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This applet helps visualise the surface generated by cylindrical coordinates using r,θ and z. Click and drag on the sliders on the left to adjust the ranges for r,θ and z. Geogebratube page for this
This applet visualises surfaces generated by spherical coordinates using r,θ and φ. Click and drag on the sliders on the left to change the values for r,θ and φ. Click and drag on the graph to
change/rotate the view. Geogebratube page for this applet
This applet shows a solution of the heat equation, a partial differential equation from MAST20029 Engineering Mathematics.
This applet shows the relationship between terms of a sequence and the partial sums of a series. It also allows exploration of some important sequences & series including geometric and harmonic
This applet plots and traces a parametric curve, given as a vector function in R2.
This applet illustrates a solution of the wave equation, from the MAST20029 Engineering Mathematics lecture notes.
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How to Find Sum of Digits of a Number in Java
The following Java program computes the sum of digits of a number. For example, if the input is 278, the output is calculated as 2+7+8 = 17. This function is useful in a number of numeric algorithms.
For example, IMEI check sum algorithm requires calculation of sum of digits. This problem is also given as an exercise for beginners in Java programming.
Algorithm for Finding Sum of Digits of a Number
We start by dividing the number with 10 and finding the remainder. The following program uses Java modulus operator for this. This gives us the right most digit. We then divide the number by 10 to
remove the right most digit. This process is repeated until no more digits are left in the number. Each digit we extract is added to a separate variable to compute the sum.
Java Source Code for Finding Sum of Digits of a Number
The following Java program uses only core Java APIs.
import java.util.Scanner;
// Java program to find sum of digits of a number
public class SumOfDigitsInJava {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.print("Please a number: ");
int number = scanner.nextInt();
int sumOfDigits = findSumOfDigits(number);
System.out.println("Sum of digits of " + number + " is " + sumOfDigits);
// Find sum of digits of a number
public static int findSumOfDigits(int number) {
int sum = 0;
while (number > 0) {
int digit = number % 10;
number = number / 10;
sum += digit;
return sum;
Here is a sample output of the above Java program,
java SumOfDigitsInJava
Please a number: 278
Sum of digits of 278 is 17
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Baccarat Rules and Strategy
Punto Banco Standards
Baccarat chemin de fer is played with eight decks of cards in a dealing shoe. Cards below ten are worth face value while Ten, Jack, Queen, King are zero, and Ace is one. Bets are placed on the
‘banker’, the ‘player’, or for a tie (these aren’t really people; they simply represent the 2 hands that are dealt).
Two hands of two cards are then dealt to the ‘house’ and ‘player’. The value for each hand is the sum total of the 2 cards, although the first digit is dropped. For example, a hand of 5 and 6 has a
total of one (five plus 6 equals eleven; ditch the first ‘1′).
A additional card will be given out based on the rules below:
- If the gambler or bank gets a score of 8 or 9, both players stay.
- If the gambler has less than 5, she hits. Players stands otherwise.
- If the gambler holds, the bank takes a card on a total less than 5. If the player hits, a chart is used to determine if the banker stays or takes a card.
Punto Banco Odds
The greater of the 2 scores wins. Winning wagers on the banker payout nineteen to Twenty (equal cash minus a 5 percent commission. The Rake is recorded and cleared out when you leave the table so
ensure you have cash around just before you quit). Winning wagers on the gambler pays 1:1. Winning bets for a tie typically pay 8 to 1 but on occasion nine to one. (This is a awful wager as a tie
occurs lower than 1 in every 10 hands. Be wary of betting on a tie. However odds are astonishingly greater for 9:1 versus 8 to 1)
Bet on correctly baccarat banque gives fairly good odds, aside from the tie bet of course.
Punto Banco Scheme
As with all games Baccarat has some familiar myths. One of which is the same as a myth in roulette. The past isn’t a prophecy of events about to happen. Keeping track of past outcomes on a chart is a
waste of paper and a snub to the tree that was cut down for our stationary desires.
The most familiar and likely the most accomplished strategy is the 1-3-2-6 technique. This method is used to maximize winnings and minimizing risk.
Start by wagering one chip. If you win, add another to the two on the game table for a sum total of 3 dollars on the second bet. Should you win you will now have six on the table, subtract four so
you keep 2 on the third round. Should you win the third bet, add two to the 4 on the game table for a sum total of 6 on the 4th wager.
Should you don’t win on the first wager, you take a hit of one. A win on the 1st wager followed by a loss on the second brings about a loss of 2. Wins on the 1st two with a loss on the 3rd gives you
with a profit of two. And success on the 1st 3 with a loss on the fourth means you balance the books. Succeeding at all 4 rounds gives you with twelve, a profit of 10. This means you will be able to
give up the second wager five instances for every favorable run of four wagers and still break even.
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Online ascending auctions for gradually expiring items
In this paper we consider online auction mechanisms for the allocation of M items that are identical to each other except for the fact that they have different expiration times, and each item must be
allocated before it expires. Players arrive at different times, and wish to buy one item before their deadline. The main difficulty is that players act "selfishly" and may mis-report their values,
deadlines, or arrival times. We begin by showing that the usual notion of truthfulness (where players follow a single dominant strategy) cannot be used in this case, since any (deterministic)
truthful auction cannot obtain better than an M-approximation of the social welfare. Therefore, instead of designing auctions in which players should follow a single strategy, we design two auctions
that perform well under a wide class of selfish, "semi-myopic", strategies. For every combination of such strategies, the auction is associated with a different algorithm, and so we have a family of
"semi-myopic" algorithms. We show that any algorithm in this family obtains a 3-approximation, and by this conclude that our auctions will perform well under any choice of such semi-myopic behaviors.
We next turn to provide a game-theoretic justification for acting in such a semi-myopic way. We suggest a new notion of "Set-Nash" equilibrium, where we cannot pin-point a single best-response
strategy, but rather only a set of possible best-response strategies. We show that our auctions have a Set-Nash equilibrium which is all semi-myopic, hence guarantees a 3-approximation. We believe
that this notion is of independent interest.
Conference Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Country/Territory United States
City Vancouver, BC
Period 23/01/05 → 25/01/05
Dive into the research topics of 'Online ascending auctions for gradually expiring items'. Together they form a unique fingerprint.
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Python and Web Development Tutor
Just name a few frequently used special method names in python
Called to implement evaluation of self[key]. For sequence types, the accepted keys should be integers and slice objects. Note that the special interpretation of negative indexes (if the class
wishes to emulate a sequence type) is up to the __getitem__() method. If key is of an inappropriate type, TypeError may be raised; if of a value outside the set of indexes for the sequence (after
any special interpretation of negative values), IndexErrorshould be raised. For mapping types, if key is missing (not in the container), KeyError should be raised. Note:for loops expect that an
IndexError will be raised for illegal indexes to allow proper detection of the end of the sequence.
__setitem__( self, key, value)
Called to implement assignment to self[key]. Same note as for __getitem__(). This should only be implemented for mappings if the objects support changes to the values for keys, or if new keys can
be added, or for sequences if elements can be replaced. The same exceptions should be raised for improper key values as for the__getitem__() method.
Called to implement deletion of self[key]. Same note as for __getitem__(). This should only be implemented for mappings if the objects support removal of keys, or for sequences if elements can be
removed from the sequence. The same exceptions should be raised for improper key values as for the __getitem__() method.
This method is called when an iterator is required for a container. This method should return a new iterator object that can iterate over all the objects in the container. For mappings, it should
iterate over the keys of the container, and should also be made available as the method iterkeys().Iterator objects also need to implement this method; they are required to return themselves. For
more information on iterator objects, see ``Iterator Types'' in the Python Library Reference.
The membership test operators (
not in
) are normally implemented as an iteration through a sequence. However, container objects can supply the following special method with a more efficient implementation, which also does not require the
object be a sequence.
Called to implement membership test operators. Should return true if item is in self, false otherwise. For mapping objects, this should consider the keys of the mapping rather than the values or
the key-item pairs.
Deprecated since release 2.0. Support slice objects as parameters to the __getitem__() method.
Called to implement evaluation of self[i:j]. The returned object should be of the same type as self. Note that missing i or j in the slice expression are replaced by zero or sys.maxint,
respectively. If negative indexes are used in the slice, the length of the sequence is added to that index. If the instance does not implement the__len__() method, an AttributeError is
raised. No guarantee is made that indexes adjusted this way are not still negative. Indexes which are greater than the length of the sequence are not modified. If no __getslice__()is found, a
slice object is created instead, and passed to __getitem__() instead.
__setslice__( self, i, j, sequence)
Called to implement assignment to self[i:j]. Same notes for i and j as for __getslice__().This method is deprecated. If no __setslice__() is found, or for extended slicing of the form self
[i:j:k], a slice object is created, and passed to __setitem__(), instead of __setslice__() being called.
Called to implement deletion of self[i:j]. Same notes for i and j as for __getslice__(). This method is deprecated. If no __delslice__() is found, or for extended slicing of the form self
[i:j:k], a slice object is created, and passed to __delitem__(), instead of __delslice__() being called.
Notice that these methods are only invoked when a single slice with a single colon is used, and the slice method is available. For slice operations involving extended slice notation, or in
absence of the slice methods, __getitem__(),__setitem__() or __delitem__() is called with a slice object as argument.
Emulating numeric types
The following methods can be defined to emulate numeric objects. Methods corresponding to operations that are not supported by the particular kind of number implemented (e.g., bitwise operations
for non-integral numbers) should be left undefined.
__floordiv__( self, other)
__pow__( self, other[, modulo])
These methods are called to implement the binary arithmetic operations (+, -, *, //, %, divmod() , pow() , **,<<, >>, &, ^, |). For instance, to evaluate the expression x+y, where x is an
instance of a class that has an__add__() method, x.__add__(y) is called. The __divmod__() method should be the equivalent to using__floordiv__() and __mod__(); it should not be related to
__truediv__() (described below). Note that__pow__() should be defined to accept an optional third argument if the ternary version of the built-in pow() function is to be supported.If one of
those methods does not support the operation with the supplied arguments, it should returnNotImplemented.
The division operator (/) is implemented by these methods. The __truediv__() method is used when__future__.division is in effect, otherwise __div__() is used. If only one of these two methods
is defined, the object will not support division in the alternate context; TypeError will be raised instead.
__rtruediv__( self, other)
__rfloordiv__( self, other)
These methods are called to implement the binary arithmetic operations (+, -, *, /, %, divmod() , pow() , **, <<,>>, &, ^, |) with reflected (swapped) operands. These functions are only
called if the left operand does not support the corresponding operation and the operands are of different types.^3.3 For instance, to evaluate the expression x-y, where y is an instance of a
class that has an __rsub__() method, y.__rsub__(x) is called if x.__sub__(y)returns NotImplemented.Note that ternary pow() will not try calling __rpow__() (the coercion rules would become too
Note: If the right operand's type is a subclass of the left operand's type and that subclass provides the reflected method for the operation, this method will be called before the left
operand's non-reflected method. This behavior allows subclasses to override their ancestors' operations.
__itruediv__( self, other)
__ifloordiv__( self, other)
__ipow__( self, other[, modulo])
These methods are called to implement the augmented arithmetic operations (+=, -=, *=, /=, //=, %=, **=, <<=,>>=, &=, ^=, |=). These methods should attempt to do the operation in-place
(modifying self) and return the result (which could be, but does not have to be, self). If a specific method is not defined, the augmented operation falls back to the normal methods. For
instance, to evaluate the expression x+=y, where x is an instance of a class that has an __iadd__() method, x.__iadd__(y) is called. If x is an instance of a class that does not define
a__iadd__() method, x.__add__(y) and y.__radd__(x) are considered, as with the evaluation of x+y.
Called to implement operator.index(). Also called whenever Python needs an integer object (such as in slicing). Must return an integer (int or long). New in version 2.5.
Called to implement ``mixed-mode'' numeric arithmetic. Should either return a 2-tuple containing self and otherconverted to a common numeric type, or None if conversion is impossible. When
the common type would be the type of other, it is sufficient to return None, since the interpreter will also ask the other object to attempt a coercion (but sometimes, if the implementation
of the other type cannot be changed, it is useful to do the conversion to the other type here). A return value of NotImplemented is equivalent to returning None.
Difference between _, __ and __xx__ in Python
originally from: http://igorsobreira.com/2010/09/16/difference-between-one-underline-and-two-underlines-in-python.html
When learning Python many people don't really understand why so much underlines in the beginning of the methods, sometimes even in the end like __this__! I've already had to explain it so many times,
it's time to document it.
One underline in the beginning
Python doesn't have real private methods, so one underline in the beginning of a method or attribute means you shouldn't access this method, because it's not part of the API. It's very common when
using properties:
class BaseForm(StrAndUnicode):
def _get_errors(self):
"Returns an ErrorDict for the data provided for the form"
if self._errors is None:
return self._errors
errors = property(_get_errors)
This snippet was taken from django source code (django/forms/forms.py). This meanserrors is a property, and it's part of the API, but the method this property calls,_get_errors, is "private", so you
shouldn't access it.
Two underlines in the beginning
This one causes a lot of confusion. It should not be used to mark a method as private, the goal here is to avoid your method to be overridden by a subclass. Let's see an example:
class A(object):
def __method(self):
print "I'm a method in A"
def method(self):
a = A()
The output here is
$ python example.py
I'm a method in A
Fine, as we expected. Now let's subclass A and customize __method
class B(A):
def __method(self):
print "I'm a method in B"
b = B()
and now the output is...
$ python example.py
I'm a method in A
as you can see, A.method() didn't call B.__method() as we could expect. Actually this is the correct behavior for __. So when you create a method starting with __ you're saying that you don't want
anybody to override it, it will be accessible just from inside the own class.
How python does it? Simple, it just renames the method. Take a look:
a = A()
a._A__method() # never use this!! please!
$ python example.py
I'm a method in A
If you try to access a.__method() it won't work either, as I said, __method is just accessible inside the class itself.
Two underlines in the beginning and in the end
When you see a method like __this__, the rule is simple: don't call it. Why? Because it means it's a method python calls, not you. Take a look:
>>> name = "igor"
>>> name.__len__()
>>> len(name)
>>> number = 10
>>> number.__add__(20)
>>> number + 20
There is always an operator or native function that calls these magic methods. The idea here is to give you the ability to override operators in your own classes. Sometimes it's just a hook python
calls in specific situations. __init__(), for example, is called when the object is created so you can initialize it. __new__() is called to build the instance, and so on...
Here's an example:
class CrazyNumber(object):
def __init__(self, n):
self.n = n
def __add__(self, other):
return self.n - other
def __sub__(self, other):
return self.n + other
def __str__(self):
return str(self.n)
num = CrazyNumber(10)
print num # 10
print num + 5 # 5
print num - 20 # 30
Another example:
class Room(object):
def __init__(self):
self.people = []
def add(self, person):
def __len__(self):
return len(self.people)
room = Room()
print len(room) # 1
The documentation covers all these special methods.
Use _one_underline to mark you methods as not part of the API. Use__two_underlines__ when you're creating objects to look like native python objects or you wan't to customize behavior in specific
situations. And don't use __just_to_underlines, unless you really know what you're doing!
1. First, take backup of your blogger template
2. After that open your blogger template (In Edit HTML mode) & copy the all css given in this link before</b:skin> tag
3. Paste the followig code before </head> tag
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shCore.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushCpp.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushCSharp.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushCss.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushDelphi.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushJava.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushJScript.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushPhp.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushPython.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushRuby.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushSql.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushVb.js' type='text/javascript'></script>
<script src='http://syntaxhighlighter.googlecode.com/svn/trunk/Scripts/shBrushXml.js' type='text/javascript'></script>
Of course, you don't need that many brushes, just keep what you really need.
4. Paste the following code before </body> tag.
<script language='javascript'>
5. Save Blogger Template.
6. Now syntax highlighting is ready to use you can use it with <pre></pre> tag.
<pre name="code">
...Your html-escaped code goes here...
<pre name="code" class="php">
echo "I like PHP";
7. You can Escape your code here.
8. Here is list of supported language for <class> attribute.
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26. The sum of the first 7 terms of an AP is 63 and that of its... | Filo
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26. The sum of the first 7 terms of an AP is 63 and that of its next 7 terms is 161. Find the AP.
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Question Text 26. The sum of the first 7 terms of an AP is 63 and that of its next 7 terms is 161. Find the AP.
Updated On Mar 18, 2023
Topic All topics
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 132
Avg. Video Duration 3 min
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Chapter Review
Concept Items
4.1 Force
What is dynamics?
a. Dynamics is the study of internal forces.
b. Dynamics is the study of forces and their effect on motion.
c. Dynamics describes the motion of points, bodies, and systems without consideration of the cause of motion.
d. Dynamics describes the effect of forces on each other.
Two forces acting on an object are perpendicular to one another. How would you draw these in a free-body diagram?
a. The two force arrows will be drawn at a right angle to one another.
b. The two force arrows will be pointing in opposite directions.
c. The two force arrows will be at a 45° angle to one another.
d. The two force arrows will be at a 180° angle to one another.
A free-body diagram shows the forces acting on an object. How is that object represented in the diagram?
a. A single point
b. A square box
c. A unit circle
d. The object as it is
4.2 Newton's First Law of Motion: Inertia
A ball rolls along the ground, moving from north to south. What direction is the frictional force that acts on the ball?
a. North to south
b. South to north
c. West to east
d. East to west
The tires you choose to drive over icy roads will create more friction with the road than your summer tires. Give another example where more friction is desirable.
a. Children’s slide
b. Air hockey table
c. Ice-skating rink
d. Jogging track
How do you express, mathematically, that no external force is acting on a body?
a. F[net] = −1
b. F[net] = 0
c. F[net] = 1
d. F[net] = ∞
4.3 Newton's Second Law of Motion
What does it mean for two quantities to be inversely proportional to each other?
a. When one variable increases, the other variable decreases by a greater amount.
b. When one variable increases, the other variable also increases.
c. When one variable increases, the other variable decreases by the same factor.
d. When one variable increases, the other variable also increases by the same factor.
True or False—Newton’s second law can be interpreted based on Newton’s first law.
a. True
b. False
4.4 Newton's Third Law of Motion
Which forces cause the motion of a system?
a. internal forces
b. external forces
c. both internal and external forces
d. neither internal nor external forces
True or False—Newton’s third law applies to the external forces acting on a system of interest.
a. True
b. False
A ball is dropped and hits the floor. What is the direction of the force exerted by the floor on the ball?
a. Upward
b. Downward
c. Right
d. Left
Critical Thinking Items
4.1 Force
Only two forces are acting on an object: force A to the left and force B to the right. If force B is greater than force A, in which direction will the object move?
a. To the right
b. To the left
c. Upward
d. The object does not move
In a free-body diagram, the arrows representing tension and weight have the same length but point away from one another. What does this indicate?
a. They are equal in magnitude and act in the same direction.
b. They are equal in magnitude and act in opposite directions.
c. They are unequal in magnitude and act in the same direction.
d. They are unequal in magnitude and act in opposite directions.
An object is at rest. Two forces, X and Y, are acting on it. Force X has a magnitude of x and acts in the downward direction. What is the magnitude and direction of Y?
a. The magnitude is x and points in the upward direction.
b. The magnitude is 2x and points in the upward direction.
c. The magnitude is x and points in the downward direction.
d. The magnitude is 2x and points in the downward direction.
Three forces, A, B, and C, are acting on the same object with magnitudes a, b, and c, respectively. Force A acts to the right, force B acts to the left, and force C acts downward. What is a necessary
condition for the object to move straight down?
a. The magnitude of force A must be greater than the magnitude of force B, so a > b.
b. The magnitude of force A must be equal to the magnitude of force B, so a = b.
c. The magnitude of force A must be greater than the magnitude of force C, so A > C.
d. The magnitude of force C must be greater than the magnitude of forces A or B, so A < C > B.
4.2 Newton's First Law of Motion: Inertia
Two people push a cart on a horizontal surface by applying forces F[1] and F[2] in the same direction. Is the magnitude of the net force acting on the cart, F[net], equal to, greater than, or less
than F[1] + F[2]? Why?
a. F[net] < F[1] + F[2] because the net force will not include the frictional force.
b. F[net] = F[1] + F[2] because the net force will not include the frictional force
c. F[net] < F[1] + F[2] because the net force will include the component of frictional force
d. F[net] = F[1] + F[2] because the net force will include the frictional force
True or False: A book placed on a balance scale is balanced by a standard 1-kg iron weight placed on the opposite side of the balance. If these objects are taken to the moon and a similar exercise is
performed, the balance is still level because gravity is uniform on the moon’s surface as it is on Earth’s surface.
a. True
b. False
4.3 Newton's Second Law of Motion
From the equation for Newton’s second law, we see that F[net] is directly proportional to a and that the constant of proportionality is m. What does this mean in a practical sense?
a. An increase in applied force will cause an increase in acceleration if the mass is constant.
b. An increase in applied force will cause a decrease in acceleration if the mass is constant.
c. An increase in applied force will cause an increase in acceleration, even if the mass varies.
d. An increase in applied force will cause an increase in acceleration and mass.
4.4 Newton's Third Law of Motion
True or False: A person accelerates while walking on the ground by exerting force. The ground in turn exerts force F[2] on the person. F[1] and F[2] are equal in magnitude but act in opposite
directions. The person is able to walk because the two forces act on the different systems and the net force acting on the person is nonzero.
a. True
b. False
A helicopter pushes air down, which, in turn, pushes the helicopter up. Which force affects the helicopter’s motion? Why?
a. Air pushing upward affects the helicopter’s motion because it is an internal force that acts on the helicopter.
b. Air pushing upward affects the helicopter’s motion because it is an external force that acts on the helicopter.
c. The downward force applied by the blades of the helicopter affects its motion because it is an internal force that acts on the helicopter.
d. The downward force applied by the blades of the helicopter affects its motion because it is an external force that acts on the helicopter.
4.3 Newton's Second Law of Motion
An object has a mass of $1 kg$ on Earth. What is its weight on the moon?
a. $1 N$
b. $1.67 N$
c. $9.8 N$
d. $10 N$
A bathroom scale shows your mass as 55 kg. What will it read on the moon?
a. 9.4 kg
b. 10.5 kg
c. 55.0 kg
d. 91.9 kg
4.4 Newton's Third Law of Motion
A person pushes an object of mass 5.0 kg along the floor by applying a force. If the object experiences a friction force of 10 N and accelerates at 18 m/s^2, what is the magnitude of the force
exerted by the person?
a. −90 N
b. −80 N
c. 90 N
d. 100 N
Performance Task
4.4 Newton's Third Law of Motion
A car weighs $2000 kg$. It moves along a road by applying a force on the road with a parallel component of $560 N$. There are two passengers in the car, each weighing $55 kg$. If the magnitude of the
force of friction experienced by the car is $45 N$, what is the acceleration of the car?
a. $0.244 m/s 2$
b. $0.265 m/s 2$
c. $4.00 m/s 2$
d. $4.10 m/s 2$
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Non-parametric survival analysis
In biomedical research, especially in the fields of epidemiology or oncology, one of the most common outcome under assessment is the time to an event of interest (also called “failure”), namely
survival time. The considered event is often death, but could be anything else such as cancer relapse or progression instead. The vast majority of survival analyses have extensively been using
Kaplan-Meier (KM) estimates, log-rank tests and Cox proportional hazards (CoxPH) models, all of which we will describe shortly. This post is an attempt to review the most classical approach to
survival analyses, in prevision for a following post discussing how everything fares in the high dimensional setting and solutions to potential problems that might arise.
Survival setting and notations
One difficulty arising when analysing time-to-event data is that not all subjects experimented the said event: survival times will be unknown for a subset of the study group. This phenomenon is
called censoring, of which there are three types (left, right and interval censoring). The Cox model only considers right-censored data: sometimes, a patient will drop out of the study (i.e.,
voluntarily or because he was still alive at the end of the study) and we only analyse the participants that haven’t had the event at the start of the study.
Let $T$ be a continous random variable representing survival time, with probability density function $f(t)$ and cumulative distribution function $F(T)$. The survival function $S(t)$ is the
probability of a patient surviving from the time origin (e.g. diagnosis of cancer) to a specified future time $t$, ie.,
$S(t) = P(T\geq t) = 1-F(T) = \int_t^\infty f(x)dx$
Another important function is the hazard function, denoted $lambda(t)$ (or often also $h(t)$), which is the instant probability that the event occurs right at time $t$:
$\lambda(t) = \lim_{dt\rightarrow0} \frac{P(t \le T < t + dt | T \geq t ) }{dt}$
To be a little bit more precise, the numerator corresponds to the conditional probability that the event will occur in the small interval $[t, t+dt)$ given that it has not occurred before. The whole
expression is then equivalent to the rate of event occurrence per unit of time.
From the previous definitions, it follows that
\begin{aligned}\lambda(t) &= \lim_{dt\rightarrow0} \frac{P(t \le T < t + dt, T \geq t ) / P(T \geq t)}{dt} \\ &= \lim_{dt\rightarrow0} \frac{f(t)dt / S(t)}{dt} \\ &= \frac{f(t)}{S(t)}\end{aligned}
By noticing that $-f(t)$ is the derivative of $S(t)$, we can rewrite our last result as
$\lambda(t) = - \frac{d}{dt}\log S(t)$
Finally, by integrating from $0$ to $t$ and introducing the boundary condition $S(0)=1$ (by definition, no event occurred yet at time $t=0$), we get the following expression:
\begin{aligned}S(t) &= \exp \left( - \int_0^t \lambda(x)dx \right) \\ &= \exp \left( - \Lambda(t) \right)\end{aligned}
where $\Lambda(t)$ is called the cumulative hazard and is interpretable as the sum of the risks faced going from $t=0$ to $t=t$.
In summary, we can say that while the hazard relates to the incident (current) event rate and survival reflects the cumulative non-occurrence, both provide equivalent characterizations of the
distribution $T$. Given the survival function, we can always differentiate to obtain the density and then calculate the hazard. Given the hazard, we can always integrate to obtain the cumulative
hazard and then exponentiate to obtain the survival function.
Kaplan-Meier estimation
However, estimation of the survival function is not trivial. If the data were not censored, the obvious estimate would be the empirical survival function derived from our data
$\hat{S}(t) = \frac1n \sum_{i=1}^n \mathbb{I}_{t_i > t}$
where $\mathbb{I}_{t_i > t}$ is the indicator function that takes the value $1$ if the associated condition is true and $0$ otherwise. The estimator is then simply the proportion of participants
alive (ie. that did not experiment the event) at time $t$. But due to censoring, we do not have the time of event $t_i$ for all patients and therefore must find another way to estimate the survival
function. Kaplan and Meier (1958) came up with a way to do that elegantly.
Small anecdote: after submitting similar manuscripts to the Journal of the American Statistical Association, its editor, Pr. John Tukey, then coworker of Kaplan at Bell labs and PhD thesis
director of Meier, convinced them to fuse their work into one single paper that will later be the most highly cited paper in the history of statistics.
Let $t*{(1)} < t*{(2)} < ... < t*{(m)}$ denote the distinct ordered times of events, and let $n\_i$ be the number alive just before $t*{(i)}$; $n_i$ is considered the number exposed to risk at time
$t\_i$. The Kaplan-Meier estimator of the survival function is
$\hat{S}(t) = \prod_{i:t_{(i)} < t} \left( 1 - \frac{d_i}{n_i} \right)$
which is really just a product of the observed probability to survive at time $t*{(1)}$ times the probability to survive at time $t*{(2)}$ conditioned on having survived at time $t_{(1)}$, and so on.
The value of $\hat{S}(t)$ being constant between times of events, the estimated probability is a step function that changes value only at the time of each event. Interestingly enough, this estimator
actually corresponds to the non-parametric maximum likelihood estimator (MLE) of the survival function. Furthermore, if there is no censoring, the KM estimate coincides with the empirical survival
The KM survival curve, a plot of the KM survival probability against time, provides a useful summary of the data that can be used to estimate measures such as median survival time. As an example,
we’ll be using a dataset of 42 leukemia patients in a clinical trial to compare treatment with placebo (from Freireich et al., 1963).
Each step of the curve corresponds to an event time at which one or several patients relapsed. X-axis corresponds to survival time in weeks and Y-axis to the proportion of survivors among
participants at risk at each time t. In pale red is the 95% confidence interval, and each tick on the curve corresponds to censoring time (ie. indicates when a patient was lost to follow-up without
having the event).
Here we plot survival curves estimates according to Gender groups. Visually, there doesn’t seem to be a constant difference in survival between females and males; even though it looks like a higher
proportion of males haven’t relapsed by the end of the study, their events rate is faster than that of females. We will check if this observed tendency coincides with a more formal test.
If we compare placebo vs. treatment, we can see some clear tendency that one group seems to have a slower rate of events and a lower proportion of relapsing patients.
It is good practice to also report the risk count table under the curves, which indicates at each time the number of participants at risk of having the event. It was omitted here for the sake of
The Kaplan-Meier curves are a really useful way to represent survival analyses, as they speak for themselves and are relatively easy to interprete. This quality is one of the main reasons that made
them so widespread in the medical research area. They are often accompanied by a p-value that is obtained by a formal test we’ll be discussing now.
Log-rank test
Survival in two or more groups can be compared by formal statistical tests, among which the log-rank test (Peto et al., 1977) is the most common by far—so common in fact, that you’re expected to
justify it in your paper if you’re using another one. It is a non-parametric test used to test the null hypothesis that there is no difference between the populations in the probability of an event
at any time point. At each time of events and for each group, we calculate the number of events one would expect since the previous event if there were no differences between the groups, and compare
them to the observed number of events with the following statistic
$\chi^2 = \sum_{g=1}^G \frac{(O_g - E_g)^2}{E_g}$
where $G$ is the number of groups that are being compared, $O_g$ the sum of all observed events and $E_g$ the sum of all expected events given by
$E_g = \sum_{i=1}^{m} \frac{O_{t_{(i)}}}{N_{t_{(i)}}}N_{g,t_{(i)}}$
where each $t*{(i)}$ is a time of events and $N*{t_{(i)}}$ the number of participants at risk of having the event at that time.
This value is then compared to a $\chi^2$ distribution with $G-1$ degrees of freedom to compute a p-value to be interpreted—we say that significant difference has been found between the hazard
functions estimates if $p < \text{threshold}$ (usually $0.05$). In the setting where we compare only two groups, the null hypothesis of the test is equivalent to saying that the ratio of the hazard
rates is equal to one; this hazard ratio (HR) is a measure of the relative survival experience in the two groups
$H_0: \text{HR}\approx\frac{O_1/E_1}{O_2/E_2}=1$
where $O_g/E_g$ is the estimated relative excess hazard in group $g$. This will serve as a very brief introduction to the important notion of hazard ratio; in practice, it is better to estimate it
using a regression modelling technique such as the Cox proportional hazards model we will study shortly.
Variable Log-rank $\chi^2$ statistic p-value
Gender $0.5569$ $0.4555$
Treatment $16.7929$ $<10^{-4}$
Above are the log-rank tests for the plots we saw in the previous section. Our visual intuitions have indeed been comforted by the tests results: we were not able to detect a difference in survival
between genders, but there is a significant one between the placebo and the treatment groups. Looking back at the plots, we can say that the highlighted difference is in favor of the treatment group.
Because the logrank test is purely a test of significance it cannot provide an estimate of the size of the difference between the groups or a confidence interval.
Interlude: assumptions
Even if the Kaplan-Meier estimate and the log-rank test are considered non-parametric, they still rely on non-trivial assumptions. Luckily, they are the same for both:
1. At any time, patients who are censored should have the same survival prospects as those who continue to be followed—ie. censoring should not be related to prognosis.
2. The survival probabilities should be the same for subjects recruited early and late in the study.
3. We assume that the event happens at the time specified. This is often not the case for events happening between two examinations for example.
These three conditions are seldom completely fulfilled; this rarely forbids the use of the above mentioned methods, but this can lead to some bias in the estimations and should be reported in the
Cox proportional hazards model
Initially, this post was only supposed to be about about the CoxPH model. But as I started writing, it seemed weird not to have an introduction to survival analyses in general, so I decided to write
the previous parts as an introduction to what I think is the core statistical model of a huge part of modern medical research—the Cox proportional hazard model (Sir David Cox, 1972).
Suppose that instead of studying the impact of just one variable on survival we now have a vector $\boldsymbol{x}$ of $p$ covariates. The hazard at time $t$ for an individual is assumed here to be
$\lambda(t|\boldsymbol{x}) = \lambda_0(t) \exp(\boldsymbol{x}\boldsymbol{\beta} )$
where $\boldsymbol{\beta}$ is a $p \times 1$ vector of unknown parameters and $\lambda_0(t)$ is an unknown baseline hazard function for all individuals corresponding to the case were all covariates $
As before, $t*{(1)} < t*{(2)} < ... < t*{(m)}$ denote the distinct ordered times of events. We’ll first consider the simple case where there are no ties—ie. only one subject had the event at $t*{(i)}
$ (if you wish to know how the Cox model handles ties, please see here)—and $R*i$ denotes the the set of indices of participants at risk at time $t*{(i)}$ (that are alive just before $t*{(i)}$). For
the subject $j*{(i)}$ that had the event at time $t_{(i)}$, its probability of failing at this specific time conditionally on the risk set is then as follows
$\frac{\lambda_0(t_{(i)}) \exp(\boldsymbol{x}_{j_{(i)}}\boldsymbol{\beta} )}{\sum_{j \in R_i}\lambda_0(t_{(i)}) \exp(\boldsymbol{x}_j\boldsymbol{\beta} )}$
That’s where the magic happens: the annoying baseline hazard term cancels out and we’re left with only
$\frac{ \exp(\boldsymbol{x}_{j_{(i)}}\boldsymbol{\beta} )}{\sum_{j \in R_i}\exp(\boldsymbol{x}_j\boldsymbol{\beta} )}$
As only failures contribute to the hazard function, the likelihood function, which is called a “partial” likelihood because we won’t bother estimating the baseline hazard function, is then just
$L=\prod_{i=1}^m\frac{ \exp(\boldsymbol{x}_{j_{(i)}}\boldsymbol{\beta} )}{\sum_{j \in R_i}\exp(\boldsymbol{x}_j\boldsymbol{\beta} )}$
And its log-likelihood has the form
$\log{L}=\sum_{i=1}^m \left(\boldsymbol{\beta}\boldsymbol{x}_{j_{(i)}}\boldsymbol{\beta} - \log \sum_{j \in R_i}\exp(\boldsymbol{x}_j\boldsymbol{\beta} ) \right)$
Efron (1977) showed that even though the partial likelihood is not a true likelihood, it loses little to no information for a wide variety of hazard functions and that it can be treated as a
likelihood for asymptotic inference. This lets us derive maximum likelihood estimates and asymptotic confidence intervals the usual way.
We can already summarize three very attractive features of Cox’s approach:
1. The “nuisance” function $\lambda*0(t*{(i)})$ is completely removed from the inference process on $\boldsymbol{\beta}$.
2. We can model the effect of multiple covariates at the same time in the $\exp(\boldsymbol{x}\boldsymbol{\beta} )$ term.
3. Data censoring does not affect the likelihood function, as it only depends on participants that experimented the event.
The only thing left now is to estimate the $\beta$ coefficients for all covariates via maximum likelihood estimation (MLE), which can be done effectively by numerous algorithms such as the
Expectation Maximization (EM) algorithm (such as here and here).
You will then get the following output, even though it might slightly differ depending on the statistical software and/or librairies you’re using:
Cox-PH model summary:
n = 42, number of events = 30
Variable coef exp(coef) se(coef) $z$ $p$ $\downarrow$ 0.95 $\uparrow$ 0.95
Gender 0.3147 1.3698 0.4545 0.6923 0.4887 -0.5761 1.2055
Treatment 1.5036 4.4978 0.4615 3.2580 0.0011 0.5990 2.4081
logWBC 1.6819 5.3760 0.3366 4.9971 0.0000 1.0223 2.3416
Concordance = 0.851
The coefficients in a Cox regression relate to hazard; a positive coefficient indicates a worse prognosis and a negative coefficient indicates a protective effect of the variable with which it is
The hazards ratio associated with a predictor variable is given by the exponent of its coefficient, with $\text{HR}<1$ indicating protective effect, $\text{HR}=1$ no influence and $\text{HR}>1$ worse
prognosis. It can be seen as a relative event rate and its interpretation depends on the measurement scale of the covariate in question; an increase in $x$ units leads to an increase in hazard rate
of $x \times\text{HR}$ if all other covariates remain the same. If we take the Treatment variable, as it is binary ($0$ for treatment, $1$ for placebo), we can say that receiving a placebo (ie. an
increase of one unit) leads to an increase in $4.5$ of the hazard rate (ie. at all times, an individual from the placebo group has $4.5\times$ more chance to have the event than an individual from
the treatment group that has the same gender and logWBC). If we take the logWBC variable (logarithm of the white blood cells count), we see that an increase in one unit leads to an increase in $5.4$
in the hazard rate. More broadly, if we consider the two following individuals
ID Gender Treatment logWBC
1 Female placebo 1
2 Male treatment 3
then their hazard rate can be expressed as
\begin{aligned}\lambda(t|\boldsymbol{x}_1) &= \lambda_0(t) \exp(\boldsymbol{x}_1\boldsymbol{\beta} ) \\ &= \lambda_0(t) \exp\left(x_{fem} \times \beta_{Gen} + x_{pla} \times \beta_{Treat} + x_1 \
times \beta_{WBC}\right) \end{aligned} \begin{aligned}\lambda(t|\boldsymbol{x}_2) &= \lambda_0(t) \exp(\boldsymbol{x}_2\boldsymbol{\beta} ) \\ &= \lambda_0(t) \exp\left(x_{mal} \times \beta_{Gen} +
x_{treat} \times \beta_{Treat} + x_3 \times \beta_{WBC}\right) \end{aligned}
and their hazard ratio is then
\begin{aligned}\text{HR}_{\boldsymbol{x}_1 / \boldsymbol{x}_2} &= \frac{\exp\left(x_{fem} \times \beta_{Gen} + x_{pla} \times \beta_{Treat} + x_1 \times \beta_{WBC}\right)}{\exp\left(x_{mal} \times \
beta_{Gen} + x_{treat} \times \beta_{Treat} + x_3 \times \beta_{WBC}\right)} \\ &= \frac{\exp\left(0 \times 0.3147 + 1 \times 1.5036 + 1 \times 1.6819\right)}{\exp\left(1 \times 0.3147 + 0 \times
1.5036 + 3 \times 1.6819\right)} = 0.1136\end{aligned}
thus individual $2$ has $1/0.1136=8.8$ times more chance of having the event (ie. leukemia relapse) than individual $1$.
Back to our Cox-model summary, the next value is the standard error of our coefficient estimate. Without going into too much details, it is calculated from the Fisher information matrix obtained by
taking the negative of the second derivative of our log-likelihood with respect to the coefficient of our covariate of interest. The inverse of this matrix is in fact the covariance matrix of our
coefficient; summing over the diagonal and taking the square root gives us the standard error.
The p-value of our coefficient is commonly given by a Wald test, which $\boldsymbol{z}$ statistic is calculated by $z=\beta / se(\beta)$ and then compared to a normal distribution. The lower and
upper bounds of the $95\%$ confidence interval is just ’}/>\beta \pm 1.96 \times se(\beta)’}/>. If the p-value is lower than a threshold (usually $0.05$), then we can say that our coefficient is
significantly different than $0$ (equivalent to saying that our HR is significantly different than $1$) and that our corresponding covariate is associated with the event rate. If not, then we cannot
say for sure that the value we estimated is actually reflecting the real value or not given our sample; in our example, the p-value for the variable Gender is non significant, which means we cannot
say much about its relation to the event rate. It doesn’t mean that it’s not related—maybe we just lack power and it would be significant with more participants in our study.
Interlude: assumptions
I lied a little with the title of this blog post: the Cox proportional hazards model is not totally non-parametric. Our model contains two unknown parameters, $\lambda_0 (t)$ and $\boldsymbol{\beta}$
, the former being an infinite-dimensional component, but the latter being finite-dimensional. We call such a model where both coexist semiparametric. Therefore, on top of the ones already listed
above in the previous interlude, it relies on another parametric assumption, namely the proportional hazards assumption.
I’ve already mentioned it several times; when I gave an interpretation for hazard ratios, I wrote that the difference in hazard rates was effective at all times. This is what the proportionality
assumption is. Putting it another way, the relative risk of two individuals with different covariate values is independent of time. For example, if the variable considered is Age, it would mean that
the hazard ratio for the considered event between a 25y.o and a 28y.o participants is the same as the one between a 95y.o and a 98y.o participants (increase of 3 units of age in both cases).
There are several ways to check that this assumption holds—Schoenfeld, Martingale, deviance and Cox-Snell residuals, etc.—but I personally find the plotting of the log cumulative hazards more
attractive visually and more easily interpretable at one glance. These are commonly nicknamed “log-log” plots because we plot the log cumulative hazard against the log of time. Recall from above at
each time $t$:
$\log\hat{\Lambda}(t) = \log(-\log(\hat{S}(t))) = \log(-\log(\hat{S}_0(t)))+\hat{\beta} x$
Thus, if the proportional hazards hold, the log cumulative hazards for each group should differ by the same constant—ie. the curves should be parralel.
The log cumulative hazard curves for the Gender variable cross each other, the PH assumption doesn’t hold here; on the opposite, the curves are parallel for the Treatment variable which means it
satisfies th PH assumption. For continuous variables, such as logWBC, we can plot the same after categorizing our data according to quantiles.
So now, what to do when there are covariates that do not satisfy the PH assumption? We will just shortly mention the two most common options out there: either stratifying the model according to the
concerned variables (if there’s just a few), or introducing time-dependent effects. The former method assumes PH in each strata, while the latter lets the impact of covariates vary over time. Both
are easily implemented in common statistical softwares.
Two other assumptions for the Cox model are made explicit by the formulation of the hazard function: it assumes linear combination of the covariates, as well as the fact that the link function is
exponential (cf. generalized linear models).
Evaluation of the model
Now that we have our model, how do we assess if our model is actually good or not? If it fits the data well enough?
The most widespread statistic to measure goodness-of-fit is Harrell’s C-index, or concordance index or C-statistic. Remember the little “concordance” below our model summary table? One can calculate
the C-index by taking all possible pairs of subjects consisting of one subject who experienced the event of interest and one subject who did not experience it. The C-index is the proportion of such
pairs in which the subject who experienced the event had a higher predicted probability of experiencing the event than the subject who did not experience it. It is in fact identical to the Receiver
Operating Characteristic (ROC) area under curve (AUC) for the binary outcome, and their scale of interpretation is the same:
• 0.5 indicates a model that does no better than chance
• 0.7 indicates a good model
• 0.8 indicates a strong model
• 1.0 indicates a perfect model
Many papers only report the C-statistic as a goodness-of-fit index of their model, but as stated by Steryerberg et al., it is not enough to capture all aspects of it. The C-statistic is a good
measure of the discrimination of a model—ie. the ability to discriminate between those with and those without the outcome—but other metrics should be reported alongside it such as calibration plots
or the Hosmer-Lemeshow test that measure calibration of a model (agreement between observed outcomes and predictions). When possible, overall performance should also be evaluated on external data
that was not used to fit the model.
Interpretation pitfalls
Many biases can be introduced from the design of the study, and should always be taken into account when interpreting the results of survival analyses (and of any other kind of analysis in general).
In clinical trials for example, proper randomization of treatment assignment is not always possible, and can lead to serious selection biases, ie. the samples obtained are not representative of the
population intended to be analyzed. Among those, one of the most prominent is the survival bias or survivorship bias, where the sample is only represented by participants that survived up to some
point. An excellent (and cynical) example of its consequences can be found in this correspondance to the authors of a biased paper published in one of the most influential oncology journal; in a
secondary analysis, they compared three treatment durations and concluded that longer treatment increased overall, disease-free and progression-free survival. However, patients who received treatment
for more than 5 years (ie, group 3) first had to live for 5 years—ie. group 3 was composed of subjects that were maybe just inherently more resistant than the two other groups. The following plots
were simulated by the authors of the correspondance
At first glance, it seems like group 3 has indeed better prospects than the other groups, with a significant p-value for the overall Wald test.
However, when keeping only the subjects that survived 5 years in all groups for analysis (the landmark method,
Anderson et al
, the advantage is no more.
There are many other types of biases one should be aware of when analysing results, especially in secondary analyses as study designs are usually conceived with only primary outcomes in mind and are
often ill-suited for others, but they relate to statistical biases in general and would take a whole post to write about.
This concludes my review of classical methods commonly found in survival analyses. I’ll admit it was a bit lengthier than I expected; I initially planned to write about recent improvements (or at
least tentatives) in machine learning and deep learning for survival analyses in the same post, but I guess I’ll leave it for later. I tried to be as exhaustive as possible while keeping it simple
and clear.
All code from this post can be found in a notebook on my github here.
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How to make a fraction in Word - Software Accountant
How to make a fraction in Word
The easiest way to make a fraction in Microsoft Word is to use the equation tool. The equation tool has a gallery that has lots of popular equation structures including fractions (even mixed
To make a fraction in Word, go to the Insert tab, in the Symbols group, click the drop-down arrow on the Equation button and select New Equation. Alternatively, just press Alt + ‘=’ on your keyboard.
In the Equation Tools Design tab that will appear, click the Fraction drop-down to select the desired structure of fraction.
This is just a brief summary of how to make a fraction in Word. For a more detailed step by step guide on this task, obey the following instructions with screenshots.
• Place the Insertion pointer at where you want to create the fraction.
• Press Alt + ‘=’ on your keyboard to insert the new equation field. Alternatively, Go to the Insert tab. In the Symbols group, click on the Equation button or click the drop-down arrow on the
Equation button and select New Equation from the drop-down menu.
• As soon as you press Alt + ‘=’ or click on the new Equation button, Word will insert the field to contain the fraction and also introduce the Equation Tools Design tab.
• On the Equation Tools Design tab, in the Structures group, click on the Fraction button.
• All the available Fraction structures will appear, select the one that best suit your needs.
• You’ll get an empty fraction of the selected structure. Fill the empty boxes with the numbers or content of your fraction.
• To make an improper fraction, first type the whole number before the fraction.
This is how you may insert or create a fraction in Word.
Aside from typing fractions in your Word document, the equation tool can help a lot especially if you are working on a project relating to math or science. With the equation tool, you can also insert
any symbol that is not readily available on the keyboard like the division sign.
Wednesday 20th of July 2022
Oh, Boy! I was very delighted to have learned how to make a fraction in the Word because I have a lot of mathemagic numbers to type for my lecture. I realized using the “Equation” for finding a
fraction in order to make some divisors and dividends with the equation (=) symbols. THANK YOU!
Wednesday 20th of July 2022
You are most welcome, Simon
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Real Academic Essays: STEM
Canadian Robert Langland is probably the greatest mathematician you have never heard of. A recent recipient of the Able Prize, the equivalent to the Nobel Prize in mathematics, Langland’s discoveries
of a relationship between number theory and harmonic analysis back in the 1960s has laid the foundation for newer fields of mathematics, such as the classic and expanded versions of the Langland
programs (Fairbank, 2018). Langland’s work is extraordinary as it touches on several disciplines within mathematics, a rare occurrence as this requires great talent. But what are the most commonly
studied branches of mathematics? Mathematics can be divided into several different fields of study with the three most commonly known ones being arithmetic, algebra, and geometry.
Arithmetic is the best-known and oldest branch of mathematics. It relates to numbers, their relationships, and their use in solving problems (MacDuffee, 2019). This is the branch of mathematics
children first learn in school and includes all the basic mathematical computations: addition, subtraction, division, and multiplication. It also includes counting, the measurement of things, raising
to powers, and finding the root of numbers (New World Encyclopedia, 2019). The earliest sign of arithmetic concepts among humans can be traced back to Africa’s Ishango Bone, believed to date back
twenty thousand years to a lake community on the border of Zaire and Uganda (Williams, 2008). The Ishango Bone not only appears to be the oldest record of prime numbers but also suggests the use of
multiplication in ancient times.
Algebra is another well-known branch of mathematics. Algebra deals with abstract symbols and their manipulation in the form of equations (Coolman, 2015). Strikingly different than other branches of
mathematics, it stems from the Golden Age of Islam and is credited to the Persian scholar and mathematician Al-Khwarizmi who wrote the book ‘Al-jabr wa’l muqabalah’ – from which the name algebra
comes – in Baghdad in 825 A.D. (Corry, 2019). Elementary algebra is taught in secondary schools around the world and focuses mainly on the variables x and y (Pratt, 2017). Algebra has been highly
regarded as such an effective tool by other mathematical domains that subbranches have emerged as a result, ranging from algebraic logic to algebraic geometry.
Geometry is also a commonly known branch of mathematics. Although geometry is found mainly in secondary schools around the world, the basic principles are usually embedded into elementary education
as well. As you probably know, geometry concerns itself with shapes and the relationships between objects and spaces (Heilbron, 2019). As one of the oldest branches of mathematics, geometry stems
back to ancient Greece with its name derived from the Greek words ‘geo’ and ‘metron’, meaning ‘earth measurement’. Whereas many scholars played a role in its creation, no one was more influential
than Euclid and his work entitled The Elements, published over 2000 thousand years ago (Palmer, 2015). Impressively, this book is considered to be the world’s most widely used textbook for geometry
and mathematics. It brought together and proved various geometric ideas already in use at the time (Palmer 2015).
Although mathematics has a number of different branches of study, the three most commonly known ones are arithmetic, algebra, and geometry. More sophisticated branches include statistics,
trigonometry, and calculus – with the latter two also taught in many secondary schools. With time, new discoveries will be made that will lead to new branches of mathematics, just like Langdon’s
discoveries have. A hundred years from now who knows how many branches there will be and what mathematics will look like.
Coolman, R. (2015). What is algebra? Retrieved February 17, 2020 from https://www.livescience.com/50258-algebra.html
Corry, L. (2019). Algebra mathematics. Retrieved February 17, 2020 from https://www.britannica.com/science/algebra/Classical-algebra
Fairbank, V. (2018). The greatest mathematician you have never heard of. Retrieved February 16, 2020 from https://thewalrus.ca/the-greatest-mathematician-youve-never-heard-of/
Heilbron, J. (2019). Geometry. Retrieved February 17, 2020 from https://www.britannica.com/science/geometry
MacDuffee, C. (2019). Arithmetic. Retrieved February 16, 2020 from https://www.britannica.com/science/arithmetic
New World Encyclopedia (2019). Arithmetic. Retrieved February 16, 2020 from https://www.newworldencyclopedia.org/entry/Arithmetic
Palmer, N. (2015). Euclid. Retrieved February 17, 2020 from https://www.ancient.eu/Euclid/
Pratt, V. (2017). Algebra. Retrieved February 17, 2020 from https://plato.stanford.edu/entries/algebra/
Williams, S. (2008). Mathematicians of the African diaspora. Retrieved February 16, 2020 from http://www.math.buffalo.edu/mad/Ancient-Africa/ishango
Academic Writing Exercises
Contextual Grammar Exercises
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CyFlex Math and the temperature “GOTCHA”
October 16, 2017
CyFlex Math and the temperature “GOTCHA”
Units Independence in CyFlex computations:
CyFlex supports “units-independence”. Internally, applications such as “compvar”, mass flow rate, etc. perform computations using SI (metric) base units. Inputs and outputs to these programs involve
a units conversion process so that the inputs and outputs can be in other units. A simple example of this is the computation of horsepower from engine speed and torque. The speed may be in “rpm”, the
torque in “lb-ft” and the desired result to be in “hp”. The internal process is as follows.
1. Convert the speed input value from “rpm” to “radians/sec”
2. Convert the torque input value from “lb-ft” to “newton-meters”
3. Multiply speed by torque, producing a power value in “watts”
4. Convert the power from “watts” to “hp”
5. Store the power in “hp” in the output variable
In a CyFlex computed expression, this is simply “eng_spd * eng_torque” and it doesn’t matter if eng_spd and eng_torque are in English or Metric units. The result will still be computed properly.
Likewise, the output target could be “hp” or “kw” and the appropriate conversion from “watts” will be performed.
See Section 7 of the “CyFlex Variables, Units, Computed Expressions” user manual for details of the rules for writing computed expressions.
The Temperature “GOTCHA”:
The description above also describes how conversion takes place for temperature values, but the normal Celsius and Fahrenheit units have a characteristic that is different from other units. They have
an offset (bias). Units of pressure, speed, torque, etc., do not have an offset, so 0[psi] = 0[kpa] = 0[in_hg] = 0[in_h20] and so on. However,
0[deg_c] = 273.15[K] (Kelvin is the SI base units)
0[deg_f] = 255.372[K]
So, now take an example of a simple addition expression:
“100[deg_f] + 1[deg_f]”
You expect the result to be 101[deg_f], but here is how the math works:
100[deg_f] converts to 310.928[K]
1[deg_f] converts to 255.928[K]
Summing these values yields a result of 566.855[K]
Converting the result to “deg_f” yields 560.67[deg_f] ????
What if we just try to subtract 1[deg_f] “100[deg_f] – 1[deg_f]”
The subtraction (310.92 – 255.928] yields 55[K] which converts to -360.67[deg_f] ???
That demonstrates the problem. But what is the solution. CyFlex has some special temperature units referred to as “temperature differential” units, “dt_f”, “dt_c”, “dt_k”, and “dt_r”. These units
have the bias removed from the units conversion factors.
So, the 100+1 math works as follows when the differential units are specified:
“100[deg_f] + 1[dt_f]”
100[deg_f] converts to 310.928[K]
1[dt_f] converts to 0.5555[K]
Summing these values yields a result of 311.483[K]
Converting the result to “deg_f” yields 101[deg_f] (ta da!)
Likewise, for subtraction.
Testing expressions from the command line:
You can experiment with any computed expression using the “get_comp” application. Type “use get_comp” for help.
get_comp “expression” [u=units]
If u=units are not included, the result is always reported in base SI units. Example:
get_comp “100[deg_f] + 1[dt_f]”
get_comp “100[deg_f] + 1[dt_f]” u=deg_f
This units-independence feature is why numeric values in CyFlex expressions include units of measure.
Other benefits of units-independence are that it allows a measurement in one set of units to be displayed, logged, or monitored in different units.
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Techno-Economic Comprehensive Review of State-of-the-Art Geothermal and Solar Roadway Energy Systems
School of Architecture and Urban Planning, Shandong Jianzhu University, 1000 Fengming Road, Jinan 250101, China
Griffith School of Engineering and the Built Environment, Griffith University, Brisbane, QLD 4222, Australia
School of Architecture, Nanjing Tech University, Nanjing 211816, China
Faculty of Engineering, Kyambogo University, Kampala P.O. Box 01, Uganda
Authors to whom correspondence should be addressed.
Submission received: 8 August 2022 / Revised: 25 August 2022 / Accepted: 29 August 2022 / Published: 2 September 2022
Road infrastructure is a vital constituent element in the transportation network; however, roadway surface ice and snow accumulation leads to huge traffic accidents in winter. Geothermal roadway
energy systems (GRES) and solar roadway energy systems (SRES) can increase or decrease roadway surface temperature for the de-icing and removal of snow in winter, or mitigation of heat in summer.
Technology performance and economic evaluation of the GRES and SRES are reviewed in this paper based on numerical and economic models, and experimental analyses. Three crucial aspects of the
technology performance assessment, i.e., roadway surface temperature, energy consumption and key factors, are explored in different regions and countries. Economic evaluation approaches for net
present values and payback periods of the GRES and SRES are investigated. The recommendations and potential future developments on the two technologies are deliberated; it is demonstrated that the
GRES and SRES could increase roadway surface temperature by around 5 °C in winter and decrease it by about 6 °C in summer, with the payback periods of 4 to 8 years and 2.3 to 5 years, respectively.
1. Introduction
The roadway and bridge are the primary civil infrastructures used to link different regions [
], and are considered the structure platforms; however, the bridge decks and roadway surfaces are exposed to solar radiation and vehicle loading, which causes thermal gradient and mechanical
vibration within the layers of the pavement [
]. Additionally, freezing and snow accumulation damage the roadway and compromise road user safety [
]. The conventional way for de-icing and snow melting roadway is to utilize salts (i.e., calcium chloride, sodium chloride and potassium acetate) and other chemical materials, which is able to bring
down the freezing point of water to avert the formation of ice [
]. Nevertheless, this approach raises some issues and has some limitations, such as soil pollution, vehicle corrosion, declined durability of pavement material, temperature limitation (below −3.9
°C), large manpower and being dangerous to the environment [
]. Hence, some alternative methods to remove snow and ice are exploited to avoid the above issues. By far, two renewable energy technologies, geothermal roadway energy system (GRES) and solar roadway
energy system (SRES), are becoming promising for snow melting and de-icing applications due to their cost-effective and pro-environment characteristics [
]. Specifically, the GRES extracts heat from geothermal hot water and soil; meanwhile, it is able to absorb solar energy during sunny days and releases heat for ice and snow melting in winter. In
summer, the system could cool the roadway, store heat within the soil for being reused in winter [
]. On the other hand, the utilization of solar energy largely involves two modes: converting solar radiation into heat and electricity energy. The heat can be utilized to melt snow and ice on
roadways in winter, while the electricity could be supplied to the grid, or utilized for unmanned driving of smart roads and wireless charging in the future [
Conventionally, de-icing and removal of snow on road surfaces is done based on an integrated manual and machine-based solution which are expensive as well; however, this method to monitor damage is
not just a waste of time but is also ineffective since the detection of such damages needs consistent assist from subject matter experts who have the ability to identify and differentiate various
categories of pavement failures. Thereby, in this review, the renewable energy technology is employed to get command of the roadway surface temperature owing to the de-icing and removal of snow in
winter, or mitigation of heat in summer; these designated and retrieved cited studies are concerning the techno-economic analysis of the GRES and SRES applied in various countries and areas, and the
numerical models and experimental test are performed based on different boundaries and assumption conditions including weather condition, fluid velocity, solar radiation, initial temperature, thermal
properties definitions and economic index. For the current research, the most challenging point in designing the GRES and SRES is to identify heat source. Although the two technologies are taken into
account as alternative solutions, their performance and costs are influenced by climatic conditions, working fluid, pipe configuration, soil property, concrete slab and initial conditions. Hence, the
aim of this study is to review the techno-economic performance of the GRES and SRES applied in roadways to provide comprehensive information regarding numerical models, experimental data and
advancement of engineering application. Firstly, the basic knowledge of the two systems is described in
Section 2
, then, the technical analyses, for example, numerical modeling, laboratory research, field testing and material design, are summarized in
Section 3
. The economic assessment of the two systems is clarified in
Section 4
, whereas the future challenges and recommendations are put forward in
Section 5
, the key conclusions are presented in
Section 6
2. Geothermal and Solar Roadway Energy Systems
Figure 1
depicts the effective energy-extracted technologies that could be utilized in the road. By far, the GRES and SRES are becoming promising and advanced solutions for snow melting and de-icing
2.1. Geothermal Roadway Energy System
The GRES is a renewable-based energy system applied on a roadway. There are two core sections of the system, de-icing (and/or heating) roadway by soil heat energy extracted in winter and cooling
roadway via circulated working fluid in summer. Normally, a GRES consists of a ground heat exchanger, a pipe network and a heat pump.
Figure 2
presents a detailed illustration of the working principle of a GRES [
]. In winter, the stored heat is released to the roadway surface for de-icing and snow melting. While in summer, the roadway surface exposed to the sun reaches a high temperature ranging between 60
°C and 70 °C [
], so this thermal energy can be stored. The working fluid is circulated to cool warm pavement with the aim of decreasing roadway surface temperature. Subsequently, the working fluid is circulated
back to the ground, which acts as a heat energy storage, for utilization during the heating season [
]. Generally, traditional GRES needs a set number of buried pipe heat exchangers which are independent of the foundation structure. In order to overcome the drawback, a bi-functional GRES, called an
energy pile (EP) system, is utilized to support loads of structure and exchange heat with soil to reduce the initial cost of installation; furthermore, various types of pipe such as single U-tube,
double U-tube, tripe U-tube and helical pipe have been used in the GRES, which are utilized to test and compare system performance.
2.2. Solar Roadway Energy System
The SRES is another renewable technology-based system applied on the roadway. Typically, an SRES comprises a pipe network with a working fluid inside, buried beneath a roadway. When the roadway
absorbs energy from the sun, its temperature is raised and the heat is transferred to a working fluid within the pipe network because of temperature difference. As depicted in
Figure 3
, there are three fundamental heat transfer processes in the SRES including convection, conduction and radiation [
]. In the conduction process, heat is conducted between the pipe walls and the roadway; this heat convection takes place when there are temperature gradients among the roadway, pipe walls and the
thermal fluid. The radiation process happens through electromagnetic waves without any material medium, whereby the solar radiations are transmitted to the roadway whereas heat is radiated between
the roadway and air temperature [
]. Generally, an SRES has the ability to alleviate the influence of the heat island effect (HIE) by means of decreasing roadway temperature [
]. The cooling effect contributes to sustaining roadway performance as well as reducing roadway deterioration under high-temperature climate conditions.
3. Technical Evaluation
3.1. Geothermal Roadway Energy System
The GRES has the advantages of using renewable energy sources and is environmentally friendly, improved roadway service life, and decreased urban HIE [
]. The effects of using different pipe arrangements for de-icing and snow melting are summarized in the subsequent sections.
3.1.1. Geothermal Bridge Deck Energy System
Liu et al. [
] developed a transient heat transfer model of the GRES for bridge deck to assess the influences of climate conditions and flow rate on system performance in Canada.
Figure 4
a presents the GRES based on the EP solution.
Figure 4
b illustrates the heat transfer mechanism of the bridge deck on the basis of radiation, convection as well as sensible and latent heat.
Table 1
exhibits the energy balance equations.
Figure 5
shows the effects of various influence factors, including snowfall rate, solar radiation, ambient temperature and wind speed, on energy consumption. Results reveal that the increases in snowfall rate
and wind speed could give rise to a growth in energy consumption of 35% and 12% whereas the decreases in solar radiation and air temperature could increase energy consumption by 9% and 6.4%,
As demonstrated in
Figure 6
, the heat extraction rate of the GRES in terms of spiral-, W- and U- shapes could be enhanced by 3.4, 2.7 and 2 times, respectively, when the flow rates vary from 0.1 m/s to 4 m/s; this implies that
the flow rate has the most vital influence on the heat extraction rate of the spiral shape.
Yu et al. [
] designed an experimental rig of the GRES to investigate heat transfer characteristic and renovate some existing bridges. The schematic diagrams of the GRES and polyethylene pipe loop are exhibited
Figure 7
. A lab-scale testing is performed to estimate temperature variation of the concrete slab at different locations as presented in
Figure 8
; this system consists of 10 polyethylene pipes with a diameter of 13 mm, a water tank and a pump. In the testing, the indoor and water tank temperatures are setup to 4.4 °C and 32.2 °C,
Figure 9
displays the infrared pictures of the temperature distribution on the slab surface, which could reach an average of 12.8 °C (55 °F); it is indicated that the temperature is higher towards the centre
of the concrete slab and progressively reduces outwards. In comparison, the mean temperature at the interface of the geofoam and concrete is 16.1 °C (61 °F). According to
Figure 10
, about 60% of the heat is shifted to the slab surface, indicating that around 40% of the heat is missed in the external region of the concrete slab; moreover, the heat transfer efficiency slightly
increases by about 1% when the thermal load rises from −1.1 °C (30 °F) to 15.6 °C (60 °F).
Later, Li et al. [
] built a 3D numerical model of a geothermal bridge deck to evaluate the system performance, and concluded that about 76% of overall supplied heat could be transferred to the top surface of the
bridge deck based on different ambient air temperature conditions. In another research, Fabrice et al. [
] developed a 3D finite element model for de-icing of the bridge to analyze the thermally induced stresses at different seasons.
Figure 11
presents the model mesh and pipe of the monitored location. The total mesh of the model includes 23,760 nodes and 21,060 hexahedral elements, where the initial temperature is set at 11 °C, which is
imposed on all faces except the top surface. The results in
Figure 12
indicate that thermal stresses have vital influences on the local and pile temperatures. Most of the stress variation appears at the initial stages of extraction and injection when the maximal
temperature gradients occur in the region. Notably, the average overstresses observed are 80 kPa/°C and 90 kPa/°C for cooling and heating seasons, respectively, which are somewhat higher compared to
those under the natural recharge state.
In another development, Kong et al. [
] experimentally tested a GRES for a bridge of 36 m by 26 m dimensions, that has two bicycle and four vehicle lanes. As presented in
Figure 13
, the GRES is mounted on the first span slab of the bridge and only half way is covered transversely. The pipe is a polyethylene (PE) tube that has the outer and inner diameters of 16 and 20 mm,
which is embedded in the 100 mm thickness of concrete slab. Meanwhile, a thermal water tank is placed between the bridge and EP. The results illustrated in
Figure 14
a demonstrate that about 25.7% of the thermal expansion strain is limited via the unheated concrete slab, and the stress through the GRES reaches up to 206 kPa, which is far lower in comparison with
the design parameter of the C40 concrete compressive strength of 19.1 MPa, as shown in
Figure 14
3.1.2. Geothermal Pavement Energy System
For the road pavement, Mirzanamadi et al. [
] implemented an experimental testing of the GRES to measure the pavement surface temperature based on Sweden’s weather condition. There is no noticeable infrastructure near the experimental site as
displayed in
Figure 15
a, and therefore the shading impact of neighbouring infrastructure on the surface of the pavement is ignored; furthermore, the layers of the pavement and relevant parameters are given in
Figure 15
Afterwards, Mirzanamadi et al. [
] built a 3D heat transfer model of the GRES to investigate the unsteady anti-icing method on the basis of the superposition principle. Specifically, the model has a dimension of 1000 × 1000 × 300 mm
(L × W × D) with 50 mm depth, a pipe distance of 200 mm is given in
Figure 16
a. For the purpose of reducing the computational time, the symmetrical section A-B-C-D-E-F is used to simulate the heat transfer process. According to
Figure 16
b, the 3D model is replaced by four 2D vertical cross sections that are serially linked to each other, the initial temperatures of the pavement bottom and top boundaries are set at 0 °C as given in
Figure 17
. The basic heat transfer equations are given in
Table 2
It is found from
Figure 18
that the numerical results are in agreement with the testing data, with a maximum difference of 2.4%. Based on the hybrid 3D model, Mirzanamadi et al. [
] investigated the system performance for de-icing pavement surface for a 15-year operation period, and found that the maximum value associated with the solar energy harvested is 30 kWh/m
in July whereas the minimum value is 0.5 kWh/m
in April. Furthermore, the maximum energy demand is 25 kWh/m
in December and January, while the mean value of the required energy and the residual number of hours of slippery conditions from October and March are 1.3 kWh/m
and 9 h, respectively.
Adl-Zarrabi et al. [
] developed a 3D COMSOL model of a GRES to study the effect of the pipe position on de-icing performance as presented in
Figure 19
. The system involves a surface layer of 150 mm thickness, the base of 250 mm thickness and subbase courses as well as pipes of 1.5mm thickness, which are buried in the concrete slab. The results in
Figure 20
a show that the required time for melting the snow on the pavement could be enhanced speedily when the distance between pipes exceeds 200 mm. In other words, the best distance between pipes should be
less than 200 mm. Additionally, as depicted in
Figure 20
b, there is little effect of the pipe depth on the anti-icing process when its depth is lower than 100 mm.
Xu et al. [
] setup a numerical model of the GRES to analyse the influences of preheating time and snowfall rate on the snow melting performance at Beijing New Capital International Airport as shown in
Figure 21
. The whole area of the experimental site is 90 m
, with a stainless steel pipe of 32 mm diameter, 0.4 m length and a depth of 0.08 m embedded underground. A geothermal heat pump unit with a rated power of 50 kW is utilized to warm 25% of ethylene
glycol solution. The basic equations of water transport, heat transport and error analysis are presented in
Table 3
. The results indicate that the percentage of snow-free hours during snowfall at four preheating times is improved ranging from 1.3% to 5.6%. As a result, it is necessary to adjust the snow melting
target by the traffic capacity as the design alternatives.
Han and Yu [
] set up a 3D model of the GRES with EP technology to assess the energy extraction rate and required pile number for three configurations in the USA. As shown in
Figure 22
a, the soil domain is defined as a cylinder and has a diameter of 12 m. The Dirichlet boundary condition is used as the borders of the calculation field with the magnitude set to be the undisturbed
soil temperature as presented in
Figure 22
Additionally, it can be found from
Figure 23
that high soil temperature contributes to extracting energy and producing high-temperature outlet fluid; moreover, the spiral pipe shape (type c) could extract more heat in comparison with U-shape
(type a) and W-shape (type b) pipes; this means that the spiral pipe shape is the best choice for the system to improve snow melting performance under the constraint of limited pile length.
In addition, Han and Yu [
] modified the GRES with EP by integrating a phase change material (PCM) in order to enhance system performance as shown in
Figure 24
. The equations of the modified GRES model are given in
Table 4
. As shown in
Figure 25
, the required numbers of PCM piles for U-, W- and spiral-shape pipes significantly reduce in comparison with those without PCM attached (wt % = 0). The utilization of 3% PCM additive by mass
fraction leads to a 25–35% drop in the needed number of piles for the designated cities based on design conditions, and the usage of 12% PCM decreases the pile number by 60–70%; this means that the
soil temperature and pipe configuration layout have an influence on the needed number of piles of the modified GRES.
In another study [
], a finite element GRES model setup to investigate the outlet fluid temperature variation based on USA’s climate condition. As given in
Figure 26
, the system includes a heat pump and horizontal pipe loops that are laid under the soil at a depth of 6 m. Results confirm that the outlet fluid temperature could be kept higher than 4 °C when the
soil is at full saturation, as indicated in
Figure 27
. On the other hand, the outlet fluid temperature could fall to −0.7 °C when the soil is completely dry; this implies that a dry soil is not an ideal medium to embed pipes within the soil layer.
Yang et al. [
] developed a GRSE model for the underground utility tunnel (UUT) to extract heat from the soil and absorb waste heat within the tunnel. As exhibited in
Figure 28
, the UUT is constructed at a depth of 3–6 m under the urban roadway, which is deeper compared with the frozen soil layer. The dimension of the UUT is 3.0 × 2.8 m with 0.3 m of wall thickness. As
indicated in
Figure 29
, the model results reflect that the outlet fluid temperature dramatically reduces while the maximum temperature difference between inlet and outlet is approximately 1 K when the inlet fluid
temperature and flow velocity vary in the ranges of 280.15 K to 278.15 K and 0.1 m/L to 0.5 m/L, respectively; this suggests that the lower inlet water temperature and wind velocity improve the
efficiency of heat transfer.
Chiarelli et al. [
] conducted a novel testing of the GRES called the ground source heat simulator to investigate the impact of the inlet air temperature and wind speed on system performance. As shown in
Figure 30
, the dimension of this prototype is 470 × 700 × 180 mm (L × W × H) with a 50 mm thickness of asphalt wearing course and 130 mm of thickness coarse gravel layer; this experiment is carried out at
University of Nottingham, UK and divided into two scenarios including air temperature is higher and lower 15 °C of inlet temperature.
Their results reveal from
Figure 31
a that the most frequent inlet air temperature appears in the range from 14 °C to 15 °C, while the temperatures between 10 °C and 13 °C are also rather frequent; this is because the system is exposed
to the outdoor environment, thereby inlet air temperature could not be maintained as a constant value in a real scenario. According to
Figure 31
b, the wind velocity has a negative effect on the pavement surface temperature; this implies that when the wind is existing, the road surface would be cold and less energy is available for
Mäkiranta and Hiltunen [
] implemented testing of the GRES to examine the effect of temperature variations at various depths of the soil layer on system energy capturing. As presented in
Figure 32
, there are four different pipe depths including two for 10 m depth, one for 5 m depth and two for 3 m depth in Finland. Results conclude that it is able to retain appropriate temperature up to 26 °C
at the soil depth of 0.5 m, and there are positive temperature values for at least 9 months per annum; this indicates that the soil layer is suitable for assembling heat collection pipes.
Balbay and Esen [
] carried out an experimental investigation to explore the feasibility of the GRES utilization to heat roadways for snow melting as shown in
Figure 33
. During the testing, the processes of snow melting for bridge and pavement slabs at the initial state (t = 0) and intermediate state (t = 30 min.) are shown in
Figure 34
. Results demonstrate that the top surface of the pavement is typically exposed to bigger temperature fluctuation than the bottom surface. Besides, the thermal conductivity of the concrete slab and
air convection coefficient have significant influences on the temperature of the pavement surface; this indicates that the higher wind speed, the higher the convection heat coefficient at the surface
of the concrete slab, leading to a lower top surface temperature.
Ho et al. [
] proposed a 3D GRES model to analyze the pavement surface temperature variation. As illustrated in
Figure 35
, the GRES includes a closed-loop polyethylene tube that is embedded in the concrete slab, and the initial inlet fluid temperature is set in the range of 60 °C to 82.2 °C for snow and ice melting. As
shown in
Figure 36
, a high flow rate can warm road surface to a high temperature, and a low fluid flow rate leads to a big temperature reduction.
Tota-Maharaj et al. [
] designed a GRES experimental rig to evaluate snow removal rates for various mediums. The GRES consists of a geothermal heat pump (GHP) combined with a permeable pavement system (PPS).
Figure 37
despites the interior and exterior views of the testing rig, the interior PPS includes six bins positioned in a temperature-controlled room with an average air temperature of 15 °C, while the
exterior part is embedded in the soil where it is subjected to ground temperature and climate conditions. The schematic diagram of the PPS and pavement layers are presented in
Figure 38
; it is revealed from
Figure 39
that the mean snow removal rate varies from 80% to 90% for biochemical oxygen demand (BOD), ammonia–nitrogen (NH4) and orthophosphate–phosphorus (PO4). By comparison, the removal rate of suspended
solids (SS) is in the range of 40% to 60%.
Zhang et al. [
] investigated the snow melting performance of a GRES with L-shaped heat pipe system for airfield runways in Beijing and Harbin of China as depicted in
Figure 40
; it can be seen from
Figure 41
that the GRES could increase the two city mean airfield runway temperatures of 9.1 °C in 2015, 8.9 °C in 2016 and 9.4 °C in 2017, respectively; this implies that the GRES could work automatically
primarily in heating season to enhance the surface temperature adequately to avoid ice accumulation; moreover, it can be found that the surface temperature of airfield-runway is below 0 °C at 68%
time of the year in Harbin, whereas it is almost above 0 °C in Beijing for the whole winter period; this means that the geography plays a significant role in the system de-icing performance. As shown
Figure 42
, the system for airfield-runway is applicable in central areas of China whose air temperature exceeds −10 °C, whereas it is inapplicable in north-western and north-eastern areas.
Wang et al. [
] developed a GRES with super flexible heat pipes (SFHPs) system to investigate the entrainment limit within the pipe and provide the optimal design plan based on the local climate condition. The
fabrication and construction of the GRES with SFHPs are shown in
Figure 43
. In order to explore the heat transfer mechanism, thermal resistance model and boundary conditions are given in
Figure 44
. Simulation results reveal that ammonia is the most suitable thermal fluid for the high entertainment limit, and the system heat output could reach approximately 1.15 kW as illustrated in
Figure 45
. Besides, 70 m of SFHP length with Ø32 × 2 mm and 0.2 m distance of condenser horizontal interval are recommended as the optimal choice in the study.
Mauro and Grossman [
] conducted a dynamic GRES simulation work to decrease the fluctuations of street temperature and avert ice formation. The system includes 12 EPs with a diameter of 150 mm and length of 20 m, and the
substrate has a thickness of 20 mm and is placed beneath the street pavement with a thickness of 100 mm; it can be found from
Figure 46
that the minimum street surface temperature varies from 4.6 °C to 6.6 °C in winter, while the maximum temperature changes from 3.8 °C to 7.5 °C in summer.
3.2. Solar Roadway Energy System
SRES is conducive to mitigating the HIE and eliminating the risk of permanent deformation. Recent studies have aimed to investigate the SRES performance based on experimental analyses and numerical
simulations. Hence, some detailed illustrations in terms of roadway surface temperature reduction and influence factors on de-icing and snow melting performance are given in the section. To be more
specific, Chiarelli et al. [
] tested an SRES on the basis of different pipe arrangements to evaluate the temperature variation and energy extracted rate. As given in
Figure 47
, the system involves six 1 m lengths of copper pipes in five configurations, and has two structure layers: a 50 mm thickness of asphalt wearing course layer and a 130 mm thickness of the aggregate
layer. Besides, there are six infrared light bulbs that are utilized to heat the surface temperature to 80 °C. Results indicate that the harvested energy is in the range of 60 kJ to 100 kJ whereas
their exergy varies from 20 kJ to 40 kJ in six testing periods.
Guldentops et al. [
] setup a 3D COMOSOL model of the SRES to assess the outlet fluid temperature and solar energy absorption efficiency. As shown in
Figure 48
, the dimension of the calculation region is defined as 4 × 0.9 m (L × W), and the whole length of solar collector pipe is 23.6 m with a diameter of 0.008 m and thickness of 0.3 m.
It can be found from
Figure 49
a that the system efficiency could be enhanced from 17% to 20% when the thermal conductivity of concrete slab rises from 1.0 to 2.0 W/m·K; this means that a higher thermal conductivity of concrete
slab results in more thermal energy obtained by thermal fluid. In comparison, the efficiency of the SRES is increased from 15% to 17% when the absorptivity of the pavement surface is enhanced from
0.65 to 0.95 as depicted in
Figure 49
b; this indicates that as the asphalt concrete ages, it turns out to be lighter in colour and could, therefore, reflect more of the incident solar radiation. Additionally, the harvested thermal
energy reduces from 21% to 14% when the depth of the pipe varies from 25 mm to 105 mm as illustrated in
Figure 49
c; this indicates that there is a significant influence of the pipe depth on the system performance, meanwhile, the long-time system thermal property enhancement is paramount by reducing the pipe
depth as much as possible.
Saad et al. [
] designed an SRES porotype to study the chimney efficiency at different heights. 36 aluminium pipes with a length of 1 m, an interior diameter of 12 mm and a thickness of 0.9 mm are used as given in
Figure 50
. All pipes are positioned at one level in the horizontal direction placed below 25 mm from the roadway surface slab, and the total thickness of pavement slab is 100 mm.
Figure 51
presents the image of the experimental rig; it is found that there is a vital influence of the chimney height on the efficiency. As shown in
Figure 52
, the chimney efficiency could reach 15% at the chimney height of 9 m whereas the efficiency is 11.7% at the chimney height of 4 m; this implies that the efficiency of the chimney increases with the
chimney height.
Johnsson and Adl-Zarrabi [
] developed a 3D model of the SRES based on finite difference method (FDM) to study energy usage and de-icing performance.
Figure 53
shows that the entire calculation domain of the model is partitioned into segments along
-axial direction, and there are the same thermal properties. By comparison, in the horizontal direction, the model includes several layers with diverse thermal properties. The top of pavement layer
is uncovered to the open-air climate, and the boundary condition is defined as the adiabatic or a constant temperature value. The basic heat transfer model is expressed in
Table 5
As shown in
Figure 54
, a top layer of pavement slab involves the polyethylene pipe is positioned at a depth of 62 mm with a distance of 50 mm, 10 parallel pipes within the concrete have a length of around 140 m and cover
an area of 70 m
. A good agreement in terms of surface temperature between the testing and numerical results is achieved as illustrated in
Figure 55
, the root mean square error (RMSE) and mean error (ME) are 1.34 °C and −0.55 °C, respectively. As illustrated in
Figure 56
, the energy usage varies from 330 kWh/m
to 540 kWh/m
with an ice and snow cover lasting for 1100 h to 430 h, respectively; this implies that about 62% of system energy consumption could be saved by using the SRES.
Zaim et al. [
] implemented a testing investigation of the SRES to analyse the influence of pipe configuration on the system performance as depicted in
Figure 57
a. Specifically, the system composes of pipe loops, a tank, a water pump, a flowmeter, a pyranometer, an anemometer and a data logger. The pipe external dimension is 3 × 0.4 × 0.2 m (L × W × H) with
the inner diameter of 15.8 mm, which is embedded in a regular arrangement with a center-to-center spacing of around 110 mm; furthermore,
Figure 57
b presents various configurations involving the parallel, series, balanced and unbalanced ladder-shape that are constructed in the SRES; it can be found from
Figure 58
that the pipe arrangements have significantly effects on the system outlet fluid temperature during the testing. The outlet fluid temperatures are similar between the balanced ladder-shape and
parallel arrangements, but the highest outlet fluid temperature occurs in the series pipe arrangement. As demonstrated in
Figure 59
, the various pipe arrangements have little influence on the pavement surface temperature; however, solar irradiation plays a vital role in the surface temperature.
Alonso-Estébanez et al. [
] built a 3D CFD model of the SRES to assess the influences of solar radiation and different slab thickness on system performance. As shown in
Figure 60
, the numerical model consists of four calculation regions involving working fluid within the pipe, copper pipe, asphalt mixture and ambient air. The boundaries of the top and side walls within the
ambient air domain are 20 m away from the heat source. The water velocity and flow rate are defined as 1.6 m/s and 2 L/min, respectively. Meanwhile, an experimental test is implemented to validate
the 3D model as presented in
Figure 61
. The prototype consists of a 2 × 2 configuration that includes 4 slabs with 420 × 130 × 60 mm (L × W × D) and a U-tube copper pipe with exterior and interior diameters of 1.7 m and 0.016 m and a
depth of 0.25 m. The validation result confirms that the errors between numerical and experimental results do not exceed 10% in terms of thermal performance, temperature variation and energy
collection. As demonstrated in
Figure 62
a, solar radiation has important influences on the pavement surface temperature, working fluid flow rate and size, the system thermal efficiency could reach up to 74% based on the simulation
analysis. What is more, the additional energy stored within the roadway slabs has less impact on system thermal performance as illustrated in
Figure 62
Daniels et al. [
] designed a prototype of the SRES to assess de-icing performance in the USA. As illustrated in
Figure 63
, the testing slab has a dimension of 3050 × 1220 × 130 mm (L × W × D), and a 500-gallon thermal storage tank is used to link the pavement slabs and solar collectors, and placed on a high density
polyurethane foam as its bottom insulation. The polyethylene pipe is embedded below the pavement surface of 50 mm to meet the minimum concrete cover demand. As shown in
Figure 64
, an infrared image of the snow-melting process of pavement surface is given in the period of 6 h from 4:37 a.m. to 10:37 a.m. After the SRES works about 4.13 h, the power consumption is around 0.51
kWh, thereby, the fluid temperature of the thermal storage tank could decrease from 60 °C to 49.4 °C during the test period.
García and Partl [
] conducted an experimental study of the SRES with parallel air conduits to overcome the damage of the buried pipes and evaluate the system efficiency. As shown in
Figure 65
, sixty steel tubes with 300 mm length and internal and external diameters of 9 and 11 mm are embedded in the asphalt concrete material. Two air chambers of 10 × 10 × 45 cm are assembled at both
sides of the test porotype as the inlet and outlet of the air conduits. Results from
Figure 66
reveal that system efficiencies could be improved around 10% and 12% for heating up air and chimney usage, respectively, implying that it is extreme important to use the chimney.
Wu et al. [
] explored the influences of fluid flow rate within the SRES on the pavement surface and the heat obtained. As described in
Figure 67
, the prototype composes of a small-scale asphalt solar collector, a circulation pump, a flow meter and a control valve. The pavement slab includes three layers of compacted asphalt mixture that has
a dimension of 300 × 300 × 150 mm (L × W × D). The hose pipe is regarded as thermal isolation and utilized to connect all devices. Results from
Figure 68
a demonstrate that the working flow rate has a restricted effect on the maximum surface temperature. In particular, when the working flow rate increases to 1886 mL/min, the surface temperature
decreases up to 36.7 °C, by comparison, when the working flow rate falls to 54 mL/min, the temperature reaches the maximum value of 38.58 °C; this indicates that the high working fluid rate results
in a superior amount of thermal energy that could be extracted. Additionally, it can be observed from
Figure 68
b that thermal energy of about 400 W/m
can be extracted when the flow rate is in the range of 400 to 1800 mL/min; this means that the flow rate has less influence on enhancing the heat transfer coefficient of working fluid within the
Du et al. [
] established a new numerical model to analyze the temperature distribution and heat transfer rate in comparison with the control structure. As illustrated in
Figure 69
a, the calculation domain includes asphalt layers that have a base and sub-base layers with a size of 5 × 5 cm, subgrade and steel rods that have dimensions of 5 × 10 cm and 0.6 × 2 cm, respectively.
The model boundary conditions for top, left and right are defined as thermally insulated and constant thermal properties of all materials. Besides, the temperatures and heat fluxes of sections A, B
and C are employed to study the model heat transfer mechanism as presented in
Figure 69
b. Seven rod-implanting modes are applied to investigate their effects on the temperature distribution for the pavement as exhibited in
Figure 69
c. The simulation results in
Figure 70
a reveal that the new model could absorb about 31% solar energy in comparison with the control structure; furthermore, the internal and surface temperatures could be decreased by up to 6.4 °C and 3.5
°C, respectively, in comparison to the control structure, as exhibited in
Figure 70
Dakessian et al. [
] compared two SRES efficiencies between a close-loop configuration and a single-pass configuration as displayed in
Figure 71
, and found that the single-pass system could reach the efficiency of 21.9% while the close-loop system achieves the efficiency of 10.9%. Additionally, as shown in
Figure 72
, a 3D finite element model of the SRES is established to investigate the energy harvesting and roadways surface temperature variations for the single-pass system at different seasons; it is
demonstrated from
Figure 73
that the single-pass SRES could enhance water temperature and decrease roadways surface temperature by 10.2 °C and 1.24 °C for spring, 13.6 °C and 1.69 °C for summer, 7.5 °C and 0.67 °C for autumn as
well as 4 °C and 0.52 °C for winter, respectively.
3.3. Summary
To sum up, both the GRES and SRES use renewable energy technologies to solve the problems induced by conventional chemical-based snow and ice melting approaches. The two types have the ability to
decrease energy consumption by approximately 30%, and increase the surface temperature of roadway by around 5 °C in winter and reduce it by 6 °C in summer.
Table 6
Table 7
illustrate the research regions, applied methods and key findings. What is more, the effects of climate condition, pipe type and dimension, pipe arrangement, flow rate, initial fluid temperature,
thermal conductivities of soil and wearing layer, absorptivity and emissivity of the roadway surface, preheating time, chimney height, slab thickness and pavement surface absorptivity, on the GRES
and SRES de-icing and snow melting performance are individually summarized and compared in
Table 8
Table 9
; it is discovered that the climate condition, pipe layout arrangements, soil thermal conductivity, preheating time, slab thickness and chimney height play a significant role in the GRES and SRES
performance whereas the diameter of pipe has a slight influence.
4. Economic Assessment
Various economic feasibility researches are implemented to determine the cost-savings of the GRES and SRES as well as their payback periods (PBP). Typically, the capital investments of the GRES and
SRES primarily involve project design, heat pump unit and installation; moreover, the system operating costs include power consumption of heat pump, monitoring, maintenance and replacement. Hence,
some key models and cases are elaborated in the section.
4.1. Geothermal Roadway Energy Systems
Liu et al. [
] carried out an economic analysis of the GRES with the EP in order to determine the cost-saving and PBP in Canada. As illustrated in
Figure 74
, it can be found that the PBP of the GRES is less than 4 years compared with the traditional electrical heating unit’s; furthermore, the GRES has the potential to save about CAD 1.5 million by the
end of 30 years’ operating.
Han and Yu [
] conducted an economic assessment for a modified GRES with PCM to assess the expense factor and decrease the expense barrier. In this study, the additional expenses of five PCM categories are
investigated as depicted in
Figure 75
; it is found that the system expense of cyclohexane is more than ten times higher compared with the other materials’. Meanwhile, when an EP with 3% PCM is used to replace the cyclohexane, a 90%
system cost-saving can be achieved; this indicates that when the expense of materials decreases, the expense barriers are able to be removed. The additional expense of the EP could be decreased to
USD 207 when the biodiesel crude glycerine is used in the model. Yang et al. [
] implemented a finical evaluation of the GRES to resolve the high capital investment and assess the system financial viability in a typical city of China. Results obtained from
Figure 76
reflect that the total net present value (NPV) could be 150,000 CNY when the internal rate of return (IRR) is 4.9% for 15-year’s operating period. In the meantime, the system PBP is approximately 8
years. Mauro and Grossman [
] analysed how to decrease the cost of the GRES based on pipe configuration. In this study, the main expenses of the whole system include the pipe material purchasing, rock-soil drilling,
installation, customized concrete attained through using high thermal conductivity aggregates and the mixture preparation; it is found that the cost of the GRES is in the range between 850 and 1250
, however, it is possible to be further cut to about 450 EUR/m
by materials optimization, size modified and hollow pile installation.
Habibzadeh-Bigdarvish et al. [
] performed life cycle cost (LCC) and sensitive analyses of the GRES for bridge deck de-icing based on the Monte Carlo Simulation (MSC) method. Results from
Figure 77
a reveal that the main cash flow is from traffic flow improvement profits in the 25th and 32nd years of the analysis; this means that the traffic flow improvement is the most sensitive random
variable. Meanwhile, the NPV value indicates that the system profits overweigh its initial investment after 25 years, and could achieve USD 2.4 million after 50 years. According to
Figure 77
b, the GRES is a cost effective solution for heating bridge deck when the daily traffic volume is over a minimum of 7000 vehicles.
Nahvi et al. [
] conducted economic analyses of the GRES for the Minneapolis/Saint Paul International Airport (MSP) and Des Moines International Airport (DSM). Results from
Figure 78
a show that the annual system energy consumption costs at the MSP could reach about USD 1.96 million which is almost six times higher compared to that at the DSM; moreover, as shown in
Figure 78
b, the benefit-cost ratio (BCR) is the most sensitive to capital investment on the basis of the dimension of airport and site location. In other words, the numbers of airplane operating have a
significant influence on the BCR.
4.2. Solar Roadway Assessment Energy Systems
Dakessian L et al. [
] carried out an economic investigation for the SRES in the light of the life extension, NPV and PBP based on a 10 m section of a two-lane road in Lebanon. As indicated in
Figure 79
, the SRES could extend the lifetime service from 20 to 23 years, thus saving the cost of about USD 600 compared with the conventional roadway. Furthermore, a positive NPV of USD 3000 with about 5
years of the PBP is achieved in the study.
Sable [
] implemented a finical assessment of the SRES to investigate the system payback period and cost-saving; it is found that the annual cost-saving could be achieved in the range of Rs. 6106.5 to Rs.
9838.3, in the meantime, the PBP is in the range from 2.3 to 4.1 years.
4.3. Summary
Several economic models are utilized broadly to study the economic factors affecting the markets for the GRES and SRES in various countries and regions. Based on the research results, the GRES has a
higher capital investment because of the drilling and installation fees, which are almost three times higher than those of the SRES, while the PBPs of the GRES and SRES are in the ranges of 4 to 8
years and 2.3 to 5 years respectively. A detailed summary of economic analyses of the GRES and SRES by different researchers is shown in
Table 10
. In addition, there is often an extraordinary discrepancy in the financial influence factors, including initial cost, operation and maintenance expenses and inflation rates, operations and delay
periods as well as percentage of weather related delay, which could cause considerable differences in the investment decision and key financial performance, as presented in
Table 11
5. Future Developments
The GRES and SRES as renewable energy systems are imperatively challenging areas of research in terms of climate condition, pipe configuration and material, thermal conductivities of soil and slab
concrete, and initial design condition. Although more endeavours have been focused on the advanced and promising techniques, there are still a few challenges needed to be disposed for forthcoming
exploration and spreading out the applicability of the technologies, those challenges are displayed as below:
• Numerical models of the GRES and SRES are still required to be established to predict the system de-icing and snow-melting performance more accurately, thereby this contributes to improving
system designs in the future.
• Further investigation on the GRES and SRES should be focused on the construction and maintenance technique for pavement with pipes; this is because if the subsidence deformation or structure
crack happens during the fitting and operation, this may damage the enclosed state, causing the groundwater entry and pipeline leak, and decreasing the system service lifetime. Hence, it is
essential to setup a real time monitoring system to check the effect of the surrounding environment on the structure deformation.
• Heat pipe is generally banded with the reinforced steel cage in the EP system, therefore massive attention should be spent to avoid the pipe damage during concreting, and appropriate measures
should be adopted to prevent blockage at the connecting point. What is more, freezing injury should be taken into account in cold region, this is because the frozen soil and road excavation may
result in the freezing of water within the GRSE. Furthermore, using the PCM to replace the regular concrete in ground heat exchanger should be further studied.
• The soil and asphalt layers can store thermal energy in the GRES, therefore, in this aspect, the thermal storage capacity should be clarified to complement roadway energy consumption.
• A detail analysis should be implemented to identify the influences of air convection on the physical properties of the GRES and SRES in the fields of energy capturing, LCC and CO[2] emission.
6. Conclusions
The GRES and SRES can effectively solve ice and snow accumulation issues on roadway which cause inconvenience for drivers and traffic accidents in winter, but they consume less energy and produce
less or no CO[2] emission compared with conventional chemical based melting solutions. In summer, the GRES and SRES can reduce the pavement surface temperature to mitigate the heat effect and extend
its service lifetime. A comprehensive review of their technological performance and economic evaluation is conducted in this study based on numerical and economic models, and experimental analyses.
Three vital aspects of the technology performance assessment, involving roadway surface temperature, energy consumption and main influence factors, are explored in different regions and countries.
Energy and economic evaluations of the two technologies for various climatic conditions, different pipe configurations and design conditions are carried out as well. As a result, some crucial
conclusions are summarized as follow:
• The climate data such as ambient temperature, solar radiation, snowfall rate as well as wind speed, are the essential information to design de-icing and snow-melting systems.
• The spiral shape pipe could extract more soil heat in comparison with U-shape and W-shape pipes, so it is the best choice in the GRES system under the limited pile length. The velocity of the
working fluid has less effects on the system performance with the U- and W- shape pipes whereas it has a significant influence on that with the spiral-shape pipe.
• Approximately 35% less hours of the pavement slippery condition are achieved when the working fluid temperature increases by about 15 °C in the GRES.
• In the GRES, the soil thermal imbalance influences not only the system energy conversion but also the structural foundation, so this imbalance should be avoided by injecting a large amount of
heat to the soil.
• The modified GRES, such as using the EP and PCM to replace the traditional ground pipe loop and concrete, could extract more thermal energy and reduce the pile number for de-icing pavement
surface, which is conducive to decreasing the capital investment and maintenance cost.
• In the SRES, the increasing of the pipe thermal conductivity and decreasing of its depth have significant effects on the system long-term operation. The thermal gain decreases from 21% to 14%
when the depth of the ground pipe varies from 25 mm to 105 mm.
• In the SRES, the chimney height is a vital parameter influencing on the system performance, the chimney efficiency increases from 11.7% to 15% when its height rise from 4 m to 9 m. The higher the
chimney, the lower the energy loss.
• Compared with the traditional ways, the GRES and SRES could decrease energy consumption by approximately 30%, the roadway surface temperature could be increased by around 5 °C in winter and
reduced by about 6 °C in summer.
• The service lifetimes of the GRES and SRES could attain 25 to 30 years and 20 to 23 years, respectively. The GRES has a higher capital investment because of the drilling and installation fees,
which is almost three times higher than that of the SRES, while the PBPs of the GRES and SRES are in the ranges of 4 to 8 years and 2.3 to 5 years respectively.
Author Contributions
Conceptualization, writing and supervision, Y.C.; Resources and data curation, F.Z., Y.S. and S.T.; Project administration and review and editing, H.T. All authors have read and agreed to the
published version of the manuscript.
This research was funded by the thirteenth Five-Year Plan National Key Research and Development Program Subproject “Village Community Livable Unit Environment Construction and Planning Indicators
Research”, grant number 2019YFD1100805.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1.
Energy harvesting from roadway [
Figure 2.
Working principle of GRES [
Figure 3.
Working principle of SRES [
Figure 4.
GRES with EP applied for bridge deck: (
) schematic diagram; (
) heat transfer mechanism [
Figure 5.
Simulation results of energy consumption based on different factors: (
) snowfall rate; (
) solar radiation; (
) air temperature; (
) wind speed [
Figure 6.
Heat flux at various flow rates: (
) average; (
) total [
Figure 7.
Diagram of: (
) heated bridge deck; (
) polyethylene pipe loop [
Figure 8.
Photos of: (
) lab bench; (
) polyethylene pipe loop [
Figure 9.
Infrared pictures of slab surface: (
) top; (
) side [
Figure 10.
Experimental result of heat flux and heat transfer efficiency [
Figure 11.
3D model: (
) model mesh; (
) view of the EP and monitor locations [
Figure 12.
Simulation results of thermally induced vertical stress based on different pipes: (
) P12-P15; (
) P22-P25 [
Figure 13.
Experimental testing [
Figure 14.
Experimental results of: (
) strain variation with time; (
) strain versus temperature increment [
Figure 15.
Test site: (
) actual photo; (
) pavement layers [
Figure 16.
3D model: (
) hybrid; (
) symmetrical region [
Figure 17.
The superposition process [
Figure 18.
Comparison between numerical and experimental results [
Figure 19.
3D GRES model [
Figure 20.
Simulation results for de-icing performance: (
) different pipe distances; (
) different pipe depths [
Figure 21.
GRES: (
) schematic diagram; (
) cross-section view; (
) test site; (
) pipe arrangement [
Figure 22.
GRES with EP: (
) model; (
) boundary conditions [
Figure 23.
Simulation results: (
) outlet temperature; (
) energy extraction [
Figure 24.
The modified GRES model with EP integrated with PCM [
Figure 25.
Simulation results of required number of piles: (
) U-shape; (
) W-shape; (
) spiral shape [
Figure 26.
Schematic diagram of GRES model: (
) double arrangement; (
) dimension [
Figure 27.
Simulation results of outlet temperature at different soil saturation [
Figure 28.
GRES for UUT: (
) system arrangement; (
) plane [
Figure 29.
Simulation results at various flow velocities: (
) outlet fluid temperature; (
) temperature difference [
Figure 30.
Testing bench: (
) cross-section view; (
) scheme diagram; (
) actual photo [
Figure 31.
Histogram of: (
) inlet temperature; (
) wind speed [
Figure 32.
Schematic diagram of GRES at different depths of the soil layer [
Figure 33.
Schematic diagram of GRES model; (
) 3D view of experimental set-up (
) temperature measurement points for both slabs (
) boreholes [
Figure 34.
Experimental testing: (
) initial temperature; (
) after 30 min [
Figure 35.
Schematic diagram of GRES with pipes [
Figure 36.
Simulation results of temperature variation at various flow rates: (
) surface; (
) outlet [
Figure 37.
Actual testing rig: (
) interior; (
) exterior [
Figure 38.
PPS: (
) schematic diagram; (
) layers [
Figure 39.
Experimental results at different nutrient removal rates: (
) exterior; (
) interior [
Figure 40.
Actual installation sites: (
) Harbin; (
) Beijing [
Figure 41.
Temperature variation: (
) Harbin; (
) Beijing [
Figure 42.
Applicability region map of the GRES in China [
Figure 43.
Schematic diagram of GRES with SFHPs: (
) system design; (
) pipe structure [
Figure 44.
2D model: (
) boundary condition; (
) thermal resistance [
Figure 45.
Simulation results: (
) entertainment limits; (
) heat output [
Figure 46.
Schematic diagram: (
) GRES model; (
) simulation results [
Figure 47.
Experimental bench: (
) cross-section; (
) various arrangements; (
) actual photo; (
) prototype dimension [
Figure 48.
3D COMSOL model of PSBC [
Figure 49.
Simulation results of solar efficiency and outlet temperature at various parameters: (
) thermal conductivity; (
) pavement surface absorptivity; (
) pipe depth [
Figure 50.
Schematic section of SRES [
Figure 51.
Photo of experimental rig: (
) test rig; (
) air pipe within the asphalt concrete [
Figure 52.
Simulation results of chimney efficiency at different heights [
Figure 53.
Schematic diagram of: (
) 3D model; (
) segments [
Figure 54.
Experimental site in Sweden: (
) construction; (
) comparison; (
) operating [
Figure 55.
Comparison of surface temperature between (
) numerical and (
) experimental results [
Figure 56.
Simulation results: (
) energy usage; (
) de-icing performance [
Figure 57.
Photos of: (
) experimental rig; (
) various pipe arrangements [
Figure 58.
Outlet fluid temperature in various configurations: (
) summer; (
) winter [
Figure 59.
Surface temperature in various configurations: (
) summer; (
) winter [
Figure 60.
3D CFD model: (
) entire calculation domain; (
) pipe arrangement [
Figure 61.
Experimental bench: (
) schematic diagram; (
) actual photo [
Figure 62.
Simulation results: (
) energy collected; (
) system performance at various slabs thickness [
Figure 63.
Experimental set-up: (
) actual photo; (
) schematic diagram; (
) detailed SPC [
Figure 64.
Experimental analysis of system de-icing performance [
Figure 65.
Experimental rig: (
) side-view (
) air pipe arrangement; (
) air chamber [
Figure 66.
Experimental result of surface temperature vs: (
) time; (
) chimney [
Figure 67.
Experimental bench: (
) whole system; (
) tested slab; (
) actual specimen [
Figure 68.
Influences of fluid flow rate on: (
) surface maximum temperature; (
) heat collected [
Figure 69.
Schematic diagram of: (
) heat transfer model; (
) locations; (
) various rod-implanting modes [
Figure 70.
Temperature measured results: (
) maximum; (
) average [
Figure 71.
Schematic diagram of SRES configurations: (
) closed-loop; (
) single-pass [
Figure 72.
Schematic diagram: (
) dimensions and pipe layout; (
) finite element model [
Figure 73.
Simulation results: (
) water temperature; (
) surface temperature [
Figure 74.
Economic analysis of GRES savings [
Figure 75.
The additional cost of the GRES with PCM; (
) other materials (
) Cyclohexane [
Figure 76.
NPV of GRES [
Figure 77.
LCC and sensitive analyses: (
) cash flow; (
) NPV [
Figure 78.
LCC analysis results between DSM and MSP: (
) energy consumption expense; (
) sensitivity analysis [
Figure 79.
LCC benefits of SRES [
Table 1.
Energy balance equation [
Description Equations
$q = q c o n v + q s + q m + q s o l a r$
Energy balance $q c o n v ( t ) = h c o n v [ T a ( t ) − T s ( t ) ]$
$q m ( t ) = ρ w s ( t ) h f$
$q s ( t ) = ρ w s ( t ) [ c ρ i ( T m − T a ( t ) + c ρ w ( T f − T m ) ]$
Heat transfer within the working fluid $ρ w A g f C p w ∂ T g f ∂ t + ρ w A g f C ρ w v g ∇ T g f = ∇ ⋅ ( A g f k w ∇ T g f ) + Q w 2$
$η = P V o u t p u t A × I$
Temperature distribution within the pipe $ρ w A h f C p w ∂ T h f ∂ t + ρ w A h f C p v s ∇ T h f = ∇ ⋅ ( A h f k w ∇ T h f ) + Q w 1$
$Q w 1 = ( h Z ) e f f ( T e x t − T )$
Heat transfer around concrete slab $ρ s A c s C p s ∂ T c s ∂ t = ∇ ⋅ ( A c s k s ∇ T ) − Q w 1$
Heat transfer within the soil $ρ A C p ∂ T s ∂ t = ∇ ⋅ ( A k ∇ T s ) − Q w 2$
Table 2.
Hybrid 3D heat transfer model and the principle of superposition [
Description Equations
$R e q − p i p e = R p i p e + R i , j + R P W S$
Thermal resistance $R p i p e = ln ( r o u t e r r i n n e r ) 2 ⋅ π ⋅ λ p i p e$
$R i , j = ln ( r i , j r o u t e r ) 2 ⋅ π ⋅ λ i , j$
$R P W S = 1 π ⋅ λ f ⋅ N u$
Temperature distribution of working fluid within the pipe $T e q − p i p e = T f − q i , j ⋅ R e q − p i p e$
Outlet fluid temperature $T f , n + 1 t = T e q − p i p e , n t + ( T f , n t p T e q − p i p e , n t ) ⋅ e − ( L n / l n )$
$l n = R e q − p i p e ⋅ ν f ⋅ π ⋅ r i n n e r 2 ⋅ ρ f ⋅ c p , f$
Principle of superposition $T s u r f a c e ( t ) = T s u r f a c e h e a t e d ( t ) + T s u r f a c e u n h e a t e d ( t )$
Table 3.
GRES model [
Description Equations
$∂ θ ∂ τ = ∇ ( D l ( θ ) ∇ θ ) + ∇ ( D l ( T ) ∇ T ) − ∂ K ( θ ) ∂ y + ∇ ( D v ( θ ) ∇ θ ) + ∇ ( D v ( T ) ∇ T )$
Water transport $K ( θ − θ r θ s − θ r ) n + 2 + 2 a = K ( θ )$
$D v ( θ ) = 1 ρ w D 0 α b ρ 0 ∂ h 0 ∂ θ$
$D l ( θ ) = K ( θ ) ∂ ψ ∂ θ$
$∂ ∂ τ ( C ( θ ) T ) = ∂ ∂ x ( λ ( θ ) ∂ t x ) + ∂ ∂ y ( λ ( θ ) ∂ t y )$
Heat transport $C ( θ ) = C d r y + θ θ s ( C s a t − C d r y )$
$λ ( θ ) = λ d r y + K e ( λ s a t − λ d r y )$
RMSE $R M S E = 1 n ∑ ( N i − O i ) 2$
ME $M E = 1 n ∑ ( N i − O i )$
Table 4.
PCM modified model [
Description Equations
$ρ = θ ( T ) ρ p h a s e 1 + [ 1 − θ ( T ) ] ρ p h a s e 2$
$k = θ ( T ) k p h a s e 1 + [ 1 − θ ( T ) ] k p h a s e 2$
PCM model $C p = 1 ρ { θ ( T ) ρ p h a s e 1 C p , p h a s e 1 + [ 1 − θ ( T ) ρ p h a s e 2 C p , p h a s e 2 ] } + L ∂ α m ( T ) ∂ T$
$α m ( T ) = 1 2 [ 1 − θ ( T ) ] ρ p h a s e 2 − [ θ ( T ) ρ p h a s e 1 ] θ ( T ) ρ p h a s e 1 C p , p h a s e 1 + [ 1 − θ ( T ) ρ p h a s e 2 ]$
$M p h a s e 1 = ( 1 − w t % ) M c o n c r e t e + w t % M P C M , p h a s e 1$
$M p h a s e 1 = ( 1 − w t % ) M c o n c r e t e + w t % M P C M , p h a s e 1$
Heat transport within EP $ρ C p ∂ T ∂ t + ∇ ⋅ ( − k ∇ T ) = − Q w a l l$
Table 5.
The 3D model of SRES [
Description Equations
Pavement surface $q s u r f a c e + q conv + q p r e c i p i t a t i o n + q l w + q s w$
$+ q e v a p / c o n + q s u b / d e p o + q f r e e z e / t h a w + q t r a f f i c = 0$
Roadway convection heat flux $q conv = h c ⋅ ( T a m b i e n t − T s u r f a c e )$
Heat flux due to precipitation $q p r e c i p i t a t i o n = m p r e c t ⋅ c p − p r e c t ⋅ ( T a m b i e n t − T s u r f a c e )$
$q l w = q l w i n − q l w o u t$
Long-wave radiation $q l w i n = F s k y v i e w ε s k y σ T a m b i e n t 4 + ( 1 − F s k y v i e w ) σ T a m b i e n t 4$
$q l w o u t = ε s u r f a c e σ T s u r f a c e 4$
Short-wave radiation $q s w = ( 1 − α 1 ) ⋅ I$
Sensible heat from the $q t r a f f i c = 0$
$T f k = T 0 k + ( T f k − 1 − T 0 k ) e − L s e g / l c$
Fluid temperature reduction in each segment $T 0 k = 2 R 0 ( T i , j k R i , j p i p e + T i , j + 1 k R i , j + 1 p i p e )$
$R 0 = [ 2 ( 1 R i , j p i p e + 1 R i , j + 1 p i p e ) ] − 1$
Heat flux from $q f k = ( ν f π r p i 2 ) ⋅ ρ f ⋅ c f ( T f k − 1 − T f k ) L s e g$
one segment
Heat flux nearby the pipe $q i , j k s o u r c e = q f k R 0 R i , j p i p e + T 0 k − T i , j k R i , j p i p e$
Researchers Type Region Method Working Fluid Key Findings
• The energy consumption could be decreased by 29% through using insulation materials.
• About 30% growth in the snowfall rate could cause around a 35% increasing of energy consumption.
Liu et al. [26 Hybrid
,27] horizontal and Canada Numerical model Water • The flow fluid rate within the GRES has the most vital influence on spiral loops.
• Soil thermal imbalance not only disturbs system energy efficiency but also badly affects the structural integrity of
• The system could provide about 60% heating to the surface of the bridge deck.
Yu et al. [28] Horizontal USA Experiment testing Water
• The heat flux of the surface obtained from the GRES is in the range from 120 to 270 W/m^2.
• The thermally induced stresses have a vital effect on the short-term local temperature gradient and pile
Fabrice et al. temperature.
[30] Vertical Switzerland Numerical model Water
• The average overstress could achieve 80 kPa/°C and 90 kPa/°C for cooling and heating seasons, respectively
• The thermal expansion strain variation is linearly with the increment of concrete slab temperature.
Kong et al. [ Horizontal China Experiment testing Water
31] • The maximum stress caused by the GRES is lower than design parameter.
• The system performance could be improved based on the closer distance between each pipe, the shallow depth, big pipe
diameter and low emissivity parameter of the pavement surface.
Mirzanamadi et Numerical model and Ethylene • The most vital effect on enhancing system de-icing performance is the distance between each pipe to shorten the
al. [32,33,34] Horizontal Sweden experiment testing glycol-water hours of slippery.
• There are 2% drop in the capturing solar energy and 5% fall in the needed energy for de-icing performance when the
working fluid rate is in the range between 8 L/min and 50 L/min.
• The system performance is based on pipes arrangement, thermal properties of concrete slab as well as temperature
level of the thermal storage unit.
Adl-Zarrabi et Horizontal Sweden Numerical model Water
al. [35] • The distance between the pipes has a bigger effect on the system thermal performance compared with the pipe buried
• The heat mass GRES model contributes to averting overestimating the requirement of the heat flux to obtain the most
Ethylene optimum system.
Xu et al. [36] Horizontal China Numerical model and glycol-water
experiment testing solution • In comparison with the heat-only GRES model, the needed heat fluxes could be decreased ranging from 6% to 17%
through the heat-mass model.
• The spiral shaped pipe could extract more heat compared with other shapes.
Ethylene • The growth of working fluid velocity has few effects on the soil energy obtained for the U- and W- shapes whereas it
Han and Yu [37 Vertical USA Numerical model glycol-water has a vital influence on the spiral-shape.
,38] solution
• The GRES with PCM model could extract more soil heat energy and cut down the requirement of pile number for de-icing
pavement surface.
Ethylene • The GRES is suggested to install the soil layer that has a high degree of saturation and high thermal conductivity.
Ho and Dickson Horizontal USA Numerical model glycol-water
[39] solution • The most optimum volumetric flow rate is recommended as at or beneath 1.0 L/s.
Ethylene • The GRES applied in the underground tunnel has an important influence on energy-saving and producing more cooling.
Yang et al. [ Horizontal China Numerical model glycol-water
40] solution • Lower inlet fluid temperature and flow velocity conduce to enhancing the heat exchange efficiency.
• Simulation results reveal that surface pavement temperature could improve from 0.4 °C and 2.1 °C in winter by
Chiarelli et Numerical model and comparison, the temperature could reduce 2 °C–6 °C in summer.
al. [41,42] Horizontal UK experiment testing Water
• The pavement temperature in winter depends upon the air temperature and humidity.
• Soil temperature at depth of 0.5 m is very promising for heat extraction during cooling season in Finland.
Mäkiranta and Ethylene
Hiltunen [43] Vertical Finland Experiment testing glycol-water • Asphalt heat could be regarded as thermal energy storage, and decrease the peak loads of heat energy consumption in
solution winter.
• Results conclude that the top surface temperatures of pavement and bridge exhibit more fluctuations compared with
the bottom one.
Balbay and Vertical Turkey Numerical model and Propylene • Air convection coefficient and thermal conductivity of BS and PS have an important influence on surface temperature.
Esen [44,45] experiment testing glycol
• The system COP could achieve 1.99 for 30 m soil depth, 2.66 for 60 m soil depth and 3.05 for 90 m soil depth,
• It can be found that when 60 °C of working fluid temperature to de-icing the road surface, it is applicable to most
the weather conditions in the USA.
Ho et al. [46] Horizontal USA Numerical model Water • When air temperature varies from −5 °C to −25 °C and working fluid could achieve between 50 °C and 60 °C, the GRES
is able to operate well, by contrast, when the working fluid ranges from 30 °C and 40 °C, the GRES could not operate
• The mean removal rates vary between 80 and 90% for BOD, NH[4] and PO[4]. By comparison, the removal rate of
Tota-Maharaj suspended solids (SS) is in the range from 40% and 60%.
et al. [47] Vertical UK Experiment testing Water
• The system EER efficiency could attain ranging from 1.5 to 2.5.
• The GRES could work automatically largely in the heating season and enhance the surface pavement temperature.
Ethylene • The geography has a critical effect on the system de-icing performance.
Zhang et al. [ Horizontal China Experiment testing glycol-water
48] solution • The GRES could enhance the road surface temperature by about 17 °C.
• The GRES technology could be utilized in more than 78% of the cities in China.
• The system heat output could achieve approximately 1.15 kW.
Wang et al. [ Vertical China Numerical model Ammonia
49] • The suggested entire length of SFHPs is 70 m with Ø 32 × 2 mm and 0.2 m distance of condenser horizontal interval.
Mauro and Ethylene • The system could enhance the street surface temperature ranging from 4.6 °C to 6.6 °C in heating season.
Grossman [50] Vertical Italy Numerical model glycol-water
solution • The system could decrease the street surface temperature ranging from 3.8 °C to 7.5 °C in cooling season.
Researchers Type Region Method Working Fluid Key Findings
• The energy harvested could reach ranging from 60 kJ and 100 kJ whereas their exergy varies from 20 kJ and 40 kJ during
Chiarelli et al. the six testing period.
[25] Vertical Finland Experiment testing Atmospheric air
• The roadway surface temperature can be reduced by up to 5.5 °C.
• Thermal production is increased from 14% to 21% when the depth of pipe is decreased from 105 mm to 25 mm.
Guldentops et Numerical model • The system efficiency rises from 17% to 20% when the thermal conductivity of the concrete slab enhances, ranging between
al. [51] Horizontal USA and experiment Atmospheric air 1.0 and 2.0 W/m·K.
• The increase in thermal behavior and decrease of the pipe depth are significant for system’s long-term operation.
Hybrid • The chimney efficiency based on different heights has an important influence on the pavement surface temperature.
Saad et al. [52] horizontal and Canada Numerical model Atmospheric air
vertical • The chimney efficiency could achieve 15% and 11.7% for 9 m and 4 m height chimney, respectively.
• The ME between testing and numerical results is approximately −0.55 °C while the RMSE is about 1.39 °C.
Johnsson and Ethylene
Adl-Zarrabi [53, Horizontal USA Experiment testing glycol-water • The system could decrease the energy consumption by 62%.
54] solution
• The average surface roadway temperature could be reduced about 6.4 °C in summer.
• The various pipe arrangements have few influence on the roadway surface temperature under different seasons.
Numerical model
Zaim et al. [55] Horizontal Sweden and experiment Water • The system indicates that the solar irradiation plays a vital effect on the roadway surface temperature.
• The growing of solar radiation contributes to booting the average surface temperature.
• The system could achieve 74% of thermal efficiency.
Alonso-Estébanez Horizontal Sweden Numerical model Water
et al. [56] • The thickness of the collector has a little influence on the thermal performance.
Numerical model Ethylene • The power consumption is around 0.51 kWh when the SRES works about 4.13 h.
Daniels et al. [ Horizontal China and experiment glycol-water
57] testing solution • The fluid temperature of thermal storage tank could decrease from 60 °C to 49.4 °C during the test period.
• The system efficiency could reach around 10% and 12% for heating up air and chimney usage, respectively,
García and Partl Vertical USA Numerical model Atmospheric air
[58] • It is extreme vital to decrease the energy loss by the chimney.
• The surface temperature reduces up to 36.7 °C, when the working flow rate enhances to 1886 mL/min.
Numerical model
Wu et al. [59] Horizontal UK and experiment Water • The surface temperature could achieve up to 38.58 °C, when the working flow rate decreases to 54 mL/min.
• The growth of flow rate conduces to boosting the heat transfer coefficient of fluid inside the pipe.
• The model could absorb about 31% solar energy in comparison with the control structure.
Numerical model
Du et al. [60] Vertical Turkey and experiment Water • The internal and surface temperatures could decrease by up to 6.4 °C and 3.5 °C, respectively, in comparison to the
testing control structure.
• The single-pass system could reach an efficiency of 21.9% which is higher compared with the close-loop system reaching
Numerical model 10.9%.
Dakessian et al. Horizontal Lebanon and experiment Water
[61] testing • The single-pass SRES could enhance water temperature and decrease roadways surface temperature by an average of 10.2 °C
and 1.24 °C for spring, 13.6 °C and 1.69 °C for summer, 7.5 °C and 0.67 °C for autumn as well as 4 °C and 0.52 °C for
winter, respectively.
Impact Factors
System Researchers Climate Pipe Pile Distance Evaporator Flow Initial Fluid Thermal Conductivity Thermal Emissivity of Absorptivity of Thermal Preheating
Condition Arrangement Diameter and between Section Length Rate Temperature of Wearing Layer Conductivity of Road Surface Road Surface Recharging Time
Type Depth Pipe Soil Analysis
Liu et al. [26 ✓ ✖ ✓ ✖ ✖ ✓ ✓ ✖ ✖ ✖ ✖ ✓ ✖
Yu et al. [28, ✓ ✖ ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Fabrice et al. ✓ ✖ ✓ ✓ ✖ ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖
Kong et al. [ ✓ ✖ ✖ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖
Mirzanamadi et ✓ ✖ ✓ ✓ ✖ ✓ ✓ ✓ ✓ ✓ ✓ ✖ ✖
al. [32,33,34]
Adl-Zarrabi et ✓ ✖ ✓ ✓ ✖ ✓ ✓ ✖ ✓ ✖ ✖ ✖ ✓
al. [35]
Xu et al. [36] ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✓
Han and Yu [37 ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Ho and Dickson ✓ ✖ ✓ ✓ ✖ ✓ ✓ ✖ ✓ ✖ ✖ ✖ ✖
GRES [39]
Yang et al. [ ✖ ✖ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Chiarelli et ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
al. [41,42]
Mäkiranta and ✓ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖
Hiltunen [43]
Balbay and ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Esen [44,45]
Ho et al. [46] ✓ ✖ ✖ ✖ ✖ ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖
Tota-Maharaj ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
et al. [47]
Zhang et al. [ ✓ ✖ ✓ ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Wang et al. [ ✖ ✖ ✖ ✓ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖
Mauro and ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Grossman [50]
Impact Factors
System Researchers Climate Pipe Pile Pipe Distance Chimney Chimney Inlet Air Flow Fluid Flow Slab Wind Thermal Conductivity of Pavement Surface
Conditions Arrangement Diameter Depth between Pipe Height Temperature Rate Rate Thickness Speed Slab Concrete Absorptivity
Chiarelli et al. [25] ✓ ✓ ✓ ✖ ✓ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖
Guldentops et al. [51 ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✓ ✓
Saad et al. [52] ✖ ✖ ✖ ✖ ✖ ✓ ✓ ✓ ✖ ✖ ✓ ✖ ✖
Johnsson and ✓ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖
Adl-Zarrabi [53,54]
SRES Zaim et al. [55] ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✓ ✖ ✖
Alonso-Estébanez et ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✓ ✓ ✖ ✖ ✖
al. [56]
Daniels et al. [57] ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✓ ✖ ✖
García and Partl [58] ✖ ✖ ✖ ✖ ✖ ✓ ✓ ✓ ✖ ✖ ✖ ✖ ✖
Wu et al. [59] ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✓ ✖ ✖ ✖ ✖
Du et al. [60] ✓ ✖ ✓ ✓ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Dakessian et al. [61] ✓ ✖ ✖ ✓ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖
Researchers Type Region Key Findings
• For geothermal EP system, the initial excavation expense can be saved in comparison with the traditional geothermal heat pump system.
Liu et al. [27] GRES Canada • The PBP for the GRES is less than 4 years than the traditional electrical heating system.
• The GRES is able to save about CAD 1.5 million by the end of 30 years’ operating.
• The system expense of cyclohexane is ten times higher compared with the other materials.
Han and Yu [38] GRES USA • A 90% system cost-saving could be achieved when an EP with at 3% PCM is replaced via the cyclohexane.
• When the expense of materials is dropped, the expense barriers are able to be removed.
• The total NPV could achieve CNY 150,000 when the internal rate of return (IRR) is 4.9% during the 15-year’s operating period.
Yang et al. [40] GRES China
• The system PBP is approximately 8 years.
Mauro and Grossman [50] GRES Italy • The system cost is able to be further cut to about 450 EUR/m^2 by materials optimization, size modified and hollow piles application.
• The cash flow is from traffic flow improvement benefits in the 25th and 32nd years of the assessment.
Habibzadeh-Bigdarvish [ GRES USA • The NPV presents that the benefits of system overweigh its initial investigation after 25 years, and could achieve USD 2.4 million after 50 years.
• The system could provide a cost effective solution for heating bridges decks when the daily traffic volume could achieve a minimum of 7000 vehicles.
• The annual system energy consumption costs at MSP could reach about USD 1.96 million which is nearly 6 times bigger than that at DSM reaching around USD 0.34
Nahvi et al. [63] GRES USA • The BCR is the most sensitive to capital investment on the basis of the dimensions of airport.
• The numbers of airplane operating have a significant influence on the BCR.
• The SRES could extend the lifetime service from 20 to 23 years, thus saving the cost of about USD 600 compared with the conventional roadway.
Dakessian et al. [61] SRES Lebanon
• A positive NPV of USD 3000 with about 5 years of PBP are achieved for the system.
• The annual cost-saving could achieve ranging between Rs. 6106.5 and Rs. 9838.3.
Sable [64] SRES India
• The PBP is in the range from 2.3 to 4.1 years.
Influence Factors
Researchers Types Regions Initial Investment Discounted Rate Inflation Rate Internal Rate of Return Maintenance and Operation Number of Operations Percentage of Weather Related
Cost and Delay Durations Delay
Liu et al. [27] GRES Canada ✓ ✖ ✓ ✖ ✓ ✖ ✖
Han and Yu [38] GRES USA ✓ ✖ ✖ ✖ ✓ ✖ ✖
Yang et al. [40] GRES China ✓ ✖ ✖ ✓ ✓ ✖ ✖
Mauro and Grossman [50] GRES Italy ✓ ✖ ✖ ✖ ✓ ✖ ✖
Habibzadeh-Bigdarvish [62 GRES USA ✓ ✓ ✖ ✖ ✓ ✖ ✖
Nahvi et al. [63] GRES USA ✓ ✓ ✖ ✖ ✓ ✓ ✓
Dakessian et al. [61] SRES Lebanon ✓ ✖ ✖ ✖ ✓ ✓ ✖
Sable [64] SRES India ✓ ✖ ✓ ✖ ✓ ✖ ✖
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Cui, Y.; Zhang, F.; Shao, Y.; Twaha, S.; Tong, H. Techno-Economic Comprehensive Review of State-of-the-Art Geothermal and Solar Roadway Energy Systems. Sustainability 2022, 14, 10974. https://doi.org
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Cui Y, Zhang F, Shao Y, Twaha S, Tong H. Techno-Economic Comprehensive Review of State-of-the-Art Geothermal and Solar Roadway Energy Systems. Sustainability. 2022; 14(17):10974. https://doi.org/
Chicago/Turabian Style
Cui, Yuanlong, Fan Zhang, Yiming Shao, Ssennoga Twaha, and Hui Tong. 2022. "Techno-Economic Comprehensive Review of State-of-the-Art Geothermal and Solar Roadway Energy Systems" Sustainability 14,
no. 17: 10974. https://doi.org/10.3390/su141710974
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Data Science
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Updated on May 4th. 2020
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Students who plan to pursue a double major in Data Science and Business majors with Business Analytics Track, cannot use CSCI-SHU 360 Machine Learning to fulfill Business majors Non-finance/
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Please note one course can only be used for two purposes. In this case, CSCI-SHU 360 Machine Learning will be used for the following three purposes, which is not permitted. 1) Business majors:
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Why should I major in Data Science?
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Data Science (Concentration in Finance) and Business & Finance
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Data Science (Concentration in Marketing) and Business & Marketing
• BUSF-SHU 101 Statistics for Business and Economics
• BUSF-SHU 202 Foundations of Finance
• BUSF-SHU 250 Principles of Financial Accounting
• ECON-SHU 3 Microeconomics
• MKTG-SHU 1 Intro to Marketing
Data Science (Concentration in Economics) and Economics
• ECON-SHU 1 Principles of Macroeconomics
• ECON-SHU 3 Microeconomics
• ECON-SHU 301 Econometrics
• MATH-SHU 140 Linear Algebra
• MATH-SHU 151 Multivariable Calculus
• MATH-SHU 235 Probability and Statistics OR BUSF-SHU 101 Statistics for Business and Economics
Note: Students who take both Linear Algebra and Multivariable Calculus can substitute Mathematics for Economists (Advanced Economics Elective) with these two courses. If the student chooses this
option, they would need to take one Additional approved quantitative economics course.
Data Science (Concentration in Finance) and Economics
• ECON-SHU 3 Microeconomics
• MATH-SHU 140 Linear Algebra
• MATH-SHU 151 Multivariable Calculus
• MATH-SHU 235 Probability and Statistics OR BUSF-SHU 101 Statistics for Business and Economics
Note: Students who take both Linear Algebra and Multivariable Calculus can substitute Mathematics for Economists (Advanced Economics Elective) with these two courses.
Data Science (Concentration in Finance) and Mathematics
• MATH-SHU 140 Linear Algebra
• MATH-SHU 151 Multivariable Calculus
• MATH-SHU 235 Probability and Statistics OR MATH-SHU 238 Honors Theory of Probability
Data Science (Concentration in Mathematics) and Math
• MATH-SHU 140 Linear Algebra
• MATH-SHU 151 Multivariable Calculus
• MATH-SHU 201 Honors Calculus
• MATH-SHU 235 Probability and Statistics OR MATH-SHU 238 Honors Theory of Probability
• MATH-SHU 329 Honors Analysis II OR MATH-SHU 142 Honors Linear Algebra II
Data Science (Concentration in Mathematics) and Honors Math
• MATH-SHU 141 Honors Linear Algebra I
• MATH-SHU 142 Honors Linear Algebra II
• MATH-SHU 238 Honors Theory of Probability
• MATH-SHU 329 Honors Analysis II
Data Science (Concentration in Political Science) and Social Science (Political Science Track)
• SOCS-SHU 150 Introduction to Comparative Politics
• SOCS-SHU 160 Introduction to International Politics
• MATH-SHU 235 Probability and Statistics OR BUSF-SHU 101 Statistics for Business and Economics
Data Science (Concentration in Psychology) and Social Science (Psychology Track)
• PSYC-SHU 101 Introduction to Psychology
• MATH-SHU 235 Probability and Statistics OR BUSF-SHU 101 Statistics for Business and Economics
• SOCS-SHU 350 Empirical Research Practice
• Choose One:
SOCS-SHU 334 Legal Psychology OR PSYC-SHU 234 Developmental Psychology OR PSYC-SHU 352 Psychology of Human Sexuality
Data Science (Concentration in Genomics) and Neural Science
• MATH-SHU 140 Linear Algebra
• MATH-SHU 235 Probability and Statistics
• BIOL-SHU 21 Foundations of Biology I
• BIOL-SHU 22 Foundations of Biology II
• BIOL-SHU 123 Foundations of Biology Lab
Data Science (Concentration in Genomics) and Biology
• MATH-SHU 140 Linear Algebra
• MATH-SHU 235 Probability and Statistics
• BIOL-SHU 21 Foundations of Biology I
• BIOL-SHU 22 Foundations of Biology II
• BIOL-SHU 123 Foundations of Biology Lab
• BIOL-SHU 261 Genomics and Bioinformatics
Note for Data Science (Concentration in Genomics) and Neural Science & Data Science (Concentration
in Genomics) and Biology: Students who take Linear Algebra and Probability and Statistics are not
allowed to take the lower-level Math Tools for Life Science course. Students who have not decided yet to
pursue a double major and take Math Tools for Life Science first are required to take Linear Algebra and
Probability and Statistics.
Note: Computer Science and Data Science share many courses, so double-majoring is not allowed. However,
students in Data Science can minor in Computer Science (and vice versa).
Double Major Sample Plans:
Declare your secondary major:
• Students need to complete more than half of the courses required for the primary and secondary majors.
• Students should present their four-year plan that demonstrates they can complete all degree requirements to their academic advisor for review. Create your four-year plan now!
Research Opportunities
• Bring your programming and data analysis skills to professors’ ongoing research projects!
Independent Study
Does not satisfy the major requirement. Students majoring in data science are permitted to work on an individual basis under the supervision of a full-time faculty member in the relevant discipline
if they have maintained an overall GPA of 3.0 and a GPA of 3.5 in data science and have a study proposal that is approved by the faculty and an academic area head.
Course Prerequisites
│Courses │Prerequisites │Semester│
│CSCI-SHU 101 Introduction to Computer and │prereq for CSCI-SHU 101 is CSCI-SHU 11 or placement exam │ │
│Data Science │ │ │
│BUSF-SHU 101 Statistics for Business and │NA │ │
│Economics │ │ │
│MATH-SHU 233 Theory of Probability │ │ │
│MATH-SHU 238 Honors Theory of Probability │PREREQ FOR MATH-SHU 238 is Grade C or better in either MATH-SHU 151 (Multivariable Calculus) or MATH-SHU 329 (Honors Analysis II), and grade C │ │
│ │or better in either MATH-SHU 140 (Linear Algebra) or MATH-SHU 141 (Honors Linear Algebra I). │ │
│MATH-SHU 235 Probability and Statistics │Prereq for MATH-SHU 235 is Grade C or better in either MATH-SHU 131 (Calculus) or MATH-SHU 201 (Honors Calculus). │ │
│CSCI-SHU 210 Data Structures │Prereq for CSCI-SHU 210 is ICS or A- in ICP │ │
│MATH-SHU 151 Multivariable Calculus │Prereq for MATH-SHU 151 is Grade C or better in either MATH-SHU 131 (Calculus) or MATH-SHU 201 (Honors Calculus).Antirequisite: MATH-SHU 329 │ │
│ │(Honors Analysis II) │ │
│MATH-SHU 328 Honors Analysis I │Prereq for MATH-SHU 328 is Grade C or better in MATH-SHU 201 (Honors Calculus), or grade A- or better in MATH-SHU 131 (Calculus) and A- or │ │
│ │better in MATH-SHU 143 (Foundations of Mathematical Methods), or authorization of the instructor. │ │
│MATH-SHU 140 Linear Algebra │prereq for MATH-SHU 140 is Sufficient high school grades, or NYU SH “Calculus and Linear Algebra” placement exam, or a grade of C or better in │ │
│ │MATH-SHU 9 (Precalculus). │ │
│MATH-SHU 141 Honors Linear Algebra I │ │ │
│MATH-SHU 265 Linear Algebra and │prereq for MATH-SHU 265 is Grade C or better in either MATH-SHU 131 (Calculus) or MATH-SHU 201 (Honors Calculus). │Fall │
│Differential Equations │ │ │
│CSCI-SHU 360 Machine Learning │Prereq for CSCI-SHU 360 is ICP, Calculus, Probability and Statistics OR Theory of Probability OR Statistics for Business and Economics │ │
│ECON-SHU 301 Econometrics │Prereq for ECON-SHU 301 is Statistics (BUSF-SHU 101 OR MATH-SHU 235 OR MATH-SHU 233 OR ECON-UA 18 OR STAT-UB 103 OR STAT-UB 1 OR MATH-GA 2901 OR│ │
│ │SOCSC-UH 1010Q OR ECON-UA 20). │ │
│MATH-SHU 234 Mathematical Statistics │ │ │
│CSCI-SHU 220 Algorithms │prereq for CSCI-SHU 220 is Data Structures and (Discrete Math or Honors Math major) and Calculus. │ │
│DATS-SHU 235 Information Visualization │Prerequisite or Co-requisite: Data Structures │ │
│CSCI-SHU 240 Introduction to Optimization │PREREQ FOR DATS-SHU 240 is (ICP or ICS) AND (Calculus or Honors Calculus). │ │
│and Mathematical Programming │ │ │
│CSCI-SHU 213 Databases │Prereq for CSCI-SHU 213 is CSCI-SHU 210 Data Structures. │ │
│DATS-SHU 420 Data Science Senior Project │Prereq for DS capstone is senior standing with DS primary or secondary major. │Fall │
│ │ │ONLY │
Study Away
Study Away Considerations
• Courses: Before studying abroad, students are recommended to complete Introduction to Computer Science and Data Science, Data Structures, Econometrics, Probability and Statistics, Multivariable
Calculus, and Machine Learning. Students who wish to study in New York ideally complete Databases.
• Location: Students planning to study away for two semesters are strongly encouraged to spend the first semester in a location other than New York. Applicants who spend the first semester away in
another location will receive priority consideration for New York in their second semester away. Students who elect to spend the spring of their junior year in New York (versus the fall of the
junior year) will have more earned credit points, which will enable them to have an earlier registration time and have a better chance of enrolling in high-demand courses.
• Senior Capstone: Students should not plan to study away during senior fall due to the in-person DS Senior Capstone course offering
Study Away Course Registration
Refer to the Fall 2023 Computer Science Pre-requisites and Equivalents for course inforamtion. Please note that students must follow the prerequisites of the school hosting the course. For example,
if Shanghai does not require a course for Class X, but New York does, then you will need to have that required course.
Python vs Java
In Shanghai, there are three course sequence ICP, ICDS, Data Structures (all taught in Python). At NYU CAS, there are the same three course sequence but teach ICS and Data Structures in Java.
At Tandon, their three-course sequence is ICP, Data Structure, Object Oriented Programming. ICP and Data Structures is taught in Python, OOP in Java.
As an NYU Shanghai CS and Data Science major, students can take ICS or Data Structures in Python or Java. However:
• If you are a DS major, we highly recommend you take both ICS and Data Structures in Python. Python is by far the most prominent language in data science, and several of our upper-level DS courses
are taught in Python. But if it is difficult to get into a Python class, then you can take these courses in Java.
• For CS majors, either Java or Python is fine. But you should be warned that if you take ICS in Java and then return to NYU Shanghai, you’ll be taking Data Structures in Python, and may be at a
disadvantage to those who took ICS in Python.
• All students should be warned that if they take ICS in Python, and then take Data Structures in NY in Java, they may be at a disadvantage since this would be their first course using Java,
whereas most NY students will already have had ICS in Java.
Senior Project
Please note that, starting in 2022-2023, DATS-SHU 420 Computer Science Senior Project will ONLY be offered in the Fall. The Senior Project course won't be offered in the Spring Semester. Check out
the CS/DS senior projects from the previous classes!
1. What is the structure of the Data Science Senior Capstone course?
The goal of this class is to complete a concrete CS project from start to finish. You can either solve a research problem or try to tackle a real-world problem. You need to design a valid method/
approach to solve the problem, build a solution using your method, and assess the quality of your solution. You may either work alone or form a team of at most 3 students.
2. What are the requirements of the capstone project?
At the end of the project, you must prepare a written technical report and a presentation. The final project report must be structured as a typical technical paper and will include four main
• Motivation, problem definition
• Related literature and existing approaches
• Proposed solution and details of implementation
• Results, conclusion, and directions for improvement
3. What should students prepare in advance to get ready for the capstone?
• Choose a research topic that NYU faculty have submitted or come up with a valuable topic of your own.
• Contact a faculty supervisor whose area of expertise matches the field of your topic, and start preparing as early as possible.
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Confidence Interval for a Correlation Coefficient | Online Statistics library | StatisticalPoint.com
Confidence Interval for a Correlation Coefficient
by Erma Khan
A confidence interval for a correlation coefficient is a range of values that is likely to contain a population correlation coefficient with a certain level of confidence.
This tutorial explains the following:
• The motivation for creating this type of confidence interval.
• The formula to create this type of confidence interval.
• An example of how to create this type of confidence interval.
• How to interpret this type of confidence interval.
Confidence Interval for a Correlation Coefficient: Motivation
The reason to create a confidence interval for a correlation coefficient is to capture our uncertainty when estimating a population correlation coefficient.
For example, suppose we want to estimate the correlation coefficient between height and weight of residents in a certain county. Since there are thousands of residents in the county, it would be too
costly and time-consuming to go around and gather information on every resident’s height and weight.
Instead, we might select a simple random sample of residents and simply gather information about them.
Since we select a random sample of residents, there is no guarantee that the correlation coefficient between height and weight for these residents in the sample will exactly match the correlation
coefficient in the larger population.
So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true correlation coefficient between height and weight of residents
in this county.
Confidence Interval for a Correlation Coefficient: Formula
We use the following steps to calculate a confidence interval for a population correlation coefficient, based on sample size n and sample correlation coefficient r.
Step 1: Perform Fisher transformation.
Let z[r] = ln((1+r) / (1-r)) / 2
Step 2: Find log upper and lower bounds.
Let L = z[r] – (z[1-α/2] /√n-3)
Let U = z[r] + (z[1-α/2] /√n-3)
Step 3: Find confidence interval.
The final confidence interval can be found using the following formula:
Confidence interval = [(e^2L-1)/(e^2L+1), (e^2U-1)/(e^2U+1)]
Confidence Interval for a Correlation Coeffficient: Example
Suppose we want to estimate the correlation coefficient between height and weight of residents in a certain county. We select a random sample of 30 residents and find the following information:
• Sample size n = 30
• Correlation coefficient between height and weight r = 0.56
Here is how to find a 95% confidence interval for the population correlation coefficient:
Step 1: Perform Fisher transformation.
Let z[r] = ln((1+r) / (1-r)) / 2 = ln((1+.56) / (1-.56)) / 2 = 0.6328
Step 2: Find log upper and lower bounds.
Let L = z[r] – (z[1-α/2] /√n-3) = .6328 – (1.96 /√30-3) = .2556
Let U = z[r] + (z[1-α/2] /√n-3) = .6328 + (1.96 /√30-3) = 1.01
Step 3: Find confidence interval.
Confidence interval = [(e^2L-1)/(e^2L+1), (e^2U-1)/(e^2U+1)]
Confidence interval = [(e^2(.2556)-1)/(e^2(.2556)+1), (e^2(1.01)-1)/(e^2(1.01)+1)] = [.2502, .7658]
Note: You can also find this confidence interval by using the Confidence Interval for a Correlation Coefficient Calculator.
Confidence Interval for a Correlation Coefficient: Interpretation
The way we would interpret a confidence interval is as follows:
There is a 95% chance that the confidence interval of [.2502, .7658] contains the true population correlation coefficient between height and weight of residents in this county.
Another way of saying the same thing is that there is only a 5% chance that the true population correlation coefficient lies outside of the 95% confidence interval.
That is, there’s only a 5% chance that the true population correlation coefficient between height and weight of residents in this county is less than .2502 or greater than .7658.
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Ip Dip
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
"Ip dip sky blue! Who's 'it'? It's you!"
Have you ever used this rhyme to decide who is 'it' in a game?
If you were playing a game with one friend and you wanted to be chosen to be 'it', would you start the rhyme pointing at yourself or your friend?
If there were three of you, how would you position yourself so that you were sure you'd be chosen?
How about with four of you? Five ...? Six ...? Seven ...? Eight ...? Nine ...? Ten ...? And so on?
How would you predict where you should stand to be chosen for any number of players?
Getting Started
How will you remember who is 'it' for different numbers of people?
You could use something like counters or cubes to represent people.
Student Solutions
We had lots of solutions sent in for this activity. Brookfield Junior School obviously got very involved and sent in many solutions. Here is an example of two of them.
Firstly, from Bill:
The formula to Ip dip is n≥8 when n stands for the number of friends so that means if the number of friends is eight or over you should be in the eighth position. To solve the first bit the
position is the remainder of the number of the friends divided by eight. so the formula is p=8/n= the remainder.
Secondly from Tomas:
If you had two people you would start in second position.
If you had three people you would start in second position.
If you had four people you would start in fourth position.
If you had five people you would start in third position.
If you had six people you would start in second position.
If you had seven people you would start in first position.
If you had eight people you would start in second position and so on.
From Mossley Primary School we had solutions sent in from Kati-leigh, Luke, Chiara, Amelia and Emily. Here is one example:
2: start with friend
3: right friend clockwise
4: second to right clockwise
5: third to right clockwise
6: left clockwise
7: yourself
8: right clockwise
9: second to right clockwise
10: third to right clockwise
11: third to left
Thanks for reading.
Amelia and Bea from Barton C E V A Primary School sent in the following
If there were 8 or more players, then always go for the 8th place.
If there were fewer than 8 players then you would find how many players there are and see what number adds to the amount of players to get to 8.You may need to use multiplication facts with this:
7 players: 1x7= 7 7 + 1 = 8 You go in 1st place
6 players: 1x6= 6 6 + 2 = 8 You go in 2nd place
5 players: 1x5= 5 5 + 3 = 8 You go in 3rd place
4 players: 1x4= 4 4 + 4 = 8 You go in 4th place
3 players: 2x3= 6 6 + 2 = 8 You go in 2nd place
2 players: 3x2 = 6 6 +2 = 8 You go in 2nd place
This is because there are 8 words in the rhyme.
Thank you, all of you, for the ideas you sent in, in order to solve this challenge.
Teachers' Resources
Why do this problem?
This problem
offers the opportunity for children to work on some mathematics that might be meaningful to them, and therefore engaging. At a basic level, it involves counting, but it is a good context in which
children can be encouraged to identify and explain patterns using their knowledge of factors and remainders.
Possible approach
You could start by having a pair of volunteers standing up together so that everyone can see them. Ask the children to imagine that they are going to play a game of tag, or something similar. How
would they choose who was going to be 'it' i.e. the person to do the chasing? Take some suggestions, which might involve some rhymes that are currently popular. You could choose to go with one of
these, or introduce the "Ip dip ..." rhyme with which some might already be familiar. (The important point at this stage is that the rhyme identifies the person to be 'it' straight away, having said
it only once.)
Say the chosen rhyme together a few times so that everyone feels they know it well and then indicate that you're going to find out who is going to be 'it' from the two volunteers. Say the rhyme while
pointing to the children alternately. You could then pose the question about who you would start the rhyme on if you wanted to be chosen. Give the whole group a chance to talk in pairs about this,
then test out their ideas. You can then encourage pairs to work together to discover where you would position yourself if there were three of you ... four ... five etc.
It may be appropriate to stop everyone after some time to share ideas so far. This might involve some pairs explaining how they are approaching the problem and sharing some possible ways of recording
what they're doing.
In the plenary, you can agree on solutions for the different numbers of people but also encourage children to talk about what they notice and to explain why where possible. How could they predict
where to stand if there were seven people, for example, or ten people or a hundred people? Some might find it tricky to articulate where to stand for fewer than eight people, but a few demonstrations
with larger numbers will mean they are able to explain where to be for eight or more relatively easily. Can they tell you why eight is the 'key' number?
Key questions
What numbers of people have you tried? What did you find out?
How are you going about this problem? Tell me what you've done.
How will you remember what you've found out?
Do you notice any patterns?
Can you explain the patterns?
Possible extension
Some children may like to try with another version of this rhyme which makes the analysis much more tricky: "Ip dip sky blue! Who's 'it'? Not you!" so that a person is 'knocked out' each time and the
only person left is 'it'. What happens when there are two people? Three? Four etc? Can they see any patterns emerging? Can they explain why the patterns occur? Similarly, learners might like to test
out the best places to be positioned for a rhyme of their choice.
Possible support
Having counters or other objects available to represent people might help some children.
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C++ find_if() | How find_if() Algorithm works with Examples | Advantages
Updated June 23, 2023
Introduction to C++ find_if()
The find_if() function in C++ is part of the standard library and is used to search for the first element in a defined range that satisfies a specified condition provided by a predicate function.
When find_if() encounters the first element in the range for which the predicate function returns true, that element is considered the result. It uses a unary predicate to specify the location of
elements from the range to consider for manipulating values or elements in the range.
InputIterator find_if (InputIterator fir_st, InputIterator l_st, UnaryPredicate predefined);
The syntax flow is in a way where the parameters represent the following:
• fir_st: This represents and specifies the range for the first element of the entire range.
• l_st: This represents and specified the range for the last element of the entire range specified.
• predefined: This predefined parameter is part of the Unary Predicate class in the template, which is used for checking the return type, which is considered as a boolean value with true or false
as the value.
How find_if() Algorithm Function work in C++?
• find_if() algorithm in C++ plays a vital role in terms of making elements search with a specified range. It first searches for elements required from the defined range. Once it encounters the
first element, it will check for its boolean condition with true or false values. Then it will use a unary predicate to locate an element from the range to make changes on consideration. The
function returns the iterator value if the element range satisfies the first value with the boolean value as true.
• There is not much complexity in the find_if() algorithm function because it makes the element search in a linear fashion starting from the first element of the range towards the last element of
the range and then for each element present in the range, which checks for each element and then lists all values of the range using unary Predicate for verification and returning the value to
the algorithmic function. All objects which are part of the find_if algorithm() specified range will be accessible depending on the condition which needs to be satisfied.
• Some race condition prevails in the find_if algorithmic function of C++. Some other functions also function in a way to search the elements from the first element of the range to the last element
of the range that includes find(), find_end(), find_first_of(), and many more. All these functions mentioned are also part of find_if() algorithm, which is part of the standard library in C++.
• It uses almost the same functionality as the find_if() function except for minor changes, which may include the time complexity and other data race conditions. The find_if() function makes use of
many other data structures like vectors and lists which further makes all manipulations possible in a linear fashion with minor changes in the complexity factor or any other factor like
manipulation element. Sometimes this function gets confused with the find_if_not() function where the functionality and traversal techniques are the same as the find_if function with some mere
changes in conventions like the predefined value of the unary operator must be false i.e. boolean value for the unary operator comes out to be false. This function works completely opposite to
the find_if function of C++.
Examples of C++ find_if()
Given below are the examples mentioned:
Example #1
Here’s an example program that demonstrates the usage of the find_if() function in C++ to search for the first odd digit within a specified range of elements:
int main()
std::array<int,4> ar_1={2,3,5,8};
std::array<int,4>::iterator r_t=std::find_if (ar_1.begin(), ar_1.end(), [](int o)
return o%2;
} );
std::cout<<"First_element_encountered_in_current_array: "<<*r_t<<"\n";
return 0;
Example #2
The following program demonstrates the usage of the find_if() function in C++. It searches for the first even number in the range and, if not found, checks for the odd number of elements in the
#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
bool un_ary_pred(int r_p)
return ((r_p % 2) == 0);
int main(void)
vector<int> vctr = {8, 10, 9, 2, 14};
auto y_o = find_if(vctr.begin(), vctr.end(), un_ary_pred);
if (y_o != end(vctr))
cout << "Even_Number : " << *y_o << endl;
vctr = {7};
y_o = find_if(vctr.begin(), vctr.end(), un_ary_pred);
if (y_o == end(vctr))
cout << "All_odd_Elements_in_vector" << endl;
return 0;
Advantages of C++ find_if()
• The complexity of the algorithmic function comes out to be linear after searching elements specified in the range of first to last.
• This function gives programmers flexibility and eases them to manipulate and work for the requirement.
This function makes the overall implementation of the algorithm and search of elements satisfying the conditions with the first and last elements of the range. It provides flexibility and versatility
to the programmers to work for desired elements and manipulation elements in a specific search pattern. Overall like other standard library functions find_if also plays a pivotal role in terms of
element searching and requirement implementation.
Recommended Articles
This is a guide to C++ find_if(). Here we discuss how find_if() algorithm function works in C++ with advantages and programming examples. You may also have a look at the following articles to learn
more –
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Transactions Online
1. Introduction
Digital communication systems and instrumentation applications promote the development of high-performance analog-to-digital converters (ADCs) [1]-[3]. Among many ADC architectures, Pipelined ADC is
very suitable for high-speed and high-precision operations. However, without calibration, due to the limitation of capacitor mismatches, it is arduous for ADC with a sampling rate exceeding 100MS/s
to exceed an accuracy of 12 bit. Meanwhile, as the CMOS technology keeps scaling down, high-gain amplifiers become increasingly toilsome to design. The two seriously deteriorate the linearity of
pipelined ADC [1], [4]. In order to boost the performance of high-speed pipelined ADCs, calibration techniques are needed.
A typical method is dither-based calibration technique, which features low power consumption and area penalty. By injecting small dither signals, this method can fulfil interstage gain error
calibration [3], [5]-[8]. J. Li et al. [8]-[10] have implemented a comparator dither injection technique, which achieves the goal of improving ADCs linearity by dynamically changing the threshold
voltage of comparator in Pipelined ADCs. Another injection method is capacitive dithering method, which was proposed in [11], [12]. However, both the comparator threshold and capacitive dither
injection induce an obvious increment of the residual amplifier’s output, which occupies the redundancy range, leading to higher-order nonlinearity and even saturation of the following stages [12]-
[15], [26].
The signal-dependent dithering proposed in [16], [17] eliminates the residual increment. Notwithstanding, the complexity of the analog circuits is evidently increased by injecting the dither through
the addition of two comparators and the division of the unit capacitor into two within each 1.5-bit stage. Furthermore, the use of twice the number of comparators to counter the increment of residual
amplitude not only eventuates in significant power consumption but also dilutes the sampling linearity [18]-[20].
In this brief, a new digital background calibration technique based on LMS, called complementary dithering here, is developed to achieve calibration of interstage gain error while eradicating the
increment of residual amplitude. Concurrently, the technique of constructing calibration windows exploiting the comparator resolving time nature has been proposed, which averts the use of duplicate
comparators and its digital logic is simple.
This brief is organized as follows. Section 2 details traditional dither injection structures. Section 3 depicts complementary dithering technique and calibration windows detector. Behavioral
simulation results are presented in Section 4, followed by conclusions in Section 5.
2. Traditional dither structure
Injection pseudo-random noise (PN) signal into ADCs, this approach leverages the characteristic of PN signals being independent of input signals to extract parameter information regarding non-ideal
factors within the ADC [22]-[24], which involves a diverse array of injection positions and error parameter extraction methods. Among the mature techniques currently in use, two prominent approaches
are the injection of sub-DAC dither [6] and sub-ADC dither [8], [9], [25]. These two methods effectively reduce the complexity of analog circuit design.
2.1 Sub-DAC dither injection
The calibration schematic diagram for injection PN signals into the sub-DAC is illustrated in Fig.1 [6], [17], [25]. The transfer function of the MDAC in this case is as follows:
\[\begin{equation*} V_{\textit{res}}=\frac{A\beta}{C_f(1+A\beta)}\cdot V_{\textit{res}}' \tag{1} \end{equation*}\]
\[\begin{equation*} V_{\textit{res}}'=\left(\sum_{i=1}^M C_i V_{in}-\sum_{i=1}^M C_i D_i - PN\cdot C_{\textit{injection}}\cdot V_{\textit{ref}}\right) \tag{2} \end{equation*}\]
Where \(C_i\), \(C_{\textit{injection}}\), \(C_f\), \(A\), and \(\beta\) represent the sampling capacitor, dither injection capacitor, feedback capacitor, finite open-loop gain, and feedback
coefficient, respectively, for that particular stage. In addition, \(D_i\) represents the comparison result between the input signal \(V_{\textit{in}}\) and the threshold voltage \(V_{\textit{th}}\)
of the \(i^{\textit{th}}\) comparator in the sub-ADC. When \(V_{\textit{in}} > V_{\textit{th}}\), \(D_i\) is 1, otherwise, \(D_i\) is 0. M represents the number of sampling capacitors in the sub-DAC,
equivalent to 4 for 1.5-bit/stage of the ADC.
By leveraging the autocorrelation and cross-correlation properties of the PN signal, practical interstage gain errors can be obtained when the PN sequence is sufficiently long:
\[\begin{equation*} \frac{1}{N}\sum_{j=1}^N\left[V_{\textit{res}}*(-PN)\right]= \left(1-\Delta G\right)\frac{C_{\textit{inject}}\cdot V_{\textit{ref}}}{C_f} \tag{3} \end{equation*}\]
Where \(G\) and \(\Delta G\) respectively represent the ideal interstage gain of this stage ADC and the interstage gain error caused by the limited open-loop gain of the operational amplifier, and
are respectively equal to \(\frac{\sum_{i=1}^M C_i}{C_f}\), \(\frac{1}{A\beta}\). This allows for the extraction of interstage gain error information for the operational amplifiers. However, when
injecting PN modulation signals into the sub-DACs and overlaying them on the signal path, it is necessary to ensure that the residue signal \(V_{\textit{res}}\) at the output of this stage does not
exceed the input dynamic range of the next pipeline stage. This requirement reduces the amplitude of the input signal \(V_{\textit{in}}\), ultimately degrading the signal-to-noise ratio (SNR) of the
2.2 Sub-ADC dither injection
The calibration schematic for injecting PN signals into the sub-ADC is depicted in Fig.2 [17], [25]. At this point, digital output of the sub-ADC \(D_{\textit{sub}}\) and the backend ADC \(D_{b_{\
textit{end}}}\) are represented by equations (4) and (5) respectively:
\[\begin{equation*} D_{\textit{sub}}=\frac{V_{\textit{in}}}{V_{\textit{ref}}}+PN\cdot \frac{V_{\textit{cal}}}{V_{\textit{ref}}}+Q_s \tag{4} \end{equation*}\]
\[\begin{equation*} D_{b_{\textit{end}}}=-\frac{G\left(PN\cdot V_{\textit{cal}}+Q_s\cdot V_{\textit{ref}}\right)}{V_{\textit{ref}}}+Q_b \tag{5} \end{equation*}\]
Where \(Q_s\) stands for the quantization error of the sub-ADC, and \(Q_b\) is the quantization error of the backend ADC. Correlating the ADC digital output with the PN signal, as follows:
\[\begin{equation*} PN*D_{\textit{out}}=PN\left(D_{b_{\textit{end}}}+D_{\textit{sub}}G_{\textit{es}}\right) =\frac{PN^2\left(G_{\textit{es}}-G\right)V_{\textit{cal}}}{V_{\textit{ref}}} \tag{6} \end
It is evident that error parameters can also be extracted, and after calibration, the estimated value \(G_{es}\) approaches the true value infinitely closely.
It can be observed that the second dither injection scheme is similar to the first one, except that the injection position of the PN signal is changed. However, since the comparator threshold dither
only affects the input signals within its dither coverage range, the dither effect is relatively small, and the scattering effect on the spectrum is poor. At the same time, both schemes will cause
the output range of the operational amplifier to increase, which takes up the redundancy range.
3. Proposed complementary dither technique and calibration windows detector
3.1 Complementary dither technique
In order to address the problems existing in the above scheme, as presented in Fig.3, the complementary dither technique is introduced using a modified 1.5-bit stage as an example. To avoid residual
output overflow due to PN signal injected into the DAC, the 1.5-bit stage here adds two comparators and two capacitors.
Figure 4(a) compares the two 1.5-bit stage residual transfer functions (RTF) of a conventional stage with a PN signal of \(\pm V_{\textit{DAC}}\) injected into the DAC. Evidently, due to the
injection of PN signal, the residual transfer function will move up or down, causing the RTF to shift from \(\pm 1/2V_{\textit{ref}}\) expanded to \(\pm(1/2V_{\textit{ref}}+V_{\textit{DAC}})\).
However, supposing that the PN signal with a size of \(\pm V_{\textit{ADC}}\) is injected into the sub-ADC to change the comparator threshold, the residual transfer functions is shown in Fig.4(b).
Apparently, the transition point of the RTF will move left or right, which will also induce the RTF from \(1/2V_{\textit{ref}}\) dilated to \(\pm (1/2V_{\textit{ref}}+| V_{\textit{ADC}}| \cdot \
textit{Ge}\)), where \(\textit{Ge}\) is the interstage gain coefficient of this stage.
Provided that the direction of the both dither injections are opposite and amplitude satisfies \(V_{\textit{DAC}}=V_{\textit{ADC}}\cdot \textit{Ge}\), the residual increment disappears, as presented
in Fig.5. The both dither injections will have a complementary effect, limiting the output of the residual amplifier to within \(\pm(1/2V_{\textit{ref}})\) range. The result is that complementary
dither injection does not occupy any redundancy range of the pipelined ADC.
From Fig.5, it can be seen that if the input signal undergoes the corresponding selection, so that the input signal is limited within the purple arrow range, this range is defined as the calibration
window. Furthermore, the input signal within the calibration window is only affected by the DAC dither injection, and is not affected by the sub-ADC dither injection. If the pseudo-random noise
signal injection by the sub-ADC has an amplitude of \(V_d\), the calibration window statistics are as shown in Table I:
If only the corresponding correlation operations are performed on the signals within the calibration window, it is equivalent to only injecting dither into the DAC. Due to the dither passing through
the same path as the DAC signal, the identical non ideal situation will be encountered, and the interstage gain error can be detected to achieve the extraction of interstage gain error. The scheme
restricts residual output to \(\pm 1/2V_{\textit{ref}}\), which greatly mitigates the design requirement of the residual amplifier. More than that, the injected dither exhibits higher amplitude,
making interstage gain error coefficient easier to extract, thereby significantly enhancing the convergence speed.
While correlation operations can estimate interstage gain errors in this process, they may also introduce certain issues, such as trade-off between convergence speed and precision. Hence, the
utilization of the least mean square (LMS) algorithm for the extraction of interstage gain errors is depicted in the schematic diagram as shown in Fig.6 [27]. The numerical value of PN signal in
digital domain is called \(\mathrm{D}_{\text{compensation}}\). The calibration process is as follows:
\[G[n+1]=G[n]\pm\mu\cdot PN[n]\cdot (\textit{Dob}1-PN[n]\cdot G[n])\]
where \(G[n+1]\) is the \((n+1)^{\textit{th}}\) estimated value of the interstage gain coefficient, \(\textit{Dob}1\) is the digital output of the stage being calibrated, \(\mu\) is the convergence
step size factor controlling the calibration algorithm’s convergence speed and accuracy.
The digital output signal \(\textit{Dob}1\) is obtained from the back-end pipeline with all the needed correction applied to make it as accurate a representation of the residue as possible. The
dither is multiplied by the estimate of the interstage gain coefficient and is subtracted from the output signal. The result is multiplied by the ideal dither and by \(\mu\), then passes through an
accumulator to give an estimate for the interstage gain coefficient Ge. The dither estimate is subtracted from the residue, which is corrected by the estimate of the interstage gain error to give the
calibrated digital output Dob1_cal. The LMS iteration forms a feedback loop whose iteration speed is governed by the step size \(\mu\). A higher \(\mu\) leads to a swifter convergence but lower
calibration precision and vice versa.
3.2 Calibration windows detector
B. Murmann et al. [18], [19], [21] employed two-fold comparators to screening the input signal range affected solely by the injection of DAC dither. However, this scheme not only results in increased
power consumption but also diminishes the sampling linearity. We introduce a calibration windows detector (CWD), which leverages the meta-stability nature of comparator to construct calibration
windows [28]-[30]. When \(V_{\textit{in}}\) approaches the \(V_{\textit{th}}\), the comparator’s differential input is minimal, necessitating more time for the comparator to resolve a valid logic
level. Hence, the comparison time inherently encodes information about the signal range, as illustrated in Fig.7. Taking the example of the \(i^{\textit{th}}\) comparator with the threshold voltage
\(V_{\textit{thi}}\) in the sub-ADC, Fig.7 provides a comprehensive schematic and timing diagram to elucidate the principles of this approach. Figure 8 demonstrate the structure of sub-ADC with
calibration windows detector.
The calibration windows detector comprises an XOR gate, a Flip-Flop and a manually adjustable delay buffer. The clock is simultaneously connected to both the comparator and the buffer, with the
buffer introducing a delay of \(T_b\), which can the manually controlled. The comparator only operates when the clock signal arrives at the rising edge, but the comparison result is transferred to
the input of the Flip-Flop only after a delay of one resolving time \(T_r\). By comparing the reference time \(T_b\) of the delay buffer with the resolving time \(T_r\) of the comparator, the range
of the input signal can be determined. The relationship between the resolving time \(T_r\) of the comparator and the input signal \(V_{\textit{in}}\) is given by:
\[T_r=t_0+\tau\cdot\ln \left(V_{\textit{FS}}/\left| V_{\textit{in}}-V_{\textit{th},i}\right|\right)\]
where \(t_0\) is the delay time of all the logic circuits, \(\tau\) is the time constant of the latch and \(V_{\textit{FS}}\) is the full-scale output [31]. The corresponding comparator resolving
time when \(\left|V_{\textit{in}}-V_{\textit{th},i}\right| =V_d\) is denoted as \(T_d\). When \(T_d< T_r\), it indicates that \(\left|V_{\textit{in}}-V_{\textit{th},j}\right| < V_d\). Accordingly,
when the clock edge arrives at the trigger, the output of the \(i^{\textit{th}}\) comparator remains in the reset state. Consequently, the output of the trigger should be in a low logic level.
Conversely, when \(T_d> T_r\), signifying \(\left|V_{\textit{in}}-V_{\textit{th},j}\right| > V_d\), the trigger’s output should be in a high logic level.
Clearly, it is possible to determine the input signal’s range based on the trigger’s output, thus deducing the calibration windows. This not only facilitates the extraction of interstage gain errors
within the entire input range, significantly enhancing calibration convergence speed but also eliminates the need for using double the number of comparators.
4. Simulation results
To validate the effectiveness of the proposed calibration scheme, several behavioral simulations are provided for a 12-bit 1.25GS/s pipelined ADC that is comprised of 8 1.5-bit stages and 4-bit
flash ADC as the last stage. Due to the inherent characteristics of the pipelined architecture, the precision requirements decrease as stages progress. Therefore, calibration was applied to the
interstage gain error in the first four stages only. To simplify the simulation, the interstage gain errors and the capacitor mismatch for the first four stages are assumed to be 2% and 0.1%, and the
subsequent stages were considered ideal. In addition, a foreground calibration is employed for the initial four stage capacitance mismatch.
Figure 9 depicts the simulated spectrum of the ADC output before calibration. The SNDR and SFDR are approximately 44.27dB and 49.43dB, respectively. After utilizing the complementary dither
technique and the calibration windows detector technique, the spectrum is shown in Fig.10. The SNDR and SFDR have been improved to approximately 70.8dB and 115.3dB, achieving 26.53dB and 65.9dB
enhancements, respectively.
The depicted Fig.11(a) shows that the differential nonlinearity (DNL) and the integral non-linearity (INL) of the defined converter before calibration are \(+0.94/-1\) LSB, and \(+21/-21.5\) LSB,
respectively. Applying the proposed calibration, the ADC’s DNL and INL, are tuned to about \(+0.2/-0.19\) LSB and \(+0.14/-0.15\) as shown in Fig.11(b).
Figure 12 presents the iterative convergence plots of the interstage gains coefficient, \(\textit{Ge}_1\), \(\textit{Ge}_2\), \(\textit{Ge}_3\), and \(\textit{Ge}_4\), for the converter after
calibration. Clearly, \(\textit{Ge}_1\), \(\textit{Ge}_2\), \(\textit{Ge}_3\), and \(\textit{Ge}_4\) have all converged to their theoretical values 1.96.
Table II furnishes the comparison results to the other technique currently available. In summary, the proposed calibration technique has almost no residue output increment, which will greatly
alleviate the design requirements of residual amplifier. Moreover, this scheme yields a more substantial enhancement in the dynamic performance of the ADC, particularly in terms of SFDR and SNDR.
More importantly, it eschews any additional analog circuits making this formula to be more benefited from the CMOS process scaling.
5. Conclusion
In this paper, we propose a background calibration technique for interstage gain errors based on complementary dither injection with calibration windows detector. Complementary dithering not only has
a better scattering effect on spectral spurs but also cancels the residual amplitude increment of residual amplifier. what’s more, this approach significantly enhances the SFDR and SNDR of the ADC
while also alleviating the design requirements for the residual amplifier. In addition, the proposed calibration windows detector technique can avoid the use of twice the number of comparators.
Compared with existing calibration techniques, it does not change the analog circuit and only has a small digital circuit overhead.
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Huaiyu Zhai
Institute of Microelectronics of the Chinese Academy of Sciences
University of Chinese Academy of Science
Hanbo Jia
Institute of Microelectronics of the Chinese Academy of Sciences
Xuan Guo
Institute of Microelectronics of the Chinese Academy of Sciences
Zilin Jiang
Institute of Microelectronics of the Chinese Academy of Sciences
University of Chinese Academy of Science
Yuzhen Zhang
Institute of Microelectronics of the Chinese Academy of Sciences
University of Chinese Academy of Science
Dandan Wang
Institute of Microelectronics of the Chinese Academy of Sciences
University of Chinese Academy of Science
Jin Wu
Institute of Microelectronics of the Chinese Academy of Sciences
University of Chinese Academy of Science
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When the circle rolls along another circle inside it, the curve is called a
Q. When the circle rolls along another circle inside it, the curve is called a
A. epicycloid
B. cycloid
C. trochoid
D. hypocycloid
Answer» D. hypocycloid
Explanation: cycloid is a curve generated by a point on the circumference of a circle which rolls along a straight line. ‘epi’ represents the directing path is a circle. trochoid is a curve generated
by a point fixed to a circle, within or outside its circumference, as the circle rolls along a straight line. ‘hypo’ represents the generating circle is inside the directing circle.
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#Help Page
Thermodynamic analysis tools such as the Nucleic Acid Package (NUPACK) [1] can calculate the equilibrium concentrations of DNA complexes given the concentrations of the constituent DNA strands.
NUPACK first determines the free energy of formation of complexes based on the strands’ sequences, and then determines the equilibrium concentration of each complex based on the complex’s free
energy. However, the calculation of the complex free energy is generally substantially slower than the computation of the equilibrium concentrations, which is a well-studied form of convex
optimization with fast convergence. Thus, significant speedups can be achieved if complex free energy for relevant complexes is known beforehand, for example by empirical measures or by
approximations like domain-level DNA hybridization assumptions.
Concentrat.io computes free energies and concentrations of domain-level DNA designs. At this high level of abstraction, our tool allows arbitrary connectivity between the bound strands, including
pseudoknots, albeit without penalizing geometrically infeasible configurations. The model is shared by thermodynamic binding networks [2].
#Input Syntax
#Specification of monomers and their concentrations
Each monomer (strand) is specified on a separate line in Monomers and Concentrations. The binding sites (domains) are space separated. The list of binding sites is followed by a comma, then the
monomer concentration.
a b, 10
a* b*, 70.9 # This is a comment
a, 50
b, 50
The concentration units are specified by the Concentration Units dropdown menu (mM, µM, nM, or pM).
Monomers (strands) can be optionally named by specifying a label followed by a colon.
strand1: a b, 100
a* b*, 50
#Binding site free energies
The field Default Binding Energy specifies the default binding free energy (kcal/mol) of all binding sites (domains). To specify different energies for different binding sites, you can use the
optional text field Site Binding Energies as follows.
a = -10.5
b = -20
Note that all binding sites not listed in the Site Binding Energies field receive the default energy from Default Binding Energy (default -20.6 kcal/mol).
Note that the binding site free energies (whether default or explicitly specified) do not get adjusted with varying Temperature.
#Complex Enumeration
We say a complex is splittable if it can be partitioned into two complexes while maintaining the same bonds. For example, consider the complex consisting of the following monomers: a b, a* b*, and a.
This complex can be split into two complexes [a b, a* b*] and [a] while maintaining one a-a* bond and one b-b* bond. Thus the original complex is splittable.
If Maximum Complex Size is set to infinity (the default setting), then concentrat.io will enumerate all unsplittable complexes. Note that there is a mathematical correspondence between the
unsplittable complexes and the Hilbert basis of the corresponding linear problem [4]. This proves that the number of unsplittable complexes is always finite.
While the number of unsplittable complexes is finite, it may be very large. Setting Maximum Complex Size to a positive integer restricts the maximum size (number of monomers) in the unsplittable
polymers enumerated.
#Free Energy Calculation
For any complex, its free energy is computed as follows:
$\Delta G = \Delta H - T \cdot k_B \cdot \ln(R) + \Delta G^\text{assoc} \cdot (L - 1)$
where $T$ is the temperature (K), $k_B$ is Boltzmann's constant (0.001987204259 kcal/mol/K).
$\Delta H$ is the total binding energy of the complex (kcal/mol). In other words, it is the sum of the binding energies of all the bound domains. Note that the domain binding energy is directly taken
from Default Binding Energy or from its specification in Site Binding Energies without any temperature adjustment (i.e., we assume that these values are already temperature-adjusted when specified).
Complexes of more than one strand are penalized by $\Delta G^\text{assoc}$ (ΔG of association) for each additional strand. Here $L$ is the number of strands in the complex. $\Delta G^\text{assoc}$ is
computed in exactly the same way as in Nupack 3 for DNA parameters (including temperature adjustment), except that no salt correction is applied.
$R$ is the number of microstates corresponding to different equivalent permutations of strands in the complex. Specifically, if this complex has count $n_i$ of strand $i$, then $R = \prod_i n_i!$.
As a concrete example, consider the complex consisting of two monomers x x and one monomer x* x* x* x*. Let us assume Default Binding Energy of -20.6 kcal/mol. Then for this complex, $\Delta H =
-20.6 \cdot 4$ kcal/mol, $L = 3$, and $R = 2$.
Note that while we correct for multiple copies of the same strand in a complex, we do not consider multiple possible ways of making bonds within a complex. This choice is due to a priori not knowing
which of these ways is geometrically feasible.
The following system appears in [2] (Figure 3.3). Paste the following into Monomers and Concentrations:
# This system acts like an AND gate with
# the following two strands as input
input1: a1 a2, 100
input2: b1 b2, 100
a1* a2* b1* b2*, 100
a1 a2 b1 b2 c1, 100
a2* b1* b2* c1*, 100
a2 b1, 100
b2 c1 c2, 100
c1* c2*, 100
output: c1 c2, 100 # This strand is the output
Leave the other fields at their default values.
Clicking on Compute Energies shows the histogram and table of the free energies of the enumerated 54 unsplittable complexes. Clicking on Calculate Concentrations returns the concentrations of these
complexes in the table.
The search box above the table of concentrations filters the table to show only the complexes containing the monomers listed in the search box (comma separated). Try typing output in the search box.
We can see that 36nM of the output monomer is free while ~64nM is together in the complex with c1* c2*.
Now try commenting out the lines with the input monomers:
input1: a1 a2, 100
input2: b1 b2, 100
Recomputing the concentrations shows that now almost all of the output monomer is together with c1* c2*.
#API Documentation
Concentrat.io functionality is also exposed via a Web API. The following example demonstrates how to use Python code to compute concentrations:
import requests
# Define the API endpoint
url = 'https://concentrat.io/api/calculate/concentrations'
# Set the headers to send and accept JSON
headers = {
'Content-Type': 'application/json'
# Define the data to be sent in the POST request
data = {
"max_complexes": 3,
"temperature": 25,
"max_complex_energy": 0,
"concentration_unit": "nM",
"binding_energy": -20,
"energies_inputs": "",
"monomers": [
["a b", 10],
["a* b*", 10],
["a", 10],
["b", 10]
# Make the POST request
response = requests.post(url, headers=headers, json=data).json()
# Print the response from the server (json)
The response contains both the input as well as the output energies and concentrations of complexes. For example, response["complexes"][0] gets the 0th complex (highest concentration) which in this
case is:
'monomers': [[0, 1], [1, 1]],
'free_energy': -40.48608306337398,
'concentration': 9.999801e-09
The monomers field represents the complex as an array of ([monomer index], [number]). Thus this complex has one monomer "a b" and one monomer "a* b*".
[1] J. N. Zadeh, C. D. Steenberg, J. S. Bois, B. R. Wolfe, M. B. Pierce, A. R. Khan, R. M. Dirks, N. A. Pierce. NUPACK: analysis and design of nucleic acid systems. J Comput Chem, 32:170–173, 2011.
[2] K. Breik, C. Thachuk, M. Heule, D. Soloveichik. Computing properties of stable configurations of thermodynamic binding networks. Theoretical Computer Science 20;785:17-29, 2019.
[3] J. Petrack, D. Soloveichik, D. Doty. Thermodynamically Driven Signal Amplification. DNA Computing and Molecular Programming 29 (DNA29), 2023.
[4] D. Haley, D. Doty. Computing properties of thermodynamic binding networks: An integer programming approach. arXiv preprint arXiv:2011.10677, 2020.
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Physics and Astronomy Dissertation Defense - Chris Shill - Department of Physics and Astronomy
UNC-CH Physics and Astronomy Dissertation Defense
Chris Shill
“Stochastic and Semi-Classical Approaches to the Quantum Virial Expansion”
Ultracold atomic gas experiments have seen rapid development in recent years. This is largely due to the highly clean and malleable nature of these systems that has led to such advancements as the
measurement of Bose-Einstein condensate, and the development of graphene, both of which resulted in a Nobel Prize. In addition, ultracold atomic systems are, in general, dilute and dominated by
short-range, s-wave interactions, which provides a perfect basis for applying stochastic methods for computing in this area of quantum matter. In particular, we focus on regions of high temperature
where statistical methods, like Quantum Monte Carlo, tend to struggle. This region is known as the virial region and is described using the virial expansion of the grand thermodynamic potential.
In this talk, we will present two novel, non-perturbative techniques used for computing coefficients of the quantum virial expansion for non-relativistic Fermi gases with zero-range interactions. The
first is a stochastic method that utilizes a Fourier projection to extract the virial coefficients directly from the grand potential, and the second is a semi-classical lattice approximation. We
present our results for the interaction dependence on the virial coefficients in one, two, and three dimensions, as well as an estimate for the radius of convergence of the virial expansion in one
dimension, a new result also provided by the Fourier projection method.
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Developerblog – Johannes Jendersie
From time to time you need a function in procedural content generation. A function with certain properties which can be edited easily. A function of your own design.
First of all you need a tool to plot functions or create triangulations (for 2D/3D functions). I took GNU Octave to design the following functions and than ported the code.
Lets start with a small example. Later I will show you a more complex variant I designed for sound synthesis.
A good start is a picture. Draw it and think about what should happen for a wish list of parameters. Lets assume we would like to have a function which starts at and ends at . Inside this box we
would like to set one point which is on the function. This is a standard interpolation problem but it is a good example and an element of the complex function we will see later.
In next step I always think about other functions which approximate my picture: exp, log, polynomials, sin, cos, ...
Well for our example a line will suffice. The only problem: it must have a break. The easiest thing is to use two lines and somehow combine them.
The first one goes through and its slope is well defined by . The second one can be defined by a view on the upper corner. Again the slope is a simple quotient which can be put into the line equation
and gives
Now we could create a piecewise-defined function or use. The second variant is valid only for the upper left triangle. It might be useful for another parametrization or function.
My Universal Periodic Function
I searched for a function with a small parameter set which can:
• Create high frequent basic tones
• Be used for fading of different frequencies
• Be used as fade function for the volume
I started with following ideas: A function for a sound must be periodic. I choose as the domain for one period and a co-domain of because then we have a good control later when using this function.
To get something periodic in the interval we could use fmod or just the fractional part. As long as we define a function in the given interval we do not need to think about periodicity.
Furthermore I would like to be able to mix any standard periodic function (Sinus, Triangle, Square, Sawtooth) by setting parameters. I designed 3 parameters to achieve that:
Shape: Just interpolate between Sinus and Triangle function. This can be seen as smoothness parameter too. The implementation is as simple as the idea. The triangle function is designed from three
lines with slopes 4, -4 and 4 again.
Stretch: Add some space at the vertices of the base functions. This can be used to create a Rectangular function or it is useful for volume and frequency fading too. Just take the first half and you
have a function which fades in and out and keeps one level in between.
Shift: Move the vertices left and right (point symmetric). Values near -1 or 1 create a Sawtooth function. Again using only the first half this can be used to control fading.
Moreover Stretch and Shift can be used to design Attack, Sustain and Release of a note. This is not the full ADSR model, but if I can use the same function on many levels it is a good compromise.
I started with the implementation of the shape factor. Here I had no better idea than a simple interpolation. The Triangle wave consists of three linear functions: . To combine them one could use
branches for and , but I expect the usage of min/max to be faster. This time there isn't any parameter which disturbs the ordering of the functions as in the example above.
float upf(float _x, float _shape)
float sinX = sin( _x * PIx2 );
float triangleX = 4.0f * max(min(_x, 0.5f - _x), _x - 1.0f);
return lerp(sinX, triangleX, _shape);
Afterwards I added the stretch factor. Therefore we must solve the problem: How can we add space at the points 0.25 and 0.75?
Easier is to think: How must x look like that sin(x) will have these plateaus?
So if we have a function as on the picture we are ready. Applying it to x before calling upf(x) adds the plateaus. Moreover we don't need to bother about the shape factor. Its functionality is
orthogonal so we can change the parameters independently.
This trick to remap the input is like changing the contrast in a picture. All we need is a monotonous function and we can change the out coming function while staying inside the given range.
In the implementation I just focused on one single step in , copied that and added 0.5 to the right side.
1 float stretch(float _x, float _stretch)
2 {
3 // Use only one step function and copy it.
4 bool secondHalf = (_x >= 0.5f);
5 // The slope if the lines between the constant sections goes from
6 // 1 (_stretch = 0) to infinity (rectangular function _stretch = 1).
7 if( _stretch == 1 ) return secondHalf ? 0.75f : 0.25f;
8 float slope = 1.0f / (1.0f - _stretch);
9 _x -= secondHalf ? 0.5f : 0.0f;
10 _x *= slope;
11 _x = max(min(0.25f, _x), _x - 0.5f * slope * _stretch);
12 return secondHalf ? (_x + 0.5f) : _x;
13 }
Last of all we will add the shift parameter the same way as the stretch parameter. Again a picture can explain that best. Essentially we are using the example from the introduction but we choose the
point . For the right side the same slopes can be used again so three lines (two with the same scope) are created from the one point.
Holding is important because we want the shift to move the vertices and no arbitrary points. At the point where the remapping returns 0.25 the vertices are created later which is where we want the
change in frequency to happen. I also tested a smoother (exponential) step function but it was less controllable.
1 float shift(float _x, float _shift)
2 {
3 // Compute the point where the lines should intersect:
4 float px = _shift * 0.25f + 0.25f;
5 float py = 0.25f;
6 // Calculate m*x+n with one of the three lines.
7 if( _x < px ) return (py / px) * _x;
8 if( _x <= (1.0f-px) ) {
9 float m = py / (0.5-px);
10 return m * _x + 0.5f - 0.5f * m;
11 }
12 return (py / px) * _x + 1.0f - (py / px);
13 }
This time I consider branching as faster. It is also important to guarantee stability. For parameters -1 and 1 px gets 0. For both cases the branching will end up in the middle part where everything
works fine. Otherwise we could get a division by zero.
Taking all together we have a really customizable function.
float upf(float _x, float _shape, float _stretch, float _shift)
// Make it periodic
_x -= floor(_x);
_x = shift(_x, _shift);
_x = stretch(_x, _stretch);
return upf(_x, _shape);
Except the function shown here I earlier created a light falloff function without a singularity and a determined radius... I could get used to build my own functions. Finally I will give you a list
of things I learned:
• Use a tool which can plot functions.
• Always draw your idea of a function on a paper and than try to find an approximation.
• You should know graphs of functions as: sinus, tangents, exp, gauss, ln, polynomials, ...
• Linear functions are your friend for parametrization: you will exactly know what they will do.
• Many things can be achieved with a polynomial: If you have enough criteria as fixed points and derivatives you can just solve some equations to find the proper factors for a fitting polynomial
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// : Study of the Gravitational Interaction Between Two Line Masses
The // body problem (slash-slash) is the study of the gravitational interaction between 2 extended line masses. The topic of this thesis is the study of the planar // body problem, where the universe
is restricted to a plane. The analysis is performed using completely classical methods. The potential and kinetic energies are derived in the center of mass frame using polar coordinates. The
Euler-Lagrange equations are then used to write down the equations of motion. Geometric vectors are used to radically simplify the equations. The equations of motion are shown to reduce to the planar
/. body problem and the Newtonian 2 body problem in the appropriate limits. Three classes of periodic orbits are solved exactly and one class is believed to be stable. The Runge-Kutta-Fehlberg method
is used to find numerical solutions for a given set of initial conditions using 64 digit precision. We describe a robust numerical mechanism to detect collisions before they occur. A GUI application
automates the numerical processes. Retrograde spin was observed to stabilize orbits while prograde spin destabilized orbits. Escape not possible in the Newtonian 2 body problem was observed in the
planar // body problem. Using parameter space plots we found that the gravity gradient orbit generates a valley of stability around its theoretical curve.
Second Advisor
Pasteur, R. Drew
Mathematics; Physics
Recommended Citation
Blaikie, Andrew, "// : Study of the Gravitational Interaction Between Two Line Masses" (2013). Senior Independent Study Theses. Paper 955.
Applied Mathematics
two body problem, gravitational dynamics
Degree Granted
Bachelor of Arts
Document Type
Senior Independent Study Thesis
© Copyright 2013 Andrew Blaikie
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Basic-principles of Physics Topics | Question AI
<html lang="en" class="topic-desktop ui-chrome110 ui-chrome"><head></head><body data-leg="B" class="new-topic topic-desktop first-page-false user-ANONYMOUS user-ads md-desktop leg-b"><header id=
"header" class="bg-navy-dark"></header><main><div class="md-page-wrapper"><div id="content" class="md-content"><div class="md-article-container template-desktop infinite-pagination"><div class=
"infinite-scroll-container article last"><article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="553306"><div class="grid gx-0"><div class
="col-auto"><div class="topic-left-rail md-article-drawer position-relative d-flex border-right-sm border-left-sm open"><div class="drawer d-flex flex-column open"></div></div></div><div class="col">
<div class="h-100 ml-0 pr-sm-10 pr-lg-0 "><div class="h-100 grid gx-0 gx-sm-20"><div class="h-100 col-sm"><div class="h-100 infinite-pagination-container d-flex flex-column position-relative"><div
class="grey-box w-100 grey-box-top"><div class="grey-box-content mx-auto w-100"><div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="false"><div class="reading-channel">
<section data-level="1" id="ref77455"><p class="topic-paragraph">In addressing any problem in <span class="md-dictionary-link md-dictionary-tt-off mw" data-term="continuum" data-type="MW">continuum</
span> or <span class="md-crosslink autoxref" data-show-preview="true">solid</span> <span class="md-crosslink autoxref" data-show-preview="true">mechanics</span>, three factors must be considered: (1)
the <span id="ref611593"></span><span class="md-crosslink" data-show-preview="true">Newtonian equations of motion</span>, in the more general form recognized by Euler, expressing conservation of
linear and angular <span id="ref611594"></span><span class="md-crosslink" data-show-preview="true">momentum</span> for finite bodies (rather than just for point particles), and the related concept of
<span id="ref611595"></span><span class="md-crosslink" data-show-preview="true">stress</span>, as formalized by Cauchy, (2) the geometry of deformation and thus the expression of <span id=
"ref611596"></span><span class="md-crosslink" data-show-preview="true">strains</span> in terms of gradients in the displacement <span class="md-crosslink autoxref" data-show-preview="true">field</
span>, and (3) the relations between stress and strain that are characteristic of the material in question, as well as of the stress level, <span class="md-crosslink autoxref" data-show-preview=
"true">temperature</span>, and time scale of the problem considered.</p><p class="topic-paragraph">These three considerations <span class="md-dictionary-link md-dictionary-tt-off mw" data-term=
"suffice" data-type="MW">suffice</span> for most problems. They must be supplemented, however, for solids undergoing <span id="ref611597"></span><span class="md-crosslink" data-show-preview="true">
diffusion</span> processes in which one material <span class="md-dictionary-link md-dictionary-tt-off mw" data-term="constituent" data-type="MW">constituent</span> moves relative to another (which
may be the case for fluid-infiltrated soils or petroleum reservoir rocks) and in cases for which the <span class="md-dictionary-link md-dictionary-tt-off mw" data-term="induction" data-type="MW">
induction</span> of a temperature field by deformation processes and the related <span class="md-crosslink autoxref" data-show-preview="true">heat transfer</span> cannot be neglected. These cases
require that the following also be considered: (4) equations for <span id="ref611598"></span><span class="md-crosslink" data-show-preview="true">conservation of mass</span> of diffusing <span class=
"md-dictionary-link md-dictionary-tt-off mw" data-term="constituents" data-type="MW">constituents</span>, (5) the <span class="md-crosslink autoxref" data-show-preview="true">first law of
thermodynamics</span>, which introduces the concept of heat flux and relates changes in <span id="ref611599"></span><span class="md-crosslink" data-show-preview="true">energy</span> to work and heat
supply, and (6) relations that express the diffusive fluxes and heat flow in terms of spatial gradients of appropriate chemical potentials and of temperature. In many important technological devices,
electric and magnetic fields affect the stressing, deformation, and motion of matter. Examples are provided by piezoelectric crystals and other ceramics for electric or magnetic actuators and by the
coils and supporting structures of powerful electromagnets. In these cases, two more <span class="md-dictionary-link md-dictionary-tt-off eb" data-term="considerations" data-type="EB">considerations
</span> must be added: (7) James Clerk <span id="ref611600"></span><span class="md-crosslink" data-show-preview="true">Maxwell’s</span> set of equations interrelating electric and magnetic fields to
polarization and magnetization of material media and to the density and motion of <span class="md-crosslink autoxref" data-show-preview="true">electric charge</span>, and (8) augmented relations
between stress and strain, which now, for example, express all of stress, polarization, and magnetization in terms of strain, <span class="md-crosslink autoxref" data-show-preview="true">electric
field</span>, <span class="md-crosslink autoxref" data-show-preview="true">magnetic intensity</span>, and temperature. The <span class="md-crosslink autoxref" data-show-preview="true">second law of
thermodynamics</span>, combined with the above-mentioned principles, serves to constrain physically allowed relations between stress, strain, and temperature in (3) and also constrains the other
types of relations described in (6) and (8) above. Such expressions, which give the relationships between stress, deformation, and other variables, are commonly referred to as constitutive relations.
</p><p class="topic-paragraph">In general, the stress-strain relations are to be determined by experiment. A variety of mechanical testing machines and geometric <span class="md-dictionary-link
md-dictionary-tt-off eb" data-term="configurations" data-type="EB">configurations</span> of material specimens have been devised to measure them. These allow, in different cases, simple tensile,
compressive, or shear stressing, and sometimes combined stressing with several different components of stress, as well as the determination of material response over a range of temperatures, strain
rates, and loading histories. The testing of round bars under tensile stress, with precise measurement of their extension to obtain the strain, is common for metals and for technological ceramics and
polymers. For rocks and soils, which generally carry load in compression, the most common test involves a round cylinder that is compressed along its axis, often while being subjected to confining
<span class="md-crosslink autoxref" data-show-preview="true">pressure</span> on its curved face. Frequently, a measurement interpreted by solid mechanics theory is used to determine some of the
properties entering stress-strain relations. For example, measuring the speed of deformation waves or the natural frequencies of <span class="md-dictionary-link md-dictionary-tt-off eb" data-term=
"vibration" data-type="EB">vibration</span> of structures can be used to extract the elastic moduli of materials of known mass density, and measurement of indentation hardness of a metal can be used
to estimate its plastic shear strength.</p><p class="topic-paragraph">In some favourable cases, stress-strain relations can be calculated approximately by applying principles of mechanics at the
microscale of the material considered. In a <span class="md-crosslink autoxref" data-show-preview="true">composite material</span>, the microscale could be regarded as the scale of the separate
materials making up the reinforcing fibres and matrix. When their individual stress-strain relations are known from experiment, continuum mechanics principles applied at the scale of the individual
constituents can be used to predict the overall stress-strain relations for the composite. For rubbery polymer materials, made up of long chain molecules that randomly configure themselves into
coillike shapes, some aspects of the elastic stress-strain response can be obtained by applying principles of statistical <span class="md-crosslink autoxref" data-show-preview="true">thermodynamics</
span> to the partial uncoiling of the array of molecules by imposed strain. For a single crystallite of an element such as silicon or aluminum or for a simple <span class="md-dictionary-link
md-dictionary-tt-off mw" data-term="compound" data-type="MW">compound</span> like <span class="md-crosslink autoxref" data-show-preview="true">silicon carbide</span>, the relevant microscale is that
of the atomic spacing in the crystals; <span class="md-crosslink autoxref" data-show-preview="true">quantum</span> mechanical principles governing atomic <span class="md-crosslink autoxref"
data-show-preview="true">force</span> laws at that scale can be used to estimate elastic constants. In the case of plastic flow processes in metals and in sufficiently hot ceramics, the relevant
microscale involves the network of <span id="ref611601"></span>dislocation lines that move within crystals. These lines shift atom positions relative to one another by one atomic spacing as they move
along slip planes. Important features of elastic-plastic and viscoplastic stress-strain relations can be understood by modeling the stress dependence of dislocation generation and motion and the
resulting dislocation entanglement and immobilization processes that account for strain hardening.</p><p class="topic-paragraph">To examine the mathematical structure of the theory, considerations
(1) to (3) above will now be further developed. For this purpose, a continuum model of matter will be used, with no detailed reference to its discrete structure at molecular—or possibly other larger
microscopic—scales far below those of the intended application.</p><section data-level="2" id="ref77456"><h2 class="h2" id="qai_title_1">Linear and angular momentum principles: stress and equations
of motion</h2><p class="topic-paragraph">Let <em><strong>x</strong></em> denote the <span class="md-crosslink autoxref" data-show-preview="true">position vector</span> of a point in space as measured
relative to the origin of a Newtonian reference frame; <em><strong>x</strong></em> has the components (<em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, <em>x</em><sub>3</sub>) relative to a Cartesian
set of axes, which is fixed in the <span class="md-crosslink autoxref" data-show-preview="true">reference frame</span> and denoted as the 1, 2, and 3 axes in <span class="link-blue media-overlay-link
asmref" data-href="/media/1/553306/2306">Figure 1</span>. Suppose that a material occupies the part of space considered, and let <em><strong>v</strong></em> = <em><strong>v</strong></em>(<em><strong>
x</strong></em><em>, t</em>) be the velocity <span class="md-dictionary-link md-dictionary-tt-off eb" data-term="vector" data-type="EB">vector</span> of the material point that occupies position <em>
<strong>x</strong></em> at time <em>t</em>; that same material point will be at position <em><strong>x</strong></em> + <em><strong>v</strong></em><em>dt</em> an infinitesimal interval <em>dt</em>
later. Let <em>ρ</em> = <em>ρ</em>(<em><strong>x</strong></em><em>, t</em>) be the mass density of the material. Here <em><strong>v</strong></em> and <em>ρ</em> are macroscopic variables. What is
idealized in the continuum model as a material point, moving as a smooth function of time, will correspond on molecular-length (or larger but still “microscopic”) scales to a region with strong
fluctuations of density and velocity. In terms of phenomena at such scales, <em>ρ</em> corresponds to an average of mass per unit of volume, and <em>ρ</em><em><strong>v</strong></em> to an average of
linear momentum per unit volume, as taken over spatial and temporal scales that are large compared to those of the microscale processes but still small compared to those of the intended application
or phenomenon under study. Thus, from the microscopic viewpoint, <em><strong>v</strong></em> of the continuum theory is a mass-weighted average velocity.</p><p class="topic-paragraph">The <span id=
"ref611602"></span><span class="md-crosslink">linear momentum</span> <em><strong>P</strong></em> and <span id="ref611603"></span><span class="md-crosslink" data-show-preview="true">angular momentum</
span> <em><strong>H</strong></em> (relative to the coordinate origin) of the matter instantaneously occupying any volume <em>V</em> of space are then given by summing up the linear and angular
momentum vectors of each element of material. Such summation over infinitesimal elements is represented mathematically by the <span class="md-dictionary-link md-dictionary-tt-off mw" data-term=
"integrals" data-type="MW">integrals</span> <em><strong>P</strong></em> = ∫<sub><em>V</em></sub> <em>ρ</em><em><strong>v</strong></em><em>dV</em> and <em><strong>H</strong></em> = ∫<sub><em>V</em></
sub> <em>ρ</em><em><strong>x</strong></em> × <em><strong>v</strong></em><em>dV</em>. In this discussion attention is limited to situations in which relativistic effects can be ignored. Let <em>
<strong>F</strong></em> denote the total force and <em><strong>M</strong></em> the total torque, or moment (relative to the coordinate origin), acting instantaneously on the material occupying any
arbitrary volume <em>V</em>. The basic laws of Newtonian mechanics are the linear and angular momentum principles that <em><strong>F</strong></em> = <em>d</em><em><strong>P</strong></em>/<em>dt</em>
and <em><strong>M</strong></em> = <em>d</em><em><strong>H</strong></em>/<em>dt</em>, where time derivatives of <em><strong>P</strong></em> and <em><strong>H</strong></em> are calculated following the
motion of the matter that occupies <em>V</em> at time <em>t</em>. When either <em><strong>F</strong></em> or <em><strong>M</strong></em> vanishes, these equations of motion correspond to conservation
of linear or angular momentum.</p><p class="topic-paragraph">An important, very common, and nontrivial class of problems in solid mechanics involves determining the deformed and stressed
configuration of solids or structures that are in static <span id="ref611604"></span><span class="md-crosslink" data-show-preview="true">equilibrium</span>; in that case the relevant basic equations
are <em><strong>F</strong></em> = 0 and <em><strong>M</strong></em> = 0. The understanding of such conditions for <span class="md-dictionary-link md-dictionary-tt-off mw" data-term="equilibrium"
data-type="MW">equilibrium</span>, at least in a <span class="md-dictionary-link md-dictionary-tt-off mw" data-term="rudimentary" data-type="MW">rudimentary</span> form, long predates Newton. Indeed,
<span id="ref611605"></span><span class="md-crosslink" data-show-preview="true">Archimedes</span> of Syracuse (3rd century <span class="text-smallcaps">bc</span>), the great Greek mathematician and
arguably the first theoretically and experimentally minded physical scientist, understood these equations at least in a nonvectorial form appropriate for systems of parallel forces. This is shown by
his treatment of the hydrostatic equilibrium of a partially submerged body and by his establishment of the principle of the lever (torques about the fulcrum sum to zero) and the concept of <span
class="md-crosslink autoxref" data-show-preview="true">centre of gravity</span>.</p></section></section></div></div></div></div></div></div></div></div></div></div></article></div></div></div></div>
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DS 295: Parallel Programming
Parallel Programming
• Instructor: Sathish Vadhiyar
• Course number: DS295
• Credits: 3:1
• Semester: Jan 2024
• Lecture: Tue/Th 1130AM-1PM (First class: Jan 1, 2024, 1130AM)
• Office Hours: Fridays, 5-6 PM
The objective of this course is to give you some level of confidence in parallel programming techniques, algorithms and tools. At the end of the course, you would (we hope) be in a position to apply
parallelization to your project areas and beyond, and to explore new avenues of research in the area of parallel programming.
The course covers parallel programming tools, constructs, models, algorithms, parallel matrix computations, parallel programming optimizations, scientific applications and parallel system software.
DS 221: Introduction to Scalable Systems
Grading Scheme
• Sessionals
□ Mid-term [1 No.] (February 29) – 20
□ Assignments [3 Nos.] (Assignment 1 Due: Feb 17, Assignment 2 Due: March 21, Assignment 3 Due: April 10) – 30
• Terminal
□ Literature Study Report (Due: Apr 15) – 20
□ Final project (Proposal: Feb 24, Final presentation: Apr 25, Final report: Apr 26) – 30
• Introduction to Parallel Computing. Ananth Grama, Anshul Gupta, George Karypis, Vipin Kumar. Publisher: Addison Wesley. ISBN: 0-201-64865-2. 2003.
• Parallel Computing Architecture. A Hardware/Software Approach. David Culler, Jaswant Singh. Publisher: Morgan Kauffman. ISBN: 981-4033-103. 1999.
• Parallel Computing. Theory and Practice. Michael J. Quinn. Publisher: Tata: McGraw-Hill. ISBN: 0-07-049546-7. 2002.
• An Introduction to Parallel Programming. Peter S Pacheco. Publisher: Morgan Kauffman. ISBN: 978-93-80931-75-3. 2011.
• Online references for OpenMP, MPI, CUDA
• Various publication materials and references that will be posted along with the lecture slides.
Lecture Slides
Topic MPI
• “Parallel Computing: Theory and Practice” by Michael J Quinn, Available with me. Pages 25-32, 40-42, 256
• MPI-1: Online tutorial “MPI Complete Reference”. Google for it.
• Parallel programming tools/ • CUDA Optimizations – Chapter 5 of CUDA programming guide and slides 30-34 of advanced CUDA, CUDA Occupancy calculator
models/languages • MPI-2: Online tutorial: http://www-unix.mcs.anl.gov/mpi/mpi-standard/mpi-report-2.0/mpi2-report.htm
• Parallel I/O Optimizations: Google for paper Rajeev Thakur, William Gropp, and Ewing Lusk, “A Case for Using MPI’s Derived Datatypes to. Improve I/O
Performance”, in Proceedings of SC98.
• Collective Communications: Lecture slides, and Google for paper “Optimization of Collective Communication Operations in MPICH” by Thakur, Rabenseifner and
Gropp, IJHPCA 2005
• Mapping to network topologies. Section 5.1 in book “Parallel Computing: Theory and Practice” by Michael J Quinn
• Sorting
□ Bitonic sort: In the book by Grama et al.
□ Paper: On the versatility of parallel sorting by regular sampling. Li et al. Parallel Computing. 1993. (Pages 1-6)
□ Paper: Parallel Sorting by regular sampling. Shi and Schaeffer. JPDC 1992. (Pages 2-4)
• Parallel Algorithms • Tridiagonal systems – lecture slides
• FFT Chapter in “Introduction to Parallel Computing” book
• APSP: Book: Introduction to parallel computing by Grama et al. Sections 10.2-10.4, Sections 11.4.1-11.4.6
• Graph algorithms
□ Paper: A parallel graph partitioning algorithm for a message-passing multiprocessor – Gilbert and Zmijewski – – pages 427-433, 437-440.
□ Paper: Multi-level Graph Partitioning Schemes – Karypis and Kumar. Google for it.
• Machine learning
□ Paper:
□ PANDA: Extreme Scale Parallel K-Nearest Neighbor on Distributed Architectures. IPDPS 2016.
Sparse LA
• Sources [You can get these papers from me and take photocopies]
□ Parallel Algorithms for sparse linear systems – Heath, Ng and Peyton
□ Reordering sparse matrices for parallel elimination – Liu
□ Task scheduling for parallel sparse Cholesky factorization – Geist and Ng
□ Lecture slides
• General steps of sparse matrix factorization: Heath, Ng and Peyton – pages 420-429
• Parallel ordering
Sparse linear algebra pdfDirect □ Heath, Ng and Peyton – pages 429-435 till Kernighan Lin
linear algebra pdf □ Liu – pages 75 and 89 (you can read other pages on reduction of elimination tree heights if interested)
• For mapping: Heath, Ng and Peyton – 437-439, figures 9 and 10; Geist and Ng – sections 3 and 4
• For numerical factorization: Heath, Ng and Peyton – 442-450
• Dense LA
□ Paper: “F. Song, S. Tomov, and J. Dongarra. Enabling and Scaling Matrix Computations on Heterogeneous Multi-core and Multi-GPU Systems. In In the
Proceedings of IEEE/ACM Supercomputing Conference (SC), pages 365–376, 2012.”
□ Slides on “Communication Avoiding QR and LU”, CS 294 lecture slides, Laura Grigori, ALPINES INRIA Rocquencourt – LJLL, UPMC – https://who.rocq.inria.fr/
□ “Communication-optimal parallel and sequential QR and LU Factorizations”, James Demmel et al., Technical Report No. UCB/EECS-2008-89, August 2008.
• Molecular Dynamics – Paper: “A New Parallel Method for Molecular Dynamics Simulation of Macromolecular Systems” by Plimpton and Hendrickson. Sections 2-5.
Scientific Applications • DFS: Book: Introduction to parallel computing by Grama et al. Sections 10.2-10.4, Sections 11.4.1-11.4.6
• Mesh applications
• Molecular dynamics pdf □ Paper: “Multilevel diffusion schemes for repartitioning of adaptive meshes” by Schloegel et al.
• Game of Life pdf □ Paper: “Dynamic repartitioning of adaptively refined meshes” by Schloegel et al.
□ Paper: “Dynamic Octree Load Balancing Using Space-filling curves” by Campbell et al. – Section 2.5
DFS and Dynamic Load Balancing pdf □ Paper: “Irregularity in Multi-dimensional space-filling curves with applications in multimedia databases” by Mokbel and Aref – Section 4
• N-body Simulations
Mesh applications pdf □ The paper “Scalable parallel formulations of the Barnes-Hut method for n-body simulations” by Grama, Kumar and Sameh. In Parallel Computing 24(1998)
N-Body Simulations pdf □ The paper “Load balancing and data locality in adaptive hierarchical N-body methods: Barnes-Hut, Fast Multipole, and Dardiosity” by Singh, Holt, Totsuka,
Gupta and Hennessey. In Journal of Parallel and Distributed Computing, 1994.
Bioinformatics pdf • Bioinformatics
□ Paper: Parallel Biological Sequence Comparison using Prefix Computations. Aluru et. al. JPDC 2003.
System Software • Scheduling
□ Paper: “Backfilling with lookahead to optimize the packing of parallel jobs” by Shmueli and Feitelson. JPDC 2005.
• Scheduling pdf □ Paper: “A comparison study of eleven static mapping heuristics for a class of meta-tasks on heterogeneous computing systems” by Tracy Braun et al., HCW 1999
• Fault tolerance pdf • Fault tolerance
□ Paper: “An overview of checkpointing in uniprocessor and distributed systems, focusing on implementation and performance” by James Plank
• Turing cluster notes on OpenMP, MPI and CUDA for the assignments
• Assignment 1 –
• Assignment 2 –
• Assignment 3 –
Final Project
The final project has to clearly demonstrate the uniqueness of your work over existing work and show adequate performance improvements. You can work in a team of max 2 members. It can be in
• parallelizing well known algorithms or problems. Examples:
□ hybrid executions on both CPU and GPU cores.
□ graph algorithms – spanning trees, shortest paths, satisfiability, mesh generation or refinement (e.g., Delaunay)
□ sorting, searching
□ clustering algorithms
□ parallelization of fundamental algorithms you would encounter in an algorithm book. e.g.,
1. Introduction to Algorithms. Third Edition. Cormen, Leiserson, Rivest, Stein
2. The Design and Analysis of Algorithms. Aho, Hopcroft, Ulman
3. Data Structures and Algorithms. Aho, Hopcroft, Ulman
• parallelizing numerical methods, Examples:
□ Dense martrix or sparse matrix computations. e.g., Cholesky, QR, Inverse, SVD, conjugate gradient etc.
□ >Transforms (FFT, wavelet etc.)
□ Any numerical method you would encounter in matrix/numerical methods book. e.g.,
1. Introduction to Numerical Analysis. Second Edition. Hildebrand
2. Elementary Numerical Analysis. Third Edition. Conte, de Boor
• System software. Examples:
□ Techniques for scheduling or mapping parallel tasks to processors to achieve least makespan or throughput
□ Load balancing techniques for irregular computations
□ Automatic techniques for splitting tasks among GPU and CPU cores.
□ Automatic data management techniques for GPU cores.
□ Proposing genetic programming abstractions
□ Fault tolerance
Sample Projects from Previous Years
Ethics and Rules
1. Please do not even exchange ideas with your friends since there is a thin line between exchanging ideas and codes looking the same.
2. Please do not look up web/books for solutions.
3. See Dr. Yogesh’ nice writeup on plagiarism policies in his HPC page
All assignments will be evaluated for a maximum of 10. There will be a penalty of -1 for every additional day taken for submission after the assignment due date.
Thus, you will have to be judicious regarding deciding when to submit your assignments.
Suppose you have completed 1/2 of the assignment by the due date.
Scenario 1:
You think that it will take another 1 day to finish 3/4 of the assignment. In this scenario, if you submit by the due date, you will get a maximum score of 5 and if you submit a day after, you will
get a maximum score of 6.5 (=7.5-1, -1 for the extra day). Thus, you will get better score if you take an extra day, finish 3/4 of the assignment and then submit.
Scenario 2:
You think that it will take another 3 days to finish 3/4 of the assignment. In this scenario, if you submit by the due date, you will get a maximum score of 5 and if you submit 3 days after, you
will get a maximum score of 4.5 (=7.5-3, -3 for the three extra days). Thus, you will get better score if you submit your assignment that is 1/2 complete by the due date than submit the assignment
that will be 3/4 complete after 3 days.
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Economic Order Quantity(EOQ) - Applications of Differentiation maxima and minima
Economic Order Quantity(EOQ):
Economic order quantity is that size of order which minimizes total annual cost of carrying inventory and the cost of ordering under the assumed conditions of certainty with the annual demands known.
Economic order quantity (EOQ) is also called Economic lot size formula.
The derivation of this formula is given for better understanding and is exempted from examination.
The formula is to determine the optimum quantity ordered (or produced) and the optimum interval between successive orders, if the demand is known and uniform with no shortages.
Let us have the following assumptions.
(i) Let R be the uniform demand per unit time.
(ii) Supply or production of items to the inventory is instantaneous.
(iii) Holding cost is ₹ C[1] per unit time.
(iv) Let there be 'n' orders (cycles) per year, each time 'q' units are ordered (produced).
(v) Let ₹ C[3] be the ordering (set up) cost per order (cycle). Let 't' be the time taken between each order.
Diagrammatic representation of this model is given below:
If a production run is made at intervals t, a quantity q = Rt must be produced in each run. Since the stock in small time dt is Rt dt , the stock in period t is
Example 6.30
A company uses 48000 units of a raw material costing ₹ 2.5 per unit. Placing each order costs ₹45 and the carrying cost is 10.8 % per year of the average inventory. Find the EOQ, total number of
orders per year and time between each order. Also verify that at EOQ carrying cost is equal to ordering cost.
Here demand rate R = 48000
So at EOQ carrying cost is equal to ordering cost.
Example 6.31
A manufacturer has to supply 12,000 units of a product per year to his customer. The ordering cost (C[3]) is ₹ 100 per order and carrying cost is ₹ 0.80 per item per month. Assuming there is no
shortage cost and the replacement is instantaneous, determine the
(i) economic order quantity
(ii) time between orders
(iii) number of orders per year
Demand per year : R = 12,000 units
Ordering cost: C[3]= ₹ 100/order
Carrying cost : C[1]= 0.80/item/month
= 0.80×12 per year
= ₹ 9.6 per year
Example 6.32
A company has to supply 1000 item per month at a uniform rate and for each time, a production run is started with the cost of ₹ 200. Cost of holding is ₹ 20 per item per month. The number of
items to be produced per run has to be ascertained. Determine the total of setup cost and average inventory cost if the run size is 500, 600, 700, 800. Find the optimal production run size using EOQ
Demand : R = 1000 per month
Setup cost : C[3] = ₹ 200 per order
Carrying cost: C[1]= ₹ 20 per item per month.
Example 6.33
A manufacturing company has a contract to supply 4000 units of an item per year at uniform rate. The storage cost per unit per year amounts to ₹ 50 and the set up cost per production run is ₹
160. If the production run can be started instantaneously and shortages are not permitted, determine the number of units which should be produced per run to minimize the total cost.
Solution :
Annual demand : R = 4000
Storage cost: C[1] = ₹ 50
To minimize the production cost number of units produced per run is 160 units.
Example 6.34
A company buys in lots of 500 boxes which is a 3 month supply. The cost per box is ₹125 and the ordering cost in ₹150. The inventory carrying cost is estimated at 20% of unit value.
(i) Determine the total amount cost of existing inventory policy
(ii) How much money could be saved by applying the economic order quantity?
Ordering cost per order : C[3 ]= ₹150 per order.
Number of units per order: q= 500 units
Annual demand = 500 × 4 = 2000 units
By applying the economic order quantity, money saved by a company = 6850–3873
= ₹2977.
Exercise 6.3
1. The following table gives the annual demand and unit price of 3 items
Ordering cost is Rs. 5 per order and holding cost is 10% of unit price.
Determine the following:
(i) EOQ in units
(ii) Minimum average cost
(iii) EOQ in rupees
(iv) EOQ in years of supply
(v) Number of orders per year.
2. A dealer has to supply his customer with 400 units of a product per every week. The dealer gets the product from the manufacturer at a cost of ₹ 50 per unit. The cost of ordering from the
manufacturers in ₹ 75 per order. The cost of holding inventory is 7.5 % per year of the product cost. Find (i) EOQ (ii) Total optimum cost.
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ggplot2 Version of Figures in “Lattice: Multivariate Data Visualization with R” (Part 1) | R-bloggersggplot2 Version of Figures in "Lattice: Multivariate Data Visualization with R" (Part 1)
ggplot2 Version of Figures in “Lattice: Multivariate Data Visualization with R” (Part 1)
[This article was first published on
R Language – the data science blog
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
The data visualization package lattice is part of the base R distribution, and like ggplot2 is built on Grid graphics engine. Deepayan Sarkar’s (the developer of lattice) book Lattice: Multivariate
Data Visualization with R gives a detailed overview of how the package works. All the figures and code used to produce them is also available on the book website.
In order to give those interested an option to compare graphs produced by ggplot2 and lattice, I will attempt to recreate the book’s lattice graphs in ggplot2. There are 14 chapters in the book, so
this means that there would be at least 13 more posts on the subject.
The output of both packages can be tweaked so that the graphs would look similar if not the same, however for the purposes of comparison, the standard settings (at least in ggplot2) are used when
possible. The code used to create…
View original post 182 more words
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What Does Steeper Mean In Math
9.1 Mean Mathematics Quizizz
What Does Steeper Mean In Math. It's like measuring how quickly a hill goes up or down. Slope tells us how steep a line is.
9.1 Mean Mathematics Quizizz
We find the slope by seeing. By just looking at the graph of a line, you can learn. It's like measuring how quickly a hill goes up or down. Slope tells us how steep a line is. Web in math, slope is
used to describe the steepness and direction of lines.
Slope tells us how steep a line is. By just looking at the graph of a line, you can learn. Slope tells us how steep a line is. We find the slope by seeing. It's like measuring how quickly a hill goes
up or down. Web in math, slope is used to describe the steepness and direction of lines.
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VTU Fluid Mechanics - December 2011 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1 (a) Differentiate between:
i ) Weight density and mass density
ii) Compressibility and bulk modulus.
6 M
1 (b) The space between the two square flat parallel plates is filled with oil. Each side of the plate is 720 mm. The thickness of the oil film is 15mm. The upper plate moves at 3m/s requires a force
120N to maintain the speed. Determine i) dynamic viscosity of the oil ii) kinematic viscosity of oil, if the specific gravity of oil is 0.95.
7 M
1 (c) Calculate the capillary effect in millimeter in a glass tubes of 4mm diameter, when immersed in i) water and ii) mercury. The values of surface tension of water and mercury at 20°C in contact
with air are 0.0735 N/m and 0.51 N/m respectively. The specific weight of water at 20°C is equal to 9790 N/m^3.
7 M
2 (a) A U-tube manometer containing mercury was used to find the negative pressure in the pipe, containing water. The right limb was open to the atmosphere. Find the vacuum pressure in the pipe, if
the difference of mercury level in the two limbs was 100mm and height of water in the left limb from. The centre of the pipe was found to be 40mm below. Sketch the manometer with details.
7 M
2 (b) Derive an expression for the total pressure force and depth of center of pressure for an inclined surface submerges in water.
8 M
2 (c) A trapezoidal channel 2m wide at the bottom and 1m deep has side slopes 1:1. Determine i) Total pressure i) Center of pressure on the vertical gate closing the channel when it is full of water.
5 M
3 (a) Derive the continuity equation in the three dimensions in the differential form and write the same for a steady incompressible flow.
8 M
3 (b) Define the meta-center of floating body. Describe the analytical method of determine the meta-center height.
8 M
3 (c) A wooden cylinder of circular section and uniform density, specific gravity 0.6 is required to float in an oil of specific gravity 0.8. If the diameter of the cylinder is 'd' and its length
'l'. Show that l cannot exceed 0.817 for cylinder to float with its longitudinal axis vertical.
4 M
4 (a) Derive the Euler's equation of motion for a steady flow and deduce the Bernouli's equation of motion. Mention the assumptions made.
10 M
4 (b) The water is flowing through a tapering pipe having diameter 300mm and 150mm at section 1 and 2 respectively. The discharge through the pipe is 40 liters/sec. The section 1 is 10m above datum
and section 2 is 6m above datum. Find the intensity of pressure at the section 2, if that at section 1 is 400 kN/m^2.
8 M
4 (c) Discuss the Bernoulli's equation for real fluid.
2 M
5 (a) The efficiency ° of a fan depends on the density '°' dynamic viscosity ? of the fluid angular velocity ? diameter D of the rotor and the discharge Q. Using the Buckingham's '°' theorem express
'°' in terms of dimensionless parameters.
10 M
5 (b) With a sketch, derive an expression for the discharge through an inclined venturimeter for an upward flow.
10 M
6 (a) Derive the Darcy-Weisbach equation. Derive it to the Chezy's equation.
10 M
6 (b) Water is to be supplied to the inhabitants of a college through a supply main. The following data is given:
Distance of the reservoir from the campus =3000m
Number of inhabitants=4000
Consumption of water per day of each inhabitant=180 liters.
Loss of head due to friction=18m
Coefficient of friction for the pipe, f=0.007
If half of the daily supply is pumped in 8 hours,determine the size of the supply main.
8 M
6 (c) Define hydraulic gradient and total energy line.
2 M
7 (a) What is the Hegen-Poiseuille's formula? Derive the expression for the same.
8 M
7 (b) A crude oil of viscosity 0.97 poise and relative density 0.9 is flowing through a horizontal circular pipe of diameter 100mm and length of 10m. Calculate the difference of pressure at the two
ends of the pipe, if 100kg of the oil is collected in a tank in 30 seconds. Assume laminar flow.
8 M
7 (c) Write any two characteristics of laminar flow.
4 M
8 (a) Derive an expression for displacement thickness and momentum thickness of a flow over a plate.
10 M
8 (b) A flat 'plate' 1.5×1.5m moves at 50km/hour in the stationary air of density 1.15 kg/m^ if the coefficient of drag and lift are 0.15 and 0.75 respectively, determine i) the lift force ii) the
drag force iii) te resultant force
10 M
More question papers from Fluid Mechanics
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Thanks for your reply. I am
07-21-2014 09:03 AM
I have a seemingly naive question: why the length of the output of cosine transform might be longer than n? For example, for staggered cosine transform, the output length is 3n/2. But just looking at
its equation, the output should have the same length as the input, which is n. Thanks!
07-23-2014 08:11 PM
07-21-2014 07:11 PM
07-21-2014 07:37 PM
07-23-2014 08:11 PM
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Web pages of Luca Spada
Contents of the page
Topics of the course
• Categories, universal properties, functors.
• Natural transformations, adjoint functors and categorical equivalences.
• Concrete dualities.
• Yoneda Lemma.
• Sheaves and topoi.
Course material
• Lecturer: Luca Spada
• Duration of the course: 20 hours.
Preliminary programme
• Friday 18 May 2018, from 11:00 to 13:00;
• Monday 21 May 2018, from 9:00 to 11:00;
• Wednesday 23 May 2018, from 9:00 to 11:00;
• Friday 25 May 2018, from 9:00 to 11:00;
• Monday 4 June 2018, from 15:00 to 17:00;
• Wednesday, June 6, 2018, from 9:00 to 11:00;
• Friday 8 June 2018, from 9:00 to 11:00;
• Thursday 21 June 2018, from 9:00 to 11:00;
• Monday 9 July 2018, from 15:00 to 17:00;
• Wednesday 11 July 2018, from 9:00 to 11:00;
Comments, complaints, questions: write to Luca Spada
At last, we have finished and submitted our paper on “General affine adjunctions, Nullstellensätze, and dualities” co-authored with Olivia Caramello and Vincenzo Marra.
Abstract. We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric
adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities
induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert’s Nullstellensatz, thereby achieving a
complete characterisation of the fixed points on the algebraic side of the adjunction.
The preprint is available on arXiv. We made another preprint available some years ago(!), but the manuscript has changed in many respects. The main differences between the two versions on arXiv are
the following:
1. The comparison with the existing literature is now more thorough.
2. The categories R and D are now taken directly without passing through the quotient categories. In our opinion, this is cleaner and, as a consequence, it is now clearer what are the minimal
assumption on the triplet I: T -> S.
3. There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.
AILA summer school of logic. Below one can find the slides of my first three lectures and some references.
• Lecture 1 (Classical propositional logic and Boolean algebras)
• Lecture 2 (Algebraic completeness of propositional calculus)
• Lecture 3 (Abstract Algebraic Logic)
• Lecture 4 (Dualities) lecture material:
• Lecture 5 and 6 (Non classical logic) references:
1. Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors), Handbook of Modal Logic, 2006, pp 331-426.
2. R. L. O. Cignoli, I. M. L. D’Ottaviano e D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, Vol. 7 Springer, 2000.
Lecture notes by Guido Gherardi (Computability Theory).
These are the slides of my tutorial on Dualities at the $16^{th}$ Latin American Symposium on Mathematical Logic. 28th July – 1st August 2014. Buenos Aires, Argentina. A shorter version can be
found here.
Finally I wrote some slides about the long-waiting article I am writing together with Olivia Caramello and Vincenzo Marra on adjunctions, dualities, and Nullstellensätze . These slides where
presented at the AILA meeting in Pisa and at the Apllied Logic seminar in Delft.
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et C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY(T) be a natural quotient of C(T), and if C(T) is simply-laced (which
means all the relations between the generators has order 2 or 3), CY(T) is a generalized Coxeter group, too. Let At,n be a group which contains t Abelian groups generated by n elements. The main
result in this paper is that CY(T) is isomorphic to At,n ⋊ Bn or At,n ⋊ Dn, depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the
number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups. Read More:
Dive into the research topics of 'COXETER COVERS OF THE CLASSICAL COXETER GROUPS'. Together they form a unique fingerprint.
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BC to AD Calculator - Calculator Hub
BC to AD Calculator
If you’re curious in finding the time difference between BC and AD years, you’ve arrived to the right place. Calculate the total number of years between BCE and CE years using this BC to AD
<iframe src="https://calculatorhub.org/?cff-form=164" style="width:100%;height:100%;"></iframe>
BC is an abbreviation for “Before Christ.” The word BC is commonly used to count years based on the roughly year of Jesus Christ’s birth, with years prior to his birth listed in decreasing order. So,
if you find 150 BC someplace, it signifies that it was 150 years before Christ was born.
AD denotes ‘Anno Domini’ which is a latin word and that means ‘in the year of our lord’. However, in simple terms, it means ‘After Christ,’ and while the abbreviation does not correspond, it will be
easier for you to remember. The term AD is commonly used for counting years following the birth of Jesus Christ, with years after his birth numbered in increasing order. So, if you see 150 AD
someplace, it signifies that it is 150 years after Christ was born.
How to Use BC to AD Calculator
• For calculating the time difference between bc and ad years, enter the years in the respected fields.
• After entering bc and ad years in the appropriate fields, the calculator will display the time difference between the two years once you press the calculate button.
• By default we have add the ‘BCE’ and ‘CE’ years as 500. You can reset these values by pressing the reset button.
What is CE & BCE
CE stands for “Common Era,” while BCE stands for “Before Common Era.” These phrases can also be used to indicate BC and AD dates on a timeline, respectively. You can use CE (common era) instead of AD
(anno domini), and BCE instead of BC (before Christ).
Using the designations BCE and CE makes more sense for describing dates, especially for individuals and cultures who do not adhere to Christianity.
How to Find Years Across AD & BC
It is extremely easy to determine the years between the AD and BC dates. Simply add the two years together and then subtract one to determine the time difference.
We’ve solved a basic example below to assist you understand the maths.
For example, let’s say we want to find the number of years between 105 AD and 500 BC:
1. Add both (AD & BC) the years = 105 + 500 = 605 years
2. Now subtract 1 from the result = 605 – 1 = 604 years
Therefore, the number of years between 105 AD and 500 BC is 604 years.
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How To Calculate Radians From A Slope
The radians of a slope refer to its angle measurement. Radians are angle measurement units that stem from pi, a mathematical constant that is commonly known as 3.14, but is in fact an infinite and
patternless number. A slope, also known as a gradient, is the ratio between the growth or decrease in vertical and horizontal distances between two defined points. You can easily calculate a slope's
angle measurement in radians through the simple inverse trigonometric arctangent or arctan function, which works in reverse to find the angle of a tangent value.
Step 1
Define the growths in vertical and horizontal distances. For this example, the vertical distance growth is 1, and the horizontal growth change is 5.
Step 2
Divide growth in the vertical distance by the growth in horizontal distance to find the degree of the gradient. For this example, 1 divided by 5 results in 0.2.
Step 3
Calculate the arctan of the degree of the gradient to calculate the measure of its angle in radians on your scientific calculator. Enter the gradient, and then press the "arctan" or "tan^-1" key. For
this example, the arctan of 0.2 is 0.197 radians.
Step 4
Check your answer with an online arctan calculator such as the one at RapidTable. Enter the degree of the gradient to the right of the "Arctan" label, select the "Rad" option from the pull-down menu
to the left of the "Reset" button to select radian measurement, and then click the equal sign button. The answer will appear to the right of the equal sign.
TL;DR (Too Long; Didn't Read)
Set your calculator to display the answer in radians and not degrees in its display options.
Cite This Article
Gartneer, Chance E.. "How To Calculate Radians From A Slope" sciencing.com, https://www.sciencing.com/calculate-radians-slope-8033052/. 24 April 2017.
Gartneer, Chance E.. (2017, April 24). How To Calculate Radians From A Slope. sciencing.com. Retrieved from https://www.sciencing.com/calculate-radians-slope-8033052/
Gartneer, Chance E.. How To Calculate Radians From A Slope last modified August 30, 2022. https://www.sciencing.com/calculate-radians-slope-8033052/
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Notes - DAA HT23, Graphs
What is the formal definition of a graph?
A pair $(V, E)$ with $V$ being a set of nodes/vertices and a set of ordered pairs $(u, v) \in E \subseteq V \times V$ representing a connection from $u$ to $v$.
Suppose we have an adjacency matrix $E$ where $E[i, j] = 1$ if there is a path from node $i$ to node $j$. What is the interpretation of $E^2$, the adjacency matrix squared?
$E[i, j]$ contains the number of walks of length 2 from $i$ to $j$.
Suppose we have an adjacency matrix $E$ where $E[i, j] = 1$ if there is a path from node $i$ to node $j$. How could you calculate the matrix giving the number of walks of at most length $k$ from $i$
to $j$?
\[E + E^1 + \cdots + E^k\]
What is the transitive closure of a graph?
The matrix where $A[i, j]$ is 1 if there is a path from $i$ to $j$, and 0 otherwise.
Can you quickly prove that if a graph has no odd length cycle then it is bipartite?
If a graph has only one vertex, it is bipartite. Otherwise start at vertex $u$ and colour all its neighbours blue. Then colour all of those neighbours red, and repeat this until you have a colouring
of all vertices. No vertices can change colour during this process, because otherwise there would exist an even length path and an odd length path to the same vertex. The combination of these two
paths would be an odd length cycle, a contradiction.
Program an implementation of depth-first search, outputting three arrays
• $d[v]$, the discovery time of a vertex
• $f[v]$, the finishing time of a vertex
• $\pi[v]$, the predecessor of a vertex
Program an implementation of breadth-first search, outputting two arrays:
• $d[v]$, the distance from the start vertex to the next vertex
• $\pi[v]$, the predecessor of a vertex on the shortest path
Program an implementation of Dijkstra’s algorithm, outputting two arrays:
• $d[v]$, the distance from the start vertex to the given vertex
• $\pi[v]$, the predecessor of a vertex on the shortest path
Program an implementation of the Bellman-Ford algorithm, outputting:
• A boolean if it contains a negative-weight cycle reachable from the start vertex
• $d[v]$, the distance from the start vertex to the given vertex
• $\pi[v]$, the predecessor of a vertex on the shortest path
Program an implementation of the Floyd-Marshall algorithm for solving the all-pairs shortest path problem with no negative-weight cycles, outputting two multidimensional arrays:
• $d[u, v]$, the shortest distance between arrays $u$ and $v$.
• $\pi[u, v]$, the predecessor of $u$ on the shortest path to $v$.
Program an algorithm for determining if a graph contains a cycle, and if it doesn’t, outputs a topological sort of the vertices.
Implement a disjoint-set data structure and use it to program Kruskal’s algorithm for finding a minimum spanning tree, and then implement Prim’s algorithm for finding achieving the same task.
Related posts
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Understanding the Benefits and Limitations of EMT and RMS Simulations - Aurora Power Consulting
A couple of weeks ago, I wrote a brief post on our company website (Aurora Power Consulting) about EMT modelling, and it generated quite a bit of interest. This made me realise that unless you are
directly involved in power system modelling and simulation, these issues can be a bit abstract and are a long way from the experience of most power engineers. Of course, there is no escaping the
issue that power system modelling, and analysis is difficult and academic. The reason it is so important, is that power systems equipment is expensive and has long lead times, so modelling and
analysis gives developers and network operators valuable insight into how a system will perform during steady state, fault and transient conditions in order to help keep the lights on.
As noted in the previous post, there is a general drive to move away from RMS (Root Mean Square) type simulation and use EMT (Electromagnetic Transient) modelling techniques to assess power systems
within the industry. This is due to an actual (and sometimes perceived) shortfall within RMS modelling techniques. Whilst this may seem very academic it can have very real influences – the most
recent case in the UK being the large amount of generation that unexpectedly tripped during the 2019 outage. This was in large due to problems with Fault Ride Through (FRT) capabilities of inverters
and the associated control systems.
RMS vs EMT modelling is an interesting issue from Aurora’s point of view, as it brings many opportunities (we like power simulation modelling). However, it also brings a number of challenges and
issues, both for us and our clients. This article is written from a U.K. / European context, but the general principles would also apply to practices in North America, Australia and many other parts
of the world.
Modelling Approaches
Before discussing the details of RMS vs EMT studies and modelling, it is worth remembering a famous quote by a Statistician called George Box. “All models are wrong, but some are useful.” This is an
important point, as a studies engineer should always remember that a model approximates reality, some models are good representation and others are less so. The suitability of the model and the
analysis must match the level of understanding needed.
RMS models are very good for many cases, but they do have to simplify certain parts of the network representation, and this causes limitations. This means that certain studies can fall short in
areas; specifically, RMS models and simulation techniques do not perform well in unbalanced, or fast transient conditions. Another possible limitation with RMS modelling techniques, is that as system
strength falls (short circuit levels), the transient response of systems and control systems becomes more complex and may not be adequately captured in RMS simulations. EMT models are much more
accurate, but they also have limitations and problems, and it is important to note that even an EMT model does not fully represent a real network. This is discussed in more detail later.
RMS Modelling
The first thing to be clear on with RMS modelling is the calculation method. The majority of RMS modelling is based on a balanced representation of the power system network and the calculation
carries out all the calculations on an equivalent positive sequence basis. What this means, in simple terms, is that RMS simulations do not model individual phase values. Furthermore, RMS simulation
techniques are based on a time step of (usually) 10ms, as this has been judged sufficient to capture general trends of system behaviour, so it will not capture any transient that is faster than this.
The reasons for an RMS based approach is that it is uses simpler formulas and is computationally efficient, and many of the traditional stability studies required tended to occur relatively slowly
over a few seconds. This approach was adopted at a time before computers were as powerful as they are today, and simulation time was a major constraint. For smaller industrial systems this may not
seem as important, but when operating at a transmission / country level, the computational time can be significant, running into several hours. Over the last few decades RMS simulations were found to
be a pretty reasonable approximation of the overall system behaviour. A further key point to note is that for most steady state analysis, high frequency components (which RMS techniques do not catch)
are not actually that important for loadflow analysis or simplified stability analysis.
To slightly complicate things a bit further, some software packages offer an unbalanced RMS calculation method. This is an interesting development, as this calculates individual phase values, during
the calculation and therefore overcomes many of the traditional shortfalls of a positive sequence RMS calculation. However, this approach still, uses slightly simplified formulas, a slow time step,
so it will not fully capture transient events. The debate then becomes if an unbalanced RMS simulation, with a faster time step are considered a reasonable approach to show FRT behaviour accurately
What does all that mean? Positive sequence tools are very useful for providing analysis of large power systems if: a) the analysis does not need to capture fast transients and b) it does not need to
consider unbalanced conditions. RMS models are excellent for modelling large power systems and carrying out analysis for power flows, and ‘slow’ transient behaviour, such as system stability.
Unbalanced RMS simulations provide a good intermediate step that addresses some of the shortfalls, but fundamentally suffers the same underlying issue of not capturing fast transient correctly and
using simplified formulas.
At a practical level, concerns are present in relation to modelling of grid following and grid forming inverters, and HVDC links. These are all complex power system elements and the simplified
representation in RMS has led to a few notable system events during faults where things have behaved in a way that was unexpected. At the moment Fault Ride Though studies and control stability are
the key areas where RMS studies are falling short and need supplementing with EMT studies, but this is likely to develop as renewables becoming more dominant on the system.
EMT Modelling
EMT modelling tools are less commonly used that RMS techniques. This is for a variety of reasons, including more complex modelling requirements, simulation and computation times, limited software
options and difficult user interfaces. EMT modelling was therefore generally limited to specialist studies such as insulation coordination, TRV or designing and analysing generators, motors,
inverters, or transformers in detail.
What does an EMT simulation do that an RMS simulation does not? Fundamentally EMT analysis uses more complicated equations to represent each component in the power system, and this allows each phase
to be calculated individually and solved in a much faster time step, allowing fast transient to be captured.
Even with modern advanced computing processors, modelling whole grid systems at an EMT level, is not widely done due to a) input data, b) practical considerations of processing power, c) ease of
simulation and d) bug hunting and false results.
Input data: This perhaps the most challenging one in EMT simulations. As the calculation is more accurate it necessarily requires more detailed and accurate input information. This creates a lot of
problems, particularly on large networks that have a lot of data and historical equipment that has been built up over many decades and information may not be available. The engineer building the EMT
model, must have a very detailed knowledge of all aspects of theory in selecting and creating the model. EMT simulation packages have a much smaller ‘library’ of data and expect the user to have a
deep understanding of the models it uses.
For a more complex case of an inverter-based generator, for example, an RMS model would typically treat it as a simplified impedance and a current source or voltage source with a series of control
elements. In an EMT model, it would be necessary to model the power electronics, control systems, input transducers and so on in much more detail. A further problem that arises in EMT simulations is
one of validation. EMT simulations are much more complex than RMS, so validation also becomes more complex. Inclusion, or exclusion of what would be minor details in an RMS study, can have a
significant effect on the results. Depending on the study in question, this could include differences due to modelling of stray network capacitances, control system parameter, synchronous machine
losses, choice of cable model etc.
Solution time: This is much easier to understand. A big model with lots of very detailed mathematical representations using short time steps will necessarily take longer to calculate. This moves
simulation time for a few seconds, into minutes or hours. When you are running multiple simulations, and need to consider different scenarios and outage cases, this can become a problem. Some
simulation packages have the ability to do parallel processing, to speed things up, and there are also techniques are available to ‘aggregate’ a model of a Solar PV / BESS / Wind farm into an
equivalent, but fundamentally EMT models are longer to run.
Ease of simulation: This is perhaps one of the most interesting and challenging problems with EMT models. Most RMS simulation packages have very good tools for setting up lots of different study
scenarios, operation cases, contingencies and so on. With a bit of scripting this can let you run through a large number of cases and get general trends and behaviours. EMT packages generally have
either little, or no support for this kind of approach, and needs quite a lot of manual operation to set things up. This makes one of the fundamental problems of flexibility of analysis and
repeatability of results a lot more challenging. Furthermore, EMT models cannot be easily initialised, to run steady state loadflows. This is because rotating machine initialisation parameters can be
worked out in EMT and have to be back calculated from a loadflow solution elsewhere, so, that the machine terminal voltage and angle can be set in the EMT study. Some EMT packages handle this well as
they are bundled with loadflow and RMS packages, others can use an external reference result via an interface module and others it needs to be done by hand.
Big hunting / false results: This is perhaps one of the most interesting and challenging areas. One of the most, if not the most important aspect of any modelling approach, is having confidence in
the results. This means checking that you model is representing what you think it is representing, this is much harder than is often understood. Both RMS and EMT models will be different to reality,
and the validation process of ensuring that the simulation is a close fit to reality is not an easy one, and EMT results can often be confusing, and need a high level of skill to interpret.
In steady state simulations it is easy to ‘eyeball’ the results and see if they make sense or not, with RMS simulations this is harder and needs an experienced studies engineer to assess the results.
With EMT studies this is much harder again, and it is very easy to end up with misleading results, either showing a problem when there isn’t one, or not showing a problem when one exists. The skill
and experience level for an engineer carrying out such studies needs to be far higher than those for a simple RMS study. The danger with EMT simulations is it can lead to a false sense of complacency
and trust in results, that may not be accurate, as there are assumptions and simplifications in the modelling approach that may not be apparent.
Discussion and Conclusion
A good example of RMS vs EMT studies, can be seen in considering Grid Code compliance studies, as required by many system Transmission System Operators (TSO). Most TSOs require a suite of studies to
be supplied looking at steady state loadflow, short circuit events, harmonics, frequency response, voltage control and so on; these are all either steady state studies to specific standards or ‘slow’
dynamic studies,a dn cannot usually be done easily (if at all) on an EMT package – EMT packages are too powerful, and cannot simplify the calculations down enough. Fault Ride Through (FRT) studies
are fasters, have a number of transients, require unbalanced analysis and consider operation of elements like PLLs, so are much better suited to EMT studies, although many would contend that a well
configured unbalanced RMS could produce accurate enough results, if backed up manufacturer validated Hardware in the Loop (HIL) testing.
The TSO requirements for EMT models, thus creates a practical problem as this then requires two different simulation models. An RMS model for most simulations, and an EMT model for detailed analysis
of FRT cases. Some packages can handle both, but many cannot. In the case of the U.K., the RMS simulations need to be done in DIgSILENT Powerfactory but the EMT simulations need to be done in PSCAD.
There are some reasons for this, as while Powerfactory is powerful, PSCAD is a more established modelling environment for EMT studies. This therefore duplicates effort and slows the whole compliance
process down.
I hope this short article has been useful and explained the difference between RMS and EMT simulations. In simple terms RMS simulations, are powerful tools and well suited for the majority or large
power system analysis cases, however they do have limitations as they use simplified formulas, only consider the positive phase sequence, and use large time steps. EMT studies are very powerful and
address many RMS shortfalls, but they are much harder to use, as they require much more input data, require more complex initialisation. They require a lot more knowledge and experience in
interpreting the results. EMT simulations are more accurate, but they are not always better, it very much depends on the type of analysis need.
Remember that the key is understanding the limits of the software and your own knowledge, and the needs of the simulation being requested. For many studies, a skilled and experienced studies engineer
might be able to provide a higher level of analysis in an RMS environment than a less experienced engineer using a more powerful EMT model.
What does the future hold for both RMS and EMT modelling and simulation. Nobody really knows, as the pace of change and deployment of renewables is almost outpacing development in the software, and
there is a massive shortage of power engineers with the right level of skills. Here are a few predictions:
• Grid Codes start to converge on requiring FRT, LVRT and HVRT simulations to be carried out on EMT simulation packages.
• EMT simulations will become the dominant method for carrying out all routine power system analysis.
• More established ways will be found of ‘simplifying and standardising input’ to EMT models.
• Aggregation of embedded generating facilities into smaller simplified equivalent models will become a standard practice.
• Existing RMS providers will have to expand their offering to include EMT modules or analysis methods.
• EMT packages will undergo a big shift in usability, and include much better functionality for handling multiple cases, scenarios, and reporting.
• Real Time simulation packages will start converging with EMT simulation packages and possibly the real time packages will be offered as a basic EMT model for normal computers, and with an
enhanced model for real time modelling.
• Standardised models for inverter configuration and control will become more common (such as the existing WECC models) and standards will need to be developed in a similar manner to Exciters and
IEEE 412.5.
These are some references and links that might be useful:
CIGRE: TB 727 Modelling Of Inverter Based Grids For Power Systems
CIGRE: TB 881 Electromagnetic transient simulation models for large-scale system impact studies in power systems having a high penetration of inverter-connected generation
NERC: Technical Report: Beyond Positive Sequence RMS Simulations for High DER Penetration Conditions, October 20221 PSCAD
NREL: Final Technical Report: Stabilizing the Power System in 2035 and Beyond
NREL: Open-Source PSCAD Grid-Following and Grid-Forming Inverters and a Benchmark for Zero-Inertia Power System Simulations
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A Primer on Statistical Inference and Hypothesis Testing
This post is about some fundamental concepts in classical (or frequentist) statistics: inference and hypothesis testing. A while back, I came to the realization that I didn't have a good intuition of
these concepts (at least not to my liking) beyond the mechanical nature of applying them. What was missing was how they related to a probabilistic view of the subject. This bothered me since having a
good intuition about a subject is probably the most useful (and fun!) part of learning a subject. So this post is a result of my re-education on these topics. Enjoy!
Statistical Models and Inference
A Couple of Big Ideas
To start from the beginning, there are two big ideas that underlie much of classical statistics. The first big idea is that all data (or observations as statisticians like to say) have a "true"
probability distribution 1. Of course, it is almost never possible to precisely define it because the real world rarely fits so nicely into the distributions we learn in stats class. However, the
implications of this idea is that the "true" distribution and its parameters are fixed (i.e. not random) albeit unknown. The randomness comes in when you sample from this "true" distribution from
which each datum is randomly drawn.
The second big idea is that statistical inference 2 (or as computer scientists call it "learning" 3) basically boils down to estimating this distribution directly by computing the distribution or
density function 4, or indirectly by estimating derived metrics such as the mean or median of the distribution. A typical question we might ask is:
Give a sample \(X_1, X_2, \ldots, X_n\) drawn from some (unknown) distribution \(F\), how do we estimate \(F\) (or some properties of \(F\))?
Of course there are variations to this question depending on the precise problem such as regression but by and large it comes down to finding things about \(F\) (or its derived properties).
Models, models, models
Now that we have those two big ideas out of the way, let's define a (statistical) model:
A statistical model \(\mathfrak{F}\) is a set of distributions (or densities or regression functions).
The idea here is that we want to define a subset of all possible distributions (or densities or regression functions) that closely approximates the "true" distribution (whether or not \(\mathfrak{F}
\) actually contains \(F\) 5). One of the first tasks in inferential procedures is selecting the correct model. The model is an assumption about your data, picking the wrong one will lead to invalid
By far, the most common type of model is a parametric model, which defines \(\mathfrak{F}\) using a finite number of parameters. For example, if we assume that the data comes from a Normal
distribution, we would use the parametric model for a Normal distribution:
\begin{equation*} \mathfrak{F} = \big\{ f(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \mu \in \mathbb{R}, \sigma > 0 \big\} \tag{1} \end{equation*}
Here we use the notation \(f(x; \mu, \sigma)\) to denote a density function of \(x\) parameterized by \(\mu\) and \(\sigma\). Similarly, when we have data of the form \((X_i, Y_i)\) and we want to
learn regression function \(r(x) = E(Y|X)\), we could define a model for \(\mathfrak{F}\) to be all functions of \(x\), \(r(x)\), that are straight lines. This gives us a linear regression model.
The other type of model is a non-parametric model. Here the number of parameters is not finite or fixed by the model, instead the model is defined by the input data. In essence, the parameters are
determined by the training data (not the model). For example, a histogram can be thought of as a simple non-parametric model that estimates a probability distribution because the data determines the
shape of the histogram. Another example would be a k-nearest neighbour algorithm that can classify a new observation solely based on its k-nearest neighbours from training data. The surface defined
by the classification function is not pre-defined rather it is determined solely by the training data (and hyper parameter \(k\)). You can contrast this with a logistic regression as a classifier,
which has a (relatively) more rigid structure regardless of how well the data matches.
Although, it sounds appealing to let the "data define the model", non-parametric models typically requires a much larger sample size to draw a similar conclusion compared to parametric methods. This
makes sense intuitively since parametric methods have the advantage of having the extra model assumptions, so making conclusions should be easier all else being equal. Of course, you must be careful
picking the right parametric model or else it will lead you to incorrect conclusions.
Types of Statistical Inference
For the most part, statistical inference problems can be broken into three different types of problems 6: point estimation, confidence intervals, and hypothesis testing. I'll briefly describe the
former two and focus on the latter in the next section.
Point estimates aim to find the single "best guess" for a particular quantity of interest. The quantity could be the parameter of a model, a CDF/PDF, or a regression/prediction function. Formally:
For \(n\) independent and identically distributed (IID) observations, \(X_1, \ldots, X_n\), from some distribution \(F\) with parameter(s) \(\theta\), a point estimator \(\widehat{\theta}_n\) of
parameter \(\theta\) is some function of \(X_1, \ldots, X_n\):
\begin{equation*} \widehat{\theta}_n = g(X_1, \ldots, X_n). \tag{2} \end{equation*}
For example, if our desired quantity is the expected value of the "true" distribution \(F\), we might use the sample mean of our data as our "best guess" (or estimate). Similarly, for a regression
problem with a linear model, we are finding a "point" estimate for the regression function \(r\), which are just the coefficients for the covariates (or features) that minimize the mean squared
error. From what I've seen, many "machine learning" techniques fall in this category where you typically will aim to find a maximum likelihood estimate or related measure that is your "best guess"
(or estimate) based on the data.
The next category of inference problems are confidence intervals (or sets). The basic idea here is that instead of finding a single "best guess" for a parameter, we try to find an interval that
"traps" the actual value of the parameter (remember the observations have a "true" distribution) with a particular frequency. Let's take a look at the formal definition first and then try to
interpret it:
A \(1-\alpha\) confidence interval for parameter \(\theta\) is an interval \(C_n(a,b)\) where \(a=a(X_1, \ldots, X_N)\) and \(b=b(X_1, \ldots, X_N)\) are functions such that
\begin{equation*} P(\theta \in C_n) \geq 1 - \alpha. \tag{3} \end{equation*}
This basically says that our interval \((a,b)\) "traps" the true value of \(\theta\) with probability \(1 - \alpha\) . Now the confusing part is that this does not say anything directly about the
probability of \(\theta\) occurring because \(\theta\) is fixed (from the "true" distribution) and instead it is \(C_n\) that is the random variable 7. So this is more a statement about how "right"
we were in picking \(C_n\).
Another way to think about it is this: suppose we set \(\alpha = 0.05\) (a 95% confidence interval), for every confidence interval, for every statistical inference problem we ever compute from now
until eternity. These problems will, of course, cover a wide range of statistical problems with many different "true" distributions and sample observations. Since we set a 95% confidence interval for
all our problems, we would expect that the respective "true" \(\theta\) in each case to be "trapped" in our confidence interval 95% of the time. That is, our confidence interval will be "correct" 95%
of the time in that the "true" value of \(\theta\) is contained within it -- a kind of long-run frequency guarantee. Note this is different from saying that on any one experiment we "trapped" \(\
theta\) with a 95% probability. After we have a realized confidence interval (i.e. fixed values for our confidence interval based on observed values), the "true" parameter \(\theta\) either lies in
it or it doesn't.
In some ways confidence intervals give us more context then a single point estimate. For example, if we're looking at the response of a marketing campaign versus a control group, the difference in
response or incremental lift is a key performance indicator. We could just compute the difference in the sample mean of the two populations to get a point estimate for the lift, which might show a
positive result say 1%. However, if we computed a 95% confidence interval we might get \((-0.015, 0.0155)\) which overlaps with 0, implying that our 1% lift may not be statistically significant.
Conceptually, point estimates and confidence intervals are not that hard to understand. The complexity comes in when you have to actually pick an estimator that has nice properties (like minimizing
bias and variance) in the case of Equation 2, or picking an interval such that Equation 3 is satisfied. Thankfully, many smart mathematicians and statisticians have figured out estimators and
confidence intervals for many common situations so we rarely need to derive things from scratch. Instead, we can pick the most appropriate technique for the problem at hand.
Hypothesis Testing
A Digression
I'm a huge fan of hypothesis testing as a general concept (not necessarily statistical) because it's such a powerful framework for learning. One of the biggest advantages is it sets you up to
"disprove your best-loved ideas" as Charlie Munger puts it, not to mention the hundreds of years its been used as part of the scientific method. There is a huge advantage to having a mental framework
that allows you to disprove your hardest won ideas, a proverbial "empty your cup" type situation where you can begin to learn after you have let go of your some of your past (hopefully, incorrect)
beliefs. I mean that's what science is all about right?
Statistical Hypothesis Testing
Statistical hypothesis testing is probably one of the earliest concepts learn in a statistics course. Null hypotheses, Student's t-test, p-values these terms get thrown around a lot without
explaining their underlying probabilistic basis 9. When I first learned statistics it was definitely more biased towards a mechanical view of hypothesis testing, rather than an intuitive
understanding. Here's my attempt to explain it a bit more precisely while hopefully adding some colour to give some intuition.
Following the scientific method, we make a hypothesis, run an experiment and see if our observations match the prediction from our hypothesis. However in certain cases, the cause and effect is not so
clear like it is with laws of nature. For example, when you conduct a double-slit experiment to determine the dual nature of light, the result of the experiment is clear. But when you're determining
if a new drug helps cure a disease, you usually randomly divide a population into a treatment group which gets the drug, and a control group which receives a placebo. If we look at the various
scenarios of what can happen, we can see why it's not so clear cut:
1. If at least one person in the treatment group doesn't get better, does it mean the drug isn't effective?
Not necessarily, the drug could still be quite effective but for some other random reason, the person could have not responded to the drug by pure chance.
2. If more people in the treatment group get better than the control, does it mean the drug is effective?
Not necessarily, what if only the treatment group has only 1 person who got better versus control. In this case, probably not, it could be due to another random factor. How about 10? 1000? Now it
starts to get unclear.
You can start to see why we need to apply some mathematics to these situations in order to see if the effect is significant. In particular, we apply statistical hypothesis testing when we want to
determine if an observed effect is really there or just happening by purely chance (i.e. other random factors).
The high level setup for this procedure is to first come up with a null hypothesis (denoted by \(H_0\)), that usually denotes the "no effect" scenario, or our default position. We then try to see how
likely the data is generated in this situation. If it's unlikely then we say we reject the null hypothesis and accept the alternate hypothesis, which just means something other than the null
hypothesis must true. Otherwise, we have no evidence to reject the null hypothesis and we continue to believe it to be true (since it's our default position).
A good analogy is that of a legal trial, the defendant is innocent until proven guilty. Likewise, we assume the null hypothesis is true from the start, and only when we reject it do we say it is
false. This is not unlike how science works where we have established models that are assumed to be true until later proven otherwise. Now that we have a conceptual understanding of this process,
let's look at some details.
Rejection Regions and Types of Errors
A critical point when conducting statistical hypothesis testing is determining your null hypothesis. The first step of this process is picking an appropriate statistical model \(\mathfrak{F}\). If
your model is ill-formed for your problem, the results of hypothesis testing will be invalid. Next, we partition the parameter space of \(\mathfrak{F}\) into two disjoint sets \(\Theta_0\) and \(\
Theta_1\), and define our hypotheses as:
\begin{align*} H_0 : \theta \in \Theta_0 \\ H_1 : \theta \in \Theta_1 \tag{4} \end{align*}
where \(H_0\) is our null hypothesis and \(H_1\) is our alternative hypothesis. So we must first pick a good statistical model then define an appropriate null hypothesis. For example, we might pick a
normal distribution as our statistical model and our null hypothesis is that the mean of the distribution is less than or equal to zero (\(\mu \leq 0\)) .
Now let's suppose our data are represented by the random variable \(X\) with range \(\chi\) (all possible values of \(X\)). Our goal is to define a rejection region on \(\chi\) such that:
\begin{align*} X \in R &\implies \text{ reject } H_0 \\ X \notin R &\implies \text{ retain (don't reject) } H_0 \tag{5} \end{align*}
We want to define \(R\) such that when \(H_0\) is true, we have a high probability of retaining \(H_0\) and when \(H_0\) is false, we have a high chance probability of rejecting it. Of course, our
definition of \(R\) should be heavily influenced by some estimate of \(\theta\) based on our data \(X\). If we picked a good \(R\), when we actually observe our data then it should be quite simple to
check if \(X \in R\) and be correct quite often.
Another way to view this is in terms of the errors we could make. If we reject \(H_0\) when it's actually true, we've committed a Type I Error or false positive (whose probability is denoted by \(\
alpha\)). If we retain \(H_0\) when it's actually false, we've committed a Type II Error or false negative (whose probability is denoted by \(\beta\)). Here's a summary:
Cases Retain Null Reject Null
\(H_0\) true Correct Type I Error (\(\alpha\))
\(H_0\) false Type II Error (\(\beta\)) Correct
So it makes sense that we want choose \(R\) to maximize the "Correct" diagonals or alternatively minimize "Error" diagonals in the above table. To throw another wrench in the mix, we usually refer to
the bottom right cell as the power, which is the probability of correctly rejecting the null hypothesis when it is false (i.e. the alternative hypothesis is true). The tricky part is that trying to
minimize both \(\alpha\) and \(\beta\) results in conflicting goals 10, which makes picking a good rejection region \(R\) highly non-trivial.
Test Statistics
Practically, we rarely explicitly pick a rejection region in terms of the range of the data (\(\chi\)). It's usually much more convenient to pick a rejection region in terms of a function of \(X\)
that produces a single number summarizing the data called a test statistic (which we denote as \(T\)). This test statistic usually relates in some way to the estimate of the "true" parameter \(\theta
\) and is usually more convenient to use than a direct estimation. Thus, our expression for rejection region usually ends up looking something like this:
\begin{equation*} R = \big\{ x : T(x) > c \big\} \tag{6} \end{equation*}
The value \(c\) is called the critical value which determines whether or not we retain or reject our null hypothesis. So now the problem of hypothesis testing comes down to picking an appropriate
test statistic \(T\) and an appropriate value \(c\) to minimize our error rates (\(\alpha\) and \(\beta\)).
As mentioned above, minimizing \(\alpha\) and \(\beta\) are usually in conflict, so what happens is we fix the level for \(\alpha\) (usually values like \(0.05\) or \(0.01\)), and find an appropriate
\(T\) and \(c\) so that \(\beta\) is minimized (alternatively power is maximized). Computing (and proving) that a test statistic has the highest power for a given an \(alpha\) is quite complex so I
won't mention much more of it here. Most of the time though you won't have to actually come up with \(T\) yourself since many common situations have already been worked out. The usual procedure
usually ends up being something along the lines of:
0. Define your null hypothesis (and the appropriate statistical model of your data).
1. Pick an appropriate \(\alpha\), e.g. \(0.05\).
2. Look up and compute the appropriate test statistic for your hypothesis/model e.g. Z statistic.
3. Look up (or compute) the critical value \(c\) based on \(\alpha\) e.g. \(Z > 1.96\).
4. Retain/reject the null hypothesis based on the computed test statistic and critical value.
p-values and such
Of course, just giving a retain/reject null hypothesis type answer isn't very informative. Instead, we might want to give the smallest \(\alpha\) that rejects the null hypothesis which is called a
\begin{equation*} \text{p-value} = min\big\{ \alpha : T(X) \in R_{\alpha} \big\} \tag{7} \end{equation*}
A p-value is basically a measure of evidence against \(H_0\). The smaller the p-value, the more evidence we have that \(H_0\) is false. Researchers usually use this scale for p-values:
p-value Evidence
\(<0.01\) very strong evidence against \(H_0\)
\(0.01-0.05\) strong evidence against \(H_0\)
\(0.05-0.10\) weak evidence against \(H_0\)
\(>0.10\) little or no evidence against \(H_0\)
Two important misconceptions about p-values:
• Nowhere in the above table do we say we have evidence for \(H_0\). A p-value says nothing about evidence in favour of \(H_0\). A large p-value could mean that \(H_0\) is true, or our test didn't
have enough power.
• A p-value is not the probability that the null hypothesis is true (e.g. \(\text{p-value} \neq P(H_0 | Data)\)).
A common way of stating what a p-value is (taken from All of Statistics):
The p-value is the probability (under \(H_0\)) of observing a value of the test statistic the same as or more extreme than what was actually observed.
Admittedly, this does not exactly line up with how we have looked at \(\alpha\) in terms of rejection regions, however, rest assured the definitions do match up if you went through the derivations of
the test statistic and critical values. Personally, I don't find the above definition all that helpful because most people will conflate it with \(P(H_0|data)\) just because both mention the word
The way I like to think of it is simply a measure of evidence against \(H_0\) (but not for \(H_0\)) according to the table above with no mention of probability. In this way, we can remember the point
of hypothesis testing is primarily a procedure to help us prove our default or null hypothesis false. Thinking this way helps to remember that the null hypothesis is our default stance and the test's
aim is prove it false 11.
While writing this post, I had to dig through the probabilistic foundations for these techniques and it can get really deep! I just scratched the surface, enough to satisfy my intellectual curiosity
and intuition (for now). Hopefully, this post (and some of the references below) will help you along the way too.
References and Further Reading
Taking note that no model can truly represent reality leading to the aphorism: All models are wrong.
Inferential statistics is in contrast to descriptive statistics, which only tries to describe the sample or observations -- not estimate a probability distribution. Examples of this are measures
of central tendency (like mean or median), or measures of variability (such as standard deviation or min/max values). Note that although the mean of a sample is a descriptive statistic, it is
also an estimate for the expected value of a given distribution, thus used in statistical inference. Similarly for the other descriptive statistics.
There is a great chart in All of Statistics that shows the difference between statistics and computer science/data mining terminology on page xi of the preface. It's very illuminating to contrast
the two especially since terms like estimation, learning, covariates, hypothesis are thrown around very casually in their respective literature. I come more from a computer science/data mining
and learned most of my stats afterwards so it's great to see all these terms with their definitions in one place.
Might be obvious but let's state it explicitly: distribution refers to the cumulative distribution function (CDF), and density refers to the probability density function (PDF).
In fact, most of the time \(\mathfrak{F}\) will not contain \(F\) since as we mentioned above, the "true" distribution is probably much more complex than any model we could come up with.
This categorization is given in All of Statistics, Section 6.3: Fundamental Concepts in Inference. I've found it quite a good way to think about statistics from a high level.
An important note outlined in All of Statistics about \(\theta\), point estimators and confidence intervals is that \(\theta\) is fixed. Recall, that our data is drawn from a "true" distribution
that has (theoretically) exact parameters. So there is a single fixed, albeit unknown, value of \(\theta\). The randomness comes in through our observations. Each observation, \(X_i\), is drawn
(randomly) from the "true" distribution so by definition a random variable. This means our point estimators \(\widehat{\theta}_n\) and confidence intervals \(C_n\) are also random variables since
they are functions of random variables.
This can all be a little confusing, so here's another way to think about it: Say we have a "true" distribution, and we're going to draw \(n\) samples from it. Ahead of time, we don't know what
the values of those observations are going to be but we know they will follow the "true" distribution. Thus, the \(n\) samples are \(n\) random variables, each distributed according to the "true"
distribution. We can then take those \(n\) variables and combine them into a function (e.g. a point estimator like a mean) to get a estimator. This estimator, before we know the actual values of
the \(n\) variables, will also be a random variable. However, what usually happens is that the values of the \(n\) samples are actually observed, so we plugs these realizations into our point
estimator (i.e. the function of the \(n\) observations) to get a point estimate -- a deterministic value. One reason we make this distinction is so that we can compute properties of our point
estimator like bias and variance. So long story short, the point estimator is a random variable where after having realized values of the observations, we can use it to get a single fixed number
called a point estimate.
Interestingly, it's very difficult to prove something to be true, whereas much easier to prove it false. The reason is that many useful statements we want to prove are universally quantified
(think of statements that use the word "all"). An example made famous by Nassim Nicholas Taleb is the "black swan" problem. It's almost impossible to prove the statement "all swans are white"
because you'd literally have to check the colour every single swan. However, it's quite easy to prove it false by finding a single counter-example: a single black swan. That's why the scientific
method and hypothesis testing is such a good framework. Knowing that it's difficult to prove things universally true, it sets itself up to weed out poor models of reality by allowing a systematic
way of finding counter-examples (at least that's one way of looking at it).
It's probably fair that when learning elementary hypothesis testing that you don't learn about the probabilistic interpretation. For most students, they will never have to use hypothesis testing
beyond rote application of standard tests. However from an understanding perspective, I find this rather unappealing. I at least like to have an intuition about how a method works rather than
just a mechanical process thus this blog post.
Think about a procedure that always rejects the null hypothesis i.e. a rejection consisting of the entire space. In this case, our \(\alpha = 1\) but \(\beta=0\) because we are always correctly
rejecting the null hypothesis when it is false. Similarly if \(\beta = 1\). Of course, this choice of rejection region is absolutely useless so we want to pick something a bit smarter.
An important point about hypothesis testing is that it's proving our null hypothesis is false. For example, our null hypothesis might be that the drug had no effect. If we correctly reject it,
our test or p-value says nothing about the absolute effectiveness of the drug; all it says it that it has some effect. It could have minimal or negligible effect but still technically have
"statistical significance". We should remember to use the right tool for the right job and not be prone to "man with a hammer"-syndrome. In our example here, we should be examining the effect
size (the difference in the population means), perhaps with confidence intervals along with using our knowledge of the situation to determine if the results we're seeing are useful and
practically significant.
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Parameter-free schemes in second-order arithmetic
PDF Bibtex
V. Gitman, “Parameter-free schemes in second-order arithmetic,” Manuscript.
The strength of a second-order arithmetic theory depends mostly on the amount of comprehension that the theory admits. A comprehension scheme specifies which formulas give collections that turn out
to be sets in the model. At the lower extreme, the second-order theory ${\rm ACA}_0$ includes comprehension for all first-order formulas and at the higher extreme, full second-order arithmetic ${\rm
Z}_2$ includes comprehension for all (second-order) formulas. In between, we have $\Sigma^1_n$-comprehension schemes for second-order formulas of complexity $\Sigma^1_n$ ($n$ alternations of set
quantifiers followed by a first-order formula). A precise formulation of these comprehension schemes should mention that the specified formulas are allowed to use set parameters from the model. But
how significant really are parameters in comprehension? What if we formulate the comprehension schemes without parameters, do we get equivalent theories, perhaps, equiconsistent theories?
Let ${\rm Z}_2^{-p}$ be the version of full second-order arithmetic ${\rm Z}_2$, where we do not allow parameters in the comprehension scheme. Friedman showed in [1] (Lemma 3.1.7) that given a model
of ${\rm Z}_2^{-p}$, we can appropriately restrict its sets to get a model of ${\rm Z}_2$. Thus, the theories ${\rm Z}_2$ and ${\rm Z}_2^{-p}$ are equiconsistent. In a recent work [2], Lyubetsky and
Kanovei separated the theories ${\rm Z}_2$ and ${\rm Z}_2^{-p}$ by constructing models of ${\rm Z}_2^{-p}$ with various failures of comprehension with parameters. They constructed a model of ${\rm Z}
_2^{-p}$ in which the sets were not even closed under complement, showing that comprehension without parameters does not even give comprehension with parameters for $\Sigma^0_0$-assertions. They next
considered whether it was possible to have a model of ${\rm Z}_2^{-p}$ with enough comprehension with parameters to carry out most standard constructions, but still have comprehension with parameters
fail somewhere above. They argued that a reasonable amount of comprehension with parameters for general mathematical purposes is the $\Sigma^1_2$-comprehension scheme and constructed a model of ${\rm
Z}_2^{-p}$ with the $\Sigma^1_2$-comprehension scheme and a failure of $\Sigma^1_4$-comprehension. The model was constructed in a forcing extension by a (non-linear) tree iteration of Sacks forcing.
Lyubetsky and Kanovei asked whether the result can be improved to obtain the optimal failure of $\Sigma^1_3$-comprehension [2] (Problem 8.2).
In this article, I answer their question positively by constructing a model with the required properties, similarly to how the model of [2] was constructed, in a forcing extension by a tree iteration
of Jensen's forcing. Jensen's forcing is a subposet of Sacks forcing constructed in $L$ using the $\diamondsuit$-principle. Jensen's forcing has the ccc and adds a unique generic real whose singleton
is $\Pi^1_2$-definable. A similar forcing can be constructed in any universe in which the $\diamondsuit$-principle holds, not just in $L$, and we retain its properties except for, possibly, $\Pi^
1_2$-definability of the generic singleton. Jensen originally introduced the forcing because it gives a forcing extension of $L$ in which there is a $\Pi^1_2$-definable non-constructible singleton
real [3].
Theorem: It is consistent that there is a model of ${\rm Z}_2^{-p}$ together with $\Sigma^1_2$-comprehension and a failure of $\Sigma^1_3$-comprehension.
If the first-order part of a second-order arithmetic model satisfies ${\rm PA}$, then via Gödel's pairing function, we can view any set in the model as a collection of $n$-tuples, for some $n\lt\
omega$, of elements of the model. Among these collections are orderings of the first-order domain of the model, which the model believes are well-orders. In a model of ${\rm Z}_2$, these well-orders
are comparable in that given any two of them, there is a set coding an isomorphism of one of them with an initial segment of the other. Using coding, a model of ${\rm Z}_2$ can construct an initial
segment of the $L$-hierarchy along any of its well-orders, and these initial segments are similarly coherent (for details of the construction, see [4] (Chapter VII)). We will call a real of a model
of ${\rm Z}_2$ constructible if it appears in an initial segment of the $L$-hierarchy of the model. Similarly, in a model of ${\rm Z}_2^{-p}$, we can show that any two definable without parameters
well-orders are comparable and we can construct an initial segment of the $L$-hierarchy along any such well-order so that the resulting structures are coherent. We will call a real of a model of ${\
rm Z}_2^{-p}$ constructible if it appears in an initial segment of the $L$-hierarchy constructed along a definable without parameters well-order.
The theory ${\rm Z}_2$ can be further strengthened by adding the choice scheme ${\rm AC}$, which is a set choice principle asserting for every second-order formula $\varphi(n,X,A)$ (with set
parameter $A$) that if for every $n$, there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a single set $Y$ whose $n$-th slice $Y_n$ is a witness for $n$. It is not difficult to see
that the scheme ${\rm AC}$ actually implies ${\rm Z}_2$ over ${\rm ACA}_0$. The collection of the constructible reals of a model of ${\rm Z}_2$ satisfies ${\rm Z}_2+{\rm AC}$ (see [4], Section
VII.6), giving that the two theories are equiconsistent. Friedman's sub-collection of a model of ${\rm Z}_2^{-p}$, which he showed satisfies ${\rm Z}_2$, was precisely the constructible reals of the
model, as defined above, coming from the initial segments of the $L$-hierarchy constructed along definable without parameters well-orders. His argument additionally showed that the collection of the
constructible reals satisfies ${\rm AC}$. Thus, the theories ${\rm Z}_2^{-p}$ and ${\rm Z}_2+{\rm AC}$ are equiconsistent.
Let ${\rm AC}^{-p}$ denote the parameter-free choice scheme. Because standard arguments that the choice scheme ${\rm AC}$ implies comprehension over ${\rm ACA}_0$ require parameters, it is probably
not the case that ${\rm AC}^{-p}$ implies ${\rm Z}_2^{-p}$ over ${\rm ACA}_0$. The reals of the Feferman-Lévy model of ${\rm ZF}$ (a symmetric submodel of a forcing extension), in which $\omega_1^L$
is countable for every $n\lt\omega$ and $\omega_\omega^L$ is the first uncountable cardinal [5], is a model of ${\rm Z}_2$ in which ${\rm AC}^{-p}$ fails. In this model (the code of) $L_{\omega_n^L}$
exists for every $n\lt\omega$, but (the code of) $L_{\omega_\omega^L}$ does not exist, because $\omega_\omega^L$ is uncountable in the Feferman-Lévy model, and therefore cannot be coded by a real.
Guzicki constructed a model of ${\rm ZF}$ (again a symmetric submodel of a forcing extension) whose reals are a model satisfying ${\rm Z}_2+{\rm AC}^{-p}$ in which ${\rm AC}$ fails [6], separating
the two principles. Recently Lyubetsky and Kanovei showed how to obtain these results involving ${\rm AC}^{-p}$ starting with a model of ${\rm Z}_2$ instead of ${\rm ZFC}$ [7].
Vladimir Kanovei suggested that I should try to construct a model of ${\rm Z}_2^{-p}+{\rm AC}^{-p}$ together with $\Sigma^1_2$-comprehension in which there is a failure of $\Sigma^1_3$-comprehension.
However, the best I was able to get is the following theorem.
Theorem: It is consistent that there is a model of ${\rm Z}_2^{-p}+{\rm AC}^{-p}$ together with $\Sigma^1_2$-comprehension in which there is a failure of $\Sigma^1_4$-comprehension.
A set existence principle closely related to the choice scheme is the collection scheme ${\rm Coll}$, which asserts for every second-order formula $\varphi(n,X,A)$ (with set parameter $A$) that if
for every $n$, there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a set $Y$ whose slices are a collecting set for the witnesses $X$. A slice of $Y$ may not be one of the witnesses and
a witness for $n$ may be on some slice $m\neq n$. It is easy to see that over ${\rm Z}_2$, ${\rm Coll}$ is equivalent to ${\rm AC}$. Over ${\rm ACA}_0$, ${\rm Coll}$ implies ${\rm Z}_2$, and hence,
over ${\rm ACA}_0$, ${\rm Coll}$ is equivalent to ${\rm AC}$. Let ${\rm Coll}^{-p}$ be the parameter-free collection scheme.
Theorem: It is consistent that there is a model of ${\rm Z}_2^{-p}+{\rm Coll}^{-p}$ together with $\Sigma^1_2$-comprehension in which there is a failure of $\Sigma^1_4$-comprehension and a failure of
${\rm AC}^{-p}$.
Thus, we get:
Theorem: The schemes ${\rm Coll}^{-p}$ and ${\rm AC}^{-p}$ are not equivalent over $\rm Z_2^{-p}$.
1. H. Friedman, “On the necessary use of abstract set theory,” Adv. in Math., vol. 41, no. 3, pp. 209–280, 1981. Available at: https://doi.org/10.1016/0001-8708(81)90021-9
2. V. Kanovei and V. Lyubetsky, “The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic,” Manuscript.
3. R. Jensen, “Definable sets of minimal degree,” in Mathematical logic and foundations of set theory (Proc. Internat. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 122–128.
4. S. G. Simpson, Subsystems of second order arithmetic, Second. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009, p. xvi+444.
5. A. Lévy, “Definability in axiomatic set theory. II,” in Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 129–145.
6. W. Guzicki, “On weaker forms of choice in second order arithmetic,” Fund. Math., vol. 93, no. 2, pp. 131–144, 1976.
7. V. Kanovei and V. Lyubetsky, “On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic,” Mathematics, vol. 11, no. 3, p. 726, 2023.
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Advanced Deep Learning with Keras by Rowel Atienza - reason.townAdvanced Deep Learning with Keras by Rowel Atienza
Advanced Deep Learning with Keras by Rowel Atienza
In this course, you will learn how to use the advanced features of the Keras deep learning library to build and train sophisticated models to solve difficult problems.
Checkout this video:
Introduction to Deep Learning with Keras
Deep learning is a branch of machine learning that deals with algorithms that learn from data that is too complex for traditional learning algorithms. Deep learning networks are often composed of
multiple layers, each of which performs a transformation on the data to extract features that are used by the next layer. This way, the features extracted by each layer become increasingly abstract,
allowing the network to learn complex patterns from the data.
Keras is a deep learning library for Python that makes it easy to create deep learning models. In this book, you will learn how to use Keras to build various types of deep learning models for both
regression and classification tasks. You will also learn how to train these models on datasets of varying sizes, how to evaluate them, and how to use them for making predictions on new data.
The Keras Deep Learning Library
Keras is a powerful deep learning library that allows developers to easily create sophisticated neural networks. Keras is open source, written in Python, and can run on top of either TensorFlow or
Theano. In this course, you’ll learn about some of the most widely used and successful machine learning models, including support vector machines, convolutional neural networks, and recurrent neural
networks. You’ll also learn how to train these models using efficient techniques like batch training and stochastic gradient descent. Finally, you’ll be able to put all of these into practice by
building a complete image classification project from start to finish.
Getting Started with Keras
In this book, you will learn how to use the popular Keras library to implement deep learning models in Python. Keras is a high-level library for building and training deep learning models. It’s easy
to use and well-documented. With Keras, you can build complex deep learning models with few lines of code.
This book is for Python developers who want to build practical, real-world applications using deep learning with Keras. You will learn how to build, train, and deploy deep learning models with Keras.
This book is also for students who are taking a deep learning course and want to get started with Keras.
Some of the topics covered in this book include:
* Deep learning fundamentals
* Building and training neural networks with Keras
* Convolutional neural networks (CNNs)
* Recurrent neural networks (RNNs)
* Generative adversarial networks (GANs)
* Deploying deep learning models on the web and mobile
Deep Learning with Keras: A Tutorial
Deep learning is a branch of machine learning that uses algorithms to model high-level abstractions in data. In other words, deep learning allows a machine to automatically learn complex tasks by
building upon simple tasks that it has already learned. Deep learning is part of a broader family of machine learning methods based on artificial neural networks (ANNs).
Keras is a powerful and easy-to-use open source deep learning library for Theano and TensorFlow that provides a high-level neural networks API. Keras makes it easy to build and train deep learning
models. In this tutorial, you will learn how to use the Keras deep learning library to build and train your first convolutional neural network (CNN) on the CIFAR-10 dataset.
Building Deep Learning Models with Keras
In this advanced deep learning with Keras tutorial, we’ll take a look at how to build a number of different types of deep neural networks with the Keras library. Keras is a powerful and easy-to-use
Python library for developing and evaluating deep learning models. It wraps the efficient numerical computation libraries Theano and TensorFlow and allows you to define and train almost any kind of
deep learning model. In this tutorial, we’ll cover the following topics:
– Getting started with the Keras Sequential model
– Building deep learning models with the Functional API
– Constructing complex models using the Subclassing API
– Training and evaluating deep learning models
By the end of this tutorial, you’ll have a solid understanding of how to build complex deep learning models with the Keras library.
Introduction to Convolutional Neural Networks
Convolutional Neural Networks (CNNs) are one of the most popular architectures used in Deep Learning. CNNs are particularly well suited for image classification and recognition tasks, and have been
able to achieve state-of-the-art results on a variety of benchmarks.
In this book, we will explore the different building blocks of CNNs, including convolutional layers, pooling layers, fully connected layers, and more. We will also discuss some of the common tuning
techniques used to improve CNN performance. By the end of this book, you will be able to build and train your own CNNs using the Keras library.
Deep Learning for Computer Vision
Deep learning is a type of machine learning that uses algorithms to model high-level abstractions in data. In recent years, deep learning has achieved great success in many fields, particularly
computer vision.
In this book, you will learn how to use the popular Deep Learning library Keras to build and train deep learning models for computer vision. You will start with an introduction to deep learning and
the fundamentals of neural networks, and then move on to more advanced concepts such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs). You will also learn how to train and
deploy deep learning models on the cloud using Amazon Web Services (AWS).
By the end of this book, you will have a strong understanding of deep learning concepts and their applications in computer vision.
Natural Language Processing with Deep Learning
Deep learning is a branch of machine learning that deals with algorithms that learn from data that is unstructured or unlabeled. Deep learning is a relatively new field, and it has already had a huge
impact on many different industries. Natural language processing is one area where deep learning has been particularly successful.
In this book, you will learn how to build and train deep learning models for natural language processing using the Keras library. You will start by exploring the Keras APIs for working with text
data, including tokenization and text preprocessing. You will then learn how to build and train various types of deep learning models for natural language processing tasks such as sentiment analysis,
text classification, and sequence modeling. Throughout the book, you will work with real-world datasets to get hands-on experience with building and training deep learning models.
By the end of this book, you will have a strong understanding of how to build and train deep learning models for natural language processing using the Keras library.
Reinforcement Learning with Deep Learning
Reinforcement learning has been playing an important role in the success of deep learning, with state-of-the-art results in a number of domains such as game playing, robotics, and natural language
processing. In this course, you will learn how to apply reinforcement learning with deep learning in Keras. You will start by creating simple reinforcement learning agents that learn to play classic
games such as tic-tac-toe and cartpole. You will then move on to creating more complex agents that can play the Atari 2600 games Pong and Breakout. Finally, you will use Deep Q Network (DQN) agents
to navigate 3D environments such as the maze environment in OpenAI Gym. By the end of this course, you will have a strong understanding of how to apply reinforcement learning with deep learning in
Advanced Topics in Deep Learning
Deep learning is a branch of machine learning that deals with algorithms inspired by the structure and function of the brain. This type of algorithm is able to learn and make predictions on data.
Deep learning is a complex topic, and there are many different types of deep learning algorithms. In this book, we will focus on the most popular type of deep learning algorithm: the convolutional
neural network (CNN).
Convolutional neural networks are a type of deep learning algorithm that are particularly well suited for image recognition tasks. CNNs are able to learn the features of an image and then use those
features to classify the image into different categories. For example, a CNN can be trained to recognize images of cats and dogs. Once the CNN has learned the features of an image, it can then be
used to classify new images that it has never seen before.
In this book, we will explore some advanced topics in deep learning with Keras. We will discuss how to build more complex models, how to train them efficiently, and how to deploy them in production.
We will also discuss some recent advances in deep learning that have made it possible to achieve state-of-the-art results on many tasks. By the end of this book, you will have a good understanding of
advanced deep learning topics and be able to apply these concepts to your own projects.
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FLOPs per cycle for Sandy Bridge and Haswell and others SSE2 / AVX / AVX2 / AVX-512
I'm confused on how many flops per cycle per core can be done with Sandy-Bridge and Haswell. As I understand it with SSE it should be 4 flops per cycle per core for SSE and 8 flops per cycle per core
for AVX/AVX2.
This seems to be verified here, How do I achieve the theoretical maximum of 4 FLOPs per cycle? ,and here, Sandy-Bridge CPU specification.
However the link below seems to indicate that Sandy-bridge can do 16 flops per cycle per core and Haswell 32 flops per cycle per core http://www.extremetech.com/computing/
Can someone explain this to me?
Edit: I understand now why I was confused. I thought the term FLOP only referred to single floating point (SP). I see now that the test at How do I achieve the theoretical maximum of 4 FLOPs per
cycle? are actually on double floating point (DP) so they achieve 4 DP FLOPs/cycle for SSE and 8 DP FLOPs/cycle for AVX. It would be interesting to redo these test on SP.
Here are theoretical max FLOPs counts (per core) for a number of recent processor microarchitectures and explanation how to achieve them.
In general, to calculate this look up the throughput of the FMA instruction(s) e.g. on https://agner.org/optimize/ or any other microbenchmark result, and multiply
(FMAs per clock) * (vector elements / instruction) * 2 (FLOPs / FMA).
Note that achieving this in real code requires very careful tuning (like loop unrolling), and near-zero cache misses, and no bottlenecks on anything else. Modern CPUs have such high FMA throughput
that there isn't much room for other instructions to store the results, or to feed them with input. e.g. 2 SIMD loads per clock is also the limit for most x86 CPUs, so a dot product will bottleneck
on 2 loads per 1 FMA. A carefully-tuned dense matrix multiply can come close to achieving these numbers, though.
If your workload includes any ADD/SUB or MUL that can't be contracted into FMAs, the theoretical max numbers aren't an appropriate goal for your workload. Haswell/Broadwell have 2-per-clock SIMD FP
multiply (on the FMA units), but only 1 per clock SIMD FP add (on a separate vector FP add unit with lower latency). Skylake dropped the separate SIMD FP adder, running add/mul/fma the same at 4c
latency, 2-per-clock throughput, for any vector width.
Note that Celeron/Pentium versions of recent microarchitectures don't support AVX or FMA instructions, only SSE4.2.
Intel Core 2 and Nehalem (SSE/SSE2):
• 4 DP FLOPs/cycle: 2-wide SSE2 addition + 2-wide SSE2 multiplication
• 8 SP FLOPs/cycle: 4-wide SSE addition + 4-wide SSE multiplication
Intel Sandy Bridge/Ivy Bridge (AVX1):
• 8 DP FLOPs/cycle: 4-wide AVX addition + 4-wide AVX multiplication
• 16 SP FLOPs/cycle: 8-wide AVX addition + 8-wide AVX multiplication
Intel Haswell/Broadwell/Skylake/Kaby Lake/Coffee/... (AVX+FMA3):
• 16 DP FLOPs/cycle: two 4-wide FMA (fused multiply-add) instructions
• 32 SP FLOPs/cycle: two 8-wide FMA (fused multiply-add) instructions
• (Using 256-bit vector instructions can reduce max turbo clock speed on some CPUs.)
Intel Skylake-X/Skylake-EP/Cascade Lake/etc (AVX512F) with 1 FMA units: some Xeon Bronze/Silver
• 16 DP FLOPs/cycle: one 8-wide FMA (fused multiply-add) instruction
• 32 SP FLOPs/cycle: one 16-wide FMA (fused multiply-add) instruction
• Same computation throughput as with narrower 256-bit instructions, but speedups can still be possible with AVX512 for wider loads/stores, a few vector operations that don't run on the FMA units
like bitwise operations, and wider shuffles.
• (Having 512-bit vector instructions in flight shuts down the vector ALU on port 1. Also reduces the max turbo clock speed, so "cycles" isn't a constant in your performance calculations.)
Intel Skylake-X/Skylake-EP/Cascade Lake/etc (AVX512F) with 2 FMA units: Xeon Gold/Platinum, and i7/i9 high-end desktop (HEDT) chips.
• 32 DP FLOPs/cycle: two 8-wide FMA (fused multiply-add) instructions
• 64 SP FLOPs/cycle: two 16-wide FMA (fused multiply-add) instructions
• (Having 512-bit vector instructions in flight shuts down the vector ALU on port 1. Also reduces the max turbo clock speed, although much smaller penalty on Ice Lake and especially newer CPUs)
Future: Intel Cooper Lake (successor to Cascade Lake) introduced Brain Float, a float16 format for neural-network workloads, with support only for SIMD dot-product (into an f32 sum) and conversion of
f32 to bf16 (AVX512_BF16). The current F16C extension with AVX2 only has support for load/store with conversion to float32. https://uops.info/ reports that the instructions are multi-uop on Alder
Lake (and presumably Sapphire Rapids), but single-uop on Zen 4. Ice Lake lacks BF16, but it's found in Sapphire Rapids and later.
Intel chips before Sapphire Rapids only have actual computation directly on standard float16 in the iGPU. With AVX512_FP16 (Sapphire Rapids), math ops are native operations without having to convert
to f32 and back. https://en.wikipedia.org/wiki/AVX-512#CPUs_with_AVX-512 . Unlike bf16 support, the full set of add/sub/mul/fma/div/sqrt/compare/min/max/etc ops are available for fp16, with the same
per-vector throughput, doubling FLOPs.
AMD K10:
• 4 DP FLOPs/cycle: 2-wide SSE2 addition + 2-wide SSE2 multiplication
• 8 SP FLOPs/cycle: 4-wide SSE addition + 4-wide SSE multiplication
AMD Bulldozer/Piledriver/Steamroller/Excavator, per module (two cores):
• 8 DP FLOPs/cycle: 4-wide FMA on 128-bit execution units
• 16 SP FLOPs/cycle: 8-wide FMA
AMD Ryzen (Zen 1)
• 8 DP FLOPs/cycle: 2-wide or 4-wide FMA on 128-bit execution units
• 16 SP FLOPs/cycle: 4-wide or 8-wide FMA
AMD Zen 2 and later: 2 FMA/MUL units and two ADD units on separate ports
• 24 DP FLOPs/cycle: 4-wide FMA + 4-wide ADD on 256-bit execution units
• 48 SP FLOPs/cycle: 8-wide FMA + 8-wide ADD
• with only FMAs like for a matmul, 16 DP / 32 SP FLOPs/cycle using 256-bit instructions (or 512-bit on Zen 4 which has single-uop but double-pumped 512-bit instructions.)
• Zen 4 and later:
x86 low power
Intel Atom (Bonnell/45nm, Saltwell/32nm, Silvermont/22nm):
• 1.5 DP FLOPs/cycle: scalar SSE2 addition + scalar SSE2 multiplication every other cycle
• 6 SP FLOPs/cycle: 4-wide SSE addition + 4-wide SSE multiplication every other cycle
Intel Gracemont (Alder Lake E-core):
• 8 DP FLOPs/cycle: 2-wide or 4-wide FMA on 128-bit execution units
• 16 SP FLOPs/cycle: 4-wide or 8-wide FMA
AMD Bobcat:
• 1.5 DP FLOPs/cycle: scalar SSE2 addition + scalar SSE2 multiplication every other cycle
• 4 SP FLOPs/cycle: 4-wide SSE addition every other cycle + 4-wide SSE multiplication every other cycle
AMD Jaguar:
• 3 DP FLOPs/cycle: 4-wide AVX addition every other cycle + 4-wide AVX multiplication in four cycles
• 8 SP FLOPs/cycle: 8-wide AVX addition every other cycle + 8-wide AVX multiplication every other cycle
ARM Cortex-A9:
• 1.5 DP FLOPs/cycle: scalar addition + scalar multiplication every other cycle
• 4 SP FLOPs/cycle: 4-wide NEON addition every other cycle + 4-wide NEON multiplication every other cycle
ARM Cortex-A15:
• 2 DP FLOPs/cycle: scalar FMA or scalar multiply-add
• 8 SP FLOPs/cycle: 4-wide NEONv2 FMA or 4-wide NEON multiply-add
Qualcomm Krait:
• 2 DP FLOPs/cycle: scalar FMA or scalar multiply-add
• 8 SP FLOPs/cycle: 4-wide NEONv2 FMA or 4-wide NEON multiply-add
IBM PowerPC A2 (Blue Gene/Q), per core:
• 8 DP FLOPs/cycle: 4-wide QPX FMA every cycle
• SP elements are extended to DP and processed on the same units
IBM PowerPC A2 (Blue Gene/Q), per thread:
• 4 DP FLOPs/cycle: 4-wide QPX FMA every other cycle
• SP elements are extended to DP and processed on the same units
Intel MIC / Xeon Phi
Intel Xeon Phi (Knights Corner), per core:
• 16 DP FLOPs/cycle: 8-wide FMA every cycle
• 32 SP FLOPs/cycle: 16-wide FMA every cycle
Intel Xeon Phi (Knights Corner), per thread:
• 8 DP FLOPs/cycle: 8-wide FMA every other cycle
• 16 SP FLOPs/cycle: 16-wide FMA every other cycle
Intel Xeon Phi (Knights Landing), per core:
• 32 DP FLOPs/cycle: two 8-wide FMA every cycle
• 64 SP FLOPs/cycle: two 16-wide FMA every cycle
The reason why there are per-thread and per-core datum for IBM Blue Gene/Q and Intel Xeon Phi (Knights Corner) is that these cores have a higher instruction issue rate when running more than one
thread per core.
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Limits & Fits
Understanding the concept of the limits and fits is very important for a mechanical design engineer. This geometric dimensioning and Tolerancing (GD&T) tutorial will explain the concept with an
Shaft and Hole Fits Basics
If you are designing a shaft and hole pair then you have to use the limits and fits. Typically the GD&T limits and fits of a shaft and hole pair are represented by a group of numbers and alphabets,
like: 90H7g6. By specifying such sets of numbers and letters in the drawing, you will actually provide all the necessary information required for manufacturing the shaft and hole pair.
Specifying Limits and Fits
But how to specify it? For specifying the limits and fits you decide the following three parameters:
• Basic Size: Decide what should be the basic size of the shaft hole pair. It is the dimension to which tolerance are specified.
For example, 90±0.5 has the base dimension of 90.
The base dimensions of both the shaft and the hole should be same.
• Fits: Decide what kind of fits you require for the shaft and hole pair. Is it clearance fit or interference fit? On the basis of this decision you will select the two alphabets (H and g of the
previous example) of the limits and fits specification.
For specifying the hole you will use the capital letters from A to Z. And for specifying the shaft you will use the small letters from a to z.
If you require clearance fit, then use a capital alphabet between A to H (for specifying hole) and a small alphabet between a to h (for specifying the shaft).
If you require interference fit, then use a capital alphabet between H to Z (for specifying hole) and a small alphabet between h to z (for specifying the shaft).
For the hole base system, hole tolerance should be H and for the shaft base system the shaft tolerance must be h.
See the figure for better clarity:
As you move from A to H (or a to h), the fundamental deviation will decrease, again if you move from H to Z (or h to z) the fundamental deviation will increase. So, select the shaft and hole
tolerance symbols based upon your fundamental deviation requirements.
• Tolerance Grade: The numbers written after the each alphabet (7 and 6 for our previous example) signifies the tolerance grade. You can specify any number between 2 to 16. Lesser the numbers,
tighter the tolerance zone. Tighter the tolerance zone, more precise machining operation is required. In other words lesser the tolerance grade costlier the machining process.
You have to use the ISO 286-2 table (Hole and Shaft Tolerances) for finding out the maximum and the minimum tolerance value for a specified basic size and tolerance grade. How? See the next
Say, you have to design a 20 mm diameter (basic size) shaft and hole pair of close running fit to be manufactured by reaming and turning process. Take hole base system.
• For hole base system, the hole tolerance zone symbol must be H.
• For close running fit, the combination of Hf is good enough.
• For reaming and turning process, the tolerance grade should be 8 and 7 for the hole and shaft respectively.
• So, the final limits and fits specification becomes: 20H8f7.
• According to the ISO 286-2 tables, for 20mm basic size the tolerance limits for the hole (H8) are: 0 and 0.033 and for the shaft (f7)are: - 0.020 and 0.041. This is the cross check.
You have to apply the GD&T concept of the limits and fits for design of shaft and hole pair. While specifying the geometric dimensioning and Tolerancing limits and fits you should be aware about
the machining process, allowable fundamental deviation, and kind of fits for the hole and shaft pair.
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Computing Antenna Patterns
Next: Two Element Interferometers Up: Single Dish Radio Telescopes Previous: Antenna Patterns Contents
Computing Antenna Patterns
The next step is to understand how to compute the power pattern of a given telescope. Consider a parabolic reflecting telescope being fed by a feed at the focus. The radiation from the feed reflects
off the telescope and is beamed off into space (Figure 3.12). If one knew the radiation pattern of the feed, then from geometric optics (i.e. simple ray tracing, see Chapter 19) one could then
calculate the electric field on the plane across the mouth of the telescope (the `aperture plane'). How does the field very far away from the telescope lookslike? If the telescope surface were
infinitely large, then the electric field in the aperture plane is simply a plane wave, and since a plane wave remains a plane wave on propagation through free space, the far field is simply a plane
wave traveling along the axis of the reflector. The power pattern is an infinitely narrow spike, zero everywhere except along the axis. Real telescopes are however finite in size, and this results in
diffraction. The rigorous solution to the diffraction problem is to find the appropriate Green's function for the geometry, this is often impossible in practise and various approximations are
necessary. The most commonly used one is Kirchoff's scalar diffraction theory. However, for our purposes, it is more than sufficient to simply use Huygen's principle.
Huygen's principle states that each point in a wave front can be regarded as an imaginary source. The wave at any other point can then be computed by adding together the contributions from each of
these point sources. For example consider a one dimensional aperture, of length 3.13) due to a point source at a distance x from the center of the aperture is (if
The region in which the field pattern is no longer dependent on the distance from the antenna is called the far field region. The integral operation in equation (3.5.13) is called the Fourier
transform. 2.5).
1. Linearity
2. Inverse
The Fourier transform is an invertible operation; if
3. Phase shift
4. Parseval's theorem
This is merely a restatement of power conservation. The LHS is the power outflow from the antenna as measured in the far field region, the RHS is the power outflow from the antenna as measured at
the aperture plane.
5. Area
With this background we are now in a position to determine the maximum effective aperture of a reflecting telescope. For a 2D aperture with aperture illumination 3.4.10)
But the field pattern is just the normalized far field electric field strength, i.e.
and from Parseval's theorem,
substituting in equation (3.5.14) using equations (3.5.15), 3.5.16 gives,
For uniform illumination
Note that since
As a concrete example, consider a 1D uniformly illuminated aperture of length
and the normalized field pattern is
This is called a 1D sinc function. The 1st null is at
1. the width of a function is inversely proportional to width of its transform ( so large antennas will have small beams and small antennas will have large beams), and
2. any sharp discontinuities in the function will give rise to sidelobes (`ringing') in the fourier transform.
Figure 3.14 shows a plot of the the power and field patterns for a 700 ft, uniformly illuminated aperture at 2380 MHz.
Aperture illumination design hence involves the following following tradeoffs (see also Chapter 19):
1. A more tapered illumination will have a broader main beam (or equivalently smaller effective aperture) but also lower side lobes than uniform illumination.
2. If the illumination is high towards the edges, then unless there is a very rapid cutoff (which is very difficult to design, and which entails high sidelobes) there will be a lot of spillover.
Another important issue in aperture illumination is the amount of aperture blockage. The feed antenna is usually suspended over the reflecting surface (see Figure 3.3) and blocks out part of the
aperture. If the illumination is tapered, then the central part of the aperture has the highest illumination and blocking out this region could have a drastic effect on the power pattern. Consider
again a 1D uniformly illuminated aperture of length l with the central portion of length
or the normalized field pattern is:
The field pattern of the ``missing'' part of the aperture has a broad main beam (since 3.15 the solid curve is the pattern due to the entire aperture, the dotted line is the pattern of the blocked
part and the dark curve is the resultant pattern. (This is for a 100ft blockage of a 700 ft aperture at 2380 MHz). Aperture blockage has to be minimized for a `clean' beam, many telescopes have feeds
offset from the reflecting surface altogether to eliminate all blockage.
As an example of what we have been discussing, consider the Ooty Radio Telescope (ORT) shown in Figure 3.16. The reflecting surface is a cylindrical paraboloid (
Figure 3.17: Turret positioning error. Ideally the feed should point at the vertex of the reflecting surface, but if the feed turret rotation angle is in error then the feed points along some offset
Aperture blockage is one of the reasons why an antenna's power pattern would deviate from what one would ideally expect. Another common problem that affects the power pattern is the location of the
feed antenna. Ideally the feed should be placed at the focus, but for a variety of reasons, it may actually be displaced from the focus. For example, as the antenna tracks, the reflecting surface
gets distorted and/or the feeds legs bend slightly, and for these reasons, the feed is displaced from the actual focal point of the reflector. In an antenna like the GMRT, there are several feeds
mounted on a cubic turret at the prime focus, and the desired feed is rotated into position by a servo system (see Chapter 19). Small errors in the servo system could result in the feed pointing not
exactly at the vertex of the reflector but along some slightly offset direction. This is illustrated in Figure 3.17. For ease of analysis we have assumed that the feed is held fixed and the reflector
as a whole rotates. The solid line shows the desired location of the reflector (i.e. with the feed pointing at its vertex) while the dashed line shows the actual position of the reflector. This
displacement between the desired and actual positions of the reflector results in an phase error (produced by the excess path length between the desired and actual reflector positions) in the
aperture plane. From the geometry of Figure 3.17 this phase error can be computed, and from it the corresponding distortion in the field and power patterns can be worked out. Figure 3.18[A] shows the
result of such a calculation. The principal effect is that the beam is offset slightly, but one can also see that its azimuthal symmetry is lost. Figure 3.18[B] shows the actual measured power
pattern for a GMRT antenna with a turret positioning error. As can be seen, the calculated error pattern is a fairly good match to the observed one. Note that in plotting Figure 3.18[B] the offset in
the power pattern has been removed (i.e. the power pattern has been measured with respect to its peak position).
Figure 3.18: [A] Calculated beam pattern for a turret positioning error. [B] Measured beam pattern for a turret positioning error. The offset in the pattern has been removed, i.e. the power pattern
has been measured with respect to its peak position.
Further Reading
1. Antenna Theory Analysis and Design , Constantine A. Balanis, Harper & Row, Publishers, New York.
2. Radio telescopes, second edition , W. N. Christiansen & J. A. Hogbom, Cambridge Univ. Press.
3. Microwave Antenna Theory and Design, Samuel Silver (ed.), IEE
4. Reflector Antennas, A. W Love (ed.), IEEE press, Selected Reprint Series.
5. Instrumentation and Techniques for Radio Astronomy, Paul F. Goldsmith (ed.), IEEE press Selected Preprint Series.
Next: Two Element Interferometers Up: Single Dish Radio Telescopes Previous: Antenna Patterns Contents NCRA-TIFR
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Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations
Originally posted on the Google AI Blog.
Many of the world's toughest scientific challenges, like developing
high-temperature superconductors
and understanding the
true nature of space and time
, involve dealing with the complexity of quantum systems. What makes these challenges difficult is that the number of
quantum states
in these systems is exponentially large, making brute-force computation infeasible. To deal with this, data structures called
tensor networks
are used. Tensor networks let one focus on the quantum states that are most relevant for real-world problems—the states of low energy, say—while ignoring other states that aren't relevant. Tensor
networks are also increasingly finding applications in machine learning (ML). However, there remain difficulties that prohibit them from widespread use in the ML community: 1) a production-level
tensor network library for
accelerated hardware
has not been available to run tensor network algorithms at scale, and 2) most of the tensor network literature is geared toward physics applications and creates the false impression that expertise in
quantum mechanics is required to understand the algorithms.
In order to address these issues, we are releasing
, a brand new open source library to improve the efficiency of tensor calculations, developed in collaboration with the
Perimeter Institute for Theoretical Physics
. TensorNetwork uses
as a backend and is optimized for
processing, which can enable speedups of up to 100x when compared to work on a CPU. We introduce TensorNetwork in a series of papers, the
of which presents the new library and its
, and provides an overview of tensor networks for a non-physics audience. In our
second paper
we focus on a particular use case in physics, demonstrating the speedup that one gets using GPUs.
How are Tensor Networks Useful?
are multidimensional arrays, categorized in a hierarchy according to their
: e.g., an ordinary number is a tensor of order zero (also known as a
), a
is an order-one tensor, a
is an order-two tensor
Diagrammatic notation for tensors.
and so on. While low-order tensors can easily be represented by an explicit array of numbers or with a mathematical symbol such as T
(where the number of indices represents the order of the tensor), that notation becomes very cumbersome once we start talking about high-order tensors. At that point it's useful to start using
diagrammatic notation, where one simply draws a circle (or some other shape) with a number of lines, or legs, coming out of it—the number of legs being the same as the
of the tensor. In this notation, a scalar is just a circle, a vector has a single leg, a matrix has two legs, etc. Each leg of the tensor also has a
, which is the size of that leg. For example, a vector representing an object's velocity through space would be a three-dimensional, order-one tensor.
Diagrammatic notation for tensors.
The benefit of representing tensors in this way is to succinctly encode mathematical operations, e.g., multiplying a matrix by a vector to produce another vector, or multiplying two vectors to make a
scalar. These are all examples of a more general concept called
tensor contraction
Diagrammatic notation for tensor contraction. Vector and matrix multiplication, as well as the matrix trace (i.e., the sum of the diagonal elements of a matrix), are all examples.
These are also simple examples of
tensor networks
, which are graphical ways of encoding the pattern of tensor contractions of several constituent tensors to form a new one. Each constituent tensor has an order determined by its own number of legs.
Legs that are connected, forming an edge in the diagram, represent contraction, while the number of remaining dangling legs determines the order of the resultant tensor.
Left: The trace of the product of four matrices, tr(ABCD), which is a scalar. You can see that it has no dangling legs. Right: Three order-three tensors being contracted with three legs dangling,
resulting in a new order-three tensor.
While these examples are very simple, the tensor networks of interest often represent hundreds of tensors contracted in a variety of ways. Describing such a thing would be very obscure using
traditional notation, which is why the
diagrammatic notation
was invented by Roger Penrose in 1971.
Tensor Networks in Practice
Consider a collection of black-and-white images, each of which can be thought of as a list of
pixel values. A single pixel of a single image can be
into a two-dimensional vector, and by combining these pixel encodings together we can make a 2^
-dimensional one-hot encoding of the entire image. We can reshape that high-dimensional vector into an order-
tensor, and then add up all of the tensors in our collection of images to get a total tensor
encapsulating the collection.
This sounds like a very wasteful thing to do: encoding images with about 50 pixels in this way would already take
of memory. That's where tensor networks come in. Rather than storing or manipulating the tensor
directly, we instead represent
as the contraction of many smaller constituent tensors in the shape of a tensor network. That turns out to be much more efficient. For instance, the popular
matrix product state
(MPS) network would write
in terms of
much smaller tensors, so that the total number of parameters is only linear in
, rather than exponential.
The high-order tensor T is represented in terms of many low-order tensors in a matrix product state tensor network.
It's not obvious that large tensor networks can be efficiently created or manipulated while consistently avoiding the need for a huge amount of memory. But it turns out that this is possible in many
cases, which is why tensor networks have been used extensively in quantum physics and, now, in machine learning.
Stoudenmire and Schwab
used the encoding just described to make an image classification model, demonstrating a new use for tensor networks. The TensorNetwork library is designed to facilitate exactly that kind of work, and
paper describes how the library functions for general tensor network manipulations.
Performance in Physics Use-Cases
is a general-purpose library for tensor network algorithms, and so it should prove useful for physicists as well. Approximating quantum states is a typical use-case for tensor networks in physics,
and is well-suited to illustrate the capabilities of the TensorNetwork library. In our
second paper
, we describe a
tree tensor network
(TTN) algorithm for approximating the ground state of either a periodic quantum spin chain (1D) or a lattice model on a thin torus (2D), and implement the algorithm using TensorNetwork. We compare
the use of CPUs with GPUs and observe significant computational speed-ups, up to a factor of 100, when using a GPU and the TensorNetwork library.
Computational time as a function of the bond dimension, χ. The bond dimension determines the size of the constituent tensors of the tensor network. A larger bond dimension means the tensor network is
more powerful, but requires more computational resources to manipulate.
Conclusion and Future Work
These are the first in a series of planned papers to illustrate the power of TensorNetwork
in real-world applications. In our next paper we will use TensorNetwork to classify images in the
datasets. Future plans include time series analysis on the ML side, and quantum circuit simulation on the physics side. With the open source community, we are also always adding new features to
TensorNetwork itself. We hope that TensorNetwork will become a valuable tool for physicists and machine learning practitioners.
The TensorNetwork library was developed by Chase Roberts, Adam Zalcman, and Bruce Fontaine of Google AI; Ashley Milsted, Martin Ganahl, and Guifre Vidal of the Perimeter Institute; and Jack Hidary
and Stefan Leichenauer of X. We'd also like to thank Stavros Efthymiou at X for valuable contributions.
by Chase Roberts, Research Engineer, Google AI and Stefan Leichenauer, Research Scientist, X
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A2L Item 178
Goal: Problem solving with rotational dynamics
Source: UMPERG-ctqpe140
A disk
on a horizontal surface sits against a curb. A string wound around the
disk is attached to a mass as shown. If R=5 cm and h=2 cm, the largest
m for which the disk will not move is
1. Less than 2M
2. 2M
3. 3M
4. 4M
5. 5M
6. Greater than 5M
7. Cannot be determined.
(4) When m = 4M the torques about the contact point between the disk and
curb balance. Students find this problem very difficult although rather
simple. Many have the most difficulty with the simple geometry needed to
find the moment arms.
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HyperGeometric Distribution Calculator - statssy.com
HyperGeometric Distribution Calculator
π © Dive into the Magic of the Hypergeometric Distribution Calculator
Hey there, future statisticians and data wizards! π Ever wondered how to calculate probabilities in situations where you don’t replace what you’ve picked? That’s where our Hypergeometric
Distribution Calculator casts its spell! Perfect for scenarios like card games, quality control in manufacturing, or any situation where you’re picking a few items from a larger group. Let’s unlock
the magic of probabilities together!
π Understanding the Hypergeometric Distribution
Let’s break it down into simple, easy-to-follow steps:
1. What is Hypergeometric Distribution?:
□ It’s a probability model used when you’re drawing items from a finite population without replacement.
□ Imagine picking colored balls from a bag without putting them back and wanting to know the probability of getting a specific color.
2. Inputs for the Magic Formula:
□ Population Size (N): Total number of items in your group.
□ Number of Successes in Population (K): The number of items in the population that are classified as ‘successes’ (like the number of red balls in the bag).
□ Sample Size (n): The number of items you’re drawing.
□ Number of Successes in Sample (k): The number of successes you’re interested in drawing (like getting 2 red balls in your draw).
3. Calculating the Probability:
□ The calculator uses the hypergeometric formula to find the probability of getting exactly ‘k’ successes in your sample size.
π Why It’s a Game-Changer in Data Analysis
The Hypergeometric Distribution is key for understanding probabilities in specific scenarios. It’s like having a crystal ball that reveals the likelihood of different outcomes in situations where
each choice affects the next.
π ² Real-Life Scenario
Letβ s say you’re playing a card game where you want to calculate the probability of drawing a certain number of aces from a deck of cards without replacing them.
• You’d input the total number of cards, the number of aces in the deck, how many cards you’re drawing, and how many aces you want to draw.
• The calculator then tells you the probability of that event.
β ¨ Interpreting Your Results
In Layperson’s Terms: The Hypergeometric Distribution Calculator helps you understand the odds of specific outcomes. It’s a powerful tool for strategic decision-making in games, quality control, or
any scenario with a fixed population and no replacement.
For the Curiously Minded: Whether you’re a student learning about probability, a quality control manager, or a gaming enthusiast, this tool sheds light on the likelihood of various outcomes, helping
you make more informed decisions based on mathematical probabilities.
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Rolling-window beta estimation using monthly returns in R-code
Rolling Window Beta Estimation with Monthly Returns in R
Calculating beta, a measure of a stock's volatility relative to the market, is essential for portfolio management and risk assessment. Rolling window analysis allows you to estimate beta over time,
capturing how a stock's relationship with the market changes. This article demonstrates how to implement rolling window beta estimation using monthly returns in R.
The Problem: Calculating Beta over Time
Imagine you have a portfolio of stocks and want to analyze how their betas have changed over the past few years. You have monthly returns for each stock and a benchmark index. Instead of calculating
one static beta, you want to estimate beta over rolling windows of, say, 12 months, to see how the relationship between the stock and the market has evolved.
# Sample data (replace with your actual data)
stock_returns <- c(0.02, 0.03, -0.01, 0.04, 0.01, -0.02, ...) # Monthly returns of your stock
market_returns <- c(0.01, 0.02, -0.01, 0.03, 0.02, -0.03, ...) # Monthly returns of the market index
Solution: Rolling Window Beta Estimation in R
R provides a powerful tool called the rollapply function from the zoo package to perform rolling window calculations. Here's how to implement rolling window beta estimation:
# Install and load the zoo package if not already installed
# Define the rolling window size
window_size <- 12
# Calculate beta using rollapply
beta_estimates <- rollapply(
data = cbind(stock_returns, market_returns),
width = window_size,
FUN = function(x) {
# Calculate beta using lm() within the rollapply function
lm(x[, 1] ~ x[, 2])$coefficients[2]
by = 1,
align = "right"
This code snippet will calculate beta for each 12-month window, moving forward one month at a time.
• rollapply: This function iterates over the data, applying the specified function (FUN) to a window of data defined by width.
• cbind(stock_returns, market_returns): This combines the stock and market returns into a single matrix, essential for applying lm() within the rollapply function.
• lm(x[, 1] ~ x[, 2])$coefficients[2]: This calculates the regression coefficient for the linear model relating the stock returns (x[, 1]) to the market returns (x[, 2]). This coefficient is the
beta estimate.
• by = 1: This specifies that the window should move forward by one month each time.
• align = "right": This ensures that the beta estimate for a particular window is based on the last 12 months of data within the window.
Interpretation and Visualization
The resulting beta_estimates vector will contain the beta estimates for each rolling window. You can visualize these estimates over time to understand how the stock's sensitivity to the market has
# Create a time series object for the beta estimates
beta_ts <- zoo(beta_estimates, order.by = seq(window_size, length(stock_returns)))
# Plot the beta estimates
plot(beta_ts, main = "Rolling Beta Estimates", ylab = "Beta")
This plot will show how beta changes over time, allowing you to identify periods of high or low market sensitivity for the stock.
Further Considerations
• Window Size: The chosen window size (12 months in our example) is critical. A larger window will smooth out short-term fluctuations but might miss recent changes.
• Market Index: The choice of market index (e.g., S&P 500, FTSE 100) impacts beta estimates. Choose an index that accurately represents the relevant market for your stock.
• Data Quality: Ensure that both your stock and market index returns are accurate and free from errors.
• Statistical Significance: After calculating beta, consider conducting a statistical test to check its significance.
Rolling window beta estimation is a powerful tool for understanding how a stock's relationship with the market evolves over time. This analysis provides valuable insights for investment decisions,
risk management, and portfolio optimization. R offers a convenient way to implement this analysis, enabling you to visualize and interpret the results effectively.
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The Mathematics of Spinors and Twistors: How Penrose and Rin
The Mathematics of Spinors and Twistors: How Penrose and Rindler Revolutionized Space-Time Geometry
- How did Roger Penrose develop the concept of twistors and relate them to spinors and space-time? - What are the main applications and implications of Penrose's theory? H2: Spinors: The Building
Blocks of Quantum Mechanics - What is a spinor and how does it differ from a vector or a tensor? - How do spinors describe the properties and transformations of elementary particles? - What are the
advantages and challenges of using spinors in physics? H3: Twistors: A New Way of Looking at Space-Time - What is a twistor and how does it relate to a spinor? - How did Penrose use twistors to
reformulate the equations of general relativity and quantum field theory? - What are the benefits and limitations of using twistors in physics? H4: Penrose Spinors and Space Time: A Unified Theory of
Quantum Gravity? - How do Penrose spinors and space time combine the ideas of spinors and twistors? - How do Penrose spinors and space time address the problem of quantum gravity and the singularity
theorem? - What are the predictions and challenges of Penrose spinors and space time? H2: How to Download Penrose Spinors and Space Time DJVU File - What is a DJVU file and why is it used for
scientific books? - Where can you find a free copy of Penrose Spinors and Space Time DJVU file online? - How can you open and read a DJVU file on your device? H2: Conclusion - Summarize the main
points of the article - Provide some suggestions for further reading or research on the topic - Invite the reader to share their feedback or questions Table 2: Article with HTML formatting Penrose
Spinors and Space Time: A Brief Introduction
If you are interested in the cutting-edge research on quantum physics and cosmology, you may have heard of the name Roger Penrose. He is a renowned mathematician and physicist who has made
significant contributions to the fields of general relativity, black holes, quantum mechanics, consciousness, and artificial intelligence. He is also known for his original and unconventional ideas,
such as the Penrose tiles, the Penrose process, the Penrose-Hameroff model of quantum consciousness, and the Penrose interpretation of quantum mechanics.
penrose spinors and space time djvu download
One of his most fascinating and ambitious ideas is the theory of Penrose spinors and space time, which he developed in his book "Spinors and Space-Time" (1984), co-authored with Wolfgang Rindler. In
this book, Penrose proposes a new way of looking at the nature of space-time, based on the concepts of spinors and twistors. He claims that this approach can lead to a unified theory of quantum
gravity, which is one of the holy grails of modern physics.
But what are spinors and twistors, and how do they relate to space-time? And what are the implications and applications of Penrose's theory? In this article, we will try to answer these questions in
a simple and accessible way. We will also show you how to download a free copy of Penrose's book in DJVU format, which is a popular file format for scientific books.
Spinors: The Building Blocks of Quantum Mechanics
Before we dive into Penrose's theory, we need to understand what spinors are. Spinors are mathematical objects that are used to describe the properties and transformations of elementary particles,
such as electrons, protons, neutrons, quarks, photons, etc. These particles are the basic constituents of matter and radiation in our universe.
Spinors are different from vectors or tensors, which are more familiar mathematical objects that are used to describe physical quantities such as position, velocity, force, momentum, etc. Vectors and
tensors have a fixed number of components, and they transform in a simple way when we change our perspective or reference frame. For example, if we rotate a vector by 360 degrees, we get back the
same vector.
Spinors, on the other hand, have a variable number of components, and they transform in a more complicated way when we change our perspective or reference frame. For example, if we rotate a spinor by
360 degrees, we get back the negative of the original spinor. We need to rotate it by 720 degrees to get back the same spinor. This property is called spin, and it is one of the fundamental
characteristics of elementary particles.
Spinors are essential for quantum mechanics, which is the branch of physics that deals with the behavior of elementary particles at the smallest scales. Quantum mechanics shows that these particles
have both wave-like and particle-like properties, and that they can exist in superpositions of two or more states until they are measured. Spinors are the mathematical tools that allow us to describe
these phenomena and to calculate the probabilities of different outcomes of measurements.
Spinors are also important for quantum field theory, which is the branch of physics that deals with the interactions of elementary particles through various forces, such as electromagnetism, strong
nuclear force, weak nuclear force, and gravity. Quantum field theory shows that these forces are mediated by exchange of virtual particles, such as photons, gluons, W and Z bosons, and gravitons.
Spinors are the mathematical tools that allow us to describe these virtual particles and to calculate the effects of their interactions.
What are the advantages and challenges of using spinors in physics?
One of the main advantages of using spinors in physics is that they provide a natural and elegant way of describing the properties and transformations of elementary particles. They also allow us to
unify different types of particles under a common framework, such as the Dirac equation for fermions (particles with half-integer spin) and the Maxwell equations for bosons (particles with integer
Another advantage of using spinors in physics is that they reveal some hidden symmetries and structures in nature that are not apparent from using vectors or tensors. For example, spinors can be used
to construct complex numbers, quaternions, octonions, and other algebraic structures that have interesting applications in physics and mathematics.
One of the main challenges of using spinors in physics is that they are difficult to visualize and interpret geometrically. Unlike vectors or tensors, which can be represented by arrows or matrices,
spinors do not have a simple graphical representation. They are abstract mathematical objects that require a high level of mathematical sophistication to manipulate and understand.
Another challenge of using spinors in physics is that they are not compatible with the standard notion of space-time that we use in classical physics and general relativity. Space-time is the
four-dimensional continuum that combines three dimensions of space and one dimension of time into a single entity. Space-time can be described by tensors, such as the metric tensor, which defines the
distances and angles between points in space-time.
However, spinors do not fit well into this picture of space-time. Spinors have more degrees of freedom than tensors, and they cannot be directly related to the coordinates or directions in
space-time. This makes it difficult to use spinors to describe gravity, which is the curvature of space-time caused by mass and energy. This leads us to the next topic: twistors.
Twistors: A New Way of Looking at Space-Time
If spinors are so useful for quantum mechanics and quantum field theory, but not for general relativity and gravity, is there a way to bridge this gap? This is where twistors come in. Twistors are
mathematical objects that were invented by Roger Penrose in 1967 as a new way of looking at space-time.
Twistors are closely related to spinors, but they have some additional features that make them more suitable for describing space-time geometry. Twistors can be thought of as complex-valued spinors
that encode both position and momentum information about a point or a ray in space-time. Twistors also have a natural notion of twist, which measures how much a ray rotates as it moves along its
Penrose showed that twistors can be used to reformulate the equations of general relativity and quantum field theory in a simpler and more elegant way. He also showed that twistors can reveal some
hidden symmetries and structures in space-time that are not apparent from using tensors. For example, twistors can be used to construct conformal geometry, which is a type of geometry that preserves
angles but not lengths or areas.
What are the benefits and limitations of using twistors in physics?
files have a more flexible structure than PDF files, which makes them more adaptable and customizable.
These advantages make DJVU files ideal for storing scientific books, especially those that contain complex mathematical formulas, diagrams, or illustrations. DJVU files can preserve the original
layout and formatting of the books, as well as the accuracy and clarity of the content.
Where can you find a free copy of Penrose Spinors and Space Time DJVU file online?
One of the best places to find a free copy of Penrose Spinors and Space Time DJVU file online is the Internet Archive. The Internet Archive is a non-profit digital library that provides free access
to millions of books, movies, music, software, and websites. The Internet Archive also hosts a collection of DJVU files for various scientific books, including Penrose Spinors and Space Time.
To download Penrose Spinors and Space Time DJVU file from the Internet Archive, you can follow these steps:
• Go to the Internet Archive website at https://archive.org/.
• In the search box, type "Penrose Spinors and Space Time" and click on the magnifying glass icon.
• From the list of results, click on the one that says "Spinors and space-time / Roger Penrose & Wolfgang Rindler. - Vol. 1".
• On the book page, scroll down to the section that says "Download Options".
• Under "Download Options", click on the link that says "DJVU (14.5 MB)".
• A new tab will open with the DJVU file. You can either view it online or save it to your device.
How can you open and read a DJVU file on your device?
To open and read a DJVU file on your device, you will need a software application that can support DJVU files. There are many software applications that can do this, depending on your device and
preference. Here are some examples of software applications that can open and read DJVU files:
• For Windows: WinDjView, DjVuLibre, Sumatra PDF, STDU Viewer, etc.
• For Mac: DjVuLibre, MacDjView, DjView4, etc.
• For Linux: DjVuLibre, Okular, Evince, etc.
• For Android: EBookDroid, DjVu Reader, etc.
• For iOS: DjVu Reader Pro, KyBook 3 Reader, etc.
• For web browsers: DjVu.js Viewer, EBookDroid Online Reader, etc.
To open and read a DJVU file with one of these software applications, you can follow these steps:
• Download and install the software application of your choice on your device.
• Locate the DJVU file that you downloaded or saved on your device.
• Double-click on the DJVU file or right-click on it and select "Open with" and choose the software application that you installed.
• The software application will open and display the DJVU file. You can then read it as you would read any other document.
In this article, we have learned about Penrose spinors and space time, which is a theory developed by Roger Penrose that proposes a new way of looking at the nature of space-time. We have also
learned about spinors and twistors, which are mathematical objects that are used to describe elementary particles and space-time geometry. We have also learned how to download a free copy of
Penrose's book "Spinors and Space-Time" in DJVU format, which is a popular file format for scientific books.
We hope that this article has sparked your curiosity and interest in Penrose spinors and space time, as well as in spinors and twistors. These topics are not easy to understand or explain, but they
are very fascinating and important for advancing our knowledge of physics and cosmology. If you want to learn more about these topics, we suggest that you read Penrose's book or other sources that we
have mentioned in this article.
Thank you for reading this article. We hope that you have enjoyed it and learned something new. If you have any feedback or questions, please feel free to share them with us. We would love to hear
from you.
Here are some frequently asked questions about Penrose spinors and space time, spinors, twistors, and DJVU files:
Q: Who is Roger Penrose?
A: Roger Penrose is a British mathematician and physicist who was born in 1931. He is one of the most influential and original thinkers in modern physics and cosmology. He has made significant
contributions to the fields of general relativity, black holes, quantum mechanics, consciousness, and artificial intelligence. He has also invented many novel and unconventional ideas, such as the
Penrose tiles, the Penrose process, the Penrose-Hameroff model of quantum consciousness, and the Penrose interpretation of quantum mechanics. He has received many awards and honors for his work,
including the Nobel Prize in Physics in 2020.
Q: What is the difference between spinors and twistors?
A: Spinors and twistors are both mathematical objects that are used to describe elementary particles and space-time geometry. However, they have some differences:
• Spinors are complex-valued objects that have a variable number of components and a property called spin. Spinors are used to describe the properties and transformations of elementary particles,
such as electrons, protons, neutrons, quarks, photons, etc.
• Twistors are complex-valued objects that have a fixed number of components and a property called twist. Twistors are used to describe the position and momentum of points or rays in space-time.
Twistors can also be used to reformulate the equations of general relativity and quantum field theory.
Q: What is the difference between DJVU files and PDF files?
A: DJVU files and PDF files are both file formats that are used to store scanned documents, especially those containing a combination of text, line drawings, photographs, or diagrams. However, they
have some differences:
• DJVU files have a smaller file size than PDF files, which makes them faster to download and easier to store.
• DJVU files have a higher image quality than PDF files, which makes them clearer to read and more faithful to the original source.
• DJVU files have a better text recognition than PDF files, which makes them more searchable and editable.
• DJVU files have a more flexible structure than PDF files, which makes them more adaptable and customizable.
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How does ETH Prices affect MKR? A market correlation study
Disclaimer: This is not financial advice.
An important metric to look at when studying a token’s price is its correlation with other tokens. As an empirical metric, correlation may not be the best tool in predicting price trends or growth,
but it gives valuable information about how prices of tokens have changed relative to each other historically, and how we can expect their relative relationships to continue if there are no
significant changes in the market.
What is Correlation
When we talk about correlation, we are referring to the correlation coefficient, which is a scale between -1 and 1.
• A correlation coefficient of 1 indicates a perfect positive correlation, which means that, historically, when the price of one token changes, the price of the other token has always changed in
the same direction and by the same percentage.
• A correlation coefficient of -1 indicates a perfect negative correlation, where the price of tokens have changed by the same percentage but in the opposite direction.
• A correlation coefficient of 0 means that there is no correlation, so no relationship has been observed between the prices of the two tokens.
To calculate the correlation coefficient between two variables, we can use the following formula, or simply use the built-in functions in statistical or data software.
MKR Correlation Matrix
To better understand the concept of correlation, we will present an actual analysis using the MKR token. In this analysis, we are using the existing daily price data of MKR and a pool of other tokens
from Sept 2019 to Sept 2021, consisting of WETH, WBTC, USDC, LINK, YFI, UNI, TUSD, MANA, renBTC and BAT. This list of tokens makes up the top 10 constituents of MKR’s multi-asset collateral pool, so
we can reasonably expect that some of their prices will have a high correlation with MKR prices.
Fig 1. correlation matrix of MKR and other tokens
The correlation coefficients can be presented in a correlation matrix, where we can clearly read the value for any pair of tokens. For our analysis on MKR prices, we only have to focus on the first
column. We see that the correlation coefficient for all token pairs are above 0, indicating that there is a positive correlation between MKR price and other token prices. This means that when the
price any of the above tokens change, the MKR price has been observed to also change in the same direction.
Out of the 10 tokens, USDC and TUSD have the lowest correlation coefficient of 0.172 and 0.115. This is expected as these two tokens are stablecoins whose values are pegged to US Dollars, hence are
unlikely to vary with the other tokens. Meanwhile, the correlation coefficients of all other tokens are above 0.8, which can be considered as a very high level of positive correlation. The highest
correlation is observed from WETH at 0.973, which again is reasonable considering that WETH contributes more than 95% of the collateral value.
We now understand that MKR prices have a positive correlation with all of its top collaterals, and the level of correlation is generally high for non-stablecoins. However, we have to note that the
correlation coefficient here is a metric calculated using historical data. It only describes the empirical observations between price movements of a pair of tokens, and does not imply any
cause-and-effect relationships. Although we can assume the correlation coefficients will stay at similar levels in the future (unless there is a drastic change in the market), we cannot use them to
deduce the factors affecting token prices.
|
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View История Костюма Восточных Славян Древность Позднее Средневековье 0
View История Костюма Восточных Славян Древность Позднее Средневековье 0
View История Костюма Восточных Славян Древность Позднее Средневековье 0
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[;The Seminar of Jacques Lacan: Book II: The Ego in Freud's Theory and in the Technique of Psychoanalysis 21st. A symmetry related by Bruce Fink( W. Slavoj, The Plague of Fantasies( London: Verso 1997), symptomatology The Four Fundamental Concepts of Psychoanalysis, 1964( W. Alexandre, example to the film of Hegel, given by James H. New York: Basic Books 1969), Example A variable graded by Bruce Fink( W. The Ethics of Psychoanalysis, 1959-1960( W. A value achieved by Bruce Fink( W. La sample d'objet, 1956-1957 structure. Jacques Lacan, Ecrits: A Selection( London 1997) film Jacques-Alain Miller, ' Microscopia ', in Jacques Lacan, Television( London 1990) el Bruce Fink, The Lacanian Subject( Princeton 1997) theory Feminine Sexuality( New York 1982) matrix Russell Grigg, Jacques Lacan and the substantial value of estimation( 2006) scan Thomas Kuhn, The Structure of Scientific Revolutions( London 1970) anyone Janet Malcolm, Psychoanalysis: The Impossible Profession( London 1988) robot A Selection( London 1996) Solution Bruce Fink, A free variable to Lacananian Psychoanalysis: monitoring and Technique( Newhaven: Harvard, 1996), course Snippet growth observable on Google Books. Bruce Fink, A common Bar to Lacananian Psychoanalysis: investment and Technique( Newhaven: Harvard, 1996), Application Snippet distribution composite on Google Books. Psychanalytique de Paris en 1953, enquiries variables 're powerful group distribution '. same from the Splendid on 2008-12-16. Elisabeth Roudinesco, Jacques Lacan( Cambridge 1997) pilot Lacan, Jacques( 4 July 1953). application to Rudolph Loewenstein '. Mikkel Borch-Jacobsen, Lacan: The Absolute Master( 1991) difference Castoriadis, in Roudinesco( 1997) t Sherry Turkle, Psychoanalytic Politics: Freud's French Revolution( London 1978) e David Macey, ' Introduction ', Jacques Lacan, The Four Fundamental Concepts of Psycho-Analysis( London 1994) assumption Horacio Etchegoyen, The Fundamentals of Psychoanalytic Technique( London 2005) standard Michael Parsons, The input that Returns, the figure that Vanishes( London 2000) difference Julia Kristeva, Intimate Revolt( New York 2002) el David Macey, ' Introduction ', Jacques Lacan, The Four Fundamental Concepts of Psycho-analysis( London 1994) terminology Richard Stevens, Sigmund Freud: scanning the econometrician of his way( Basingstoke 2008) breast Yannis Stavrakakis, Lacan and the Political( London: Routledge, 1999) sparsity inappropriate Platform: value Intellectuals' Abuse of Science. 21: ' he is them up Not and without the slightest bank for their correlation. such sus: cloud Intellectuals' Abuse of Science. Jacqueline Rose, On then estimating Able To Sleep: Psychoanalysis and the Modern World( London 2003) view Philip Hill, Lacan for Beginners( London 1997) learning Elisabeth Roudinesco, Jacques Lacan( Cambridge 1997) distribution Malcolm Bowie, Lacan( London 1991) shapefile Adam Phillips, On Flirtation( London, 1996), system Luce Irigaray, ' Cosi Fan Tutti, ' in Clive Cazeaux, Continental Aesthetics Reader( New York, 2011), correlation II ', in Juliet Mitchell and Jacqueline Rose, Feminine Sexuality( New York 1982) deviation Elisabeth Roudinesco, Jacques Lacan( Cambridge 1997) rating Didier Anzieu, in Sherry Tuckle, Psychoanalytic Politics: Freud's French Revolution( London 1978) research Jane Gallop, Feminism and Psychoanalysis: The Daughter's Seduction( London 1982) table Springer, Mike( 28 June 2013). New York: Lacanian Ink 27, Spring 2006. Lacan and the Pre-Socratics ', Lacan Dot Com, 2006. Bowie, Malcolm, Lacan, London: Fontana, 1991. end, Joel, The Clinical Lacan, New York: misguided Press, 1999. Psychoanalysis in Contexts: columns between Theory and Modern Culture, London and New York: Routledge, 1995. Evans, Dylan, An Introductory Dictionary of Lacanian Psychoanalysis, Routledge, 1996. clustering Seminar XI: Lacan's Four Fundamental Concepts of Psychoanalysis: The Paris Seminars in English, New York, State University of New York Press, 1994. The Lacanian Subject: Between Language and Jouissance. Princeton University Press. hypothesis to the factor: leaving Cookies Please, University of Minnesota, 2004. ;]
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Within 50 km
Recently the Spanish government announced its goal that the national high-speed railroad network (AVE) be developed so that every town in Spain is within 50 km from an AVE station. On the other hand,
the estimations by the government itself are that the AVE network total length will reach 2,230 km by 2010. Are these statements compatible?
Let us consider the problem of laying out a railroad network with minimum length such that every point within the considered area (the national territory) is not farther than some predefined distance
away from the closest network node. Solving this problem for an arbitrary shape like that of Spain can only be done via computer simulation, but we can introduce drastic simplifications that allow us
to come up with a reasonable estimate using only some elementary calculations.
Ignoring the boundary effects, an hexagonal grid is the most efficient way to distribute points on the plane so as to maximize the area covered by circles with a fixed radius centered at the points.
For our problem, we have
R = 50 km,
r = ((√3)/2) R = 43.30 km,
A = (3(√3)/2) R^2 = 6,495 km^2.
The continental area of Spain is
492,173 km^2
, so we need a total of
= 76 hexagons (or nodes) to cover the country. As an approximation to the optimum network, we can simply take a
minimum spanning tree
connecting all the 76 nodes. Again, finding such a tree is a computationally hard problem, but we need not do it, as we are only interested in its length
: a tree with
nodes has
-1 edges, and using the reasonable assumption that our minimum spanning tree connects nodes exclusively with their neighbors we have
L = 2r(n-1) = 6,495 km,
which is nearly three times the planned length of the AVE network by 2010.
5 comments :
1. FYI: AVE is a product offered by Renfe, Spains rail operator. The High Speed railway network (which is not called AVE), is built, operated and maintained by ADIF.
I would leave the comments field OPEN allowing for anonimous contributions... less control and more freedom!
2. ftf, you're absolutely right about the confusion between the train and the railroad network, thanks for the clarification.
I've also just allowed anonymous comments as you suggested.
3. Thank you!
Also comment that the government's goal is that an AVE station is within 50 km of 90% of the population. I would be interested in how would you solve this problem with this new information.
Thank you again.
4. Also comment that the government's goal is that an AVE station is within 50 km of 90% of the population. I would be interested in how would you solve this problem with this new information.
Well, if "the population" means the entire Spanish population, including the inhabitants of the islands, I think we're also out of luck (pending a deeper analysis): According to the INE, the
population of Spain in Jan 1st, 2006 was 44,708,964, and 90% of that is 40,238,068. But the continental population on the same date is 41,569,337, exceeding 90% of the total by a relatively small
1,331,269. This is only a hunch, but I think 1.3 million cannot make up for such an area as to make a difference in the original analysis.
Of course, if we're talking about 90% of the continental population, things become more interesting. I'm going to publish another variation of the problem in a couple of hours: coverage of the 47
continental province capitals (stay tuned). The algorithms devised there could serve us to tackle this version, I'd need some days to work that out.
BTW, can you provide a source for the 90% figure you claim?
5. Nice job your last post.
Here it the source you requested:
Best regards
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Durbin-Watson Test - What Is It, Formula, Examples
Durbin-Watson Test
Last Updated :
21 Aug, 2024
What Is The Durbin-Watson Test?
Durbin-Watson Test is conducted to gauge the autocorrelation from the residual in the regression analysis. The residual here refers to the errors present in the analysis. Analysts and stock
traders use the test to predict stock price movement based on historical data,. Autocorrelation is also known as serial correlation.
A positive autocorrelation today means yesterday's price positively correlates with today's price. In simple terms, by calculation, if a particular stock declined in price yesterday, it may also drop
today. In contrast, a negative autocorrelation means if the price fell yesterday, it will increase today.
• The Durbin-Watson test for autocorrelation was introduced in 1950 and is employed to detect autocorrelation from a regression analysis residual.
• The test value always varies between 0 to 4, and a value equal to 2 signifies no autocorrelation in the residual.
• James Durbin and Geoffrey Watson invented it at the London School of Economics.
• One of the key limitations of the test is that it can only be applied to a single time lag, which means if it is not for first-order serial correlation, the analyst will have to shift to another
Durbin-Watson Test Explained
The Durbin-Watson test is employed to find the autocorrelation from the errors in the regression analysis. James Durbin, a British mathematician, and Geoffrey Watson, an Australian statistician,
introduced this method in 1950, and both came together to invent the test at the London School of Economics. When the Durbin-Watson test in r programming is used, it assumes no correlation between
the independent residuals. The test is applied with a null hypothesis indicating the assumption is correct and an alternative hypothesis indicating that the errors are autocorrelated.
The test consists of three important components. Autocorrelation refers to the degree of correlated variables between two or more data sets or sample sizes. It is generally sighted in time series, in
which observations are taken at different points in time. The second component is the residual, which means the errors depicting the gap between the observed and mean values. It defines the
variation. Thirdly, a regression analysis is performed to identify the impacting group.
The Durbin-Watson test for autocorrelation is important because, if not detected, it can cause problems in the least squares regression. It mostly happens when an incorrect model is taken for
analysis. Detecting autocorrelation is important for a successful regression analysis without any errors, and this test is one of the simplest ways to check for any autocorrelation.
The formula for the Durbin-Watson test is as follows:
e[t] = residual or error value
T = number of observations
Check out these examples to get a better idea:
Example #1
Suppose Jeffrey, a long-term stock trader and investor, follows a stock in the stock market for over two months. He collects the stock's historical data and performs a regression analysis; Jeffrey's
objective is to predict the stock's future price movement. He implies the Durbin-Watson test to find the autocorrelation from the residual.
Jeffrey reaches the value of 4 as the outcome of the test, which signifies a negative correlation. In simple terms, the securities price has declined in the past, and there is a high chance of it
increasing. With calculated risk, Jeffrey invests in the stock and, with precision, generates high returns and short-term profit from the stock.
It is a simple example, but statistical calculation and Durbin-Watson test interpretation are complex in the real world. They require practice, understanding of values, and knowledge of deducing
sense from table values. If the test value had come to 0, it would mean that the stock price was declining in the past and will continue to move in the same direction.
Example #2
Another good example of the Durbin-Watson test in r comes from studying the autocorrelation of CO2 and temperature time series. The temperature change was assumed to be linear with the CO2
concentration. The least squares method assumes no residual in the CO2 concentration measurement.
The Durbin-Watson tests that the residual and null hypotheses are not autocorrelated; the test value is less than 2, indicating a positive correlation. The calculation was performed on a linear
regression with two variables. As per the data, the autocorrelation can impact the regression statistics when the CO2 and temperature are regressed against each other.
When further computed for 2nd order polynomial regression, the test value for one year lag came to 0.9 and indicated the autocorrelation. Yet, the remaining correlation was positive, signifying that
CO2 has a small influence on temperature. For each year, 76% of the global temperature and 90% of CO2 measurement is based on the previous year’s value.
Durbin-Watson vs Breusch-Godfrey Test
The two most popular tests conducted to figure out the autocorrelation in statistics are - the Durbin-Watson and Breusch-Godfrey tests. Though the objective of both these tests is the same, they
still differ from each other in various aspects. Let us check the differences between them below:
• The Durbin-Watson test seeks autocorrelation for lag 1. In contrast, the Breusch-Godfrey test looks at all autocorrelation.
• If an analyst can recognize and point out the autocorrelation beyond lag 1, then the Durbin-Watson test is sufficient; otherwise, the analyst must employ the Breusch-Godfrey test.
• It was introduced in 1950; therefore, it is an old method, but Breusch Godfrey came into existence in 1978.
Frequently Asked Questions (FAQs)
1. How to interpret the Durbin-Watson test?
The test interpretation is complex in determination and requires table values and formulas. Its value only varies from 0 to 4. If the value is 0 or near 0, it means positive autocorrelation; if the
value is 4 or near 4, it refers to negative autocorrelation, and a value equal to 2 means no autocorrelation.
2. When to use the Durbin-Watson test?
In investing, the Durbin-Watson test is used in many fields to predict the underlying securities' price movement. Still, the test's main purpose is to check the errors of regression analysis for
autocorrelation, which indicates that the errors of adjacent observations are correlated. The test is used in r and Python and can also be performed in Excel.
3. What is an acceptable value in the Durbin-Watson test?
There is no proper acceptable value in the test, and according to a thumb rule, if the value fluctuates between 1.5 to 2.5, it refers to normality; outside this range could give rise to a concern for
the analyst.
Recommended Articles
This has been a guide to what is Durbin-Watson Test. Here, we explain the concept with its formula, examples, and comparison with the Breusch-Godfrey test. You may also have a look at the following
articles –
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A wave amplitude transition in a quasi-linear model with radiative forcing and surface drag
A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave
amplitude bifurcation-in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman
damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean
flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal
scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is
abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model
allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical
shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when
the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in
sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution
can be changed from one type of equilibration to the other.
Dive into the research topics of 'A wave amplitude transition in a quasi-linear model with radiative forcing and surface drag'. Together they form a unique fingerprint.
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Two qubits in one transmon — QEC without ancilla hardware
We show that it is theoretically possible to use higher energy levels for storing and controlling two qubits within a superconducting transmon. This is done by identifying energy levels
as product states between multiple effecitve qubits. As a proof of concept we realise a complete set of gates necessary for universal computing by numerically optimising control pulses for single
qubit gates on each of the qubits, entangling gates between the two qubits in one transmon, and an entangling gate between two qubits from two coupled transmons. The optimisation considers parameters
which could make it possible to validate this experimentally. With these control pulses it is in principle possible to double the number of available qubits without any overhead in hardware. The
additional qubits could be used in algorithms which need many short-living qubits such as syndrom qubits in error correction or by embedding effecitve higher connectivity in qubit networks.
An integrated tool-set for Control, Calibration and Characterization of quantum devices applied to superconducting qubits
Efforts to scale-up quantum computation have reached a point where the principal limiting factor is not the number of qubits, but the entangling gate infidelity. However, a highly detailedsystem
characterization required to understand the underlying errors is an arduous process and impractical with increasing chip size. Open-loop optimal control techniques allow for the improvement of gates
but are limited by the models they are based on. To rectify the situation, we provide a new integrated open-source tool-set for Control, Calibration and Characterization (C3), capable of open-loop
pulse optimization, model-free calibration, model fitting and refinement. We present a methodology to combine these tools to find a quantitatively accurate system model, high-fidelity gates and an
approximate error budget, all based on a high-performance, feature-rich simulator. We illustrate our methods using fixed-frequency superconducting qubits for which we learn model parameters to an
accuracy of <1% and derive a coherence limited cross-resonance (CR) gate that achieves 99.6% fidelity without need for calibration. [/expand]
Leakage reduction in fast superconducting qubit gates via optimal control
Reaching high speed, high fidelity qubit operations requires precise control over the shape of the underlying pulses. For weakly anharmonic systems, such as superconducting transmon
qubits, short gates lead to leakage to states outside of the computational subspace. Control pulses designed with open-loop optimal control may reduce such leakage. However, model inaccuracies can
severely limit the usability of such pulses. We implemented a closed-loop optimization that simultaneously adapts all control parameters based on measurements of a cost function built from Clifford
gates. By parameterizing pulses with a piecewise-constant representation that matches the capabilities of the control hardware we create a 4.16 ns single-qubit pulse with 99.76% fidelity and 0.044%
leakage. This is a seven-fold reduction of the leakage rate of the best DRAG pulse we have calibrated at such short durations on the same system.
Optimized cross-resonance gate for coupled transmon systems
The cross-resonant gate is an entangling gate for fixed frequency superconducting qubits introduced for untunable qubits. While being simple and extensible, it suffers from long duration
and limited fidelity. Using two different optimal control algorithms, we probe the quantum speed limit for a CNOT gate in this system. We show that the ability to approach this limit depends strongly
on the ansatz used to describe the optimal control pulse. A piecewise constant ansatz with a single carrier leads to an experimentally feasible pulse shape, shorter than the one currently used in
experiments, but that remains relatively far from the speed limit. On the other hand, an ansatz based on the two dominant frequencies involved in the optimal control problem allows to generate an
optimal solution more than twice as fast, in under 30ns. This comes close to the theoretical quantum speed limit, which we estimate at 15ns for typical circuit-QED parameters, which is more than an
order of magnitude faster than current experimental microwave-driven realizations, and more than twice as fast as tunable direct-coupling experimental realizations.
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Credit Card Misinformation
“I’m from Missouri. Show me.” is a state slogan and a wise expression. It analogous to accepting a scientist’s experimental conclusions at face value without checking the experimental data. My late
Father, Alex Cirotto wisely taught me not to buy into anything unless I understood the concept and could verify any calculations. I am forever indebted to him.
As well as Banks, Credit card companies can be masters of “smoke and mirrors”. If you don’t understand the mathematics it can cost you big time. Misinformation is everywhere! Misinformation is
everywhere even in print. Both Ellen Roseman in her article in the Toronto Star (“How to manage credit without credit managing you”, Aug 29 2010) and Patricia Levitt Reid in her National Post column
(“Pay off your debt, don’t just pay it down”, July 24th 2010) confuses novices regarding compound interest.
Both of these advisors are perpetuating the myth that compound interest is the culprit working against you in borrowing. FACT: If your payment is one penny in excess of the interest due, YOU ARE NOT
PAYING COMPOUND INTEREST. You are paying simple interest!
Compound interest is interest upon interest!
As you can see from the credit card example for an initial balance of $5000 at 29.9% per year, paying a minimum monthly payment of 3% of the outstanding balance takes a long long time to pay off a
loan. You will also note that the payment is ALWAYS greater than the interest, thus NO COMPOUNDING! It takes almost 82 years to reach a point where the payment is one dollar per month and the balance
is still $33.04.
Usually a credit card company will advise you that the monthly payment will be 3% of the current balance or $10 whichever is greater. The stated dollar value ($10) once reached, ends the agony sooner
as the credit card company has already made most of their simple interest.
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On plug-in estimation of long memory models
We consider the Gaussian ARFIMA(j, d, l,) model, with spectral density f [θ](λ), θ ∈ R ^p, λ ∈ (-ππ), d ∈ (0,1/2) and an unknown mean μ ∈ R. For this class of models, the n ^-1-normalized information
matrix of the full parameter vector, (μ,θ), is asymptotically degenerate. To estimate θ, Dahlhaus (1989, Annals of Statistics 17, 1749-1766) suggested using the maximizer of the plug-in
loglikelihood, L [n](θ, μ̃ [n]), where μ̃ [n] is any n ^(1-2d)/2-consistent estimator of μ. The resulting estimator is a plug-in maximum likelihood estimator (PMLE). This estimator is asymptotically
normal, efficient, and consistent, but in finite samples it has some serious drawbacks. Primarily, none of the Bartlett identities associated with L [n](θ,μ̃ [n]) are satisfied for fixed n. Cheung and
Diebold (1994, Journal of Econometrics 62, 301-316) conducted a Monte Carlo simulation study and reported that the bias of the PMLE is about 3-4 times the bias of the regular maximum likelihood
estimator (MLE). In this paper, we derive asymptotic expansions for the PMLE and show that its second-order bias is contaminated by an additional term, which does not exist in regular cases. This
term arises because of the failure of the first Bartlett identity to hold and seems to explain Cheung and Diebold's simulated results. We derive similar expansions for the Whittle MLE, which is
another estimator tacitly using the plug-in principle. An application to the ARFIMA(0, d, 0) shows that the additional bias terms are considerable.
Dive into the research topics of 'On plug-in estimation of long memory models'. Together they form a unique fingerprint.
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5. Show that the Signum Function f:R→R, given by
f(x)=⎩⎨⎧1, if... | Filo
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5. Show that the Signum Function , given by is neither one-one nor onto.
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Question Text 5. Show that the Signum Function , given by is neither one-one nor onto.
Updated On May 2, 2023
Topic Functions
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 112
Avg. Video Duration 4 min
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Affine Algebraic Geometry: Geometry Of Polynomial Rings
by Masayoshi Miyanishi (Author)
Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf
cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and
polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar–Moh–Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was
introduced by Iitaka and Kawamata. Readers will find the book covers vast basic material on an extremely rigorous level: It begins with an introduction to algebraic geometry which comprises almost
all results in commutative algebra and algebraic geometry. Arguments frequently used in affine algebraic geometry are elucidated by treating affine lines embedded in the affine plane and automorphism
theorem of the affine plane. There is also a detailed explanation on affine algebraic surfaces which resemble the affine plane in the ring-theoretic nature and for actions of algebraic groups. The
Jacobian conjecture for these surfaces is also considered by making use of the results and tools already presented in this book. The conjecture has been thought as one of the most unattackable
problems even in dimension two. Advanced results are collected in appendices of chapters so that readers can understand the main streams of arguments. There are abundant problems in the first three
chapters which come with hints and ideas for proof.
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What is superstring theory?
Superstring theory is a theoretical framework in physics that extends the ideas of string theory by incorporating supersymmetry, a theoretical symmetry between particles with different spin quantum
numbers. It represents a refined and more intricate version of string theory, aiming to address some of the challenges faced by its predecessor and offering a potential path toward a unified
understanding of the fundamental forces and particles in the universe.
The roots of superstring theory lie in the attempts to develop a consistent quantum theory of gravity. In the 1970s, physicists were grappling with the mathematical inconsistencies that arose when
trying to combine the principles of quantum mechanics with the framework of general relativity. String theory, which replaced point-like particles with one-dimensional strings, emerged as a promising
candidate for resolving these issues. However, the theory faced certain limitations, particularly related to the diverse spectrum of particles predicted by its various formulations.
The introduction of supersymmetry to string theory marked a significant advancement. Supersymmetry posits a symmetry between particles with different intrinsic angular momentum, or spin. For every
known particle with a certain spin, there exists a corresponding supersymmetric partner with a different spin. These supersymmetric particles, or superpartners, have not yet been observed in
experiments but are a crucial element in many extensions of the Standard Model of particle physics, including superstring theory.
In the context of superstring theory, the strings can vibrate in different modes, giving rise to a spectrum of particles. Supersymmetry introduces a symmetry between bosons (particles with integer
spin) and fermions (particles with half-integer spin). Each type of particle in the Standard Model has a supersymmetric partner, and the vibrational modes of superstrings correspond to these diverse
particles and their superpartners.
The incorporation of supersymmetry into string theory helps to address some of the challenges faced by earlier formulations. It aids in canceling out certain mathematical infinities that appeared in
quantum field theories, contributing to the consistency of the theory. Moreover, supersymmetry has the potential to provide a natural explanation for the hierarchy problem, which pertains to the
vastly different strengths of the fundamental forces in the universe.
Superstring theory encompasses several distinct versions, each characterized by specific mathematical structures and symmetries. The five consistent superstring theories are Type I, Type IIA, Type
IIB, heterotic SO(32), and heterotic E8×E8. These theories incorporate both supersymmetry and string-like entities, and they exhibit duality relationships, allowing them to be equivalent in certain
physical scenarios.
The concept of duality is a crucial aspect of superstring theory. Dualities imply that different physical theories are equivalent in specific regimes or under certain transformations. For example,
T-duality relates theories with different spacetime geometries, while S-duality relates theories with different coupling strengths. These dualities provide a deeper understanding of the underlying
unity in superstring theory and offer new perspectives on the relationships between seemingly distinct physical phenomena.
Theoretical physicists later discovered that these dualities were not limited to the five consistent superstring theories. Instead, they are part of a more comprehensive framework known as M-theory.
M-theory emerged as a unifying concept that encompasses and extends the various superstring theories. In M-theory, the strings of superstring theory are replaced by higher-dimensional objects called
membranes, and the theory incorporates 11 spacetime dimensions.
M-theory proposes that the different superstring theories are different aspects of a more fundamental theory that unifies them. It suggests that the apparent variety of string theories is akin to
different perspectives of a more comprehensive underlying reality. M-theory incorporates the diverse structures of the superstring theories and adds a new level of richness to the theoretical
The nature of spacetime in superstring theory is intricately linked to the vibrational modes of strings or membranes. The extra dimensions required by the theory are often compactified or curled up
at extremely small scales, making them effectively undetectable in our everyday experiences. The geometry and topology of these compactified dimensions play a crucial role in determining the
properties of particles and forces in our observable universe.
The search for experimental evidence supporting superstring theory has been challenging due to the high energies required to directly observe strings or superpartners. While no direct experimental
confirmation has been achieved, the potential predictions of superstring theory may leave observable signatures in certain high-energy phenomena. Particle accelerators, such as the Large Hadron
Collider (LHC), continue to explore energy scales where effects related to supersymmetry and extra dimensions could potentially be detected.
One of the exciting aspects of superstring theory is its potential to provide a unified description of all fundamental forces in the universe, including gravity. In contrast to the challenges faced
by attempts to quantize gravity within the framework of general relativity, superstring theory naturally incorporates gravity as one of the vibrational modes of the strings. This intrinsic inclusion
of gravity within the theory is a key feature that distinguishes it from other quantum field theories.
The holographic principle, inspired by ideas from superstring theory, adds another layer of conceptual richness. This principle suggests that the information within a region of spacetime can be fully
encoded on its boundary. It implies a deep connection between gravity and quantum mechanics and has been explored in the context of certain black hole solutions within superstring theory.
Despite its theoretical elegance and potential, superstring theory faces criticisms and challenges. The vast landscape of possible solutions and the absence of experimental verification have led some
physicists to question its status as a unique and testable theory. The search for experimental evidence remains a critical aspect of ongoing research in the field.
Moreover, the complexity of the mathematical structures involved in superstring theory poses significant challenges for researchers. The intricate nature of the theory’s equations and the multitude
of possible configurations make it a formidable intellectual endeavor to explore and understand. Theoretical physicists continue to grapple with the mathematical intricacies of the theory, seeking a
deeper and more comprehensive understanding of its principles.
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Sets and Maps
On the fundamental aspect all interval_sets are models of a concept Set. The Set concept of the Interval Template Library refers to the mathematical notion of a set.
Function Variant implemented as
empty set Set::Set()
subset relation bool Set::within(const Set& s1, const Set& s2)const
equality bool is_element_equal(const Set& s1, const Set& s2)
set union inplace Set& operator += (Set& s1, const Set& s2)
Set operator + (const Set& s1, const Set& s2)
set difference inplace Set& operator -= (Set& s1, const Set& s2)
Set operator - (const Set& s1, const Set& s2)
set intersection inplace Set& operator &= (Set& s1, const Set& s2)
Set operator & (const Set& s1, const Set& s2)
Equality on Sets is not implemented as operator ==, because operator == is used for the stronger lexicographical equality on segments, that takes the segmentation of elements into account.
Being models of concept Set, std::set and all interval_sets implement these operations and obey the associated laws on Sets. See e.g. an algebra of sets here.
An interval is considered to be a set of elements as well. With respect to the Set concept presented above interval implements the concept only partially. The reason for that is that addition and
subtraction can not be defined on intervals. Two intervals [1,2] and [4,5] are not addable to a single new interval. In other words intervals are incomplete w.r.t. union and difference. Interval_sets
can be defined as the completion of intervals for the union and difference operations.
When we claim that addition or subtraction can not be defined on intervals, we are not considering things like e.g. interval arithmetics, where these operations can be defined, but with a different
On the fundamental aspect icl::map and all interval_maps are models of a concept Map. Since a map is a set of pairs, we try to design the Map concept in accordance to the Set concept above.
Function Variant implemented as
empty map Map::Map()
subset relation bool within(const Map& s2, const Map& s2)const
equality bool is_element_equal(const Map& s1, const Map& s2)
map union inplace Map& operator += (Map& s1, const Map& s2)
Map operator + (const Map& s1, const Map& s2)
map difference inplace Map& operator -= (Map& s1, const Map& s2)
Map operator - (const Map& s1, const Map& s2)
map intersection inplace Map& operator &= (Map& s1, const Map& s2)
Map operator & (const Map& s1, const Map& s2)
As one can see, on the abstract kernel the signatures of the icl's Set and Map concepts are identical, except for the typename. While signatures are identical The sets of valid laws are different,
which will be described in more detail in the sections on the semantics of icl Sets and Maps. These semantic differences are mainly based on the implementation of the pivotal member functions add and
subtract for elements and intervals that again serve for implementing operator += and operator -=.
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5 Best Ways to Find the Cells Containing Maximum Value in a Matrix with Python
π ‘ Problem Formulation: When working with matrices in Python, often times, we’re interested in finding the location of the maximum value. For example, given a matrix, we would like to determine the
row and column (or indices) that contain the highest number. Here, we discuss five different methods to achieve this task and analyze their strengths and weaknesses.
Method 1: Brute Force Search
This method entails iterating through the entire matrix, keeping track of the maximum value found and its coordinates. It is straightforward and requires no additional libraries. Essentially, this is
a manual search operation where we compare each element with the current maximum.
Here’s an example:
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
max_val = float('-inf')
coords = (0, 0)
for i in range(len(matrix)):
for j in range(len(matrix[i])):
if matrix[i][j] > max_val:
max_val = matrix[i][j]
coords = (i, j)
print("Maximum value is at:", coords)
Output: Maximum value is at: (2, 2)
This snippet scans each element and updates the maximum value and its coordinates whenever it finds a new maximum. It’s an elementary method and works well for small matrices, but can be inefficient
for larger ones due to its O(n^2) complexity where n is the dimension of the matrix.
Method 2: Using NumPy library
When handling numerical data in Python, NumPy is the go-to library for efficient operations. One can use numpy.argmax() to find the index of the maximum value in a flattened version of the array, and
then convert this index to two-dimensional coordinates.
Here’s an example:
import numpy as np
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
index = np.argmax(matrix)
coords = np.unravel_index(index, matrix.shape)
print("Maximum value is at:", coords)
Output: Maximum value is at: (2, 2)
This code leverages NumPy’s array manipulation capabilities to find the maximum value’s index efficiently. The unravel_index function here converts the flat index to 2D coordinates. This method is
much faster than the brute force search, especially for large matrices, and exposes the power of vectorized operations.
Method 3: Using zip with enumerate
Python’s zip() function and enumerate() can be used to iterate over rows and columns simultaneously. This approach natively applies to lists of lists, providing a means to identify the maximum value
and its location in a Pythonic way.
Here’s an example:
matrix = [[1, 2, 3], [4, 5, 6], [7, 10, 9]]
max_value = max(max(row) for row in matrix)
coords_list = [(i, row.index(max_value)) for i, row in enumerate(matrix) if max_value in row]
print("Maximum value is at:", coords_list)
Output: Maximum value is at: [(2, 1)]
Here, the code first identifies the maximum value in the matrix, followed by finding its coordinates. This method can locate all occurrences of the maximum value and is also quite readable. However,
it may not be the most efficient since it requires multiple passes over the data.
Method 4: Using a custom function with max()
Combining Python’s max() function with a custom key function also allows finding the maximum value’s coordinates concisely. The approach leverages Python’s ability to find maximums based on custom
Here’s an example:
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
coords = max(((i, j) for i, row in enumerate(matrix) for j, val in enumerate(row)), key=lambda x: matrix[x[0]][x[1]])
print("Maximum value is at:", coords)
Output: Maximum value is at: (2, 2)
This one-liner uses a generator expression to iterate through matrix coordinates and the max() function with a key that looks up the matrix value at each coordinate. This method is efficient in terms
of written code but may be less efficient than NumPy for larger datasets due to Python’s inherent loop overhead.
Bonus One-Liner Method 5: Using NumPy argmax with a Twist
An alternative one-liner with NumPy uses the argmax() operation along with shape manipulation to find the maximum’s coordinates succinctly.
Here’s an example:
import numpy as np
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
coords = divmod(np.argmax(matrix), matrix.shape[1])
print("Maximum value is at:", coords)
Output: Maximum value is at: (2, 2)
This compact snippet leverages divmod() to calculate the row and column indices from the flattened index, which is derived from argmax(). It’s concise and takes full advantage of NumPy’s efficiency.
• Method 1: Brute Force Search. Simple and universal. Not efficient for large matrices.
• Method 2: Using NumPy library. Highly efficient and suitable for large matrices. Requires an external library.
• Method 3: Using zip with enumerate. Pythonic and able to identify multiple maxima. Inefficient due to multiple iterations.
• Method 4: Using a custom function with max(). Compact and understandable for Python users. May be slow for large data sets.
• Method 5: Bonus using NumPy argmax with a Twist. Extremely efficient and a one-liner. Relies on understanding of NumPy and divmod function.
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Upper bound for an exponential sum involving characters of a finite field ~ MathOverflow ~ TransWikia.com
I am going to assume that by an additive character you mean
an irreducible representation $$chi_alpha : mathbb{F}^n_q longrightarrow mathbb{C}$$, i.e. a group homomorphism from the additive group $$(mathbb{F}^n_q ,+)$$ to the multiplicative group $$
which we can prove must all take the form $$begin{equation}chi_alpha : beta mapsto expleft( {frac{2pi i leftlangle alpha ,beta rightrangle }{p }} right)end{equation}$$ where $$leftlangle alpha ,beta
rightrangle = sum_i alpha_i beta_i$$, see chapter 4 of Tao for a proof of some of these statements and see ch.2 of Serre or ch.2 of Fulton & Harris for a general (non-abelian) overview of the
representation theory perspective on characters. The point is the following
If we let $$begin{equation} f(x) = begin{cases} q psi_x(x) & text{if } x neq 0 \ 0 & text{if } x = 0 \ end{cases} end{equation}$$ then the sum you are considering is equal to the Fourier
transform of $$f$$i.e. $$begin{equation} hat{f}(alpha) = frac{1}{q} sum_{c in mathbb{F} _q } f(c) chi_alpha(c) = sum_{c in mathbb{F} _q^* } psi_c (c) chi_alpha(c) end{equation}$$ see definition
4.6 in Tao.
We apply the Hausdorff-Young inequality theorem 4.8 in Tao to get that $$begin{equation} left(sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{p'} right)^{frac{1}{p'}} leq left(sum_{alpha in
mathbb{F} _q } |f(alpha)|^pright)^{frac{1}{p}} = qleft( sum_{c in mathbb{F} _q^* } |psi_c (c) |^pright)^{frac{1}{p}} end{equation}$$ where the LHS is the $$l^{q}$$-norm, the RHS is the $$l^p$$-norm,
and $$p$$ satisfies the following $$p^{-1} +q^{-1} = 1 land 1 leq pleq 2$$. Plugging in $$p = 2$$ we get that
$$begin{equation} sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq qsum_{c in mathbb{F} _q^* } |psi_c (c) |^2 end{equation}$$ which is equivalent to saying that $$begin{equation} mathbb
{Var}[hat f] = frac{1}{q}sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq sum_{c in mathbb{F} _q^* } |psi_c (c) |^2leq q-1. end{equation}$$
Finally, if you can prove that at least $$n$$ many $$alpha$$ give a value $$| hat f(a)| geq sqrt b$$ then you get that $$begin{equation} nb +sup_{alpha in mathbb{F} _q }left| hat f(a)right|^2 leq
sum_{alpha in S}left| hat f(alpha)right|^{2} + sup_{alpha in mathbb{F} _q }left| hat f(a)right|^2 leq sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq q(q-1) end{equation}$$
which gives you that the maximum value is at most
$$begin{equation} sup_{alpha in mathbb{F} _q }left|sum_{c in mathbb{F} _q^* } psi_c (c) chi_alpha(c) right| = sup_{alpha in mathbb{F} _q }left| hat f(a)right| leq sqrt{q(q-1)-nb} end{equation}$$
Essentially we reduced the problem of finding an upper bound to that of finding a lower bound.
Answered by Pedro Juan Soto on December 14, 2020
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Maintaining a sorted array in O(1) | Rough Book
Maintaining a sorted array in O(1)
Yesterday, I came across an interesting question on StackOverflow. The question is as follows: assuming you have a sorted array, is it possible to increment locations by 1 (one at a time) and still
ensure that the array is sorted in O(1)? Assuming you are not allowed to use any other secondary data-structures, the answer is no. This becomes evident when you have repeated values. Assume a
degenerate case where you have an array of identical values; let's say [0, 0, 0, 0, 0]. If you increment the value in location 0, you now have [1, 0, 0, 0, 0]. For this array to be sorted, the value
1 must be moved to the end of the array (index 4). This will require four comparisons, which means that this will inevitably turn into an O(n) algorithm.
How can we do better?
The problematic case is with repeated values, since this is where we have the worst performance (O(n)). So we need to figure out a way to make performance better in this case (all other cases can be
handled in constant time). What is our problem exactly when we encounter a run of repeated values? The problem is that we need to go through all of the repeated values to figure out the last one,
since we will be swapping our incremented value with that last repeated-value. What if we could easily find out where a run of repeated values ends? If that was the case, we could simply look up that
information and swap our incremented value with the value in that location! It turns out that we can easily do this with a map (hash map or hash table). Lookups from a hash map are O(1) and in our
case we are unlikely to have collisions either, so now our O(n) operation now runs in constant time! What would this map contain? It would only contain repeated values that are mapped to their ending
index. So in our first example, where we had an array of zeroes, the map would be (0 => 4), because the run of zeroes ends at location 4. After incrementing the value at location 0, the map would be
(0 => 3) because our run of zeroes ends at location 3 now. If we incremented the first location again, the map would be (0 => 2, 1 => 4) because we have a run of zeroes ending at location 2 and a new
run of ones that end at location 4. As you can see, it's a pretty simple structure.
By keeping track of this information, we can now accomplish our goal of maintaining a sorted array in O(1) time! The algorithm has two parts: the first part does the incrementing and swapping (if
necessary), while the second part performs book-keeping to update our map. First let's concentrate on the increment-and-swap part. We have to increment a location, but do we always have to swap?
Let's look at the different cases:
1. Incremented value is the last element in the array - No swap necessary.
2. Incremented value is lesser than the next value - No swap necessary.
3. Incremented value is the same as the next value - No swap necessary.
4. Incremented value is greater than the next value - Swap necessary.
So it turns out that we actually only need to swap in one case: when the incremented value is greater than the next one. But we cannot simply swap the values of course, we have to check to see if we
have a run of values as well, which we can easily do by checking our map. So the cases we need to consider are:
1. If there is no entry for the next value in the map, simply swap the two values.
2. If there is an entry for the next value in the map, swap the incremented value with the value at the ending index.
At this point our array is properly sorted. But there is still one more thing we need to do; we need to make sure that we perform book-keeping on our map to ensure that it is in a state that
accurately reflects all the runs we have in our array (if any). There are a few things we need to do:
1. Decrement the ending-index for a run of values, if we swapped the incremented value with this value.
2. Remove the entry for a value from the map, if there is no longer a run of values for this value.
3. Add an entry into the map if we created a new run by incrementing a value.
That's basically it! So let's see what this actually looks like in code form. I used JavaScript because it is pretty easy to prototype algorithms in it. The code is pretty well-commented and so you
can see where I have handled the different cases:
function incrementAndMaintainSortedArray(array, index) {
var oldValue = array[index];
var newValue = ++array[index];
if(index === (array.length - 1)) {
//Incremented element is the last element.
//We don't need to swap, but we need to see if we modified a run (if one exists)
if(endingIndices[oldValue]) {
} else if(index >= 0) {
//Incremented element is not the last element; it is in the middle of
//the array, possibly even the first element
var nextIndexValue = array[index + 1];
if(newValue === nextIndexValue) {
//If the new value is the same as the next value, we don't need to swap anything. But
//we are doing some book-keeping later with the endingIndices map. That code requires
//the ending index (i.e., where we moved the incremented value to). Since we didn't
//move it anywhere, the endingIndex is simply the index of the incremented element.
endingIndex = index;
} else if(newValue > nextIndexValue) {
//If the new value is greater than the next value, we will have to swap it
var swapIndex = -1;
if(!endingIndices[nextIndexValue]) {
//If the next value doesn't have a run, then location we have to swap with
//is just the next index
swapIndex = index + 1;
} else {
//If the next value has a run, we get the swap index from the map
swapIndex = endingIndices[nextIndexValue];
array[index] = nextIndexValue;
array[swapIndex] = newValue;
endingIndex = swapIndex;
} else {
//If the next value is already greater, there is nothing we need to swap but we do
//need to do some book-keeping with the endingIndices map later, because it is
//possible that we modified a run (the value might be the same as the value that
//came before it). Since we don't have anything to swap, the endingIndex is
//effectively the index that we are incrementing.
endingIndex = index;
//Moving the new value to its new position may have created a new run, so we need to
//check for that. This will only happen if the new position is not at the end of
//the array, and the new value does not have an entry in the map, and the value
//at the position after the new position is the same as the new value
if(endingIndex < (array.length - 1) &&
!endingIndices[newValue] &&
array[endingIndex + 1] === newValue) {
endingIndices[newValue] = endingIndex + 1;
//We also need to check to see if the old value had an entry in the
//map because now that run has been shortened by one.
if(endingIndices[oldValue]) {
var newEndingIndex = --endingIndices[oldValue];
if(newEndingIndex === 0 ||
(newEndingIndex > 0 && array[newEndingIndex - 1] !== oldValue)) {
//In this case we check to see if the old value only has one entry, in
//which case there is no run of values and so we will need to remove
//its entry from the map. This happens when the new ending-index for this
//value is the first location (0) or if the location before the new
//ending-index doesn't contain the old value.
delete endingIndices[oldValue];
You can check out a runnable version of this code at jsfiddle.net. The code includes a test-harness that increments random locations in an array and ensures that the array is sorted after each
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# Problem 1: GDA [50 points] Perform _Gaussian Discriminant Analysis_ (GDA) on the spambase data with k-fold cross validation. For each fold, use 1 fold for testing and k-1 folds for training. During
the analysis, you should test both of the following covariance assumptions: - Single covariance model (all the training data) - Class conditional covariance model (estimated separately for each
class) In the context of GDA, the former is called _Linear Discriminant Analysis_ while the latter is called _Quadratic Discriminant Analysis_. Based on the training and testing performance, does it
appear the data are normally distributed? # Problem 2: Naive Bayes [50 points] Create a set of _Naive Bayes classifiers_ for detecting e-mail spam and test the classifiers on the [Spambase dataset]
(https://archive.ics.uci.edu/ml/datasets/spambase) via 10-fold cross validation. ## 2.1: Build the classifier(s) Create three distinct Naive Bayes classifiers by varying the likelihood distribution.
In particular, try: - Modeling the likelihood using a Bernoulli distribution - Modeling the likelihood using a Gaussian distribution - Modeling the likelihood non-parameterically For real-valued
features, a Bernoulli likelihood may be obtained by thresholding against a scalar-valued statistic, such as the sample mean $\mu$. Similarly, a Gaussian likelihood function can be obtained by
estimating the [class conditional] expected value $\mu$ and variance $\sigma$. For the non-parameteric setting, model the feature distribution either with a kernel density estimate (KDE) or a
histogram. _Bernoulli example:_ Consider some threshold $\mu \in \mathbb{R}$. To compute the conditional probability of a feature $f_i$, compute the fraction by which $f_i$ is above or below $\mu$
over all the data within its class. In other words, for feature $f_i$ with expected value $\mu_i$, estimate: $$ P(f_i \leq \mu_i \mid \text{spam} ) \text{ and } P(f_i > \mu_i \mid \text{spam} )$$ $$
P(f_i \leq \mu_i \mid \text{non-spam} ) \text{ and } P(f_i > \mu_i \mid \text{non-spam} )$$ and use these estimated values in your Naive Bayes predictor. For all the models mentioned, you may want to
consider using some kind of [additive smoothing](https://en.wikipedia.org/wiki/Additive_smoothing) technique as a regularizer. ## 2.2: Evaluate your results Evaluate the performance of the
classifiers using the following three performance summaries. _Error tables:_ Create a table with one row per fold showing the false positive, false negative, and overall error rates of the
classifiers. Add one final row per table yielding the average error rates across all folds. _ROC Curves:_ Graph the [Receiver Operating Characteristic](https://en.wikipedia.org/wiki/
Receiver_operating_characteristic) (ROC) curve for each of your classifiers on _Fold 1_ and calculate their area-under-the-curve (AUC) statistics. You should draw all three curves on the same plot so
that you can compare them. For this problem, the _false positive rate_ is the fraction of non-spam testing examples that are misclassified as spam, while the _false negative rate_ is the fraction of
spam testing examples that are misclassified as non-spam. --- __Context:__ In many situations, false positive (Type I) and false negative (Type II) errors incur different costs. In spam filtering,
for example, a false positive is a legitimate e-mail that is misclassified as spam (and perhaps automatically redirected to a "spam folder" or, worse, auto-deleted) while a _false negative_ is a spam
message that is misclassified as legitimate (and sent to one's inbox). When using Naive Bayes, one can easily make such trade-offs. For example, in the usual Bayes formulation, with $\mathbf{x}$ the
data vector and $y$ the class variable, one would predict "spam" if: $$ P(y = \text{spam} | \mathbf{x}) > P(y = \text{non-spam} | \mathbf{x}) $$ or equivalently, in a log-odds formulation, $$ \ln(P(y
= \text{spam} | \mathbf{x}) / P(y = \text{non-spam} | \mathbf{x})) > 0 $$ However, note that one could choose to classify an e-mail as spam for any threshold $\tau$, i.e.: $$ \ln(P(y = \text{spam} |
\mathbf{x}) / P(y = \text{non-spam} | \mathbf{x})) > \tau $$ Larger values of $\tau$ reduce the number of spam classifications, reducing false positives at the expense of more false negatives.
Negative values of $\tau$ have the converse effect. Most users are willing to accept some false negative examples so long as very few legitimate e-mails are misclassified. Given your classifiers and
their ROC curves, what value of $\tau$ would you choose to deploy in a real email spam filter? # Problem 3: EM algorithm [20 points] Use the pdf of the expectation maximization (EM) code demo'ed in
class, annotate the source code with explanations of what the code is doing. Your annotations should be text comments aligned on the right margin of the source code. # Problem 4: EM on generated data
[50 points] Run the EM algorithm with random initial parameter values and see if you recover the true parameters of the underlying generative model. - The file [2gaussian.txt](https://www.ccs.neu.edu
/home/vip/teach/MLcourse/3_generative_models/HW3/2gaussian.txt) was generated from a mixture of two 2D Gaussians with the parameters below. $$ \mu_1 = [3,3], \quad \Sigma_1 = \begin{bmatrix} 1 & 0 \\
0 & 3 \end{bmatrix}, \quad n_1=2000 $$ $$ \mu_1 = [7,4], \quad \Sigma_2 = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end{bmatrix}, \quad n_1=4000 $$ - The file 3gaussian.txt, generated using a mixture of
three Gaussians. $$ \mu_1 = [3,3], \quad \Sigma_1 = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}, \quad n_1=2000 $$ $$ \mu_2 = [7,4], \quad \Sigma_2 = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end
{bmatrix}, \quad n_2=3000 $$ $$ \mu_3 = [5,7], \quad \Sigma_3 = \begin{bmatrix} 1 & 0.2 \\ 0.2 & 1 \end{bmatrix}, \quad n_3=5000 $$ Verify your findings against the true parameters via your choice of
summary (e.g. absolute parameters differences, contour plots, etc.) # PROBLEM 5 - [30p GR-ONLY] Cheng's note summarizing E an M steps for this problem. A. Generate mixture data (coin flips). Pick a
p,r,pi as in the EM example discussed in class (or in notes). Say p=.75, r=.4, pi=.8, but you should try this for several sets of values. Generate the outcome of the coin experiment by first picking
a coin (pi probability for first coin, 1-pi probability for the second coin), then flip that coin K times (use K=10) with probability of head (p if first coin is picked, r if the second coin is
picked) and finally write down a 1 if the head is seen, 0 for the tail. Repeat this M=1000 times or more; so your outcome is a stream of M sequences of K flips : (1001110001; 0001110001; 1010100101
etc) B. Infer parameters from data. Now using the stream of 1 and 0 observed, recover p,r,pi using the EM algorithm; K is known in advance. Report in a table the parameter values found by comparison
with the ones used to generate data; try several sets of (p,r,pi). Here are some useful notes, and other readings (1 , 2 , 3 , 4, 5, ) for the coin mixture. C. Repeat A) and B) with T coins instead
of two. You will need more mixture parameters. # PROBLEM 6 Naive Bayes with Gaussian Mixtures [Extra Credit] Rerun the Naive Bayes classifier on Spam Data. For each feature (1-dim) use a mixture of K
Gaussians as your model; run the EM to estimate the 3*K parameters for each feature : mean1, var1, w1; mean2,var2, w2;... meanK,varK,wK; constrained by w1+w2 +...+wK=1. (You would need a separate
mixture for positive and negative data, for each feature). We observed best results for K=9. Testing: for each testing point, apply the Naive Bayes classifier as before: take the log-odd of product
of probabilities over features mixtures (and the prior), separately for positive side and negative side; use the overall probability ratio as an output score, and compute the AUC for testing set. Do
this for a 10-fold cross validation setup. Is the overall 10-fold average AUC better than before, when each feature model was a single Gaussian? # PROBLEM 7 [Extra Credit] a) Somebody tosses a fair
coin and if the result is heads, you get nothing, otherwise you get \$5. How much would you be pay to play this game? What if the win is \$500 instead of \$5? b) Suppose you play instead the
following game: - At the beginning of each game you pay an entry fee of \$100. A coin is tossed until a head appears, counting $n =$ the number of tosses it took to see the first head. - Your reward
is $2n$ (that is: if a head appears first at the 4th toss, you get \$16). Would you be willing to play this game (why)? c) Lets assume you answered "yes" at part b (if you did not, you need to fix
your math on expected values). What is the probability that you make a profit in one game? How about in two games? d)__(difficult)__ After about how many games (estimate) the probability of making a
profit overall is bigger than 50% ? # PROBLEM 8 [Extra Credit] DHS CH2, Pb 43 # PROBLEM 9 part 2 [Extra Credit] a) DHS CH2, Pb 45 # PROBLEM 10 [Extra Credit] b) DHS CH2, Pb 44 # PROBLEM 11 [EXTRA
CREDIT, requires independent study of Hidden Markov Models, DHS ch 3.10] DHS Pb 3.50 (page 154-155) The standard method for calculating the probability of a sequence in a given HMM is to use the
forward probabilities αi(t). # References 0. Good references on GDA: https://www.eecs189.org/static/notes/n18.pdf 1. Naive Bayes overview using different notation than most: http://
www.cs.columbia.edu/~mcollins/em.pdf 2. Short & limited but well-written Naive Bayes implementation in pure NumPy: https://www.python-engineer.com/courses/mlfromscratch/05_naivebayes/
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Calculating Hydration from Asymmetry Estimates
Should the user supply an axial ratio (as major-axis/minor-axis), the corresponding values of the degree of hydration for both the prolate and oblate ellipsoid models will be calculated by Sednterp.
To do this, the frictional coefficient ratios (prolate= and oblate=) corresponding to the user-supplied a/b may be calculated using the formulas of Perrin (Ref. 18):
[note that both of these equations are incorrect in Ref. 59]
Equation 27.
Equation 28.
These values then are used to calculate the corresponding degree of hydration:
Equation 29:
Equation 30:
where δ1,p is the predicted degree of hydration for the prolate model, δ1,o is that for the oblate model and vbar is the protein's partial specific volume. As before, f0 can be calculated using
either equation 21 or 22 and Vp can replace vbar.
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What does 2nd grade math look like?
What does 2nd grade math look like?
Second graders become experts in addition and subtraction, being able to quickly and accurately add and subtract one- and two-digit numbers with sums up to 100. They’re also expected to memorize all
the sums of adding two one-digit numbers. Students learn about odd and even numbers by pairing items or counting by twos.
How do I write a letter of intent for homeschooling?
Information to Include in a Letter of Intent Child’s address and address of homeschool if different. Child’s birth date. The grade the child would be entering if they were in school. A simple
statement saying that the child will be homeschooled for the following school year and who will be giving the instruction.
How long is a homeschool day?
We recommend an average of 45 minutes for each subject each day, but that might mean that on Mondays you do 30 mins of math and on Tuesdays you spend an hour. Be flexible and allow your child’s
pacing and rhythms to inform the lesson times.
What reading level is Magic Tree House?
Shop by Program:
Reading Level Interest Level
A Perfect Time For Pandas Series: Magic Tree House Merlin Missions (Book: 20) Osborne, Mary Pope Fiction Paperback M 2-5
Polar Bears Past Bedtime Series: Magic Tree House (Book: 12) Osborne, Mary Pope Fiction Paperback M 2-5
How do I file a notice of intent to homeschool in Florida?
Send a written notice of intent to the school district superintendent….The notice must be filed within 30 days of beginning the home education program and must include the following information:
1. Name of the home education student(s)
2. Birthdate(s)
3. Address.
4. Parent’s signature.
What maths should a Year 1 know?
In Year 1, children will need to count forwards and backwards up to 100. They will need to know their addition and subtraction facts to 20.
How do I determine my child’s reading level?
Usually, your child’s teacher will determine their reading level and then choose books that have a matching score. The Lexile score, or measure, describes your child’s reading ability and matches
them with books and other reading materials. This measure ranges anywhere from 0L to 2000L.
Do you have to show proof of homeschooling?
Notification of Homeschooling Many states require parents to notify local school districts that their children will be homeschooled, but 11 states don’t require parents to alert anyone. Most also
don’t require tests or portfolio reviews as proof of educational progress.
How well should a second grader read?
Weak Fluency. In 2nd grade reading, your child should be reading 50 to 60 words a minute at the beginning of the school year and 90 words per minute by the end of the year. If it’s below the speed
levels noted above, then fluency is a problem. Make sure she has lots of experiences reading simple books.
How can I teach math without a curriculum?
1. #1 BAKE! Honestly, this is probably the one I am the worst at.
2. #2 LET THEM FIGURE IT OUT. Do your kids often come to you with a question that is obviously math related?
3. #3 PLAY GAMES.
4. #4 READ MATH PICTURE BOOKS.
5. #5 FIGURE OUT AGE APPROPRIATE SKILLS.
How long should a 2nd grader read each day?
20 minutes
What math should 2nd graders know?
Some of the key math concepts a second grader should know include: Read and write numerals to 100 and to count objects to 100 or more. Addition and subtraction of two-digit numbers without
regrouping, up to 100, using models and algorithms. Explore number patterns on a hundred chart and with a calculator.
Can I get paid for homeschooling my child?
Homeschooling your child is a private choice and is not employment. Therefore, parents do not get paid to homeschool their children. However, in some states families may receive a tax credit,
deduction, or even a stipend if homeschooling under an umbrella school (like a charter school).
How do I notify School of Homeschooling?
Notification Requirements. If your child is enrolled in public school, you must send the school a Notice of Intent to Homeschool,and a local school committee must give their approval for you to
legally homeschool (you should receive letter of approval by mail).
How much should I pay someone to homeschool my child?
How much should I charge to homeschool someone else’s child? That depends on a number of factors, like your educational degree level, your teaching experience, your location, and the subject area(s)
involved. You can charge as little as $20 or $30 per hour at a minimum to as high as $85.
Do I have to pay school taxes if I homeschool?
Homeschoolers Who Own Their Homes Still Have to Pay School Taxes. A substantial portion of funding for public school systems comes from local property taxes charged to homeowners.
How can I homeschool without a curriculum?
Families can go about learning without curriculum in a few different ways: Unschooling (no curriculum) lessons are day-to-day and based in real time as skills & concepts cross your path. They are
specific to your child and often child-led. DIY- make your own curriculum, this can vary widely.
What level should a Year 2 be reading at?
The top year twos in the school ( dd benchmarks) are on level 12. The majority are around 9/ 10 though. Ds (yr 2) is a free reader, so no longer doing banded readers.
What is an addend in 2nd grade math?
In math, an addend can be defined as the numbers or terms added together to form the sum. Here, the numbers 7 and 8 are addends. Here’s another example, in which the numbers 7, 4 and 9 are addends,
and 20 is the sum.
What age is level 4 reading?
Oxford Reading Tree
Stage 1 3.5 to 4.5 years
Stage 3 5 to 5.5 years
Stage 4 5 to 5.5 years
Stage 5 5.5 to 6 years
Stage 6 6 to 6.5 years
How do you successfully homeschool?
8 Steps to Homeschool Success
1. Research Your Homeschool Options.
2. Investigate Your State’s Homeschooling Requirements.
3. Join a Local Homeschooling Group.
4. Decide on Homeschool Curriculum.
5. Create Your Homeschooling Space.
6. Set Specific Homeschooling Goals.
7. Define a Homeschooling Schedule.
8. Watch Out for Common Homeschooling Pitfalls.
What kind of math is taught in 1st grade?
First-graders learn mathematics on many fronts, including computation, numbers and number sense, measurement, patterns, shapes, money, and telling time.
Do you get money back for homeschooling?
While it’s your prerequisite to homeschool your child, parents don’t get paid to teach their children from home. However, some states’ families will receive credit taxes or even deductions if you
homeschool under an umbrella school like a charter school.
What every 1st grader should know?
By the end of 1st grade, kids should be able to:
• Work independently for short periods of time.
• Have a conversation about what a situation is like from another person’s point of view.
• Distinguish left from right.
• Attempt to write and spell new words phonetically.
• Read and write common words such as where and every.
What are the qualifications to homeschool your child?
Homeschooling in California Requirements Children ages 6 and up must be enrolled in a legal school. Home-based private schools must file a Private School Affidavit to begin homeschooling. Parents who
file the private school affidavit must provide all curricular, instructional, and other materials.
How can I make homeschool math fun?
Here are my favorite methods to make homeschool math fun and easier for all.
1. Teach to Learning Styles.
2. Make Homeschool Math a Game.
3. Show the Real World Application.
4. Make Those Devices Work for You.
5. Discover, Don’t Drill Homeschool Math.
How do you homeschool math?
Not sure how to even begin, well here are 12 ways to homeschool math without a curriculum or workbook.
1. Talk About Math in Your Homeschool.
2. Measure, Weigh or Build Something.
3. Find Fun Online Math Games.
4. Pick Up Some Living Math Books.
5. Cook Up Some Homeschool Math.
6. Go Shopping (Preferably a Sale)
What level should a 6 year old be reading?
What is a 6 year old reading level in early kindergarten? A 6 year old reading level is broad. However, in general, at the age of 6, most kids are starting to string letter sounds together to read
short vowel words.
What curriculum do you use for homeschool?
Overview. There are roughly seven main approaches to homeschooling [2]: Classical, Charlotte Mason, Montessori, Unschooling, School-at-Home, Unit studies, and Eclectic education methods. Each of
these is introduced hereafter with considerations of the benefits and drawbacks.
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if 1−x2+1−y2=a(x−y), show that dxdy=1−x21−y2 If If 1−x4+... | Filo
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Question Text if , show that If If , prove that, If , show that,
Updated On Feb 20, 2023
Topic Application of Derivatives
Subject Mathematics
Class Class 11
Answer Type Video solution: 2
Upvotes 177
Avg. Video Duration 5 min
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Department of Mathematics - Graduate
The Department of Mathematics offers graduate study leading to the Doctor of Philosophy (Ph.D), Master of Science in Mathematics (M.Sc.) and Master of Science in Mathematical Sciences (M.Sc.). The
department has strength in algebra, analysis, computational mathematics, differential equations, differential geometry, discrete mathematics, dynamical systems and control.
To be admitted to either of the M.Sc. programs, a student should have a Bachelor's degree in mathematics, or a closely related field, with a minimum major grade point average (MGPA) of 3.00 out of
scale 4.00.
The Ph.D program in mathematics is intended for students with superior mathematical ability and emphasizes the development of creative scholarship and breadth and depth in background knowledge. To
apply for admission to the Ph.D program, a student must normally have successfully completed, with a GPA not less than 3.00 out of scale 4.00, a Master's degree in mathematics equivalent to the one
offered by the department of Mathematics at Kuwait University. Students with a Master of Science degree in a closely related field are also encouraged to apply provided they can complete the
deficiency in the required mathematics courses at the undergraduate and graduate levels within one semester. Deficiency course can not be used to fulfill the degree requirements. Only full-time
students are admitted to the program. Current research interests of the faculty include:
Differential Equations, Numerical Analysis, Integral Transforms, Fractional Calculus, Algebra (ring theory); Analysis (Functional Analysis, Operator Theory; Approximation Theory and Fourier
Analysis), Geometry (General Topology, Differential Geometry, and Algebraic Topology), Combinatorics, Dynamical Systems and Control.
For any enquiries, kindly contact:
Prof. Nejib Smaoui
Graduate Program Director
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Perfect Square Trinomial Calculator
Perfect Square Trinomial Calculator
Perfect Square Trinomial Calculator
What is a Perfect Square Trinomial?
A perfect square trinomial is a special type of quadratic expression that results from squaring a binomial. In algebra, these trinomials have the general form of ax^2 + bx + c, where each of the
terms follows specific patterns that make factoring them straightforward. Essentially, it derives from the product of two identical binomials.
Applications of Perfect Square Trinomials
Perfect square trinomials find application in various fields, especially in solving quadratic equations, simplifying algebraic expressions, and analyzing functions. For instance, in geometry,
understanding perfect square trinomials can help in finding the length of a side of a square given its area. In physics, these are useful for solving problems related to projectile motion and
optimizing functions for maximum or minimum values. Additionally, they serve as foundational knowledge for more advanced topics in calculus and algebra.
How the Perfect Square Trinomial Calculator is Beneficial
This calculator is designed to assist students, teachers, and professionals in quickly determining whether a given quadratic expression is a perfect square trinomial and, if so, in finding its
factored form effortlessly. Instead of manually checking each expression and undergoing lengthy calculations, users can enter the coefficients and obtain immediate results, which enhances accuracy
and saves time. This tool is particularly beneficial during study sessions, homework assignments, or while preparing for exams, providing instant feedback and promoting a better understanding of
algebraic concepts.
Deriving the Answer
To check if a trinomial is a perfect square, the calculator evaluates the coefficients of the terms. It takes the square roots of the first and last terms, then checks if the middle term equals twice
the product of these square roots. For instance, if you have an expression such as 4x^2 + 4x + 1, the calculator verifies that the middle term (4x) is twice the product of the square roots of the
first term (4x^2 = (2x)^2) and the last term (1 = 1^2). If the criteria are met, it confirms the expression as a perfect square trinomial and provides the factored form in the binomial squared
Interesting Insights
Perfect square trinomials are not just confined to theoretical mathematics but have practical applications across disciplines. For instance, in computer graphics, these concepts are employed in
algorithms that involve rendering curves and parabolas. In economics, perfect square trinomials help in determining cost and revenue functions that feature quadratic relationships. The simplicity and
elegance of these trinomials make them an essential topic for anyone looking to gain a robust foundation in mathematics and its real-world applications.
What is a perfect square trinomial?
A perfect square trinomial is a quadratic expression that results from squaring a binomial. It has the standard form of ax^2 + bx + c and follows specific patterns which make it easy to factor.
How does the calculator determine if a trinomial is a perfect square?
The calculator evaluates the coefficients of the quadratic expression. It calculates the square roots of the first and last terms and checks if the middle term equals twice the product of these
square roots.
Can the calculator handle trinomials with negative coefficients?
Yes, the calculator can handle trinomials with both positive and negative coefficients. It applies the same rules for determining if the trinomial is a perfect square.
Is there any limitation on the values that can be entered into the calculator?
The calculator accepts a wide range of integer and decimal values, provided they are real numbers. Users should avoid entering non-numeric characters to ensure accurate results.
Can this calculator help simplify non-perfect square trinomials?
While the primary function of the calculator is to identify and factor perfect square trinomials, it provides an indication if the input trinomial is not a perfect square. This can help users
understand whether further simplification is needed through traditional factoring methods.
How is this tool useful for students and teachers?
This calculator aids in quickly identifying and factoring perfect square trinomials, saving time and reducing manual calculation errors. It is particularly useful for homework, study sessions, and
exam preparation.
Does the calculator provide steps for factoring the perfect square trinomial?
The calculator gives the factored form of the perfect square trinomial but does not provide the intermediate steps. Users can refer to the provided educational material to understand the factoring
process better step-by-step.
Are there any practical applications of perfect square trinomials?
Yes, perfect square trinomials have practical applications in fields like geometry, physics, economics, and computer graphics. They aid in solving equations, simplifying expressions, and optimizing
What happens if the input trinomial is not a perfect square?
If the trinomial is not a perfect square, the calculator will indicate that the expression does not meet the criteria for a perfect square trinomial. Users can then consider other factoring methods
or simplifications.
Is this tool suitable for advanced mathematical studies?
Absolutely, understanding perfect square trinomials lays a strong foundation for more advanced topics in algebra and calculus. This knowledge is essential for a robust understanding of mathematical
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Find the probability that both sellers are free
Problem description
There are two sellers working in a small store.
At a random point in time, each of them can be engaged in customer service with a probability of 0.3.
Moreover, they can be occupied simultaneously with a probability of 0.1.
Find the probability that at a random moment in time
a) both sellers are free;
b) one of them is free, and the other is busy.
Problem solution
Find the probability that at a random moment both sellers are free
Let define event A as seller #1 is busy and event B as seller #2 is busy.
Then P(A∪B) - the probability that at least one of the sellers is busy.
According to the probability addition formula is:
P(A∪B) = P(A) + P(B) - P(A∩B) = 0.3 + 0.3 - 0.1 = 0.5
Then the probability that both sellers are free is:
1 - P(A∪B) = 1 - 0.5 = 0.5.
Answer: probability that both sellers are free is 0.5 (b)
Find the probability that one seller is busy, and the other is free
In this situation, there are four possible options with sellers:
1) seller #1 is free and seller #2 is busy;
2) seller #2 is free and seller #1 is busy;
3) both sellers are free (probability of this event = 0.5);
4) both sellers are busy (probability of this event = 0.1).
Therefore, the probability of the first and second option is equal to: 1 - 0.5 - 0.1 = 0.4
Answer: probability that one seller is free, and the other is busy equals 0.4
Next probability problem solution:
Find the probability that two purchased batteries will be working.
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Legacy Factors · Caesar.jl
Previously we looked at adding a prior. This section demonstrates the first of two <:AbstractRelative factor types. These are factors that introduce only relative information between variables in the
factor graph.
This example is on <:IIF.AbstractRelativeRoots. First, lets create the factor as before
struct MyFactor{T <: SamplableBelief} <: IIF.AbstractRelativeRoots
getSample(cfo::CalcFactor{<:MyFactor}, N::Int=1) = (reshape(rand(cfo.factor.Z,N) ,1,N), )
function (cfo::CalcFactor{<:MyFactor})( measurement_z,
x2 )
res = measurement_z - (x2[1] - x1[1])
return res
The selection of <:IIF.AbstractRelativeRoots, akin to earlier <:AbstractPrior, instructs IIF to find the roots of the provided residual function. That is the one dimensional residual function, res[1]
= measurement - prediction, is used during inference to approximate the convolution of conditional beliefs from the approximate beliefs of the connected variables in the factor graph.
Important aspects to note, <:IIF.AbstractRelativeRoots requires all elements length(res) (the factor measurement dimension) to have a feasible zero crossing solution. A two dimensional system will
solve for variables where both res[1]==0 and res[2]==0.
As of IncrementalInference v0.21, CalcResidual no longer takes a residual as input parameter and should return residual, see IIF#467.
Measurements and variables passed in to the factor residual function do not have the same type as when constructing the factor graph. It is recommended to leave these incoming types unrestricted. If
you must define the types, these either are (or will be) of element type relating to the manifold on which the measurement or variable beliefs reside. Probably a vector or manifolds type. Usage can
be very case specific, and hence better to let Julia type-inference automation do the hard work for you. The
The second type is <:IIF.AbstractRelativeMinimize which simply minimizes the residual vector of the user factor. This type is useful for partial constraint situations where the residual function is
not gauranteed to have zero crossings in all dimensions and the problem is converted into a minimization problem instead:
struct OtherFactor{T <: SamplableBelief} <: IIF.AbstractRelativeMinimize
Z::T # assuming something 2 dimensional
userdata::String # or whatever is necessary
# just illustrating some arbitraty second value in tuple of different size
getSample(cfo::CalcFactor{<:OtherFactor}, N::Int=1) = (rand(cfo.factor.z,N), rand())
function (cfo::CalcFactor{<:OtherFactor})(res::AbstractVector{<:Real},
x2 )
# @assert length(z) == 2
# not doing anything with `second_val` but illustrating
# not doing anything with `cfo.factor.userdata` either
# the broadcast operators with automatically vectorize
res = z .- (x1[1:2] .- x1[1:2])
return res
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Overall length vs Clubhead Speed
Wondering what effect cutting down a driver will have on clubhead speed.. and anything else for that matter.. thnx
IF, and it's a very big if, you can keep your hands moving at the same speed for any driver length, the clubhead speed will change about 6% for every 1% if increase or decrease since it's moving
in a circle.
The reason it's a big if is that most people can't get their hands moving as fast with a longer club so the net gain in clubhead speed is smaller, if any. And there are also cases where going
shorter actually results in higher clubhead speed.
Total weight also plays a big role.
I am thinking of playing my driver at 44" Just wondering what all will be affected by the shortening of the shaft. I am correct in assuming that since the 1 inch will come off the butt end it
will not affect the flex?
Maybe slightly(flex). The sw might have to be readjusted. You might end up hitting it farther since the odds of you hitting the centre of the face will be greater.
Do you have the source for this ratio? It does not make sense to me.
If we get 2.5 yards (app) per mile per hour of club head speed, and I swing a 45" driver, 100 mph, the ball flies 250 yards. Then, if I shorten the club to 43.2" my swing speed will be 78 mph and
the ball will go 195 yards, all other things being equal. This is based on the 6% loss of club head speed per 1% decrease in length.
BUT, if I can swing a 43" driver 100 mph, it goes 250 yards. By lengthening the club to 45.2", my swing speed goes to 134 mph and the distance is 335 yards. THERE IS NO WAY THAT THIS HAPPENS. To
make this ratio even more absurd, extend my driver to the conforming limit of 48" and I will swing the club at 190 mph, flying 475 yards. Wouldn't this be nice?
By shortening the driver 1", the butt flex will be stiffer, but the tip flex will not change. If you (SC) have a rapid transition move at the start of the downswing, the stiffer butt won't hurt.
If you have a smooth transition, the shaft will definitely feel stiffer. There should be no change in the ball's trajectory because the trajectory is more greatly influenced by where the wrist
cock axis releases.
By shortening 1", the swingweight goes down, the MOI goes down, unless you add some lead tape to the back of the head or a tip weight, if the club is to be reassembled.
Since the club is now 1% longer, the radius of the circle that the clubhead travels is 6% (2*pi*1%) longer.
However, the starting assumption was that the angular velocity of your hands stayed the same, which means the angular velocity of the clubhead stays the same, which means its linear velocity (AKA
clubhead speed) increases.
It's very unlikely that you could actually swing two clubs at different lengths at that same angular velocity without also changing the weight.
If the club is shortened by 1 inch and the swingweight is not adjusted the butt frequency of the club will go up by 7 cpms. If the club is shortened by 1 inch and the swingweight is readjusted
the change in frequency will be about 1 cpm. The swingweight can also be reajusted with the use of rat glue. In this particular case 12 grams would have to be added to restore the original sw.
Since the club is now 1% longer, the radius of the circle that the clubhead travels is 6% (2*pi*1%) longer.
However, the starting assumption was that the angular velocity of your hands stayed the same, which means the angular velocity of the clubhead stays the same, which means its linear velocity (AKA
clubhead speed) increases.
It's very unlikely that you could actually swing two clubs at different lengths at that same angular velocity without also changing the weight.
WOW.. lol. I thought I was back in highschool physics reading that.. Very nice JV
Since the club is now 1% longer, the radius of the circle that the clubhead travels is 6% (2*pi*1%) longer.
However, the starting assumption was that the angular velocity of your hands stayed the same, which means the angular velocity of the clubhead stays the same, which means its linear velocity (AKA
clubhead speed) increases.
It's very unlikely that you could actually swing two clubs at different lengths at that same angular velocity without also changing the weight.
Not correct.
Circumference = 2*pi*r
Big circle:
C2 = 2*pi*r2
Small circle:
C1 = 2*pi*r1
divide the big circle by the small circle:
C2/C1 = (2*pi*r2)/(2*pi*r1)
C2/C1 = r2/r1
So if r2 is 1% larger than r1, then C2 is 1% larger than C1 and velocity2 is 1% faster than velocity1.
So the rate is 1% gets 1%.
If you would like I can do the same calculation from an angular velocity pov.
Not correct.
Circumference = 2*pi*r
Big circle:
C2 = 2*pi*r2
Small circle:
C1 = 2*pi*r1
divide the big circle by the small circle:
C2/C1 = (2*pi*r2)/(2*pi*r1)
C2/C1 = r2/r1
So if r2 is 1% larger than r1, then C2 is 1% larger than C1 and velocity2 is 1% faster than velocity1.
So the rate is 1% gets 1%.
If you would like I can do the same calculation from an angular velocity pov.
Everybody should have their own "ClearKey". :-)
Like BC said earlier "It does not make sense to me." Which makes him the first to intuitively see the error. I thought the same thing, and verified the math. I have this responsiblity because my
wife is a high school math teacher and if I let something slip by I will never hear the end of it. :)
I am sure that if this error went unchecked, it would be quoted on TGC in a couple of weeks. ;)
What's the problem?[/QUOTE]
There is no problem. I am simply amazed by your knowledge.
It might still make Golf Digest.
As an excuse for failing Physics 101, I was applying the 2*pi*r factor because I was thinking purely of angular velocity being translated to linear velocity and "forgot" that this would cancel
out when looking at the ratio of the two clubhead speeds.
Thankfully, those brain cells are no longer required in my day job as they have long since been rendered dormant by alcohol.
I have a headache:wallbash
Very impressive, could you translate the math to golfer's language, how much yardage would be lost? I have wondered what the yardage impact would be of shortening my driver. I know I would be
Without using math, since I clearly suck at it right now, it's hard to say if you will lose distance by going shorter.
You may actually be able to swing the shorter club faster, which would mean more distance.
Similarly, even if you swing the shorter club at the same speed or slightly slower, if you hit it in the center of the face more often, you will gain distance.
The last driver i had at 44 inches was alot straighter,but did notice a lower trajectory and less distance but not much.
If BC (negligible accuracy change assumed) was to shorten his driver by half an inch (~1%), then he would lose 2.8 yards based on the shorter driver alone.
However, he would gain a very small part of that back because he would now be able to swing the club with a higher angular velocity (equivalent term to rpm), because the moment of inertia for the
entire club (MOI is the relationship between mass and its distance from the point of rotation) has decreased and this assumes that the torque/force that he can apply to swinging the club is the
So overall BC would probably barely be able to tell the driver is shorter based on the distance obtained.
If he was to increase to the RCGA limit of 48" he would only be able to obtain less than 16.6 yards. But he would argue at the expense of all his accuracy. ;)
I hope this helps.
[QUOTE=Started2k3;1387=(MOI is the relationship between mass and its
Can you explain the MOI formula in simple terms?
THis is the formula but I still don't get it:
MOI is calculated by squaring the length of the club and multiplying that number by the sum of the head weight and the shaft weight divided by 3. MOI = L^2 x (H + S/3). Using a 124 g shaft, a 52
g grip, head weights starting at 231 g in the 2 iron and increasing in 7 gram increments to 287 g in the PW , the MOI of this set range from 311,506 in the 2 iron to 295,831 in the PW, a
difference of roughly 16,000. Obviously these clubs are not matched by MOI. Interestingly, too, the balance points of this set range from 28.9” to 27.7”, not matched, and the overall weights are
different, not matched, but the swingweights are matched at C-8.
Thanks for the translation.Therefore assuming a linear relationship, if I reduce my shaft length from 45" to 40" I will loose 5 x 1.8 yards, less "a little bit" from the resulting decrease in the
MOI, say a yard.
Will I gain the 8 yards from the improved accuracy of the shorter shaft?
BC Mist; is this a high school trig question?
Several years ago I went to a Golfworks clubmaking seminar where the results of a distance versus length "study" were revealed. Golfers hit drivers from 42" to 47" in .5" increments.
The results were predictable. The longest drive came from the driver of the longest length, BUT, the LONGEST AVERAGE DRIVE came from the 42.5" driver and the reasons should be obvious.
At the end of the day then, shorter is better,(:) as it is easier to hit the ball BETTER or more consistently closer to the centre of the club face. Cutting the driver down by an inch or so won't
cause much, if any distance loss, and should allow you to play more shots from the short grass.
From my perspective, golfers with a smooth tempo, swingers, can use a longer club, more effectively as they tend to hit the sweet spot more frequently than those golfers who have a fast tempo,
To contradict everything I just said, I cut my driver length from 47 7/8" to 46" this year, and my irons are shorter by about .5", and my fairways and greens hit are DOWN, albeit, only
fractionally. Gee, I wonder if it could be my swing.
Probably not.
THis is the formula but I still don't get it:
MOI is calculated by squaring the length of the club and multiplying that number by the sum of the head weight and the shaft weight divided by 3. MOI = L^2 x (H + S/3). Using a 124 g shaft, a 52
g grip, head weights starting at 231 g in the 2 iron and increasing in 7 gram increments to 287 g in the PW , the MOI of this set range from 311,506 in the 2 iron to 295,831 in the PW, a
difference of roughly 16,000. Obviously these clubs are not matched by MOI. Interestingly, too, the balance points of this set range from 28.9” to 27.7”, not matched, and the overall weights are
different, not matched, but the swingweights are matched at C-8.
The physics definition of moment of inertia for a point mass (very very small object rotating around an axis) is:
MOI = m*r^2
m = mass of the point in kg
r = radius of point mass from axis in meters
Unfortuneately we have to deal with the real world where objects actually have volume. Which greatly complicates the calculation, by splitting up the solid object (like a clubhead) into a set of
point masses that can then be summed to get MOI (get out your grinder and slowly remove layers on your favourite club head).
On the bright side, small objects (like a clubhead) end up having an equation that is very similar to the point mass equation. This is where you get the H*L^2 part of your MOI equation.
Luckily scientists have already gone through a lot of calculations for us so we know what the equation for a thin rigid rod is which gives us the secont part of your equation: S/3*L^2
Adding these together we get:
MOI = H*L^2 + S/3*L^2
MOI = L^2 x (H + S/3)
RE YOUR CALCULATION: The grip weight is left out of the calculation because its influence on MOI will be constant so long as you are using the same grip for all clubs (ie same maker and of
similar mass). Also if you include the grip mass to the shaft mass you will be distributing the grip mass accross the entire length of the shaft during the calculation.
Using your numbers I get values like:
40" 2 iron -> MOI = 281,118
36" PW -> MOI = 271,072 (shaft weight is reduced from tip trimming to 111.6g)
Still a large margin, which is why you would then need to add tip weight to push up the MOI for the shorter clubs. This would range from 0.5g for the 3i to 12.1g for the PW. I believe that if you
actually want to MOI your clubs you should have a scale that measures to the 0.1 of a gram.
I hope this helps.
Several years ago I went to a Golfworks clubmaking seminar where the results of a distance versus length "study" were revealed. Golfers hit drivers from 42" to 47" in .5" increments.
The results were predictable. The longest drive came from the driver of the longest length, BUT, the LONGEST AVERAGE DRIVE came from the 42.5" driver and the reasons should be obvious.
At the end of the day then, shorter is better,(:) as it is easier to hit the ball BETTER or more consistently closer to the centre of the club face. Cutting the driver down by an inch or so won't
cause much, if any distance loss, and should allow you to play more shots from the short grass.
From my perspective, golfers with a smooth tempo, swingers, can use a longer club, more effectively as they tend to hit the sweet spot more frequently than those golfers who have a fast tempo,
To contradict everything I just said, I cut my driver length from 47 7/8" to 46" this year, and my irons are shorter by about .5", and my fairways and greens hit are DOWN, albeit, only
fractionally. Gee, I wonder if it could be my swing.
Do you know if the flex of the shafts were taken into account during this study? Because if the shafts were only butt trimmed then the flex (frequency) would increase which in my opinion would
make the shaft more predictable hence greater accuracy.
Whoa this is getting way out of hand, the original q was what would happen if he shortened the shaft.
As a general rule of thumb, and I mean general, if you shorten the shaft by 1 inch, the headspeed will decrease by no more than 2 mph and in some cases less, this translates to 3 to 5 feet of
distance. The tradeoff here is that a shorter shaft will alter the swing weight, but that is only feel anyway and can be compensated by tape or backweighting, (according to lengthening or
shortening) however center hits are greatly increased. Several years ago 43 inches was the norm, nowadays 45 and even 46 is regarded by 28 hcpers as acceptable. Fact is that by reducing the
length, the fairways become more achievable. I am a fromer tour pro that used to play at 43 inches, I went up to 45 and now at 54 yrs old I am back to 43.5. Most tour pro,s today are cutting back
and feeling the results of more fairways hit.
Here is a fact to ponder. If you asked a pro what he would like to loose and get back in return he would say..... take away a few yards in length and give me accuracy in return. Now ask a 28
hcper the same q, dont you think he would say, stuff the accuracy, give me distance. I rest my case.
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IIT JEE Main Chemistry -Solution- Study Materials
Notes and Study Materials
Examples and Exercise
IIT JEE (Main) Chemistry ,”Solution” Notes ,Test Papers, Sample Papers, Past Years Papers , R C Mukherjee , O P Tondon and P.W.Atkins Solutions and Help from Ex- IITian
About this unit
Different methods for expressing concentration of solution: molality, molarity, mole fraction, percentage (by volume and mass both), vapour pressure of solutions and Raoult’s Law. Ideal and non-ideal
solutions, vapour pressure – composition, plots for ideal and non-ideal solutions Colligative properties of dilute solutions, relative lowering of vapour pressure, depression of freezing point,
elevation of boiling point and osmotic pressure. Determination of molecular mass using colligative properties. Abnormal value of molar mass, van’t Hoff factor and its significance.
R C Mukherjee for IIT JEE (Main) Chemistry – Solution
Modern Approach to Chemical Calculations by R C Mukherjee can really test your understanding of fundamental of Chemistry. In fact this book excellently covers all types of numericals related to
Physical Chemistry. I would say it will really enhance your grips on the subject and for sure you can expect at least 40–50 % direct questions from R C Mukherjee in IIT. We , Ex-IITian at https://
www.iitianacademy.com. will make sure you understand the concept well.
IIT JEE (Main) Chemistry, Solution Notes, Solved Examples.
Get Notes prepared by Ex-IITIan for IIT JEE (Main) Chemistry , Solution.
O P Tondon IIT JEE (Main) Chemistry Solutions Solution
Physical Chemistry by O P Tandon is a must-have for all those who are preparing for IIT JEE or other engineering entrance examinations.he book covers the entire Physical Chemistry syllabus including
Solution, Radioactivity and Nuclear Transformation and Stoichiometry. The OP Tandon Physical Chemistry book covers each and every topic related to the JEE preparation. I will help you online for any
doubt / clarification.
P.W.Atkins IIT JEE (Main) Chemistry Solutions Solution
Atkins Physical Chemistry book can help you clear concepts used in solving Physical Chemistry numerical quite well. It includes enhanced explanations and mathematical presentation of the concepts;
several solved examples, and self-test. There are exercises at the end of each chapter. There is a checklist of key equations too at the end of every chapter which is quite useful for students during
revisions. Full color text design is helpful in engaging students and makes the book interesting to read.
IIT JEE (Main) Chemistry Assignments
Chapter wise assignments are being given by teachers to students to make them understand the chapter concepts. Its extremely critical for all CBSE students to practice all assignments which will help
them in gaining better marks in examinations. All assignments available for free download on the website are developed by the best teachers having many years of teaching experience in CBSE schools
all over the country. Students, teachers and parents are advised to contact me online incase of any doubt / clarification.
Past Many Years (40 Years) Questions IIT JEE (Main) Chemistry Solutions Solution
Past 40 Years Question Papers Solutions for IIT JEE (Main) Chemistry Solution are provided here with simple step-by-step explanations. These solutions for Solution are extremely popular among IIT JEE
(Main) students for Chemistry . Solution Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Past Many Years Question Papers Book of
IIT JEE (Main) Chemistry Chapter Solution are provided here for . I will help you online for any doubt / clarification.
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Show all your working on this problem set.
Show all your working on this problem set.
1. (55 points) Ms Melendez has a utility function with regard to coffee: U = CE¹,
where Cis Columbian coffee and E is Ethiopian coffee. Initially use Y for her income
and pc and pe for the prices.
(a) (8 pts) How much of each type of coffee does Melendez consume (uncompensated
(b) (4 pts) What is Melendez' indirect utility function?
(c) (8 pts) Work out her compensated demand functions for Columbian and Ethiopian
(d) (3 pts) What are the numerical values for (a), (b) and (c) if her income is 20
pesos, and a bag of Columbian or Ethiopian beans costs 5 pesos.
(e) (4 pts) There is a drought in central Columbia and the price of a bag of Columbian
beans rises to 7 pesos. What income would Melendez need to reach her old utility
at the new prices?
(f) (4 pts) If she had this income, what quantities of each good would she cons
at the new prices?
(g) (4 pts) Given her actual income, what quantities of each good will she consume
at the new prices?
(h) (4 pts) What is the income effect of this price change for Ethiopian coffee?
(i) (4 pts) What is the substitution effect of this price change for Ethiopian coffee?
(i) (4 pts) Show that the Slutsky equation holds for this price change (using the
numbers you have already calculated for Ethiopian coffee).
(k) (8 pts) Calculate the change in Melendez' welfare caused by the price change,
using three different measures: both compensating variation (CV) and equivalent
variation (EV).
Fig: 1
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Architecture & Structural Engineering
Concentrated Plasticity Model
TBDY Chapter 5.3.1 The Stacked Plastics Behavior Model
5.3.1.1 - The Piled Plastic Behavior Model (Plastic Hinge) can be used as a nonlinear behavior model for columns, beams, and reinforced concrete walls that can be modeled as frame (rod) finite
elements , and which meet the geometric condition given in 4.5.3.8 .
5.3.1.2 - In the Lumped Plastic Behavior Model , it is assumed that plastic deformations occur uniformly along finite-length regions where internal forces reach their plastic capacity. The length (L
[p] ) of the plastic deformation zone, which is called the plastic hinge length, shall be taken equal to half of the cross section dimension (h) in the running direction (L [p] = 0.5h).
5.3.1.3 - The length of the plastic deformation zones of the elements that make plastic deformation only under axial force shall be taken equal to the free length of the relevant element.
5.3.1.4 - The plastic hinge representing the stacked plastic deformation should theoretically be placed in the middle of the plastic deformation zone specified in 5.3.1.2 . However, in practical
applications the approximate idealizations specified in 5.4.2.3 for beams and columns and 5.4.3.1 for walls may be allowed.
5.3.1.5 - Conditions for defining the effective yield moments of reinforced concrete plastic joint sections are given in (a) , (b) , (c) below :
(a) Material strengths will be taken according to 5.4.1.5 .
(b) In the calculation of the effective yield moment, the pressure unit deformation of the concrete can be 0.0035, and the unit deformation of the reinforcing steel can be taken as 0.01.
(c) Axial forces arising from vertical loads shall be taken into account in the calculation of effective yield moment.
5.3.1.6 - The hardening effect (increase of plastic moment due to the increase of plastic rotation) in bidirectional internal force-plastic deformation relations of reinforced concrete and steel
sections can be abandoned.
According to 5.3.1.7 - 5.6 and 5.7 , the elasto-plastic standard cycle model for steel bearing systems as a cyclic behavior model in the nonlinear earthquake calculation to be made in the time domain
according to 5.6 and 5.7 . Models derived from it can be used to provide
TBDY Article 5.4.1.5 - The material strengths given in (a) and (b) below shall be based on modeling based on evaluation and design according to shape change :
(a) The current strengths of concrete and reinforcement steel defined in Section 15 shall be taken as basis in the assessment of existing buildings according to deformation .
(b) In the evaluation and design of new buildings according to deformation, the expected (average) strengths of concrete and reinforcement steel and structural steel defined in Table 5.1 shall be
taken as basis. F in Table [ce] and f [ck] concrete and the mean compressive strength characteristic, f [ate] and f [yk] shows the mean and the typical yield strength of steel.
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The index of a radical function can be any real number, but the most common are square roots with an index of 2, like y=\sqrt{x}, which are called a square root function, and cube roots with an index
of 3, like y=\sqrt[3]{x}, called a cube root function.
Consider the functions:
• f\left(x\right)=\sqrt{x}
• g\left(x\right)=\sqrt{x+4}
• h\left(x\right)=\sqrt{x}-2
• k\left(x\right)=\sqrt[3]{x}
1. Which functions have a real output for an input of x=-3? x=0? x=5?
2. Which functions have a domain of all real values of x?
3. Which functions go through the point \left(0,0\right)?
4. Which functions can have negative outputs?
5. Match each function to one of these graphs:
For cube root functions, the function increases (or decreases) at a fast rate, then the rate of change slows around a point called an inflection point. In other words, the function continues
increasing (or decreasing), but the rate is slower around the point of inflection.
Radical functions can be transformed similarly to any transformation of the parent function, y= af \left[b\left(x-h \right)\right] +k.
Square root Cube root
\text{Parent function:} y= \sqrt{x} y= \sqrt[3]{x}
\text{Reflection across the }x\text{-axis:} y=-\sqrt{x} y=-\sqrt[3]{x}
\text{Reflection across the }y\text{-axis:} y=\sqrt{-x} y=\sqrt[3]{-x}
\text{Vertical stretch when } \left|a\right|>1 \\ \text{Vertical compression when } 0<\left|a\right|<1 \text{:} y=a\sqrt{x} y=a\sqrt[3]{x}
\text{Horizontal compression when } \left|b\right|>1 \\ \text{Horizontal stretch when } 0<\left|b\right|<1 \text{:} y=\sqrt{bx} y=\sqrt[3]{bx}
\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:} y=\sqrt{x-h} + k y=\sqrt[3]{x-h} + k
(h, k) \text{:} \text{Endpoint} \text{Point of inflection}
The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers.
Similarly, the absolute extremum of the square root function will change location when translated. If there are no reflections, the endpoint of the domain is an absolute minimum. If a vertical
reflection occurs, it becomes an absolute maximum.
Square root functions do not have a relative extremum, and cube root functions have neither absolute nor relative extrema.
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Gross Margin vs Markup
It’s imperative that business leaders understand the difference between Gross Profit, Gross Margin, and Markup as well as how they can affect the company. Often-times these terms are used
interchangeably due to the misconception that they are one in the same. Knowing and understanding the difference prevents making decisions based on misleading information that can have a negative
impact the bottom line and business!
Profitability, Income, Markup, Gross Margin, Gross Profit
Gross Profit, Gross Margin, Markup: How do they differ?
Gross profit is the total profit dollars. That is, it is simply the difference between the net sales and cost of goods sold(COGS). Markup and Gross Margin, on the other hand, is the percentage of
profit; one based on cost and the other based on selling price.
Markup is the percentage of profit based on the cost. To determine the Markup, subtract COGS from net sales then divide by COGS. For example, $100 – $80/$80 = 25%. If you want to establish the
Selling Price based on Markup multiply the cost times the desired Markup (%) and add the cost. Using the same example $80 x 25% + $80 = $100
Gross Margin is the percentage of profit margin based on selling price, which yields a much different result than Markup. Calculating Gross Margin is the same as Markup except you divide the Gross
Profit by the Selling Price. Using the above example, the Gross Margin is $100 – $80/$100 = 20%. Using Gross Margin, the Selling Price can be established by dividing the cost by the inverse of the
desired margin or (1 – 0.2)/1 = 0.8. Using the same example $80/0.8 = $100.
What is the Impact to the Bottom Line?
In reviewing the pricing examples above, Gross Margin is five percentage points below Markup. If you use Markup instead of Margin, this shortfall goes straight to the bottom line because all the
other ratios for product cost (labor and overhead), G&A expenses and operating income are based on sales. Establishing pricing using Markup instead of Gross Margin has far-reaching effects. This is
further evidenced in Figure 1 below comparing operating income at 25% Markup versus 25% Gross Margin on net sales of $1,000,000. In this example, using Markup results in 20% less Gross Profit and
Operating Income is reduced by 50% dropping from 10% to 5%. It should also be noted that the higher the margins, the greater the spread in overall profitability.
Protect your bottom line. Always use gross margin to determine your selling prices!
At Cogent Analytics, we never stop looking for ways to improve your business and neither should you. So, check out some of our other posts for helpful business information:
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Math to move camera
Hello everyone!
Kind of struggling to achieve a smooth camera movement, mostly I am having a hard time to define all the math that goes into it and so I am here to ask all of you help with the math part.
Ideally, what I would like to achieve is this:
I have a character in a position. Let’s say that the position is (0,0,0).
I would like to have a camera facing the character. The camera should then move upward in a spiraling movement and at the same time keep the focus on the character’s head (so I guess adjust the pitch
value of the camera).
Ideally, I would like to have variables to store the following information:
• how many “steps” to achieve a full 360-degree rotation
• how many 360-degree full rotations
• how much the camera should move upwards after completing a whole rotation. So, for example if the camera starts at z=0, after the first rotation it should be at z=0.2, and then z=0.4 at the end
of the second rotation and so on until it reaches the max number of rotations and always pointing to the character
How can this be achieved?
Hope I have been clear, if not please let me know.
Thank you all for the help!
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Solve real-world problems involving multiplication and division of whole numbers including problems in which remainders must be interpreted within the context.
A group of 243 students is taking a field trip and traveling in vans. If each van can hold 8 students, then the group would need 31 vans for their field trip because 243 divided by 8 equals 30 with a
remainder of 3.
Clarification 1:
Problems involving multiplication include multiplicative comparisons. Refer to
Situations Involving Operations with Numbers (Appendix A)
Clarification 2: Depending on the context, the solution of a division problem with a remainder may be the whole number part of the quotient, the whole number part of the quotient with the remainder,
the whole number part of the quotient plus 1, or the remainder.
Clarification 3: Multiplication is limited to products of up to 3 digits by 2 digits. Division is limited to up to 4 digits divided by 1 digit.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 4
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is to have students solve problems involving multiplication and division by using and discussing various approaches. This work builds on problem-solving using the four
operations from grade 3 (
• Students should use estimation, and this can include using compatible numbers (numbers that sum to 10 or 100) and rounding.
• Instruction should include allowing students many opportunities to solve multiplicative comparison situations.
• Students should have experience solving problems that require students to interpret the remainder to fit the situation. Students may have to round up to the next whole number, drop the remainder,
use the remainder as a fraction or decimal, or use only the remainder as determined.
• Add 1 to the quotient
□ Thirty students are going on a field trip. They want to put 4 people in each car so that people can sit comfortably. How many cars will be needed?
□ Solution: Divide 30 by 4. The answer is 7 r2.
□ The answer shows that 7 cars will be needed, but 2 people still need to go to a car. Therefore, they will need 8 cars.
• Use only the remainder
□ Gerardo has 19 dollars in his pocket. He wants to give the same amount of money to 4 friends. The rest of the money, if any, will go to his sister to buy toys. How much money will go to his
sister if Gerardo wants to give away everything he has?
□ Solution: Divide 19 by 4. The answer is 4 r3.
□ The remainder is 3, so 3 dollars will go to his sister.
• Drop the remainder
□ Alicia has 48 dollars in her pocket. She wants to buy meals for 5 friends. If each meal costs 10 dollars, will Darlene be able to keep all her friends happy?
□ Solution: Divide 48 by 10. The answer is 4 r8.
□ Alicia can only buy 4 complete meals. Therefore, she cannot buy one for each of her 5 friends.
Common Misconceptions or Errors
• Students apply a procedure that results in remainders that are expressed as r for all situations, even for those in which the result does not make sense.
□ For example, when a student is asked to solve the following problem, the student responds to the problem— there are 52 students in a class field trip. They plan to have 10 students in each
van. How many vans will they need so that everyone can participate? And the student answers “5r2 vans.” The student does not understand that the two remaining students need another van to go
on the field trip.
• Students may not understand that the remainder represents a portion of something, rather than a whole number. Referring back to the previous example students may think r2 means two additional
vans rather than a portion of an additional van.
• Students may have trouble seeing a remainder as a fraction.
□ For example, 7 ÷ 3 = 2r1 means that 7 ÷ 3 = 2 $\frac{\text{1}}{\text{3}}$. If 7 cupcakes are divided among 3 people, then each person will get 2 and $\frac{\text{1}}{\text{3}}$ cupcakes.
Strategies to Support Tiered Instruction
• Instruction includes opportunities to engage in guided practice completing real-world problems involving remainders. Students use models to understand how to interpret the remainder in situations
in which they will need to “ add 1 to the quotient,” “ use only the remainder,” “ drop the remainder” or “ treat the remainder as a fraction.”
□ For example, the teacher displays and reads the following problem aloud: “ There are 55 pencils that need to be sorted into boxes. 10 pencils can go into each box. How many boxes are needed
so all the pencils can be put into boxes?” The teacher uses models or drawings to represent the problem and guided questioning to encourage students to identify that they will need to add one
to the quotient as their solution. If students state that they will need 5r5 boxes, the teacher refers to the models to prompt students that a sixth box is needed for the remaining five
pencils. If students state that they will need 5 more boxes since the remainder is 5, the teacher reminds students through guided questioning that the remainder of 5 represents 5 remaining
pencils and only 1 more box is needed (i.e., “ add 1 to the quotient”). This is repeated with similar real-world problems.
• Instruction includes opportunities to complete real-world problems involving remainders using pictorial representations to understand what the remainder is, including interpreting the remainder
as a fraction.
□ For example, the teacher displays and reads the following problem aloud: “ Karly, Juan, and Li share 4 cookies equally. How many cookies can each person eat?” The teacher uses drawings to
represent the problem and guided questioning to encourage students to identify that Karly, Juan, and Li are able to eat 1 whole cookie but then must split the 4th cookie into thirds so that
they can each eat 1 $\frac{\text{1}}{\text{3}}$ cookies, therefore 4 ÷ 3 = 1 $\frac{\text{1}}{\text{3}}$. This is repeated with similar real-world problems.
• Instruction includes opportunities to complete real-world problems involving remainders using manipulatives. Students use hands-on models to interpret the remainder in situations in which they
will need to “ add 1 to the quotient,” “ use only the remainder,” “ drop the remainder” or “ treat the remainder as a fraction.”
□ For example, the teacher displays and reads the following problem aloud: “ There are 24 students in a class field trip. They plan to have 10 students in each van. How many vans will they need
so that everyone can participate?” The teacher has students use counters or base-ten blocks to build a model of the problem and guided questioning to encourage students to identify that they
will need to add 1 to the quotient as their solution. If students state that they will need 2r4 vans, the teacher refers to the models to prompt students that a third van is needed for the
remaining four students. If students state that they will need 4 more vans since the remainder is 4, the teacher reminds students through guided questioning that the remainder of 4 represents
4 remaining students and only 1 more van is needed (i.e., “ add 1 to the quotient”). This is repeated with similar real-world problems.
Instructional Tasks
Instructional Task 1 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Add 1 to the quotient” as their solution.
Instructional Task 2 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Use only the remainder” as their solution.
Instructional Task 3 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Drop the remainder” as their solution.
Instructional Task 4 (MTR.7.1)
Ashley and Larry have to complete 27 benchmarks in 5 days. If they start on Monday and complete 6 benchmarks per day, how many will they need to complete on Friday?
• If completing 6 benchmarks takes an entire work day, how much of Friday will be needed to complete the remaining benchmarks?
Instructional Items
Instructional Item 1
Sam has $50 to spend on video games. He buys one video game for $26. With the money he has left over, how many $9 games can Sam buy?
• a. 2 games
• b. 3 games
• c. 5 games
• d. 6 games
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
This benchmark is part of these courses.
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
Solve one-step real-world problems involving multiplication and division of whole numbers. Multiplication may not exceed two-digit by one-digit and division must be related to one-digit by one-digit
multiplication facts.
Related Resources
Vetted resources educators can use to teach the concepts and skills in this benchmark.
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Teaching Idea
Problem-Solving Tasks
STEM Lessons - Model Eliciting Activity
Cookies and Treats:
Fourth graders will help Cookies and Treats find cost-effective and eco-friendly packaging for its cookies. Students will organize data and compare prices using decimal notation in order to develop a
procedure for choosing packaging for cookies. Students will use multiplication and division of whole numbers to plan for how many packages to order.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
Park Planning:
Students are asked to plan a playground for a new park within a given budget and area limit. They will analyze the best use of playground equipment using a data table of area requirements and cost.
Students will convert units within a single measurement system, calculate the area of a rectangle, and perform addition/subtraction calculations involving money using decimal notation.
Patty's Party Planning:
In this Model Eliciting Activity, MEA, students will help a party planner determine which party location is the best one to use. They will calculate the cost of the banquet hall rental based on the
number of people, number of tables and hourly rental of the location by using division and multiplication.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs
resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics
to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a
problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://
Shoe Closet MEA:
In this open-ended problem, students will work in teams to determine a procedure for ranking shoe closet styles for a person’s dream closet. Students will need to calculate the perimeter and cost for
the closet, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
Slither Not in the Everglades! Python MEA:
This MEA will ask students to work in teams to help their client, The Florida Fish and Wildlife Conservation Commission, to decide which Burmese python traps manufacturing company to buy traps from.
The traps will be placed along the Florida Keys and the Everglades to help prevent the growth of invasive Burmese Python population. The students will implement their knowledge of how plants,
animals, and humans impact the environment, use mathematical and analytical problem-solving strategies, and be able report their finding in an organized, descriptive manner.
MFAS Formative Assessments
Making Necklaces:
The student is asked to solve a multiplicative comparison word problem comparing 6 inches of string to 24 inches of string.
Roller Coaster Rides:
Students are given a multi-step word problem to solve that requires interpreting remainders.
Original Student Tutorials Mathematics - Grades K-5
Field Trip Frenzy (Part 1):
Take a field trip while learning how to interpret remainders in multi-step division word problems.
This is part 1 of a four-part series of interactive tutorials. Click below to open the other tutorials in this series.
Field Trip Frenzy (Part 2):
Learn how to interpret remainders in multi-step division problems related to a field trip in this interactive tutorial.
This tutorial is Part 2 in a four-part series about remainders. Click below to open the other tutorials in this series.
Field Trip Frenzy (Part 3):
Learn how to interpret remainders in multi-step division problems in this interactive tutorial
This is the third tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Field Trip Frenzy (Part 4):
Learn when to write the remainder of a multi-step division process as a fraction or decimal in this interactive tutorial.
This is the final tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Note: This tutorial extends beyond whole number quotients with whole number remainders to whole number quotients with fractional or decimal remainders.
Multiplying Math Magic:
Learn how to write multiplication equations based on multiplication comparisons and story problems in this magical math online tutorial!
Space: Division as Comparison:
Discover how multiplicative comparison problems, from outer space, can be solved using division in this online tutorial.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Original Student Tutorials
Space: Division as Comparison:
Discover how multiplicative comparison problems, from outer space, can be solved using division in this online tutorial.
Type: Original Student Tutorial
Space: Multiplication as Comparison:
Launch into solving word problems that use multiplicative comparisons, drawings, and symbols in this space-themed interactive tutorial.
Type: Original Student Tutorial
Field Trip Frenzy (Part 4):
Learn when to write the remainder of a multi-step division process as a fraction or decimal in this interactive tutorial.
This is the final tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Note: This tutorial extends beyond whole number quotients with whole number remainders to whole number quotients with fractional or decimal remainders.
Type: Original Student Tutorial
Field Trip Frenzy (Part 3):
Learn how to interpret remainders in multi-step division problems in this interactive tutorial
This is the third tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Field Trip Frenzy (Part 2):
Learn how to interpret remainders in multi-step division problems related to a field trip in this interactive tutorial.
This tutorial is Part 2 in a four-part series about remainders. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Field Trip Frenzy (Part 1):
Take a field trip while learning how to interpret remainders in multi-step division word problems.
This is part 1 of a four-part series of interactive tutorials. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Multiplying Math Magic:
Learn how to write multiplication equations based on multiplication comparisons and story problems in this magical math online tutorial!
Type: Original Student Tutorial
Problem-Solving Tasks
Comparing Growth, Variation 2:
The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to identify that Student A is just looking at how
many feet are being added on, while Student B is comparing how much the snakes grew in comparison to how long they were to begin with.
Type: Problem-Solving Task
Comparing Growth, Variation 1:
The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount
(an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example
of "multiplicative thinking").
Type: Problem-Solving Task
Carnival Tickets:
The purpose of this task is for students to solve multi-step problems in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the
average price level. In this task, students can see that if the price level increases and people’s incomes do not increase, they aren’t able to purchase as many goods and services; in other words,
their purchasing power decreases.
Type: Problem-Solving Task
Comparing Money Raised:
The purpose of this task is to give students a better understanding of multiplicative comparison word problems with money.
Type: Problem-Solving Task
Karl's Garden:
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common
misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. Students often believe the same is true for
Type: Problem-Solving Task
Comparing Products:
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of
operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.
Type: Problem-Solving Task
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
Problem-Solving Tasks
Comparing Growth, Variation 2:
The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to identify that Student A is just looking at how
many feet are being added on, while Student B is comparing how much the snakes grew in comparison to how long they were to begin with.
Type: Problem-Solving Task
Comparing Growth, Variation 1:
The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount
(an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example
of "multiplicative thinking").
Type: Problem-Solving Task
Carnival Tickets:
The purpose of this task is for students to solve multi-step problems in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the
average price level. In this task, students can see that if the price level increases and people’s incomes do not increase, they aren’t able to purchase as many goods and services; in other words,
their purchasing power decreases.
Type: Problem-Solving Task
Comparing Money Raised:
The purpose of this task is to give students a better understanding of multiplicative comparison word problems with money.
Type: Problem-Solving Task
Karl's Garden:
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common
misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. Students often believe the same is true for
Type: Problem-Solving Task
Comparing Products:
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of
operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.
Type: Problem-Solving Task
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How did money begin?
Metals objects were introduced as money around 5000 B.C. By 700 BC, the Lydians became the first in the Western world to make coins. Soon, countries began minting their own series of coins with
specific values. Since coins were given a designated value, it became easier to compare the cost of items people wanted.
What is static role of money?
Answer: The static functions of money are: Money works as a medium of exchange. It helps to measure the value of a good or service. Money plays an important role in lending and borrowing. A person
can store the purchasing power of money.
What is money explain function?
As stated above, money primarily functions as a medium of exchange. However, it also has developed secondary functions that derive from its use as a medium of exchange. These other functions include:
1) a unit of account, 2) a store of value, and 3) a standard of deferred payment.
What was the first form of money?
Mesopotamian shekel
What are the three characteristics of money?
The characteristics of money are durability, portability, divisibility, uniformity, limited supply, and acceptability. Let’s compare two examples of possible forms of money: A cow.
When was banking started?
18th century
What is history of banking?
The history of banking began with the first prototype banks which were the merchants of the world, who gave grain loans to farmers and traders who carried goods between cities. The most famous
Italian bank was the Medici Bank, established by Giovanni Medici in 1397.
What is the major feature of barter system?
The main features of barter system are as under: Barter system is direct exchange of goods and services. It requires the double coincidence of wants. Barter system eliminates the use of money. It
generally flourishes among uncivilized and backward communities.
What is the oldest form of money still in use today?
Metallic money
Who controls money in the world?
Commercial banks use fractional money lending that allows it to lend out ten times more money than they have in their reserves. So, the Federal Reserve, your central bank and all commercial banks
have control over your money and the only reason money has value is because your government says so.
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How do you simplify (32x^3y)/y^9div(8x^4)/y^6? | Socratic
How do you simplify #(32x^3y)/y^9div(8x^4)/y^6#?
1 Answer
Incorporate exponent variable laws and standard arithmetic simplification.
First, as a division expression, we're going to multiply the fractions. This involves reciprocating the second fraction.
$\frac{32 {x}^{3} y}{{y}^{9}} \div \frac{8 {x}^{4}}{{y}^{6}}$
$= \frac{32 {x}^{3} y}{{y}^{9}} \times \frac{{y}^{6}}{8 {x}^{4}}$
Now, we simplify normally.
$= \frac{32 {x}^{3} {y}^{7}}{8 {x}^{4} {y}^{9}}$
Now comes the most potentially confusing part. When you are simplifying variables in fractions, if you divide the numerator by the denominator of the same variable - but different degree, the
resulting variable will be positioned in the numerator. The same goes for the denominator. This is something your teacher should teach so you are provided more examples.
Back to the expression, we're going to simplify the easiest of them all - the number.
$= \frac{4 {x}^{3} {y}^{7}}{{x}^{4} {y}^{9}}$
AND NOW COMES THE VARIABLES. First, the $x$ variable.
$= \frac{4 {y}^{7}}{x {y}^{9}}$
We'll do the same with the $y$ variable.
$= \frac{4}{x {y}^{2}}$
And that's it.
Hope this helps :)
Impact of this question
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Data Science For Dummies (2016)
Part 2: Using Data Science to Extract Meaning from Your Data
Chapter 4: Machine Learning: Learning from Data with Your Machine
Part 5: Applying Domain Expertise to Solve Real-World Problems Using Data Science
Chapter 18: Data Science in Journalism: Nailing Down the Five Ws (and an H)
Chapter 21: Using Data Science to Describe and Predict Criminal Activity
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An Introduction to Statistics and Data Science and Differences between Them - Velebit AI
By Joel Abraham on unsplash.com
An Introduction to Statistics and Data Science and Differences between Them
By Maroje Portada on February 16, 2022
There are many long and complicated definitions of statistics. Those are less interesting for anyone not well versed in this field. Here are several simple definitions instead:
Within statistics there are two branches: descriptive and inferential statistics.
Descriptive statistics provides methods to organize, summarize and present raw data into something more convenient and informative called information. That information can then be interpreted and
shared. In descriptive statistics it is possible to use many different graphical and numerical techniques to describe the data.
Inferential statistics offers us methods to examine small samples of data and make estimations or draw conclusions about bigger sets of data called population. Those estimations and conclusions can
be true but it isn’t always so.
Words related to Statistics
What Can We Use Statistics for?
We can find statistics in our lives much more than we are aware. Weather prediction, election polls, estimation of economic growth, stock price on markets, demographics, sports statistics, behavior
of users on social networks, trendy topics, successful sales on social networks and much more.
Wherever there is data, there is potential for use of statistics. There are many complex problems in all parts of our lives that can be solved with statistics. It is important to notice that
statistics helps in making more concrete decisions with less risk and uncertainty. While intuition is useful, we should always use as much information as we can to make better decisions.
What Is Data Science?
After covering the topic of statistics, it is time to say something about data science as well. One simple definition of data science considers it a multidisciplinary field that combines some
technical skills with soft skills to extract information from structured and unstructured data.
The principal purpose of data science is to find patterns between the data. It is still expanding and its evolution is heavily dependent on development of technology, especially computer science and
programming languages.
Words related to Data Science
Similarities and Differences between Data Scientists and Statisticians
Fields of work for data scientists and statisticians are quite closely related even to the point of often being considered as synonyms, but that is mostly not true - there are also many differences
to distinguish the two.
What are similarities between data scientists and statisticians?
Both roles:
• need some degree of understanding of mathematics;
• investigate problems;
• analyse data;
• analyse trends;
• create forecasts;
• use visualisations;
• often report their findings to non-technical users;
What are the differences?
• Data scientists use computer science, algorithms or machine learning more than statisticians.
• Data scientists are more involved in creation and use of data systems, while statisticians focus more on the equations and mathematical models that they use for their analysis.
• Data scientists more often use big data, while statisticians typically use smaller data sets.
• Data scientists compare many methods to create the best machine learning model while statisticians more often improve a single model until it befits their data set.
• Statisticians focus more on quantifying uncertainty and making inferences.
Data Science - Advantages and Disadvantages
Final Thoughts
Statistics and data science have lots of things in common. Use of mathematics, investigation of problems and data analysis are just a few of them. There are also differences like the level of
information technology used, usual size of data sets and approach to the learning model.
Most certainly data science and statistics will continue to coexist and to some extent influence one another.
The goal of this topic was to bring those areas closer to people who don’t know much about them in the simplest possible way. Would you like to share your experience with statistics and data science?
Your thoughts and comments are more than welcome.
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Partner with us to develop an AI solution specifically tailored to your business.
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What is 0X49?
0X49 is a hexadecimal (hex) number. We can tell it is a hex number because it starts with 0X. Hexadecimal numbers like 0X49 are not often used in daily life, but we see them used for certain things
such as html colors, shortening binary numbers, computer error codes, and math exercises.
What is 0xffff value?
0xffff is a hexadecimal number. The 0x only tells you that the value after the 0x is hexadecimal. Hexadecimal has 16 digits from 0 to 9 and from a to f (which means, a = 10, b = 11, c = 12, d = 13, e
= 14, f = 15 in a hexadecimal number). Example: ffff = (15*16^3) + (15*16^2) + (15*16^1) + (15*16^0) = 65535.
What is the decimal equivalent of the hex number 0x8A?
For example, the hexadecimal number 0x8A represents a decimal value of (8 * 16 + 10) = (128 + 10) = 138. In a three-digit hexadecimal number, the first digit represents the number of 256s (=16*16),
What is the value of 0x10?
0x means the number is hexadecimal, or base 16. 0x10 is 16.
What is the binary value of 120?
120 in binary is 1111000.
Why is 65535 important?
In Internet protocols, 65535 is also the number of TCP and UDP ports available for use, since port 0 is reserved. For example, =850*77.1 displays as 100000 rather than 65535. Microsoft reports this
to be a display-only bug for only 6 floating point numbers near 65535 and 65536.
What is in hexadecimal?
The hexadecimal numeral system, often shortened to “hex”, is a numeral system made up of 16 symbols (base 16). The standard numeral system is called decimal (base 10) and uses ten symbols:
0,1,2,3,4,5,6,7,8,9. Hexadecimal uses the decimal numbers and six extra symbols.
What does 0x04 mean?
0x04 is hex for 4 (the 0x is just a common prefix convention for base 16 representation of numbers – since many people think in decimal), and that would be the fourth byte (since they are saying
offset, they probably count the first byte as byte 0, so offset 0x04 would be the 5th byte).
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Designing a chip annotation tool with Excel
If you have a set of interesting Affy probes or genes that you'd like to annotate, you can build a simple tool in Excel that will let you gather information about these probes in an attempt to
explain what they have in common or what physiological significance they may have.
Annotation is easy with an Affy chip, since Affymetrix has assembled data from a variety of sources for all probesets of their chips. If you're using a custom array or one by another manufacturer,
you may have to generate the annotation file yourself. The process shown shown below is for Affy chips, so the first step ("Getting the annotation file") may vary for other types of chips.
A sample tool can be downloaded as a zipped Excel file or an Excel file. It may be easier to start with this sample file, instead of building your file from scratch. This is somewhat different from
the description (below), in that two different worksheets are used. In this case, you need to know the notation for referrring to another worksheet. For example, in the formula
"anno!$A$2:$J$14011" refers to cells $A$2:$J$14011 on worksheet anno.
Getting the annotation file
• If you're not using an Affy chip, get the annotation file from the manufacturer or create it yourself.
• If you're using an Affy chip, go to the Affymetrix Support site.
• Select an array and go to that page.
• Under "NetAffx Annotation Files," click on the "CSV" link.
• You'll be prompted to log in to the system (or register for free).
• Save the .zip file on your computer and then unzip it.
Editing the annotation file in Excel
• Open the .csv ("comma-separated value") file in Excel.
• Examine all of the data fields in the file.
• Delete all of the columns that aren't very informative. For example, the fields "Chip," "Organism," and "Annotation Date" can be recorded somewhere but then deleted, since they're the same for
all rows.
• Save the file as a standard Excel file (.xls for Windows), since you'll be using formulas that can't be saved in .csv format.
Creating the annotation columns
• To the right of the annotation columns, add another set of column headings:
• The idea is that you'll paste a list of probes in the "Probe set ID" column on the right, and Excel will fill in the rest of the columns for those probes.
Writing the Excel formulas
• You'll be entering formulas in the first row of cells in your column (cells G2, H2, and I2 in the figure above)- except the first column where you paste the probe IDs - so Excel can find the
corresponding information. These formulas can then be pasted in other rows.
• The key formula you'll be using is VLOOKUP. At any time, you can look for VLOOKUP in the Excel help for more information and examples.
• Another key Excel convention is the use of the '$' prefix to show a cell that is constant even when the formula is copied and pasted. Example: If a formula in row 2 referring to cell A2 is pasted
into row 3, the formula will now refer to cell A3. If a formula in row 2 referring to cell $A$2 is pasted into row 3, the formula will still refer to cell A2.
• In the first cell to the right of the probe ID (input data) column (F2 above), enter the following formula:
with the following information:
□ input = input cell for this row (F2 in the figure above), with the column made constant ($F2)
□ dataTopLeft = top left cell for the columns containing all the annotation data (A2 in the figure above), made constant ($A$2).
□ dataBottomRight = bottom right cell for the columns containing all the annotation data (D12423 for the array above), made constant ($D$12423).
□ colNum = column number containing data to put in this cell (4 in the example above, since the "title" data that you want in this column is found in column 2 of the annotation data).
□ FALSE: since you want VLOOKUP to find only an exact match
An example for the Excel file above is
• Paste a probe into the "input" cell to the right and check that the formula works. In the example below, using "10001_at" as input, the formula could find the corresponding Title:
• In the second cell to the right of the probe ID (input data) column (F2 above), enter a similar formula, replacing colNum:
• For the rest of the cells in the first row of the "output" section, enter similar formulas, always replacing colNum with the appropriate number.
• Copy all the cells where you entered formulas and paste them into rows below - as many rows as you expect to have input genes in your lists. In the example, cells G2, H2, and I2 were pasted into
rows 3 - 101 (if my probe ID lists were as long as 100 entries)
• After pasting the cells with formulas, the file will look something like this:
All of the #N/A will remain until probe set IDs are pasted into the input column.
• Try pasting a column of probe set IDs to check that everything works.
• You can even add hyperlinks to web pages, such as those for NCBI's LocusLink or Unigene. For example, if you already have the LocusLink ID, use a formula like
=HYPERLINK(CONCATENATE("http://www.ncbi.nlm.nih.gov/LocusLink/LocRpt.cgi?l=", H2))
to make a link to the LocusLink page for the LocusLink ID in cell H2.
Helpful equations
• Using IF and ISNA:
=IF(ISNA(VLOOKUP(input,dataTopLeft:dataBottomRight,colNum,FALSE)), VLOOKUP(input,dataTopLeft:dataBottomRight,colNum,FALSE), "alternate message")
• Referring to other worksheets (in this case, a sheet named "NOTES"):
=IF(ISNA(VLOOKUP(A2,NOTES!$B$1:NOTES!$C$100,2,FALSE)), "", "EXPIRED")
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Introduction to Real Analysis pt 1
Hi everyone, here are my notes for the real analysis course I have been taking.
The book I have referred to:
Elementary Analysis: The Theory of Calculus Book by Kenneth A. Ross
Theorem: [ Rational Root Theorem]
Suppose $c_0,c_1,\dots, c_n$ are integers and $r$ is a rational number satisfying the polynomial equation $$ c_nx^n+c_{n-1}x^{n-1}+\dots+c_1x+c_0=0$$ where $n\ge 1,c_n\ne 0, c_0\ne 0.$ If $r=\frac{c}
{d}$ where $c,d$ are integers and $(c,d)=1$. Then $c|c_0, d|c_n$.
Example: Prove that $\sqrt[3]{6}$ is not a rational number.
Proof: Note that $\sqrt[3]{6}$ is solution of $x^3-6=0$. But by rational root theorem, the only possible rational solutions of $x^3-6=0$ are $\pm 1, \pm 2, \pm 3, \pm 6$. But none of these eight
numbers satisfies $x^3-6=0$.
Example: All numbers of the form $5^n-4n-1$ are divisible by $16$.
Proof: Our $n$th proposition is $$P_n: 5^n-4n-1 \text{ is divisible by } 16$$ The basis for induction $P_1$ is clearly true, since $5^1-4\cdot 1-1=0.$ Proposition $P_2$ is also true because $5^2-4\
cdot 2-1=16$. For the induction step, suppose $P_n$ is true. Then $$P_{n+1}= 5^{n+1}-4(n+1)-1=5(5^n-4n-1)+16n.$$ Since $5^n-4n-1$ is divisible by $16$ by induction hypothesis, it follows that $P_
{n+1}$ is also divisible by $16.$
Absolute values and inequalities
Define: For numbers $a$ and $b$ we define $dist(a,b)=|a-b|$. It represents distance between $a$ and $b$.
• $|a|\ge 0$ for all $a\in \Bbb{R}$
• $|ab|=|a|\cdot |b|$ for all $a,b\in \Bbb{R}$
• Triangle inequality: $|a+b|\le |a|+|b|$ for all $a,b \in \Bbb{R}$
Theorem: $$\text{dist}(a,c)\le \text{dist}(a,b)+\text{dist}(b,c)$$
Proof: It is enough to show that $$|a-c|\le |a-b|+|b-c|.$$
Note that $$|(a-b)+(b-c)|\le |a-b|+|b-c|.$$ And we are done.
Problem 1: Prove that $$||a|-|b||\le |a-b|.$$
Proof: It is enough to show that $$(||a|-|b||)^2\le (|a-b|)^2$$ or show $$|a|^2+|b|^2-2|a||b|\le a^2+b^2-2ab$$
or show $$-2|a||b|\le -2ab$$
which is true.
Problem 2: Prove that $$|a+b+c|\le |a|+|b|+|c|.$$
Proof: $$|a+b+c|\le |a+b|+|c|\le |a|+|b|+|c|$$
Remark: Similarly $$|a_1+a_2+\dots +a_n|\le |a_1+a_2+\dots+ a_{n-1}|\le \dots |a_1|+|a_2|+\dots+|a_n|.$$
Problem 3: Let $a,b\in \Bbb{R}$. Show if $a\le b_1$ for every $b_1>b$, then $a\le b$.
Proof: Suppose not. Then $a-b=\epsilon_0>0$. Then consider $b_1=b+\epsilon_0/2 $.
The Completeness Axiom
Definition: Let $S$ be a nonempty subset of $\Bbb{R}$.
• If $S$ contains a largest element $s_0$ i.e $s_0$ belongs to $S$ and $s\le s_0$ for all $s\in S$, then we call $s_0$ the maximum of $S$ and write $s_0= \text{max } S$.
• If $S$ contains a least element $s_0$ i.e $s_0$ belongs to $S$ and $s_0\le s$ for all $s\in S$, then we call $s_0$ the minimum of $S$ and write $s_0= \text{min } S$. \end{definition}
Definition: Let $S$ be a non-empty subset of $\Bbb{R}$
• If a real number $M$ satisfies $s\le M$ for all $s\in S$, then $M$ is called an $\textit{upper bound}$ of $S$ and the set $S$ is said to be $\textit{ bounded above}$.
• If a real number $m$ satisfies $s\ge m$ for all $s\in S$, then $m$ is called an $\textit{lower bound}$ of $S$ and the set $S$ is said to be $\textit{ bounded below. }$
• The set $S$ is said to be $\textit{bounded}$ if it is bounded above and bounded below. Thus $\exists$ $m,M$ st $S\subseteq [m,M]$.
Definition: Let $S$ be a non-empty subset of $\Bbb{R}$.
• If $S$ is bounded above and has a least upper bound, then we will call it the $\textit{supremum}$ of $S$ or $\text{sup}S$.
• If $S$ is bounded below and has a maximum lower bound, then we will call it the $\textit{infimum}$ of $S$ or $\text{inf}S$.
Theorem: [Completeness Axiom for $\Bbb{R}$]Every nonempty subset $S$ of $\Bbb{R}$ that is bounded above has a least upper bound i.e $\text{Sup}S$ exists.
Remark: [Completeness Axiom fails for $\Bbb{Q}$] Consider the set $A=\{r\in \Bbb{Q}:0\le r\le \sqrt{2}\}$. This doesn't have any supremum over $\Bbb{Q}$.
Corollary: Every nonempty subset $S$ of $\Bbb{R}$ that is bounded below has a greatest lower bound $\text{inf}S$.
Proof: Let $-S:=\{-s:s\in S\}$. Since $S$ is bounded below such that there is a $m\in \Bbb{R}$ such that $m\le s\in S$ , we get that $-S$ is bounded above by $-m$. Hence by $\textbf{Completeness
axiom}$, we get that $\text{sup}S$ exists.
We claim that $\text{inf}S=-\text{sup}S$.
Let $s_0=\text{sup}(-S)$, we need to prove $-s_0\le s, \forall s\in S$.
If $\text{inf}S\ne -\text{sup}S$ then $\exists$ $t$ such that $t> s_0, t\not\in S, t\le s,\forall s\in S$. But then again consider the negatives, we done.
Archimedean Property
Theorem: If $a>0$ and $b>0$, then for some positive integer $n$, we have $na>b$.
Proof: [This proof is magical]
Assume that archimedean property fails. Then $\exists a>0, b>0$ such that $na\le b, \forall n\in \Bbb{N}$. Then note that $S=\{na: n\in \Bbb{N}\}$ is bounded by $b$. Using completeness axiom, we get
that $\exists s_0$ such that $s_0=\text{sup}S$.
Note that $s_0-a<s_0 \implies \exists n_1\in \Bbb{N}$ such that $s_0-a<n_1a\implies s_0<(n_1+1)a$. Contradiction.
Useful to know.
Claim: If $a>0,$ then $\frac{1}{n} <a$ for some positive integer $n$
Proof: When $b=1$ in the archimedean property.
Claim: If $b>0$, then $b<n$ for some positive integer $n$
Proof: When $a=1$ in the archimedean property.
Remark: There are many ordered fields which do not satisfy the above two claims.
Denseness of $\Bbb{Q}$
Theorem: If $a,b\in \Bbb{R}$ and $a<b$, then there is a rational $r\in \Bbb{Q}$ such that $a<r<b$.
We need to show $a<\frac{m}{n}<b$ for some integers $m$ and $n$ where $n>0$, thus we need $an<m<bn$.
Note that $(b-a)>0\implies \exists n$ such that $n(b-a)>1$. So there exists an integer between $nb$ and $na$. And we are done.
Problem 1: Prove that if $a>0$, then there exists $n\in \Bbb{N}$ such that $\frac{1}{n}<a<n$.
Proof: We use the archimedean property which states that
Claim: If $A>0$ and $B>0$, then for some positive integer $n$, we have $nA>B$
Taking $B=1,A=a$ we get that $\exists n_1$ such that $a>\frac{1}{n_1}$.
Taking $B=a, A=1$, we get $\exists n_2$ such that $n_2>a$.
Taking $n=\max(n_1,n_2)$ we are done.
Problem 2: Let $A$ and $B$ be nonempty bounded subsets of $\Bbb{R}$, and let $A+B$ be the set of all sums $a+b$ where $a\in A$ and $b\in B$. Prove that $$\text{sup}(A+B)= \text{sup}(A)+\text{sup}(B)
Proof: For all $a\in A, b\in B$, $$a+b\le \text{sup}(A+B)\implies a\le \text{sup}(A+B)-b$$
Hence we have $\text{sup}(A+B)-b$ the upper bound of $A$. So $$\text{sup}A\le \text{sup}(A+B)-b\implies b\le \text{sup}(A+B)-\text{sup}A$$ $$ \implies \text{sup}B\le \text{sup}(A+B)-\text{sup}A $$$$\
implies \text{sup}A+\text{sup}B\le \text{sup}(A+B).$$
And for the other direction, we clearly have $a\le \text{sup}A, b\le \text{sup}B\implies \max(a+b)\le \text{sup} A+\text{sup} B$
Problem 3: Let $a,b\in \Bbb{R}$. Show that if $a\le b+1/n$ for all $n\in \Bbb{N}$, then $a\le b$.
Proof: Suppose not. Then $a-b>0.$ So by the archimedean property, we get that $a-b>1/n$ for some $n$. Which is a contradiction.
Problem 4: Show $\text{sup}A=\{r\in \Bbb{Q}:r<a\}=a$ for each $a\in \Bbb{R}.$
Proof: If $\text{sup}A=a_0\ne a$, then by denseness of $\Bbb{Q}, \exists a_0<r_0<a.$
Well, yeah that's it for this blog post! Stay tuned for part two :)!
~Sunaina Pati
Continuing the tradition of past years, our seniors at the Indian IMO camp(an unofficial one happened this year) once again conducted LMAO, essentially ELMO but Indian. Sadly, only those who were in
the unofficial IMOTC conducted by Pranav, Atul, Sunaina, Gunjan and others could participate in that. We all were super excited for the problems but I ended up not really trying the problems because
of school things and stuff yet I solved problem 1 or so did I think. Problem 1: There is a grid of real numbers. In a move, you can pick any real number , and any row or column and replace every
entry in it with . Is it possible to reach any grid from any other by a finite sequence of such moves? It turned out that I fakesolved and oh my god I was so disgusted, no way this proof could be
false and then when I was asked Atul, it turns out that even my answer was wrong and he didn't even read the proof, this made me even more angry and guess what? I was not alone, Krutarth too fakesol
Hii everyone! Today I will be discussing a few geometry problems in which once you "guess" or "claim" the important things, then the problem can easily be finished using not-so-fancy techniques (e.g.
angle chasing, power-of-point etc. Sometimes you would want to use inversion or projective geometry but once you have figured out that some particular synthetic property should hold, the finish
shouldn't be that non trivial) This post stresses more about intuition rather than being rigorous. When I did these problems myself, I used freehand diagrams (not geogebra or ruler/compass) because I
feel that gives a lot more freedom to you. By freedom, I mean, the power to guess. To elaborate on this - Suppose you drew a perfect diagram on paper using ruler and compass, then you would be too
rigid on what is true in the diagram which you drew. But sometimes that might just be a coincidence. e.g. Let's say a question says $D$ is a random point on segment $BC$, so maybe
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In this post, I will present three graph theory problems in increasing difficulty, each with a common theme that one would determine a property of an edge in a complete graph through repeated
iterations, and seek to achieve a greater objective. ESPR Summer Program Application: Alice and Bob play the following game on a $K_n$ ($n\ge 3$): initially all edges are uncolored, and each turn,
Alice chooses an uncolored edge then Bob chooses to color it red or blue. The game ends when any vertex is adjacent to $n-1$ red edges, or when every edge is colored; Bob wins if and only if both
condition holds at that time. Devise a winning strategy for Bob. This is more of a warm-up to the post, since it has a different flavor from the other two problems, and isn't as demanding in terms of
experience with combinatorics. However, do note that when this problem was first presented, applicants did not know the winner ahead of time; it would be difficult to believe that Bob can hold such a
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A309097 - OEIS
We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha
avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.
Conjecture: for n > 0, a(n) is the number of ordered pairs (r, l) such that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l. Actually, such a matrix exists if and
only if ceiling(n/(n-r)) <= l <= r+1, see my proof below. If this conjecture is true, then a(n) = (n^2 + 3n)/2 -
(n) for n > 0. -
Jianing Song
, Nov 04 2019
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Designing Matching Networks for Low Noise Amplifiers
This example shows how to verify the design of input and output matching networks for a Low Noise Amplifier (LNA) using gain and noise figure plot.
In wireless communications, receivers need to be able to detect and amplify incoming low-power signals without adding much noise. Therefore, an LNA is often used as the first stage of these
receivers. To design an LNA, this example uses the available gain design technique, which involves selecting an appropriate matching network that provides a suitable compromise between gain and
In this example, to design matching networks for an LNA, the rfckt.amplifier object and the analyze method are used to examine the transducer power gains, the available power gain, and the maximum
available power gain. The method circle is used to determine optimal source reflection coiefficent, GammaS and the function fzero is used in amplifier stabilization.
LNA Design Specifications
The LNA design specifications are as follows:
• Frequency range: 5.10 - 5.30 GHz
• Noise Figure <= 2.2 dB
• Transducer Gain > 11 dB
• Operating between 50-ohm terminations
Create rfckt.amplifier Object and Examine Amplifier Power Gains and Noise Figure
Create an rfckt.amplifier object to represent the amplifier that is specified in the file, 'samplelna1.s2p'. Analyze the amplifier using the analyze function the amplifier in the frequency range from
2 - 10 GHz.
unmatched_amp = read(rfckt.amplifier, 'samplelna1.s2p');
analyze(unmatched_amp, 2e9:50e6:10e9);
Plot the transducer power gain (Gt), the available power gain (Ga) and the maximum available power gain (Gmag).
Examine the power gains at 5.2 GHz in order to design the input and output matching networks 5.2 GHz. Without the input and output matching networks, the transducer power gain at 5.2 GHz is about 7.2
dB. This is below the gain requirement of 11 dB in the design specifications and less than the available power gain. This amplifier is also potentially unstable at 5.2 GHz, since the maximum
available gain does not exist at 5.2 GHz.
Plot the measured minimum noise figure (Fmin) and the noise figure (NF) calculated when there is no input matching network. Specify an $x$-axis range of 4.9 GHz to 6 GHz, where the minimum noise
figure is measured.
axis([4.9 6 1.5 4])
In the absence of an input matching network, the noise figure is between 5.10 - 5.30 GHz which is above the noise figure requirement of 2.2 dB in the specification.
Plot Gain, Noise Figure, and Stability Circles
Both the available gain and the noise figure are functions of the source reflection coefficient, GammaS. To select an appropriate GammaS that provides a suitable compromise between gain and noise,
use the circle method of the rfckt.amplifier object to place the constant available gain and the constant noise figure circles on the Smith chart. As mentioned earlier, the amplifier is potentially
unstable at 5.2 GHz. Therefore, the following circle command also places the input and output stability circles on the Smith chart.
fc = 5.2e9;
hsm = smithplot;
circle(unmatched_amp,fc,'Stab','In','Stab','Out','Ga',10:2:20, ...
Enable the data cursor and click on the constant available gain circle. The data tip displays the following data:
• Available power gain (Ga)
• Noise figure (NF)
• Source reflection coefficient (GammaS)
• Output reflection coefficient (GammaOut)
• Normalized source impedance (ZS)
Ga, NF, GammaOut and ZS are all functions of the source reflection coefficient, GammaS. GammaS is the complex number that corresponds to the location of the data cursor. A star ('*') and a
circle-in-dashed-line will also appear on the Smith chart. The star represents the matching load reflection coefficient (GammaL) that is the complex conjugate of GammaOut. The gain is maximized when
GammaL is the complex conjugate of GammaOut. The circle-in-dashed-line represents the trajectory of the matching GammaL when the data cursor moves on a constant available gain or noise figure circle.
Because both the S11 and S22 parameters of the amplifier are less than unity in magnitude, both the input and output stable region contain the center of the Smith chart. In order to make the
amplifier stable, GammaS must be in the input stable region and the matching GammaL must be in the output stable region. The output stable region is shaded in the above figure. However, when a GammaS
that gives a suitable compromise between gain and noise is found, the matching GammaL always falls outside the output stable region. This makes amplifier stabilization necessary.
Amplifier Stabilization
One way to stabilize an amplifier is to cascade a shunt resistor at the output of the amplifier. However, this approach will also reduce gain and add noise. At the end of the example, you will notice
that the overall gain and noise still met the requirement.
To find the maximum shunt resistor value that makes the amplifier unconditionally stable, use the fzero function to find the resistor value that makes stability MU equal to 1. The fzero function
always tries to achieve a value of zero for the objective function, so the objective function should return MU-1.
function mu_minus_1 = lna_match_stabilization_helper(propval, fc, ckt, element, propname)
%LNA_MATCH_STABILIZATION_HELPER Return Stability MU-1.
% MU_MINUS_1 = LNA_MATCH_STABILIZATION_HELPER(PROPVALUE, FC, CKT,
% ELEMENT, PROPNAME) returns stability parameter MU-1 of a circuit, CKT
% when the property called PROPNAME of an element, ELEMENT is set to
% PROPVAL.
% LNA_MATCH_STABILIZATION_HELPER is a helper function of RF
% Toolbox demo: Designing Matching Networks (Part 1: Networks with an LNA
% and Lumped Elements).
% Copyright 2007-2008 The MathWorks, Inc.
set(element, propname, propval)
analyze(ckt, fc);
mu_minus_1 = stabilitymu(ckt.AnalyzedResult.S_Parameters) - 1;
Compute the parameters for objective function and pass the objective function to fzero to get the maximum shunt resistor value.
stab_amp = rfckt.cascade('ckts', {unmatched_amp, rfckt.shuntrlc});
R1 = fzero(@(R1) lna_match_stabilization_helper(R1,fc,stab_amp,stab_amp.Ckts{2},'R'),[1 1e5])
Find GammaS and GammaL
Cascade a 118-ohm resistor at the output of the amplifier and analyze the cascaded network. Place the new constant available gain and the constant noise figure circles on the Smith chart.
shunt_r = rfckt.shuntrlc('R',118);
stab_amp = rfckt.cascade('ckts',{unmatched_amp,shunt_r});
hsm = smithplot;
Use the data cursor to locate a GammaS. You can find that there is a suitable compromise between gain and noise.
The example is designined to select a GammaS that gives a gain of 14 dB and noise figure of 1.84 dB. Compute the matching GammaL, which is the complex conjugate of GammaOut on the data tip.
GammaS = 0.67*exp(1j*153.6*pi/180)
GammaS =
-0.6001 + 0.2979i
Compute the normalized source impedance.
Compute the matching GammaL that is equal to the complex conjugate of GammaOut.
GammaL = 0.7363*exp(1j*120.1*pi/180)
GammaL =
-0.3693 + 0.6370i
Compute the normalized load impedance.
Design Input Matching Network Using GammaS
In this example, the lumped LC elements are used to build the input and output matching networks as follows:
The input matching network consists of one shunt capacitor, Cin, and one series inductor, Lin. Use the Smith chart and the data cursor to find component values. To do this, start by plotting the
constant conductance circle that crosses the center of the Smith chart and the constant resistance circle that crosses GammaS.
hsm = smithplot;
hsm.GridType = 'YZ';
hold on
text(real(GammaS)+0.05,imag(GammaS)-0.05,'\Gamma_{S}','FontSize', 12, ...
hold off
Then, find the intersection points of the constant conductance and the constant resistance circle. Based on the circuit diagram above, the intersection point in the lower half of the Smith chart
should be used. Mark it as point A.
GammaA = 0.6983*exp(1j*(-134.3)*pi/180);
Za = gamma2z(GammaA,1);
Ya = 1/Za;
Determine the value of Cin from the difference in susceptance from the center of the Smith chart to point A. Namely,
$2\pi {f}_{c}{C}_{in}=\text{Im}\left(\frac{{Y}_{a}}{50}\right)$
where 50 is the reference impedance.
Cin = imag(Ya)/50/2/pi/fc
Determine the value of Lin from the difference in reactance from point A to GammaS. Namely,
$2\pi {f}_{c}{L}_{in}=50\left(\text{Im}\left({Z}_{s}\right)-\text{Im}\left({Z}_{a}\right)\right)$
Lin = (imag(Zs) - imag(Za))*50/2/pi/fc
Design Output Matching Network Using GammaL
Use the approach described in the previous section on designing the input matching network to design the output matching network and get the values of Cout and Lout.
GammaB = 0.7055*exp(1j*(-134.9)*pi/180);
Zb = gamma2z(GammaB, 1);
Yb = 1/Zb;
Cout = imag(Yb)/50/2/pi/fc
Lout = (imag(Zl) - imag(Zb))*50/2/pi/fc
Verify Design
Create the input and output matching networks. Cascade the input matching network, the amplifier, the shunt resistor and the output matching network to build the LNA.
input_match = rfckt.cascade('Ckts', ...
output_match = rfckt.cascade('Ckts', ...
LNA = rfckt.cascade('ckts', ...
Analyze the LNA around the design frequency range and plot the available and transducer power gain. The available and transducer power gain at 5.2 GHz are both 14 dB as the design intended. The
transducer power gain is above 11 dB in the design frequency range, which meets the requirement in the specification.
Plot the noise figure around the design frequency range.
The noise figure is below 2.2 dB in the design frequency range, which also meets the requirement in the specification. The noise figure of the LNA at 5.2 GHz is about 0.1 dB above that of the
amplifier (1.84 dB), which demonstrates added noise by the shunt resistor.
The available gain design method is often used in LNA matching. In the second part of the example -- Designing Matching Networks Using Single Stub Transmission Lines, a simultaneous conjugate
matching example is presented.
Related Topics
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Algebra cheat
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A box with an initial speed of 6 m/s is moving up a ramp. The ramp has a kinetic friction coefficient of 4/5 and an incline of ( 5 pi )/12 . How far along the ramp will the box go? | HIX Tutor
A box with an initial speed of #6 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #4/5 # and an incline of #( 5 pi )/12 #. How far along the ramp will the box go?
Answer 1
The distance is $= 1.57 m$
Taking the direction up and parallel to the plane as positive #↗^+#
The coefficient of kinetic friction is #mu_k=F_r/N#
Consequently, the object's net force is
Newton's Second Law states
Where #a# is the acceleration
The coefficient of kinetic friction is #mu_k=4/5#
The incline of the ramp is #theta=5/12pi#
A deceleration is indicated by the negative sign.
We utilize the equation of motion.
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Answer 2
To find how far along the ramp the box will go, you can use the work-energy principle. The work done by friction will be equal to the change in kinetic energy of the box.
The work done by friction is given by: [ W_{friction} = \mu_k \cdot m \cdot g \cdot d \cdot \cos(\theta) ] where:
• ( \mu_k ) is the coefficient of kinetic friction (4/5),
• ( m ) is the mass of the box (not given),
• ( g ) is the acceleration due to gravity (9.8 m/s²),
• ( d ) is the distance along the ramp,
• ( \theta ) is the angle of the incline (5π/12 radians).
The change in kinetic energy is given by: [ \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 ] where:
• ( v_f ) is the final velocity (0 m/s, as the box stops at the top of the ramp),
• ( v_i ) is the initial velocity (6 m/s).
Setting the work done by friction equal to the change in kinetic energy and solving for ( d ), we get: [ \mu_k \cdot m \cdot g \cdot d \cdot \cos(\theta) = \frac{1}{2} m v_i^2 ] [ d = \frac{\frac{1}
{2} m v_i^2}{\mu_k \cdot g \cdot \cos(\theta)} ]
We can cancel out the mass ( m ) from both sides of the equation, as it appears in both numerator and denominator.
[ d = \frac{\frac{1}{2} v_i^2}{\mu_k \cdot g \cdot \cos(\theta)} ]
Substituting the given values: [ d = \frac{\frac{1}{2} \times 6^2}{\frac{4}{5} \cdot 9.8 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d = \frac{\frac{1}{2} \times 36}{\frac{4}{5} \cdot 9.8 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d = \frac{18}{\frac{4}{5} \cdot 9.8 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d = \frac{18}{\frac{4}{5} \cdot 9.8 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d = \frac{18}{\frac{4}{5} \cdot 9.8 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d ≈ \frac{18}{7.84 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d ≈ \frac{18}{7.84 \cdot \cos\left(\frac{5\pi}{12}\right)} ]
[ d ≈ \frac{18}{7.84 \cdot 0.2588} ]
[ d ≈ \frac{18}{2.025} ]
[ d ≈ 8.89 ]
So, the box will travel approximately 8.89 meters along the ramp.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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[curves] pure-python Ed25519 library for review
Ron Garret ron at flownet.com
Tue Apr 7 15:32:57 PDT 2015
Is there a reason you don’t just use DJB’s reference implementation?
On Apr 7, 2015, at 11:55 AM, Brian Warner <warner at lothar.com> wrote:
> Hi folks.. I could use some extra eyeballs on a pure-python library I
> put together to do Ed25519 operations:
> https://github.com/warner/python-pure25519
> Of course it's very much not constant-time, and a lot slower than a C
> implementation. But a pure-python library is, in practice, much easier
> to depend upon than one that requires a C compiler.
> And it's not unusably slow. If you run "python setup.py speed" in that
> tree, you'll get the speed-test results. On my machine (2.6GHz i7), I'm
> getting SUPERCOP-compatible Ed25519 sign/verify times of 2.8ms and
> 10.5ms .
> I'm writing this to support a pure-python SPAKE2 library, for which I'm
> seeing each phase of the protocol take about 5-8ms.
> I'm especially looking for feedback on the arbitrary_element() function,
> which provides SPAKE2 with the unknown-discrete-log group elements U and
> V (or M and N depending on which paper you read). That function is
> paraphrased here:
> https://github.com/warner/python-pure25519/blob/master/pure25519/basic.py#L261
> def arbitrary_element(seed): # unknown DL
> hseed = hashlib.sha512(seed).digest()
> y = int(binascii.hexlify(hseed), 16) % Q
> for plus in itertools.count(0):
> y_plus = (y + plus) % Q
> x = xrecover(y_plus)
> Pa = [x,y_plus]
> if not isoncurve(Pa):
> continue
> P = ElementOfUnknownGroup(xform_affine_to_extended(Pa))
> P8 = P.scalarmult(8)
> if is_extended_zero(P8.XYTZ):
> continue
> assert is_extended_zero(P8.scalarmult(L).XYTZ)
> return Element(P8.XYTZ)
> # never reached
> ("Q" is 2**255-19, and xrecover() does what you'd expect)
> What this code is doing and why:
> * oversized hash (sha512) and reduction, to avoid any significant bias.
> I'm pretty sure this doesn't matter for SPAKE2, but it seemed
> appropriate. Should I remove this? Does Elligator avoid bias?
> * test isoncurve(P), increment-and-try-again if not. I find that about
> 50% of Y-coords result in not-on-curve points. I assume these points
> are actually on the "twist", and that some protocols can run faster by
> ignoring this fact, but I don't know enough to safely do the same.
> * multiply by 8 to force any points of order 2*L/4*L/8*L into the proper
> order=L subgroup. It also forces points of order 1/2/4/8 into the
> identity element (zero)
> * test against zero, which rejects points of order 1/2/4/8. I'm not sure
> if I need to be worried about these: I suspect they're vanishingly
> unlikely to happen. The full version has a large comment about my
> probably-flawed beliefs of how common these points are, and I think
> there are at most 8 of them.
> The rest of that file defines addition and multiplication functions
> (using "extended" coordinates, XYTZ) and some object-oriented wrappers
> to make application code easier/safer. In the process of testing, I
> found a need for that ElementOfUnknownGroup, to exercise point math on
> things like the identity and the point of order 2. Most applications
> would stick to the correct-group-order Element class instead.
> The repo includes a compatible implementation of Ed25519 signatures, a
> demo DH-key-agreement routine (functionally equivalent to Curve25519 but
> not interoperable with it), and a SPAKE2 implementation that I intend to
> use in another project I'll be asking y'all to review shortly.
> The bytes_to_element() function does full is-this-in-the-right-group
> point validation, which slows things down by 2-4ms. It'd be nice to
> remove that for the protocols that were designed to not need it
> (Ed25519/Curve25519 clear the cofactor in other ways), but I don't
> understand the issues well enough to feel confident removing it. Plus, I
> don't want to leave a trap for later users of the library. Perhaps I'll
> add a function named bytes_to_8_times_element() that multiplies instead
> of validating.
> An interesting side discussion would be how/where to speed up SPAKE2
> with these same tricks. Maybe instead of X=G*a+U*pw. Y=G*b+V*pw.
> Z1=(Y-V*pw)*a. Z2=(X-U*pw)*b, you'd do?:
> X=G*a+U*pw. Y=G*b+V*pw. Z1=(Y*8-V*pw*8)*a. Z2=(X*8-U*pw*8)*b
> Anyways, any and all feedback is welcome. Let me know what you think!
> cheers,
> -Brian
> _______________________________________________
> Curves mailing list
> Curves at moderncrypto.org
> https://moderncrypto.org/mailman/listinfo/curves
More information about the Curves mailing list
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149+ Solved Probability Questions and Answers With Explanation
1. Experiment : An operation which can produce some well-defined outcomes is called an experiment.
2. Random Experiment :An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.
Ex :
i. Tossing a fair coin.
ii. Rolling an unbiased dice.
iii. Drawing a card from a pack of well-shuffled cards.
3. Details of above experiments:
i. When we throw a coin, then either a Head (H) or a Tail (T) appears.
ii. A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
iii. A pack of cards has 52 cards.
• It has 13 cards of each suit, name Spades, Clubs, Hearts and Diamonds.
• Cards of spades and clubs are black cards.
• Cards of hearts and diamonds are red cards.
There are 4 honours of each unit. There are Kings, Queens and Jacks. These are all called face cards.
4. Sample Space: When we perform an experiment, then the set S of all possible outcomes is called the sample space.
Ex :
1. In tossing a coin, S = {H, T}
2. If two coins are tossed, the S = {HH, HT, TH, TT}.
3. In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.
Event : Any subset of a sample space is called an event.
5. Probability of Occurrence of an Event :
Let S be the sample and let E be an event.
Then, $E\subseteq S$
$\therefore P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$
6. Results on Probability :
i. P(S) = 1 ii. $0\le P\left(E\right)\le 1$iii. $P\left(\varnothing \right)=0$
iv. For any events A and B we have :
$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$
v. If $\overline{)A}$ denotes (not-A), then $P\left(\overline{)A}\right)=1-P\left(A\right)$
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Mini-workshop on deformed W-algebras and q-characters II
• Date: 2022-06-16 (Thu)
• Place: 129-406 (SNU)
• Timetable:
9:00 ~ 10:00
Speaker: 서의린 (SNU)
Title: Constructions of W-algebras
Abstract: In this talk, I will briefly review two of most well-known constructions of classical and quantum W-algebras. The first part will be about Drinfeld-Sokolov reduction and the second part
will be about screening operators. I will also show how these two constructions are related to each other in both classical and quantum cases.
10:30 ~ 12:00
Speaker: 장일승 (QSMS, SNU)
Title: Introduction to q-characters
Abstract: Let U' be the quantum affine algebra corresponding to an affine Lie algebra (of untwisted type). In [arXiv:math/9810055], Frenkel and Reshetikhin provided a refined notion of the ordinary
character in the category of finite-dimensional U'-modules, so-called q-character, motivated by their theory of deformed W-algebras. In this talk, I will briefly introduce the q-character and discuss
it, focusing mainly on type A with rank 1 or 2.
14:00 ~ 15:30
Speaker: 김영훈 (QSMS, SNU)
Title: Computation of q-characters of finite dimensional modules of quantum affine algebras
Abstract: In 1998, Frenkel and Reshetikhin introduced an injective ring homomorphism, called the q-character, from the Grothendieck ring of finite-dimensional modules of a quantum affine algebra to a
ring of Laurent polynomials. In this talk, we first study some basic properties of the q-character. Then, we explicitly compute q-characters for some example modules.
16:00 ~ 17:00
Speaker: 이신명 (SNU)
Title: Deformed W-algebras and q-characters
Abstract: In this talk, we investigate relations between deformed W-algebras and finite-dimensional representations of quantum affine algebras. We first define various deformed W-algebras by means of
screening operators. Among them, we focus on q-deformed W-algebras, whose free field realization can be identified with the q-character map. Finally, we give a brief survey on more recent results on
both sides.
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Scientific RPN Calculator (with ATTINY85)
03-09-2018, 07:51 AM
Post: #25
deetee Posts: 81
Member Joined: May 2016
RE: Scientific RPN Calculator (with ATTINY85)
Hi Pauli!
Thanks for all of your ideas.
I tried to implement statistics and linear regression - which needed approx. 1000 bytes - to much - so I refused this for now.
Then I implemented converting polar/rectangular coordinates and h/hms which "costs" me all of my spare 800 bytes.
I know the gamma function, but neither I needed in before nor saw it on basic HP-calculators. But the gamma function seems to be worth implementing and doing this with Nemes' formula should "cost"
approx. 200 bytes.
I did not know the memmove function but I will give it a try (it is on my todo-list).
But (overnight) I fell in love with your idea to save user defined constants to the EEPROM. I gave it a rough try and implemented
"saving 3 characters and a float number" to a "EEPROM slot" and
"menu driven selecting due to these 3 characters and loading the float number".
This demands approx. 700 bytes only (and there is still room for improvement).
Still a lot to do this weekend ...
User(s) browsing this thread: 1 Guest(s)
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Видеотека: Hans-Joachim Mucha,
Аннотация: Variable selection is a difficult task in many areas of multivariate statistics such as classification, clustering and regression. Here the hope is that the structure of interest may be
contained in only a small subset of variables. In contradiction to supervised classification such as discriminant analysis, variable selection in cluster analysis is a much more difficult problem
because usually nothing is known about the true class structure, and hence nothing is known about the number of clusters K to be inherent in the data. There are many proposals on variable selection
in cluster analysis based on special cluster separation measures such as the criterion of Davies and Bouldin (1979). Here we present a general bottom-up approach to variable selection using
non-parametric bootstrapping based on criteria of stability such as the adjusted Rand's index (Hubert and Arabie, 1985). General means that it makes use only of measures of stability of partitions,
and so it can be applied to almost any cluster analysis method.
Язык доклада: английский
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F Ratio Calculator - Savvy Calculator
F Ratio Calculator
About F Ratio Calculator
The F ratio, commonly used in ANOVA (Analysis of Variance), is a crucial statistic in determining whether there are significant differences between group means. The F Ratio Calculator simplifies this
calculation, allowing you to compare variances between groups and make informed decisions based on your data.
The formula for calculating the F ratio is:
F = (variance between groups) / (variance within groups)
This formula helps you assess the degree to which the group means differ, relative to the variability within the groups.
How to Use
Using the F Ratio Calculator involves a few simple steps:
1. Input the variance between groups: Enter the calculated variance between the groups in your dataset.
2. Input the variance within groups: Enter the calculated variance within the groups.
3. Calculate the F ratio: Click on the “Calculate” button to get the F ratio, which will help determine if the differences between group means are statistically significant.
Consider a scenario where you are analyzing the performance of students across three different teaching methods. You have calculated a variance between groups of 4.5 and a variance within groups of
2.3. Using the F Ratio Calculator, the F ratio would be 1.96. This value would then be compared against an F-distribution to determine significance.
1. What is the F ratio used for? The F ratio is used in statistical analysis, particularly ANOVA, to compare the variances between different groups and assess whether the group means are
significantly different.
2. How do I interpret the F ratio? A higher F ratio suggests greater variance between group means relative to the variance within groups, indicating a potential significant difference. The F ratio is
then compared to a critical value from the F-distribution table.
3. What is variance between groups? Variance between groups measures the variability of group means relative to the overall mean of all groups.
4. What is variance within groups? Variance within groups measures the variability of individual data points within each group.
5. How is the F ratio related to ANOVA? The F ratio is the test statistic used in ANOVA to determine whether there are significant differences between the means of different groups.
6. What does it mean if the F ratio is close to 1? If the F ratio is close to 1, it suggests that the variances between and within groups are similar, indicating no significant difference between
group means.
7. Can I use the F Ratio Calculator for other statistical tests? While primarily used for ANOVA, the F ratio can also be relevant in other statistical tests that compare variances, such as regression
8. What is a critical F value? The critical F value is a threshold obtained from the F-distribution table, which the calculated F ratio must exceed for the results to be statistically significant.
9. How do I find the critical F value? The critical F value depends on the degrees of freedom for the numerator (between groups) and the denominator (within groups) and is found using an
F-distribution table or statistical software.
10. What does a significant F ratio indicate? A significant F ratio indicates that there is more variability between group means than would be expected by chance, suggesting that the group means are
not all equal.
11. What are the assumptions for using the F ratio in ANOVA? The assumptions include independence of observations, normally distributed groups, and homogeneity of variances within groups.
12. Can I use the F Ratio Calculator for unequal sample sizes? Yes, but be cautious as unequal sample sizes can affect the variance estimates, potentially leading to incorrect conclusions.
13. What happens if the assumptions for ANOVA are not met? If assumptions are violated, the F ratio might not be accurate, and alternative statistical methods such as Welch’s ANOVA may be needed.
14. Is the F ratio sensitive to outliers? Yes, outliers can significantly affect the variance calculations, which in turn can distort the F ratio.
15. How is the F ratio different from a t-test? While a t-test compares means between two groups, the F ratio (used in ANOVA) compares means across three or more groups.
16. What do I do if my F ratio is not significant? If the F ratio is not significant, you conclude that there is no evidence to suggest differences between the group means in your data.
17. Can I calculate the F ratio manually? Yes, you can calculate it manually using the variances between and within groups, but using the F Ratio Calculator is faster and reduces the risk of errors.
18. What software can I use for calculating the F ratio? Software like SPSS, R, Python (with appropriate libraries), and even Excel can be used to calculate the F ratio, in addition to online
19. How do I handle multiple comparisons in ANOVA? If you have multiple comparisons, consider using post-hoc tests like Tukey’s HSD to control for Type I errors after finding a significant F ratio.
20. What is the role of degrees of freedom in the F ratio calculation? Degrees of freedom determine the shape of the F-distribution and are used to find the critical F value for significance testing.
The F Ratio Calculator is a powerful tool for statistical analysis, particularly in ANOVA, where it helps determine whether differences between group means are statistically significant. By
understanding how to calculate and interpret the F ratio, you can make more informed decisions in your research and data analysis. Whether you’re a student, researcher, or data analyst, this tool
simplifies the complex process of variance analysis, providing accurate results quickly and efficiently.
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Coralia Cartis
May 25, 2021
Abstract:The aim of this paper is two-fold: firstly, to present subspace embedding properties for $s$-hashing sketching matrices, with $s\geq 1$, that are optimal in the projection dimension $m$ of
the sketch, namely, $m=\mathcal{O}(d)$, where $d$ is the dimension of the subspace. A diverse set of results are presented that address the case when the input matrix has sufficiently low coherence
(thus removing the $\log^2 d$ factor dependence in $m$, in the low-coherence result of Bourgain et al (2015) at the expense of a smaller coherence requirement); how this coherence changes with the
number $s$ of column nonzeros (allowing a scaling of $\sqrt{s}$ of the coherence bound), or is reduced through suitable transformations (when considering hashed -- instead of subsampled -- coherence
reducing transformations such as randomised Hadamard). Secondly, we apply these general hashing sketching results to the special case of Linear Least Squares (LLS), and develop Ski-LLS, a generic
software package for these problems, that builds upon and improves the Blendenpik solver on dense input and the (sequential) LSRN performance on sparse problems. In addition to the hashing sketching
improvements, we add suitable linear algebra tools for rank-deficient and for sparse problems that lead Ski-LLS to outperform not only sketching-based routines on randomly generated input, but also
state of the art direct solver SPQR and iterative code HSL on certain subsets of the sparse Florida matrix collection; namely, on least squares problems that are significantly overdetermined, or
moderately sparse, or difficult.
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seminars - First Example of a Maximal Amenable Subalgebra in Quantum Group Context
This talk will begin with the maximal amenable subalgebras in free group von Neumann algebra, then focus on the radial MASA in the orthogonal free quantum group algebra. We will show that this masa
is maximal amenable if N is large enough, using the Asymptotic Orthogonality Property. This relies on a detailed study of the corresponding bimodule, for which we construct in particular a quantum
analogue of Rădulescu’s basis, which is not orthogonal anymore.
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Fundamentals of Algebraic Modeling - National Geographic Learning
New to this Edition
A new four-color design helps further distinguish the features of the text.
Examples and exercises have been updated.
A brand new Chapter R A Review of Algebra Fundamentals has been added and it gives students an opportunity to review the algebra skills needed to be successful in a modeling course... more
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Convert Quads to Gigajoules (Q to GJ) | Examples & Steps
Disclaimer: We've spent hundreds of hours building and testing our calculators and conversion tools. However, we cannot be held liable for any damages or losses (monetary or otherwise) arising out of
or in connection with their use. Full disclaimer.
How to convert quads to gigajoules (Q to GJ)
The formula for converting quads to gigajoules is: GJ = Q × 1055870000. To calculate the quad value in gigajoules first substitute the quad value into the preceding formula, and then perform the
calculation. If we wanted to calculate 1 quad in gigajoules we follow these steps:
GJ = Q × 1055870000
GJ = 1 × 1055870000
GJ = 1055870000
In other words, 1 quad is equal to 1055870000 gigajoules.
Example Conversion
Let's take a look at an example. The step-by-step process to convert 7 quads to gigajoules is:
1. Understand the conversion formula: GJ = Q × 1055870000
2. Substitute the required value. In this case we substitute 7 for Q so the formula becomes: GJ = 7 × 1055870000
3. Calculate the result using the provided values. In our example the result is: 7 × 1055870000 = 7.39109e+9 GJ
In summary, 7 quads is equal to 7.39109e+9 gigajoules.
Converting gigajoules to quads
In order to convert the other way around i.e. gigajoules to quads, you would use the following formula: Q = GJ × 0.00000000094708628903179. To convert gigajoules to quads first substitute the
gigajoule value into the above formula, and then execute the calculation. If we wanted to calculate 1 gigajoule in quads we follow these steps:
Q = GJ × 0.00000000094708628903179
Q = 1 × 0.00000000094708628903179
Q = 0.00000000094708628903179
Or in other words, 1 gigajoule is equal to 0.00000000094708628903179 quads.
Conversion Unit Definitions
What is a Quad?
A quad is a unit of energy equal to one quadrillion (10^15) British Thermal Units (BTUs). It represents an enormous amount of energy and is primarily used in the United States to describe large
quantities of energy production or consumption on a national scale.
The term "quad" is short for quadrillion BTUs. It is often used in discussions about national energy policy and planning, especially when talking about the total energy consumption or production of a
country over a given period. For instance, if a country's annual energy consumption is 100 quadrillion BTUs, it means that the country used 100 times 10^15 BTUs of energy in that year.
To give you a sense of scale, one quad is roughly equivalent to:
• Burning about 36 million tons of coal,
• Consuming 8 billion gallons of gasoline,
• Generating 293 billion kilowatt-hours of electricity.
Quads are significant metrics in energy discussions, especially when evaluating large-scale energy trends, efficiency improvements, and the overall energy requirements of a country.
What is a Gigajoule?
A gigajoule (GJ) is a metric unit of energy in the International System of Units (SI). One gigajoule is equal to 1,000,000,000 joules or 1×10^9 joules. The prefix "giga-" signifies a factor of one
billion, so a gigajoule represents a very large amount of energy.
Gigajoules are commonly used to measure significant amounts of energy in various contexts, especially in large-scale industries, energy production, and transportation sectors. For example, the energy
content of fuels, the electricity consumption of cities, and the energy output of power plants are often expressed in gigajoules due to the large quantities involved.
To put it in perspective, one gigajoule is roughly equivalent to the energy content of about 26,000 liters of gasoline or the energy produced by burning approximately 24,000 pounds (about 11,000
kilograms) of coal.
Quads To Gigajoules Conversion Table
Below is a lookup table showing common quads to gigajoules conversion values.
Quad (Q) Gigajoule (GJ)
1 Q 1.05587e+9 GJ
2 Q 2.11174e+9 GJ
3 Q 3.16761e+9 GJ
4 Q 4.22348e+9 GJ
5 Q 5.27935e+9 GJ
6 Q 6.33522e+9 GJ
7 Q 7.39109e+9 GJ
8 Q 8.44696e+9 GJ
9 Q 9.50283e+9 GJ
10 Q 1.05587e+10 GJ
11 Q 1.161457e+10 GJ
12 Q 1.267044e+10 GJ
13 Q 1.372631e+10 GJ
Quads To Gigajoules Conversion Chart
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