrived at that state. Introduction . Generate a MDP example based on a simple forest management scenario. If you enjoyed this post and want to see more don’t forget follow and/
or leave a clap. Henry AI Labs 1,323 views. Before uploading and sharing your knowledge on this site, please read the following pages: 1. At each time, the agent gets to make some (ambiguous and
possibly noisy) observations that depend on the state. 18.4 by two probability trees whose upward branches indicate moving to state-1 and whose downward branches indicate moving to state-2. Note that
the sum of the probabilities in any row is equal to one. As a management tool, Markov analysis has been successfully applied to a wide variety of decision situations. In this blog post I will be
explaining the concepts required to understand how to solve problems with Reinforcement Learning. Our goal is to maximise the return. The state-value function v_π(s) of an MDP is the expected return
starting from state s, and then following policy π. State-value function tells us how good is it to be in state s by following policy π. Content Filtration 6. All states in the environment are
Markov. Markov Decision Process (MDP) Toolbox for Python¶ The MDP toolbox provides classes and functions for the resolution of descrete-time Markov Decision Processes. Terms of Service 7. (The Markov
Property) zInventory example zwe already established that s t+1 = s t +a t-min{D t, s t +a t} can’t end up with more than you started with end up with some leftovers if demand is less than inventory
end up with nothing if demand exceeds inventory i 0 isa pj ∞ =+ ⎪ ⎪ ⎨ = ⎪ ⎪ Pr | ,{}s ttt+1 == ==js sa a∑ depends on demand ⎪⎩0 jsa>+ ⎧pjsa Account Disable 12. The probability that the machine is in
state-1 on the third day is 0.49 plus 0.18 or 0.67 (Fig. 1. It results in probabilities of the future event for decision making. Solving the above equation is simple for a small MRPs but becomes
highly complex for larger numbers. Actions incur a small cost (0.04)." Markov Decision Process (MDP) is a mathematical framework to describe an environment in reinforcement learning. If the machine
is out of adjustment, the probability that it will be in adjustment a day later is 0.6, and the probability that it will be out of adjustment a day later is 0.4. The action-value function q_π(s,a) is
the expected return starting from state s, taking action a, and then following policy π. Action-value function tells us how good is it to take a particular action from a particular state. If the
machine is in adjustment, the probability that it will be in adjustment a day later is 0.7, and the probability that … Report a Violation 11. Perhaps its widest use is in examining and predicting the
behaviour of customers in terms of their brand loyalty and their switching from one brand to another. a sequence of random states S1, S2, ….. with the Markov property. for that reason we decided to
create a small example using python which you could copy-paste and implement to your business cases. Read the TexPoint manual before you delete this box. Put it differently, Markov chain model will
decrease the cost due to bad decision-making and it will increase the profitability of the company. The above Markov Chain has the following Transition Probability Matrix: For each of the states the
sum of the transition probabilities for that state equals 1. Python code for Markov decision processes. The probabilities are constant over time, and 4. Markov Decision Processes (MDPs) Notation and
terminology: x 2 X state of the Markov process u 2 U (x) action/control in state x p(x0jx,u) control-dependent transition probability distribution ‘(x,u) 0 immediate cost for choosing control u in
state x qT (x) 0 (optional) scalar cost at terminal states x 2 T An example in the below MDP if we choose to take the action Teleport we will end up back in state Stage2 40% of the time and Stage1
60% of the time. q∗(s,a) tells which actions to take to behave optimally. We explain what an MDP is and how utility values are defined within an MDP. Markov Process. If the machine is in adjustment,
the probability that it will be in adjustment a day later is 0.7, and the probability that it will be out of adjustment a day later is 0.3. Make learning your daily ritual. The agent only has access
to the history of rewards, observations and previous actions when making a decision. Essays, Research Papers and Articles on Business Management, Behavioural Finance: Meaning and Applications |
Financial Management, 10 Basic Managerial Applications of Network Analysis, Techniques and Concepts, PERT: Meaning and Steps | Network Analysis | Project Management, Data Mining: Meaning, Scope and
Its Applications, 6 Main Types of Business Ownership | Management. Meaning of Markov Analysis 2. Don’t Start With Machine Learning. A model for scheduling hospital admissions. In a Markov Decision
Process we now have more control over which states we go to. Graph the Markov chain and find the state transition matrix P. 0 1 0.4 0.2 0.6 0.8 P = 0.4 0.6 0.8 0.2 5-3. A simple Markov process is
illustrated in the following example: Example 1: A machine which produces parts may either he in adjustment or out of adjustment. An example sample episode would be to go from Stage1 to Stage2 to Win
to Stop. Below is a representation of a few sample episodes: - S1 S2 Win Stop- S1 S2 Teleport S2 Win Stop- S1 Pause S1 S2 Win Stop. The following results are established for MDPs An optimal policy
can be found by maximising over q∗(s, a): The Bellman Optimality Equation is non-linear which makes it difficult to solve. Motivating Applications • We are going to talk about several applications to
motivate Markov Decision Processes. It fully defines the behaviour of an agent. • One of the items you sell, a pack of cards, sells for $8 in your store. Compactification of Polish spaces 18 2. 2.1
Markov Decision Process Markov decision process (MDP) is a widely used mathemat-ical framework for modeling decision-making in situations where the outcomes are partly random and partly under
con-trol. 1. Huge Collection of Essays, Research Papers and Articles on Business Management shared by visitors and users like you. For example, what about that order = argument in the markov_chain
function? We can also define all state transitions in terms of a State Transition Matrix P, where each row tells us the transition probabilities from one state to all possible successor states. A
policy π is a distribution over actions given states. Forward and backward equations 32 3. A Markov Reward Process is a Markov chain with reward values. Plagiarism Prevention 5. All states in the
environment are Markov. Image Guidelines 4. cost Markov Decision Processes (MDPs) with weakly continuous transition probabilities and applies these properties to the stochastic periodic-review
inventory control problem with backorders, positive setup costs, and convex holding/backordering costs. MDPs were known at least as early as the 1950s; a core body of research on Markov decision
processes … using markov decision process (MDP) to create a policy – hands on – python example ... some of you have approached us and asked for an example of how you could use the power of RL to real
life. • These discussions will be more at a high level - we will define states associated with a Markov Chain but not necessarily provide actual numbers for the transition probabilities. : AAAAAAAA
... •Example applications: –Inventory management “How much X to order from Privacy Policy 9. State Transition Probability: The state transition probability tells us, given we are in state s what the
probability the next state s’ will occur. The optimal action-value function q∗(s,a) is the maximum action-value function over all policies. S₁, S₂, …, Sₜ₋₁ can be discarded and we still get the same
state transition probability to the next state Sₜ₊₁. If we can solve for Markov Decision Processes then we can solve a whole bunch of Reinforcement Learning problems. Copyright 10. Example on Markov
Analysis 3. Gives us an idea on what action we should take at states. You have a set of states S= {S_1, S_2, … Transition functions and Markov semigroups 30 2.4. Python: 6 coding hygiene tips that
helped me get promoted. In mathematics, a Markov decision process is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where
outcomes are partly random and partly under the control of a decision maker. A simple Markov process is illustrated in the following example: A machine which produces parts may either he in
adjustment or out of adjustment. In the above Markov Chain we did not have a value associated with being in a state to achieve a goal. Suppose the machine starts out in state-1 (in adjustment), Table
18.1 and Fig.18.4 show there is a 0.7 probability that the machine will be in state-1 on the second day. The MDPs need to satisfy the Markov Property. In value iteration, you start at the end and
then work backwards re ning an estimate of either Q or V . A Partially Observed Markov Decision Process for Dynamic Pricing∗ Yossi Aviv, Amit Pazgal Olin School of Business, Washington University,
St. Louis, MO 63130 aviv@wustl.edu, pazgal@wustl.edu April, 2004 Abstract In this paper, we develop a stylized partially observed Markov decision process (POMDP) An example in the below MDP if we
choose to take the action Teleport we will end up back in state Stage2 40% of the time and Stage1 60% of the time. Since we take actions there are different expectations depending on how we behave.
When studying or using mathematical methods, the researcher must understand what can happen if some of the conditions imposed in rigorous theorems are not satisfied. Markov Decision Processes and
Exact Solution Methods: Value Iteration Policy Iteration Linear Programming Pieter Abbeel UC Berkeley EECS TexPoint fonts used in EMF. State Value Function v(s): gives the long-term value of state s.
It is the expected return starting from state s. How we can view this is by saying going from state s and going through various samples from state s what is our expected return. Markov analysis is a
method of analyzing the current behaviour of some variable in an effort to predict the future behaviour of the same variable. The eld of Markov Decision Theory has developed a versatile appraoch to
study and optimise the behaviour of random processes by taking appropriate actions that in uence future evlotuion. Example: Dual-Sourcing State Set: X = R RL R + R L E + I State [i ,(y 1,..., L R) z
1 L E)] means:: I current inventory level is i 2R I for j = 1,...,L R, an order of y j units from the regular source was placed j periods ago I for j = 1,...,L E an order of z j units from the
expedited source was placed j periods ago Action Sets: A(x) = R + R + for all x 2X Content Guidelines 2. When the system is in state 0 it stays in that state with probability 0.4. A very small
example. Polices give the mappings from one state to the next. This procedure was developed by the Russian mathematician, Andrei A. Markov early in this century. MDPs are useful for studying
optimization problems solved via dynamic programming and reinforcement learning. (Markov property). The steady state probabilities are often significant for decision purposes. V. Lesser; CS683, F10
Example: An Optimal Policy +1 -1.812 ".868.912.762"-1.705".660".655".611".388" Actions succeed with probability 0.8 and move at right angles! MDP policies depend on the current state and not the
history. 2. Contribute to oyamad/mdp development by creating an account on GitHub. In a later blog, I will discuss iterative solutions to solving this equation with various techniques such as Value
Iteration, Policy Iteration, Q-Learning and Sarsa. The value functions can also be written in the form of a Bellman Expectation Equation as follows: In all of the above equations we are using a given
policy to follow, which may not be the optimal actions to take. Numerical example is provided to illustrate the problem vividly. Markov Decision Process - Reinforcement Learning Chapter 3 - Duration:
12:49. So far we have learnt the components required to set up a reinforcement learning problem at a very high level. : AAAAAAAAAAA Since we have a simple model above with the “state-values for MRP
with γ=1” we can calculate the state values using a simultaneous equations using the updated state-value function. Markov Property: requires that “the future is independent of the past given the
present”. Given an initial state x 0 2X, a Markov chain is de ned by the transition proba-bility psuch that p(yjx) = P(x t+1 = yjx t= x): (2) Remark: notice that in some cases we can turn a
higher-order Markov process into a Markov process by including the past as a new state variable. An Introduction to Reinforcement Learning, Sutton and Barto, 1998. In a Markov process, various states
are defined. Prohibited Content 3. The list of algorithms that have been implemented includes backwards induction, linear programming, policy iteration, q-learning and value iteration along with
several variations. Stochastic processes 3 1.1. Want to Be a Data Scientist? The value function can be decomposed into two parts: We can define a new equation to calculate the state-value function
using the state-value function and return function above: Alternatively this can be written in a matrix form: Using this equation we can calculate the state values for each state. Assumption of
Markov Model: 1. The optimal state-value function v∗(s) is the maximum value function over all policies. He first used it to describe and predict the behaviour of particles of gas in a closed
container. Note: Since in a Markov Reward Process we have no actions to take, Gₜ is calculated by going through a random sample sequence. If gamma is closer 0 it leads to short sighted evaluation,
while a value closer to 1 favours far sighted evaluation. A MDP is a discrete time stochastic control process, formally presented by a … Hands-on real-world examples, research, tutorials, and
cutting-edge techniques delivered Monday to Thursday. Uploader Agreement. Markov Process / Markov Chain: A sequence of random states S₁, S₂, … with the Markov property. mdptoolbox.example.forest(S=3,
r1=4, r2=2, p=0.1, is_sparse=False) [source] ¶. We can take a sample episode to go through the chain and end up at the terminal state. It is generally assumed that customers do not shift from one
brand to another at random, but instead will choose to buy brands in the future that reflect their choices in the past. The probability of moving from a state to all others sum to one. Read the
TexPoint manual before you delete this box. That is for specifying the order of the Markov model, something that relates to its ‘memory’. Two groups of results are covered: It tells us the maximum
possible reward you can extract from the system. In a discrete-time Markov chain, there are two states 0 and 1. If you know q∗ then you know the right action to take and behave optimally in the MDP
and therefore solving the MDP. Markov processes are a special class of mathematical models which are often applicable to decision problems. Example if we have the policy π(Chores|Stage1)=100%, this
means the agent will take the action Chores 100% of the time when in state Stage1. Value Iteration in Deep Reinforcement Learning - Duration: 16:50. Cadlag sample paths 6 1.4. 1/3) would be of
interest to us in making the decision. Take a look, Noam Chomsky on the Future of Deep Learning, Python Alone Won’t Get You a Data Science Job, Kubernetes is deprecating Docker in the upcoming
release. 12:49. The probabilities apply to all system participants. Random variables 3 1.2. Stochastic processes 5 1.3. In this post, we will look at a fully observable environment and how to
formally describe the environment as Markov decision processes (MDPs). I created my own YouTube algorithm (to stop me wasting time). with probability 0.1 (remain in the same position when" there is a
wall). Examples in Markov Decision Processes is an essential source of reference for mathematicians and all those who apply the optimal control theory to practical purposes. 5.3 Economical factor The
main objective of this study is to optimize the decision-making process. In order to solve for large MRPs we require other techniques such as Dynamic Programming, Monte-Carlo evaluation and
Temporal-Difference learning which will be discussed in a later blog. Now, consider the state of machine on the third day. Calculations can similarly be made for next days and are given in Table 18.2
below: The probability that the machine will be in state-1 on day 3, given that it started off in state-2 on day 1 is 0.42 plus 0.24 or 0.66. hence the table below: Table 18.2 and 18.3 above show
that the probability of machine being in state 1 on any future day tends towards 2/3, irrespective of the initial state of the machine on day-1. Figure 12.13: Value Iteration for Markov Decision
Processes, storing V Value Iteration Value iteration is a method of computing the optimal policy and the optimal value of a Markov decision process. It assumes that future events will depend only on
the present event, not on the past event. A model for analyzing internal manpower supply etc. This series of blog posts contain a summary of concepts explained in Introduction to Reinforcement
Learning by David Silver. It tells us what is the maximum possible reward you can extract from the system starting at state s and taking action a. A Markov process is a memory-less random process,
i.e. decision process using the software R in order to have a precise and accurate results. The key goal in reinforcement learning is to find the optimal policy which will maximise our return. The
Markov assumption: P(s t 1 | s t-, s t-2, …, s 1, a) = P(s t | s t-1, a)! Below is an illustration of a Markov Chain were each node represents a state with a probability of transitioning from one
state to the next, where Stop represents a terminal state. After reading this article you will learn about:- 1. If we let state-1 represent the situation in which the machine is in adjustment and let
state-2 represent its being out of adjustment, then the probabilities of change are as given in the table below. The corresponding probability that the machine will be in state-2 on day 3, given that
it started in state-1 on day 1, is 0.21 plus 0.12, or 0.33. A partially observable Markov decision process (POMDP) is a combination of an MDP and a hidden Markov model. If I am in state s, it maps
from that state the probability of taking each action. In a Markov Decision Process we now have more control over which states we go to. Keywords: Markov Decision Processes, Inventory Control,
Admission Control, Service Facility System, Average Cost Criteria. The return Gₜ is the total discount reward from time-step t. The discount factor γ is a value (that can be chosen) between 0 and 1.
Markov analysis has come to be used as a marketing research tool for examining and forecasting the frequency with which customers will remain loyal to one brand or switch to others. We want to prefer
states which gives more total reward. This probability is called the steady-state probability of being in state-1; the corresponding probability of being in state 2 (1 – 2/3 = 1/3) is called the
steady-state probability of being in state-2. In order to keep the structure (states, actions, transitions, rewards) of the particular Markov process and iterate over it I have used the following
data structures: dictionary for states and actions that are available for those states: 5-2. Transition probabilities 27 2.3. When the system is in state 1 it transitions to state 0 with probability
0.8. For example, if we were deciding to lease either this machine or some other machine, the steady-state probability of state-2 would indicate the fraction of time the machine would be out of
adjustment in the long run, and this fraction (e.g. Keywords inventory control, Markov Decision Process, policy, optimality equation, su cient conditions 1 Introduction This tutorial describes recent
progress in the theory of Markov Decision Processes (MDPs) with in nite state and action sets that have signi cant applications to inventory control. Decision-Making, Functions, Management, Markov
Analysis, Mathematical Models, Tools. 3. I have implemented the value iteration algorithm for simple Markov decision process Wikipedia in Python. Each month you order items from custom manufacturers
with the name of town, the year, and a picture of the beach printed on various souvenirs. Markov model is a stochastic based model that used to model randomly changing systems. 8.1.1Available modules
example Examples of transition and reward matrices that form valid MDPs mdp Makov decision process algorithms util Functions for validating and working with an MDP We will now look into more detail
of formally describing an environment for reinforcement learning. A Markov Decision Process is an extension to a Markov Reward Process as it contains decisions that an agent must make. This function
is used to generate a transition probability ( A × S × S) array P and a reward ( S × A) matrix R that model the following problem. 18.4). Property: Our state Sₜ is Markov if and only if: Simply this
means that the state Sₜ captures all the relevant information from the history. Other applications that have been found for Markov Analysis include the following models: A model for assessing the
behaviour of stock prices. Decided to create a small cost ( 0.04 ). or 0.67 ( Fig Markov... Model that used to model randomly changing systems, there are two 0!, S₂, … with the Markov model,
something that relates to its ‘ memory ’ are! Plus 0.18 or 0.67 ( Fig the steady state probabilities are constant over,. We want to prefer states which gives more total reward we behave s₁. 18.4 by
two probability trees whose upward branches indicate moving to state-2 0.4 0.8! Numerical example is provided to illustrate the problem vividly, S₂, …, Sₜ₋₁ can be discarded and still... 0.2 0.6 0.8
0.2 5-3 in making the Decision, research, tutorials, and 4 associated being... Create a small MRPs but becomes highly complex for larger numbers to motivate Markov Decision Process ( MDP ) for... We
did not have a value closer to 1 favours far sighted,! When '' there is a Markov Decision Process is a distribution over actions given.... A policy π is a stochastic based model that used to model
randomly changing systems management by! That depend on the state of machine on the past given the present ” • are. Me get promoted Processes, Inventory control, Admission control, Admission control,
Service Facility system Average... Two states 0 and 1 state 1 it transitions to state 0 with 0.4. Model for assessing the behaviour of systems under consideration from the system starting at state s,
a pack cards., you start at the terminal state for the resolution of descrete-time Markov Decision Processes increase the of! Service Facility system, Average cost Criteria: requires that “ the
future is of! Environment for Reinforcement Learning site, please read the TexPoint manual before you delete this box tool, Markov include... Only has access to the next state Sₜ₊₁ up at the terminal
state state probability!, tutorials, and cutting-edge techniques delivered Monday to Thursday a state achieve... Will increase the profitability of the past event with the Markov property: requires
that “ the future for. Moving from a state to the next state Sₜ₊₁ MDP Toolbox provides classes and functions the... Solving the MDP Toolbox provides classes and functions for the resolution of
descrete-time Markov Decision then. Learn about: - 1, mathematical models, Tools without a precise knowledge of their impact on behaviour..., Tools optimal state-value function v∗ ( s, it maps from
state! This study is to find the optimal action-value function q∗ ( s is... Memory-Less random Process, i.e decrease the cost due to bad decision-making it... At states Articles on business
management shared by visitors and users like you an on. In Deep Reinforcement Learning is to find the state and implement to your business cases Essays research. Assessing the behaviour of particles
of gas in a Markov Decision Theory in,... Q∗ ( s ) is the maximum action-value function over all policies at the terminal state return! And whose downward branches indicate moving to state-1 and
whose downward branches indicate moving to and! Due to bad decision-making and it will increase the profitability of the past given the event! Using python which you could copy-paste and implement to
your business cases to Stage2 to Win to Stop ( Stop... A simple forest management scenario is the maximum possible reward you can extract from the system is in s. Study is to find the optimal policy
which will maximise our return to one chain we did have... Learning - Duration: 16:50 from a state to the history of rewards, observations previous. Estimate of either Q or V the agent gets to make
some ( ambiguous possibly. As a management tool, Markov Analysis, mathematical models which are often significant for Decision purposes goal. To one that used to model randomly changing systems
state-1 and whose downward branches moving. Mdp ) Toolbox the MDP Toolbox provides classes and functions for the resolution of descrete-time Markov Process! Actions to take to behave optimally a
distribution over actions given states a Decision ) source!, Sₜ₋₁ can be discarded and we still get the same position when '' is. Mdps are useful for studying optimization problems solved via dynamic
programming and Learning! A value associated with being in a Markov Decision Processes for $ 8 in your store model! S ) is the maximum possible reward you can extract from the system starting at s!
What action we should take at states source ] ¶ on what action we should take at.! Ning an estimate of either Q or V r2=2, p=0.1, is_sparse=False ) [ source ].. Which are often made without a precise
knowledge of their impact on future behaviour of systems under consideration stock! And users like you on how we behave 0.6 0.8 P = 0.4 0.6 0.8 5-3! By creating an account on GitHub stochastic
control Process source ] ¶ we to... To oyamad/mdp development by creating an account on GitHub any row is equal to one optimize! On what action we should take at markov decision process inventory
example the Russian mathematician, Andrei Markov! A value closer to 1 favours far sighted evaluation for that reason we decided to create small. An account on GitHub management, Markov chain and end
up at end... Me wasting time ). will now look into more detail of formally describing an environment Reinforcement! Mdptoolbox.Example.Forest ( S=3, r1=4, r2=2, p=0.1, is_sparse=False ) [ source ] ¶
0.49 0.18... And we still get the same position when '' there is a Markov Process, various states defined!, Sutton and Barto, 1998 states we go to cards, sells $... The above Markov chain and end up
at the terminal state as it contains decisions an. By the Russian mathematician, Andrei A. Markov early in this blog post I will be explaining concepts... Machine is in state-1 on the state optimal
action-value function q∗ ( s, a is! Maximum action-value function over all policies before uploading and sharing your knowledge on this site, please read TexPoint! And users like you and behave
optimally gamma is closer 0 it stays in that state probability! And then work backwards re ning an estimate of either Q or V order the... Of formally describing an environment for Reinforcement
Learning by David Silver solved via dynamic programming Reinforcement. Is provided to illustrate the problem vividly Process - Reinforcement Learning problems on what we... Service Facility system,
Average cost Criteria provided to illustrate the problem.. All others sum to one account on GitHub … with the Markov,... A discrete-time stochastic control Process, Tools own YouTube algorithm ( to
Stop over actions given.! Following models: a sequence of random states S1, S2, ….. with the Markov property 0.8! Probabilities of the company learn about: - 1 Learning is to find the optimal policy
which will maximise return. Associated with being in a state to achieve a goal events will depend only the! Behaviour of stock prices of descrete-time Markov Decision Process we now have control...
Whole bunch of Reinforcement Learning is to find the optimal policy which maximise... The profitability of the past event policies depend on the third day can extract from the is! You can extract
from the system is in state 0 it stays in that with! This site, please read the following models: a model for assessing the behaviour of systems consideration..., while a value closer to 1 favours
far sighted evaluation, while a associated... Memory-Less random Process, various states are defined within an MDP is and how utility values are within... And it will increase the profitability of
the items you sell, a Markov Decision Process ( MDP Toolbox! Creating an account on GitHub source ] ¶ Sutton and Barto, 1998 the maximum reward... Applicable to Decision problems applicable to
Decision problems Barto, 1998 which will maximise our return 0.6... Using python which you could copy-paste and implement to your business cases when making a Decision the state. Process ( MDP )
Toolbox the MDP Toolbox provides classes and functions for resolution...: 12:49 Stage1 to Stage2 to Win to Stop a simple forest scenario. Tutorials, and 4 in the same position when '' there is markov
decision process inventory example Markov Decision.. Descrete-Time Markov Decision Process is an extension to a wide variety of Decision.... A closed container S=3, r1=4, r2=2, p=0.1, is_sparse=False
) [ source ] ¶ or... You enjoyed this post and want to see more don ’ t forget follow leave! S ) is the maximum possible reward you can extract from the system is in state-1 on third. Key goal in
Reinforcement Learning, Sutton and Barto, 1998 Toolbox provides classes functions! Will depend only on the past event to find the optimal policy which maximise., Sutton and Barto, 1998 the
probabilities are often made without a precise knowledge their... To see more don ’ t forget follow and/or leave a clap 1. Over actions given states this site, please read the following pages: 1
Reinforcement... R1=4, r2=2, p=0.1, is_sparse=False ) [ source ] ¶ sells for $ 8 in your.... Used it to describe and predict the behaviour of stock prices example is provided to illustrate the
problem vividly its! State to the next function q∗ ( s, a ) tells which to.
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ACO Seminar
The ACO Seminar (2017–2018)
Sep. 14, 3:30pm, Wean 8220
Freddie Manners, Stanford
Sums of permutations
Suppose G is an abelian group of order N, e.g. 𝔽[2]^n, and π[1], π[2]: {1,...,N} are permutations (i.e., bijections) chosen uniformly at random. Consider the function π[1]+π[2]: {1,...,N} → G. How
closely does it resemble a random function? In particular, what is the chance that it is again a permutation? What about π[1] + π[2] + π[3]?
The middle question, thought of as a free-standing counting problem, was the subject of a long-running conjecture due to Vardi. The outer two have applications in security and pseudorandomness,
connecting pseudorandom functions (PRFs) and pseudorandom permutations (PRPs).
I will discuss joint work with Sean Eberhard and Rudi Mrazović, as well as extensions due to Eberhard, which give precise answers to some of these questions.
Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220. | {"url":"https://aco.math.cmu.edu/abs-17-18/sep14.html","timestamp":"2024-11-05T06:05:52Z","content_type":"text/html","content_length":"2506","record_id":"<urn:uuid:f583a91a-0385-4538-a839-2416f85f734a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00405.warc.gz"} |
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1.1 Background of the study
Mathematical ability is crucial for the economic success of societies (Lipnevich, MacCann, Krumm, Burrus, & Roberts, 2011). It is also important in the scientific and technological development of
countries (Enu, Agyman, & Nkum, 2015). This is because mathematics skills are essential in understanding other disciplines including engineering, sciences, social sciences and even the arts (Patena &
Dinglasan, 2013; Phonapichat, Wongwanich, & Sujiva, 2014; Schofield, 1982). Abe and Gbenro (2014) point out that mathematics plays a multidimensional role in science and technology of which its
application outspread to all areas of science, technology as well as business enterprises. Due to the importance that mathematics engulfs, the subject became key in school curriculum. According to
Ngussa and Mbuti (2017), the mathematics curriculum is intended to provide students with knowledge and skills that are essential in the changing technological world.
Factors that can influence mathematics performance are demonstrated by Kupari and Nissinen (2013); Yang (2013); Tshabalala and Ncube (2016), when they show that poor performance in mathematics is a
function of cross-factors related to students, teachers and schools. Among the students’ factors, attitude is regarded by many researchers as a key contributor to higher or lower performance in
mathematics (Mohamed & Waheed, 2011; Mata, Monteiro & Peixoto, 2012; Ngussa & Mbuti, 2017). Attitude refers to a learned tendency of a person to respond positively or negatively towards an object,
situation, concept or another person (Sarmah & Puri, 2014). Attitudes can change and develop with time (Syyeda, 2016), and once a positive attitude is formed, it can improve students’ learning
(Akinsola & Olowojaiye, 2008; Mutai, 2011). On the other hand, a negative attitude hinders effective learning and consequently affects the learning outcome henceforth performance (Joseph, 2013).
Therefore, attitude is a fundamental factor that cannot be ignored. The effect of attitude on students’ performance in mathematics might be positive or negative depending on the individual student.
In response to this problem, this study seeks to investigate students’ attitudes towards learning mathematics in Nigeria.
In accordance with Syyeda (2016) attitude has three main components: affect, cognition and behaviour. The components are interrelated and involve several aspects contributing to the overall attitude
towards learning mathematics. We draw from the ABC (Affective, Behavioural and Cognitive) model (Ajzen, 1993) to investigate the students’ attitude towards mathematics and the Walberg’s theory of
productivity (Walberg, Fraser, & Welch, 1986) to interpret results about the factors influencing a like or dislike of mathematics and those impacting students’ performance. Walberg’s theory
postulates that individual students’ psychological attributes and the psychological environments surrounding them influences cognitive, behavioural and attitudinal learning outcomes. This theory will
be relevant for this study because it is a suitable lens through which we can explain the reasons for the attitudes that students form towards mathematics. In line with the ABC model, our study
focuses on investigating attitude aspects, including: students’ self-confidence in their mathematics ability, mathematics anxiety, mathematics enjoyment, perception about the usefulness of
mathematics and intrinsic motivation. Questions that guided the study are as follows:
1. What are the students’ attitudes towards learning geometry and mathematics?
2. Why do students acquire a like or a dislike towards geometry in mathematics?
3. What is the relationship between each attitude aspects and students’ performance (grades) in mathematics and geometry?
4. What is the relationship between attitude and student grades in mathematics and geometry?
This work is relevant since poor performance in Science Technology Engineering and Mathematics (STEM) particularly in mathematics is seen as a barrier towards achieving economic and social
development, both at the individual and national level. In Nigeria, like any other country within Sub Saharan Africa (SSA), students consistently perform poorly in mathematics and science, which
makes Nigeria lose economic advantages over other countries. Students’ achievement in countries within SSA is ranked far below the average point in international assessments (Bethell, 2016; 38).
Bethell further points out that the long-run development of countries in SSA requires significant improvements in STEM education if they are to benefit in a competitive global economy driven by new
technologies. In this regard, it is important to find ways to improve students’ performance in the subject. The study of students’ attitudes towards mathematics with associated factors and their
connection to academic performance is certainly worth examining. The results will provide teachers, students, parents, and other education stakeholders with information that will help to develop
strategies to improve students’ learning of mathematics.
Laboratory facilities brings to live abstract learnings in mathematics through experiments and experimental procedures. These performed experiments gives series of visual teachings with makes it
easier for students to understand. By laboratory practicals in geometry secondary school students are able to have a better grasp of the realities of geometry and have deeper insights into the
possibilities of geometry, and this could trigger their imaginative faculties to explore their potential in mathematics in a broader sense. This whole psycho-educational facor could in a great way
improve on students attitude towards geometry and mathematics in general.
According to Umeh (2006) material resources are structured facilities that are used to ensure affective teaching and learning such as the laboratories
All science laboratories have certain general features and requirements in addition to which each separate science has its own special demand which requires a special laboratory and facilities. For
example all modern laboratories need to have a preparation room with storage facilities and shelves for chemicals, tools as well as work benches for the preparation of solutions. They must have
sufficient space for free movement
1.2 Statement of the problem
lack of mathematics laboratory and mathematics teachers’ non-use of laboratory technique in teaching mathematics is one of the major factors that contribute to poor academic achievement in
mathematics by secondary school students. The West African Examinations Council (WAEC), Chief examiners’ Reports (2010, 2011, 2012& 2018) affirmed that candidates lack requisite practical skills to
answer the questions raised in number and numeration, which point to the fact that the most desired technological, scientific and business application of mathematics cannot be sustained.
This makes it paramount to seek for an approach for teaching Geometrythat aims at improving its understanding and academic achievement by students. Ogunkunle(2000) asserted that lack of mathematics
laboratory and mathematics teachers’ non-use of laboratory technique in teaching Geometryis one of the major factors that contribute to poor academic achievement of students in Geometryat junior
secondary school.
Despite the fact that Mathematics Laboratories serve as fundamental sources of creative thinking, skill development and problem solving for Junior Secondary School students, its facilities are seldom
used by Mathematics teachers (Shreedevi & Asha, 2014). Well-equipped Mathematics Laboratory at the basic school level has given birth to scientific and technological growth in manpower among
developed nations (Imoko & Isa, 2015).Beyond availability of Mathematics laboratory materials, Malik (2017) opined that adequate use of Mathematics laboratory prepares students for a useful and
meaningful living, because Mathematics is the language and key to everyday activities of mankind in the world of science and technology. For the fact that Mathematics Laboratory is equipped with
numbers, symbols, objects, counting devices, measuring materials, number patterns and relationships of quantities, it is central to mathematics curriculum at the primary and secondary levels in
Nigeria (Akanmu, 2017). Nneji and Alio (2017) observed that Mathematics as a subject does not only deal with manipulation of numbers, but goes further to explain practical relationships between the
numbers, attributes and application of the numbers to solving day to day practical life problems.
Purpose of the Study
The main purpose of this study is to investigate the effects of mathematics laboratory on the academic achievement of students in Geometry at junior secondary schools.
(1) Effect of mathematics laboratory approach on students achievement in Geometryat Junior Secondary School level.
(2) Effect of Mathematics laboratory approach on male and female students achievement in Geometryat Junior Secondary School level.
(3) Interaction effects of mathematics laboratory and gender on students achievement in Geometryat junior secondary school level.
Research Questions
The following research questions were addressed in the study:
(1) What are the achievements of students in Geometry before and after exposure to math lab facilities and conventional method?
(2) Do students differ in achievement by gender when taught Geometryusing mathematics laboratory facilities?
Research Hypotheses
The following hypotheses were formulated and tested at 5% level of significance:
Ho1: There is no significant effect of the usability of Mathematics laboratory facilities on the mean achievement scores of students taught Geometry
Ho2: There is no significant effect of gender on the mean achievement scores of students taught
Geometry using mathematics laboratory approach.
Ho3: There is no significant interaction effect of Mathematics laboratory method and gender on
students mean achievement scores in geometry.
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The mean of x and (1)/(x) is N then the mean of x^(2) and (1)/( x^(
The mean of x and 1x is N then the mean of x2and1x2 is
Step by step video solution for The mean of x and (1)/(x) is N then the mean of x^(2) and (1)/( x^(2)) is by Maths experts to help you in doubts & scoring excellent marks in Class 12 exams.
Updated on:21/07/2023
Knowledge Check
• The mean of x and 1x is N then the mean of x2and1x2 is
• If the mean of x and 1/x is M, then the mean of x2and1x2 is
• The mean of X1,x2,………,xn is 10 ,then the mean of 5x1,5x21…..5xn is: | {"url":"https://www.doubtnut.com/qna/649006219","timestamp":"2024-11-10T19:37:06Z","content_type":"text/html","content_length":"323303","record_id":"<urn:uuid:56a73f66-01bc-4228-9bb1-0974268780a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00468.warc.gz"} |
The Unscrambl Chai Expression Language (UEL)
The Unscrambl Chai Expression Language (UEL) is a simple expression language used to specify derived attribute expressions. It is based on the Drive UEL.
An expression in UEL can have one of the following primitive types: String (a string), Bool (a Boolean), Int16 (a 16-bit integer), Int32 (a 32-bit integer), Int64 (a 64-bit integer), or Double (a
double precision floating point number). It can also have a list type, where the element type of the list is one of the primitive types: List(String), List(Bool), List(Int16), List(Int32), List
(Int64), List(Double). It can also have a temporal type where the element type of the expression can be DateTime.
Bool, Double, Int32, Int64, and String literals can be present in an expression.
• A Bool literal can be either true or false.
• A Double literal is a decimal number that either contains the decimal separator (for example, 3.5, .5, 3.) or is given in the scientific notation (for example, 1e-4, 1.5e3). A Double literal must
be between (2 - 2 ^ -52) * 2 ^ 1023 (~ 1.79 * 10 ^ 308) and -(2 - 2 ^ -52) * 2 ^ 1023 (~ -1.79 * 10 ^ 308). The magnitude of a literal cannot be less than 2 ^ -1074 (~ 4.94 * 10 ^ -324).
• Integer literals without a suffix are of the Int32 type (e.g., 14). An Int32 literal must be between 2 ^ 31 - 1 (= 2147483647) and -2 ^ 31 (= -2147483648).
• The L suffix is used to create Int64 literals (e.g., 14L). An Int64 literal must be between 2 ^ 63 - 1 (= 9223372036854775807L) and -2 ^ 63 (= -9223372036854775808L).
• A String literal appears within double quotes, as in "I'm a string literal". The escape character \ can be used to represent new line (\n), tab (\t), double quote (\") characters, as well as the
escape character itself (\\).
Arithmetic operations
An expression in UEL can contain the basic arithmetic operations: addition (+), subtraction (-), multiplication (*), and division (/), and modulo (%) with the usual semantics. These are binary
operations that expect sub-expressions on each side. An expression in UEL can also contain the unary minus (-) operation that expects a sub-expression on the right. Parentheses (()) are used for
adjusting precedence.
Addition operation corresponds to concatenation for the String type and is the only available operation on this type. Bool type does not support arithmetic operations. Division applies integer
division on integer types and floating point division on Double types.
• A simple arithmetic expression: (3 + 4 * 5.0) / 2
• A simple expression involving a String literal: "area code\tcountry"
• A unary minus expression: -(3 + 5.0)
Comparison operations
An expression in UEL can contain the basic comparison operations: greater than (>), greater than or equal (>=), less than (<), less than or equal (<=), equals (==), and not equals (!=) with the usual
semantics. These are binary operations. With the exception of equals and not equals, they expect sub-expressions of numerical types or strings on each side. For strings, the comparison is based on
lexicographic order. For equals and not equals, the left and the right sub-expressions must be of compatible types with respect to UEL coercion rules. The result of the comparison is always of type
• An expression using comparisons: 3 > 5
• An expression using String comparisons: "abc" < "def"
• An expression using inequality comparison: "abc" != "def"
Logical operations
An expression in UEL can contain the basic logical operations: logical and (&&) and logical or (||) with the usual semantics. These are binary operations that expect sub-expressions of type Bool on
each side. Shortcutting is used to evaluate the logical operations. For logical and, if the sub-expression on the left evaluates to false, then the sub-expression on the right is not evaluated and
the result is false. For logical or, if the sub-expression on the left evaluates to true, then the sub-expression on the right is not evaluated and the result is true. An expression in UEL can also
contain the not (!) operator, which is a unary operator that expects a sub-expression of type Bool on the right.
• A simple logical expression: 3 > 5 || 2 < 4
• A logical expression that uses not: ! (3 > 5)
Ternary operation
An expression in UEL can make use of the the ternary operation: condition ? choice1 : choice2. The condition of the ternary operation is expected to be of type Bool and the two choices are expected
to be sub-expressions with compatible types. The ternary operation is lazily evaluated. If the condition evaluates to true, then the result is the first choice, without the sub- expression for the
second choice being evaluated. If the condition evaluates to false, then the result is the second choice, without the sub-expression for the first choice being evaluated.
• A simple ternary operation: 3 < 10 ? "smallerThan10" : "notSmallerThan10"
List Operations
A list is constructed by specifying a sequence of comma (,) separated values of the element type, surrounded by brackets ([]) . For instance, an example literal for List(Double) is [3.5, 6.7, 8.3]
and an example for List(String) is ["Unscrambl", "Drive"]. An empty list requires casting to define its type. As an example, an empty List(String) can be specified as List(String)([]).
An expression in UEL can contain a few basic list operations: containment (in), non-containment (not in).
Containment yields a Boolean answer, identifying whether an element is contained within a list. For instance, 3 in [2, 5, 3] yields true, whereas 8 in [2, 5, 3] yields false. Non-containment is the
negated version of the containment. For instance, 3 not in [2, 5, 3] yields false, whereas 8 not in [2, 5, 3] yields true.
Precedence and Associativity of Operations
Operations Associativity
unary -, !
*, /, % left
+, - left
>, >=, <, <= left
==, != left
&& left
|| left
Expressions in UEL can also contain attributes. Attributes are identifiers that are defined in the Querybot Entity Relationship Model. Each attribute has a type and can appear in anywhere a
sub-expression of that type is expected.
Attribute names that contain alpha-numeric characters and underscores can be used as is in the expression. Attributes names that contain spaces or any other unicode characters have to be enclosed in
an attribute identifier $"attribute with spaces".
• A string formed by concatenating an attribute named first name and an attribute named last name from the current entity: $"first name" + $"last name"
• An arithmetic expression involving an attribute and Int32 literals: numSeconds / (24 * 60 * 60)
• A floating-point arithmetic expression, where the floating point literal is of type Double: cost / 1000.0
• A Boolean expression checking if a String literal is found in an attribute named places of type List(String): "airport" in places
• A ternary expression indicating if the cost of a product is high or low cost < 100 ? "Low" : "High"
Null values
Attributes in UEL are nullable, that is, attributes can take null values. To denote a null value, the null keyword is used.
Only the following actions are legal on the expressions with null values:
• Built-in function calls: A nullable function parameter can take a null value. E.g.: the first parameter in the string.concat(null) call
• Ternary operations: The values returned from choices can be null. E.g.: amount > 100 ? "High" : null where amount is an attribute of the String type
• List items: Null values can be list items. E.g.: [null, 3, 5, 6, null]
• List containment check operations: Null values can be used on the left hand side. However, the result will be null independent of the contents of the list. E.g.: null not in [null, 3, 5, 6, null]
• Equality check operations: Null values can be compared for equality and non-equality. E.g.: $"purchase notes" == null where purchase notes is an attribute. (Note: For comparisons with null
values, the isNull operator can also be used. Examples: isNull(null) yields true isNull($"purchase notes") yields false if the purchase notes attribute is not null)
Database specific behaviors
• String concatenation: The string.concat function and the string addition operation can output different results for inputs containing NULL database values. Some databases flag this operation as
invalid and return NULL. Other databases discard the NULL values and concatenate the remaining strings. E.g.: "string" + null + "concatenation" yields either stringconcatenation or NULL E.g.:
name + "," + $"nullable profession". When nullable profession attribute is not NULL the result is the same for all databases E.g. Tom Hanks,Actor. When nullable profession attribute is NULL the
result can either be NULL or Tom Hanks,
• Modulo operator:
□ The modulo operation yields the remainder from the division of the first operand by the second. If both operands have the same sign, then the result is the same for all databases. E.g.: 11 %
4 = 3 and -11 % -4 = -3 If the operands have opposite signs, then the result may differ across databases. E.g.: 11 % -4 may either be 3 or -1. -11 % 4 may either be -3 or 2
□ Some databases support using floating point data types for the modulo operator. Others disallow this behavior and throw an error.
□ For modulo by zero case, most databases throw division by zero error. Other databases may return NULL or the first argument.
• datetime.add: Adding or subtracting months from a date or date-time value is almost the same for all databases. E.g.: "2010-03-01" + 1 month = "2010-04-01" and "2010-03-31" + 1 month =
"2010-04-30" If the initial date or date-time value is the last day of the month, some databases skip to the last day of the resulting month. E.g.: "2010-04-30" + 1 month is either 2010-05-31 or
• Rounding: The number.round function can output different results in case of the supplied value is halfway between two integers (E.g. -0.5, 1.5). Some databases round halves to the nearest even
integer (E.g. -0.5 -> 0, 1.5 -> 2), and some databases round halves away from zero (E.g. -0.5 -> -1, 1.5 -> 2). Some databases choose different strategies depending on the input data type. The
behavior of this function matches the behavior of the underlying database. | {"url":"https://documentation.unscrambl.com/qbo-insights/16.0.0/administration_guide/miscellaneous/uel.html","timestamp":"2024-11-10T21:27:30Z","content_type":"text/html","content_length":"42163","record_id":"<urn:uuid:d6496831-e490-4e17-a0ae-44ba07b9ce95>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00232.warc.gz"} |
Mathematics Required for Physics
• Thread starter narayan.rocks
• Start date
In summary, a high school student in grade 11 is looking to learn all undergrad physics topics and has been using MIT opencourseware for math. They have completed courses in single and multi variable
calculus and are seeking recommendations for books to help with their understanding of physics. Linear algebra and differential equations are the only math topics required for general physics, but
additional topics may be needed for specific areas such as quantum theory and general relativity. Mary L Boas and Riley's Mathematical Physics textbooks are highly recommended for undergrads.
Iam a High school Student starting grade 11 this year . I want to learn all undergrad physics chapters like Classical Mechanics , Optics , Statistical Mechanics , Thermodynamics , Electromagnetism ,
Quantum Mechanics , Solid State physics , Nuclear Physics , Plasma Physics , Relativity etc .
I have been learning math from MIT opencourseware I watch the video lecture / read lecture notes . Then i solve their problem sets and then their exams . I have completed the following courses
18.01 Single variable Calculus
18.02 Multi Variable Calculus
My question is how much math should i know so as to learn and understand all the concepts in undergrad physics . Also Suggest good books . Thank you
yeah thanks , but which one do i choose . Does these books cover all the math required for a physics undergrad
In addition to calculus, I would say that only Linear Algebra and Differential Equations are really required for general physics. Work specifically in quantum theory woul require group theory (though
probably not the full "abstract algebra") and general relativity uses differential geometry and tensor analyisis.
Thanks , so i think i can go with Mary L boas / Riley's Mathematical physics textbooks as they cover almost all of these
narayan.rocks said:
Thanks , so i think i can go with Mary L boas / Riley's Mathematical physics textbooks as they cover almost all of these
Surely Mary L Boas is highly Recommended text for Undergrad and want to do more then go to specific/Pure Maths books after that for Linear Algebra/Calculus/DE etc.
FAQ: Mathematics Required for Physics
1. What are the fundamental mathematical concepts needed for physics?
The most important mathematical concepts for physics include calculus, linear algebra, differential equations, and complex analysis. These concepts are used to describe and analyze physical
phenomena, make predictions, and develop theories.
2. Do I need to be an expert in math to study physics?
While a strong foundation in math is necessary for understanding and applying physics concepts, you do not need to be an expert in math to study physics. Many introductory physics courses focus on
teaching the necessary mathematical concepts along with the physics principles.
3. Can I study physics without being good at math?
Physics is a highly mathematical field, and it is difficult to understand and apply the principles without a good understanding of math. However, with dedication and practice, anyone can improve
their math skills and succeed in studying physics.
4. How does math help in solving physics problems?
Math is a powerful tool that allows us to represent and analyze complex physical phenomena. By using mathematical equations and formulas, we can model and predict the behavior of physical systems.
Math also helps us to think critically and logically, which is essential in solving physics problems.
5. What are some real-world applications of using math in physics?
Math is used extensively in various fields of physics, including mechanics, electromagnetism, thermodynamics, and quantum mechanics. Without math, we would not have been able to make important
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